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Novel backlighting techniques for high dynamic range displays Emmel, Jakob 2013

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NOVEL BACKLIGHTING TECHNIQUES FOR HIGH DYNAMIC RANGE DISPLAYS   by Jakob Emmel   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  December 2013   ? Jakob Emmel, 2013 ii Abstract High dynamic range (HDR) displays use an array of ultra-bright individually tunable light emitting diodes (LEDs) as a backlight for common liquid crystal displays (LCDs). Through local dimming of LEDs, this combination can show images with very bright highlights while maintaining very low luminance values in dark areas. Current HDR displays, however, have limitations associated with displaying images that have spatially uniform luminance levels: a periodic pattern arising from the underlying pattern of the LEDs behind the LCD is perceptible in the image. The effect of new point spread functions (PSF) on the uniformity and contrast of HDR displays was analyzed. A PSF shaped like a certain type of spline could theoretically create a uniform brightness backlight, as well as producing linear and quadratic gradients, while being capable of showing very high contrast. A practical way to produce such a PSF was used to build an experimental device that achieved a non-uniformity of only ?0.8%, while enabling a contrast ratio of 5:1 and 33:1 over distances of one and two unit cell spacings, respectively. The implementation of a third light modulator in HDR displays, in addition to the backlight LEDs and the front liquid crystal display (LCD) was studied: liquid crystal cells were combined with reflective polarizers to act as light valves, either transmitting or reflecting light. In theory, these reflective light modulators are supposed to decrease power consumption through light recycle effects. The power consumption of displays using the two discussed backlights was simulated. It was found that their power consumption is less than that of a common LCD by a factor of 2-5, but it iii is about 11 percentage points higher than for a standard HDR display. However, the image quality and contrast are improved compared to both state of the art displays.  The advantages of the backlight with the new PSF may help to make HDR displays more useful and competitive in a wide range of applications requiring faithful luminance rendering such as discerning consumer markets, medical imaging, and motion picture editing.  iv Preface All of the work presented henceforth was conducted in the Sustainability Solutions Applied Physics Laboratory at the University of British Columbia, Point Grey campus, which provided equipment, knowledge, and previously unpublished optical design concepts.  The research described henceforth is a detailed scientific evaluation of some of those optical design concepts. A version of Chapters 3 and 4 has been submitted for publication [Emmel J, Whitehead L. Modified point spread function for efficient high dynamic range LED backlight capable of high uniformity, high contrast and smooth gradients.] I was the lead investigator, responsible for design and assembly of test apparatuses and methods, data collection and analysis, and manuscript composition. Lorne Whitehead was co-author and was involved throughout the project as my research supervisor and in providing editorial guidance for the manuscript composition. I was the lead investigator for the research presented in Chapters 5 and 6 (unpublished) for which I was responsible for design and assembly of test apparatuses and methods, data collection and analysis.   Lorne Whitehead also provided editorial guidance for my thesis and additional guidance was provided by the members of my departmental supervisory committee. v Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ...........................................................................................................................v List of Tables ................................................................................................................................ ix List of Figures .................................................................................................................................x List of Abbreviations ................................................................................................................. xxi Acknowledgements .................................................................................................................. xxiii Dedication ................................................................................................................................. xxiv Chapter  1: Introduction ...............................................................................................................1 Chapter  2: Background ................................................................................................................4 2.1 Aspects of the Human Visual System ............................................................................. 4 2.1.1 Dynamic Range ........................................................................................................... 4 2.1.2 Local Contrast ............................................................................................................. 5 2.1.3 Contrast Sensitivity ..................................................................................................... 6 2.2 Dynamic Range and Static Contrast in Displays ............................................................ 9 2.3 High Dynamic Range Display ...................................................................................... 10 2.3.1 Perceptually Accurate Dynamic Range .................................................................... 11 2.3.2 Current Problems of HDR Displays ......................................................................... 12 2.4 State of the Art of Displays ........................................................................................... 13 2.4.1 Liquid Crystal Displays ............................................................................................ 13 2.4.2 Low Dynamic Range Direct-LED LCD Displays .................................................... 15 vi 2.4.3 Field Emission Backlit LCD Displays ...................................................................... 15 2.4.4 Plasma Displays ........................................................................................................ 15 2.4.5 Organic Light Emitting Diode Displays ................................................................... 16 2.4.6 Recent Developments in High Dynamic Range Technology ................................... 17 Chapter  3: Point Spread Function that Creates a Uniform Backlight and Ideal Gradients ......................................................................................................................................19 3.1 Novel Point Spread Function in 1D and 2D ................................................................. 21 3.2 Comparison to Lambertian Point Spread Function ....................................................... 31 3.3 Experimental Implementation of Backlight with Novel PSF ....................................... 35 3.4 Measurements ............................................................................................................... 40 3.4.1 Measurement Procedure............................................................................................ 40 3.4.2 Measurement Results ................................................................................................ 44 3.5 Summary and Conclusions ........................................................................................... 49 Chapter  4: Establishing the Feasibility of an Efficient Backlight with a Special Point Spread Function .................................................................................................................50 4.1 Backlight Model............................................................................................................ 50 4.2 Simulation Setup ........................................................................................................... 51 4.3 Experimental Verification of a Practical PSF ............................................................... 54 4.3.1 Absorptive Filter ....................................................................................................... 54 4.3.2 Reflective Filter ........................................................................................................ 64 4.3.3 Single LED Backlight ............................................................................................... 68 4.3.4 Mechanical Description of Efficient Multi-LED Backlight ..................................... 74 4.4 Measurements and Simulations .................................................................................... 76 vii Chapter  5: Reflective Polarizer Liquid Crystal Devices in an HDR System.........................83 5.1 Dielectric Mirrors as Reflective Polarizers ................................................................... 85 5.2 Scale Model of Backlight Module with Uniform Brightness Backlight....................... 87 5.2.1 Measurements ........................................................................................................... 89 5.2.2 Verification with Ray Tracing Software ................................................................... 92 5.2.3 Numerical Solution for Flux ..................................................................................... 93 5.3 Setup with Single LED as Backlight ............................................................................ 95 5.3.1 Numerical Solution for Flux Output ......................................................................... 98 Chapter  6: Simulation of Power Consumptions of Various Backlights...............................101 6.1 HDR Image Preparation .............................................................................................. 102 6.2 Power Consumption of Standard LCD with Uniform Brightness Backlight .............. 103 6.3 Power Consumption of HDR Display with Reflective Polarizer LCDs ..................... 104 6.3.1 Matching Luminance Values of Highlights ............................................................ 107 6.4 Power Consumption of HDR Displays with Different Point Spread Functions ......... 109 6.4.1 Deconvolution of HDR Images .............................................................................. 109 6.4.1.1 Modified PSF Measured off Experimental Backlight .................................... 110 6.4.1.2 Simulation of Standard PSF of HDR Backlight ............................................. 111 6.4.1.3 Results for Different Point Spread Functions ................................................. 113 6.4.2 Matching Luminance Values of Highlights ............................................................ 115 6.5 Summary and Conclusions ......................................................................................... 119 Chapter  7: Conclusion ..............................................................................................................122 Bibliography ...............................................................................................................................124 Appendices ..................................................................................................................................131 viii Appendix A ............................................................................................................................. 131 Appendix B ............................................................................................................................. 135 Appendix C ............................................................................................................................. 137  ix List of Tables Table 1: Measurement and simulation results for seven different reflectance patterns (see Figure 65). .................................................................................................................................... 92 Table 2: Measured and simulated reflectance, transmission and absorption values of RP-LCDs at 560nm. .............................................................................................................................. 93 Table 3: Measured and simulated photometric RTA values for RP-LCDs in a backlight module with one LED. ................................................................................................................. 100  x List of Figures  Figure 1: Comparison of display and eye introduced blur at contrast boundaries of 1 cd/m2 to 10,000 cd/m2, 10,000 cd/m2 to 5,000 cd/m2 and 5,000 cd/m2 to 1 cd/m2. Data from [3]. ... 6 Figure 2: Example of typical contrast sensitivity function. Measured data from [18], fit curve from [19]. ............................................................................................................................ 8 Figure 3: Example of an acrylic light guide (not to scale) with an LED light source which can illuminate a LCD screen (not shown). The light is contained in the light guide by a specular reflector on one side and total internal reflection at the top side, if the angle of incidence larger than the critical angle (around 42? for acrylic). ..................................... 14 Figure 4: A cross section of the radiation from Lambertian surface, which is directly proportional to the cosine of the angle between the observer's position and the surface normal. The number of photons per second directed into any wedge is proportional to the area of the wedge [35]. ....................................................................................................................... 20 Figure 5: A triangle PSF with cut-off of 1 and integrated luminance of 1. .................................. 22 Figure 6: 1D blending of seven LEDs (located at position 1-7) with a triangle point spread function shown in Figure 5. A uniform luminance level (blue) can be achieved as well as smooth linear (red) gradient. When the output of each LED is increased quadratically (green) small dis-continuities in the slope are found at the position of each LED (e. g. at 1 unit spacing). ..................................................................................................................... 23 Figure 7: Spatial derivative of luminance when output of each LED is increased quadratically from one LED to the next. Discontinuities are observed at every LED position (i.e. at integer unit spacings) ........................................................................................................ 24 xi Figure 8: Comparison of proposed PSF (solid) and Lambertian PSF (dotted) with the same full-width at half-maximum and integral. ................................................................................ 25 Figure 9: Example of fitting discrete data points (circles) with splines to yield a continuous curve, for interpolation or differentiation. The key value of this approach is that the curve at any point is entirely determined by a few closest points, it passes through those points exactly, and the curve is continuous in both first and second derivatives. ....................... 26 Figure 10: Creation of linear gradient through the convolution of the quadratic B-spline with linear increasing LED outputs at the LED locations (integer unit spacings). The colour curves are the scaled PSFs of seven LEDs and the black curve is the linear gradient yielded by the summation over the contributing LEDs. ................................................... 27 Figure 11: 1D blending of seven LEDs with the new point spread function. A constant luminance level (blue) can be achieved as well as perfectly linear (red) and quadratic (green) gradients. ........................................................................................................................... 28 Figure 12: Spatial derivative of luminance when the new PSF is used and the output of each LED is increased quadratically from one LED to the next. The gradient is perfectly smooth, unlike in Figure 7. ............................................................................................................. 28 Figure 13: 2D point spread function in 3D. The area has a size of 3 ? 3 unit spacings. .............. 30 Figure 14: 2D point spread function is not rotationally invariant. The area has a size of 3 ? 3 unit spacings. ............................................................................................................................ 30 Figure 15: Output of seven LEDs with new PSF (black) and Lambertian PSF (red). Goal is a uniform brightness screen and a linear gradient. .............................................................. 32 Figure 16: Comparison of luminance fall-off from 90% to 10% between new PSF (1.1 unit spacings) and Lambertian (3.1 unit spacings)................................................................... 32 xii Figure 17: Simulation of 10 ? 10 LED arrays with the new PSF (left) and Lambertian PSF (right): Uniform brightness screen (top), linear gradient at 28? (centre) and quadratic gradient at 28? (bottom). ................................................................................................... 34 Figure 18: Schematic of experimental setup that created the new PSF. ....................................... 35 Figure 19: Emission spectra of LED for different angles: 0? (blue), 45? (red) and 90? (green). The curves were normalized to have the same area. It shows that for larger angle the spectrum shifts to larger wavelengths, which causes a colour shift. ................................ 37 Figure 20: Photograph of diffuser cap (white) with a radius of 10 mm placed on top of an LED............................................................................................................................................ 37 Figure 21: Sketch of the backlight structure. The hexagonal holes are there to position the LEDs, the baffles are visible as well as the grid structure to attach the filters. The screen (not shown) sits ontop of the side walls. .................................................................................. 38 Figure 22: Photograph of backlight structure with 20 LEDs separated by black baffles. The LED spacing was 10 cm. ........................................................................................................... 38 Figure 23: A slowly varying luminance pattern on a diffuser masked by black cardboard with holes. ................................................................................................................................. 41 Figure 24: Luminance calibration of CCD camera: MATLAB grey scale value (0-255) vs product of luminance and exposure time shown on a log-log plot. A power function (formula given in graph) was fitted to the data, which was used in a MATLAB procedure to determine luminance values from image data. ............................................................. 42 Figure 25: Example picture of a PSF taken with the CCD camera. The red line indicates the location of the line scan shown in Figure 26. ................................................................... 43 xiii Figure 26: Line scan through a picture of a large PSF (see Figure 25) taken with the CCD camera. .............................................................................................................................. 43 Figure 27: Measured line scan across uniform brightness screen with the new PSF (red) and the Lambertian PSF (blue). First derivative of the data is shown on the right. The respective goals are shown in black. .................................................................................................. 44 Figure 28: Measured line scan of linear gradient with the new PSF (red) and the Lambertian PSF (blue). First derivative of the data is shown on the right. The respective goals are shown in black. ............................................................................................................................. 45 Figure 29: Measured line scan of quadratic gradient with the new PSF (red) and the Lambertian PSF (blue). First derivative of the data is shown on the right. The respective goals are shown in black. ................................................................................................................. 46 Figure 30: Line scans across screen with two LEDs on (at the screen location of 0.5 and 1.5 unit spacings): Lambertian PSF (blue), the new PSF (red) and a Lambertian PSF with walls (green). In the graph on the right, luminance is plotted on a logarithmic axis. ................ 47 Figure 31: Curve that approximately follows the measured data of the contrast sensitivity function (CSF) (black) from [14]: Data points higher than the black curve indicate that the corresponding patterns are invisible to the human eye. With filters (circles) the deviation on the screen at 0.26 cycles/degree is invisible. ............................................... 48 Figure 32: First idea of an efficient backlight setup that might be able to create a suitable PSF. 51 Figure 33: Schematic side view of simulation setup. E(x,y) described the pixelated light emission surface with the width of one unit spacing. I(x,y) described the pixelated intensity pattern at distance d. The width of I(x,y) was three unit spacings. Opaque baffles surrounded E(x,y) with the height of d/2. ............................................................................................ 52 xiv Figure 34: Example of an E(x,y) with dimensions 10 ? 10. The 15 independent variables are indicated in red. ................................................................................................................. 53 Figure 35: Experimental setup of backlight module with absorptive filter (10 cm ? 10 cm) on top of uniform brightness backlight. The baffles have a height of 4 cm. ............................... 55 Figure 36: Luminance measurement on screen with absorptive filter (green) and simulation result (red). .................................................................................................................................. 56 Figure 37: A 10 ? 10 matrix with horizontal, vertical and diagonal symmetry has 15 independent variables (shown in Figure 34). Based on these variables, 15 filters were produced with positions of high transmission pixels (white) and zero transmission pixels (black). ........ 57 Figure 38: Line scans of five examples of sub-patterns measured with filters shown in Figure 37............................................................................................................................................ 59 Figure 39: Luminance gradient measured off a grey scale pattern on a film of transparency and a sheet of paper in series. ..................................................................................................... 60 Figure 40: Grey scale of a printed pattern versus the luminance measured off the pattern. A sixth order polynomial is fitted to it, which is used in a MATLAB procedure for calibration. 61 Figure 41: Line scan of luminance measurement on screen with an absorptive filter with varying transmission (circles) and simulation result (solid line). .................................................. 62 Figure 42: Setup with 5 ? 6 filters (25 mm ? 25 mm each) separated by black baffles. Filters are printed on the side facing the backlight and the white side of the paper is facing up (left). When the backlight is on, the filter patterns can be seen (right). ...................................... 63 Figure 43: Measurement data (black) on the screen created by the setup shown in Figure 42. A cosine function (red) with amplitude of 0.8% was fitted to the measurement data. The RMS error was determined to be 0.7%. ............................................................................ 63 xv Figure 44: Measurement results of linear (blue) and quadratic (red) gradients along with fitted polynomials of first and second order (equations are shown in graph). The RMS errors were 3% and 2% for the linear and the quadratic gradient, respectively. ......................... 64 Figure 45: Schematics of the experimental setup with reflective filter out of stacked diffuser sheets on top of uniform brightness backlight. In this example, the transmission decreases from the centre towards the sides. The width of the setup was 10 cm, the baffles were 4 cm high and each diffuser film has a thickness of 1 mm. ................................................. 65 Figure 46: Luminance across the reflective filter: goal (black) and measurement before adding absorptive transparency (red). ........................................................................................... 66 Figure 47: Line scan across screen for setup with reflective filter (circles) and theoretical PSF (red curve). ........................................................................................................................ 68 Figure 48: Experimental setup with LED in recycle cavity underneath an acrylic diffuser, which is surrounded by retro-reflective baffles. .......................................................................... 69 Figure 49: Schematics of different baffle materials: a) opaque, b) specular reflective, c) diffuse reflective and d) retro-reflective. ...................................................................................... 70 Figure 50: Picture of the experimental setup: retroreflective baffles on an acrylic diffuser, which is lit from underneath by an LED in a reflective cavity, but is masked by a filter with four holes (see Figure 37, top centre filter) to measure the sub-pattern of this filter. .............. 71 Figure 51: Experimental setup with LED in recycle cavity with a reduced size underneath an acrylic diffuser, which is surrounded by retro-reflective baffles. The bottom side of the diffuser around the reflective cavity is covered with ESR to reflect light that would exit the acrylic diffuser otherwise. ........................................................................................... 73 xvi Figure 52: Reflective cavity (5mm ? 5mm, red arrow) cut into a piece of acrylic and covered with ESR next to a Canadian penny with a diameter of 19 mm. ...................................... 73 Figure 53: Diagram of optical paths Part 1: The photons are emitted by the LED; the photons are reflected and directional randomized in the reflective cavity; the photons are scattered in the acrylic diffuser. ........................................................................................................... 74 Figure 54: Diagram of optical paths Part 2:  The photons transmitted by the acrylic diffuser can hit the retro-reflective baffles and are reflected back; the other photons hit the acrylic diffuser screen; these photons are scattered by the diffuser and are either reflected or transmitted......................................................................................................................... 75 Figure 55: Picture of efficient multi-LED backlight without screen and with all LEDs switched on....................................................................................................................................... 75 Figure 56: Variation on the uniform brightness screen of the backlight. The measured data (black) has an RMS error of 0.6%. A cosine function (red) was fitted to the data to determine the contrast modulation perceived by a human eye: 0.8%. .............................. 76 Figure 57: Data points extracted from line scans across the screen of backlight, showing the linear gradient (circles) and the quadratic gradient (crosses). The black lines are fits using first (linear) and second (quadratic) order polynomials, respectively. The RMS errors between the complete line scans and the fits are 1.6% (linear) and 1.8% (quadratic). ..... 77 Figure 58: Line scan demonstrating a decrease of light level with increasing distance to LED (dotted, red). The active LED is positioned at ?0?. The solid black line is plotted on a logarithmic scale shown on the axis on the right-hand side. The linearity of the solid curve suggests an exponential drop of the light level. ...................................................... 78 xvii Figure 59: False colour image of a photograph of the experimental setup with 3 ? 5 LED modules and retro-reflective baffles. The flux in each square was determined and normalized to the centre module. 6.7% of the light spread to each sideways adjacent module, while 3.1% spread to each diagonal module. ...................................................... 79 Figure 60: Based on measurements of the small backlight, the flux transfer from one LED module to the next was determined, and a larger display was simulated. The given scenario was that one LED at position '0' was switched on and the amount of flux, which traveled to a distant module is shown in the graph. The ?triangle? data points show the flux transfer for a backlight with opaque baffles, the ?cross? data points are for the same backlight with retro-reflective baffles, and the ?circle? data shows the ray tracing simulation results for a backlight without baffles. ............................................................ 81 Figure 61: Transmission and reflection state of RP-LCDs for orthogonal linear polarizations 1 and 2. Polarization 1 is either transmitted or reflected, whereas Polarization 2 is always reflected. Usually, a recycle cavity is used that randomizes the polarization of the reflected light sufficiently, so that all light can be transmitted in the transmission state. 83 Figure 62: Theoretical working principle of RP-LCDs as additional light modulators. .............. 84 Figure 63: Reflectance (R), transmission (T) and absorption (A) of the RP-LCD in the transmission mode. ........................................................................................................... 90 Figure 64: Reflectance (R), transmission (T) and absorption (A) of the RP-LCD in the reflectance mode. .............................................................................................................. 90 Figure 65: Seven reflectance patterns with RP-LCDs. Grey RP-LCDs are in the reflective state. Gain factors were determined for the white RP-LCDs in the transmission state. ............ 91 xviii Figure 66: Fractional efficiency of RP-LCD backlight module with 2nd order polynomial fit, which is needed for modelling the system. ....................................................................... 95 Figure 67: Picture of 3 ? 3 RP-LCD array with up-scaled LED. ................................................. 96 Figure 68: Picture of the screen of the backlight module with nine RP-LCDs and one LED. ..... 97 Figure 69: Measurement setup for backlight module with one LED............................................ 98 Figure 70: Schematic of the used down-sampling method. A 6 ? 4 matrix (left) with indices from a to x is reduced in size to a 3 ? 2 matrix (right). The entries of the new down-sampled matrix are determined by averaging over the values (va - vx) in the corresponding 2 ? 2 areas of the original matrix. ............................................................................................ 103 Figure 71: Average relative power consumption of backlight with RP-LCDs and different numbers of LEDs. Numbers correspond to the LED resolution. .................................... 105 Figure 72: Average RMS error of  RP-LCD backlights with different numbers of LEDs. ........ 106 Figure 73:  Line scan through one picture. The luminance values of the original picture are shown in black, the simulated luminance values are shown in red (RMS optimization) and green (maximum luminance optimization). ............................................................. 107 Figure 74: Average relative power consumption of backlight with RP-LCDs and different numbers of LEDs, which always produces enough light to show highlights (diamonds) compared to a backlight with minimal RMS error (crosses). ......................................... 108 Figure 75: Average RMS error of RP-LCD backlights with different numbers of LEDs, which always produces enough light to show highlights compared to backlight (diamonds), compared to a backlight with minimal RMS error (crosses). ......................................... 109 xix Figure 76: Normalized and down-sampled modified point spread function measured off the experimental backlight. Every cell represents one LED module and the number in each cell represents the flux in this LED module coming from the central LED. .................. 111 Figure 77: Schematics of the simulation setup to determine the PSF of an HDR display that consists only of LEDs and a diffuser screen. ESR is used as a reflective backplane. The distance between the ESR plane and screen was 20 mm. ............................................... 112 Figure 78: Normalized and down-sampled point spread function simulated for a standard HDR setup. Every cell represents one LED module and the number in each cell represents the flux in this LED module coming from a central LED. ................................................... 113 Figure 79: Relative power consumption for 20 different HDR images for a uniform brightness backlight and a set of different backlights with different numbers of LEDs: 32 ? 18, 48 ? 27, 96 ? 54, 192 ? 108, 384 ? 216, 1920 ? 1080 (with efficiency 1). Every data series has 20 data points, which correspond to the 20 images. ....................................................... 114 Figure 80: Average power consumption of backlights showing 20 HDR images: backlight with modified PSF (diamonds) and standard PSF (squares) with different numbers of LEDs (label). ............................................................................................................................. 114 Figure 81: Average RMS error of backlights having either modified PSF (diamonds) or standard PSF (squares) with different numbers of LEDs. ............................................................. 115 Figure 82: Line scan through one image. Shown are the luminance values of the original picture (black), and line scans through images created by a convolution of the deconvoluted image with modified PSF (red) and with standard PSF (green). .................................... 116 xx Figure 83: Line scan through one picture created with matched luminance goal. Shown are the luminance values of the original picture (black), the luminance values after the iterative increase of the LED values with modified PSF (red) and with standard PSF (green).... 117 Figure 84: Average power consumption of backlights showing 20 HDR images: backlight with modified PSF (diamonds) and standard PSF (squares) with different numbers of LEDs. The relative power consumptions yielded with basic deconvolution are shown as black crosses. ............................................................................................................................ 118 Figure 85: Average RMS error of backlights matching the maximum luminance values of the goal images, having either modified PSF (diamonds) or standard PSF (squares) with different numbers of LEDs. The RMS errors yielded with basic deconvolution are shown as black crosses. .............................................................................................................. 119 Figure 86: Relative power consumption of three different HDR backlights, which provide enough light for highlights. ............................................................................................. 120 Figure 87: RMS errors of three different HDR backlights, which provide enough light for highlights......................................................................................................................... 121 Figure 88: Grey scale HDR image with full resolution: 1920 ? 1080. It was taken by Ryan Whitehead. ...................................................................................................................... 137 Figure 89: Down-sampled image: 96 ? 54. The square root of every pixel value is taken. ....... 138 Figure 90: Down-sampled image: 16 ? 9. The square root of every pixel value is taken. ......... 139    xxi List of Abbreviations  CCD ? Charge-coupled Device: Device that is used as image sensor in digital cameras. CSF ? Contrast Sensitivity Function: Contrast sensitivity is a measure of the ability to perceive a luminance pattern in a static image. The contrast sensitivity function is a function of spatial frequency of the pattern. ESR ? Enhanced Specular Reflector: Highly specular reflective film based on dielectric multilayers distributed by 3M Company [1]. HDR ? High Dynamic Range: In this context HDR describes a large range of luminance values covering at least 4 orders of magnitude. LC ? Liquid Crystal: Optically active material that can rotate the polarization axis of incident linearly polarized light.  LCD ? Liquid Crystal Device (Display): A display that uses liquid crystals sandwiched between polarizers as light modulators. LED ? Light Emitting Diode: A device that uses semiconductor materials to create a p-n transition that emits light when a voltage is applied. LEP ? Light Emitting Plane: Expression is used in this thesis to describe a surface that emits light, in contrast to an LED, which, in macroscopic terms, can be thought of being a point light source. OLED - Organic Light Emitting Diode: A device that uses organic semiconductor materials to create a p-n transition that emits light when a voltage is applied. PSF ? Point Spread Function: The luminance pattern on a screen created by one backlight LED. RMS ? Root Mean Square: Statistical measure of the magnitude of varying quantity. xxii RP ? Reflective Polarizer: An optical element that transmits light of a certain linear polarization and reflects light with a linear polarization perpendicular to the first polarization. RP-LCD ? Reflective Polarizer - Liquid Crystal Device: A cell filled with liquid crystal material sandwiched between two films of reflective polarizer. Depending on the applied voltage signal, it either transmits or reflects most of the light. TN ? Twisted Nematic: Type of liquid crystal alignment in displays. In this case a twisted configuration (helical structure) of nematic liquid crystal molecules is formed between the two electrodes. xxiii Acknowledgements  I owe my gratitude to all those people who have made this dissertation possible.   First, I would like to express my gratitude to my supervisor Prof. Lorne Whitehead who guided me with his immense knowledge and ideas. Thank you for your support. He opened my eyes to the exciting world of applied research.   I would like to thank my PhD committee members, Dr. Andrzej Kotlicki, Prof. Doug Bonn, Prof. Jeff Young and Prof. Birger Bergersen, for their guidance and support during my research project and the writing of my thesis.   I would like to thank the SSAP lab manager Dr. Michele Mossman for her scientific input and for her successful efforts to always provide a productive work environment.   Thanks also to all of my lab mates, especially Sepideh Khosravi, Jason Radel and Angel Valerio, for the fruitful discussions during every day's lunch and coffee breaks, and for all the fun we have had in the last four years.  Finally, I would like to thank my family and in-laws for their encouragement and support from afar and my friends for always cheering me up and making my life in Vancouver as pleasant as it could be. xxiv Dedication  I dedicate this dissertation to my wife Carmen for her unshakeable love and support.   1 Chapter  1: Introduction  The purpose of many displays is to show images that reproduce the luminance and color range of real scenes.  From a colour standpoint this has been almost achieved: the colour range (or gamut) of displays covers most of the colours perceived by humans. Also, displays are getting larger and larger to be able to show content in a more realistic size and create an immersive effect previously only attainable in movie theatres. However, the luminance range shown by common displays is dramatically lower than the dynamic range of the human visual system - typically by two to three orders of magnitude [2].  High dynamic range (HDR) images can contain a large range of luminance values, spanning the entire dynamic range of the human visual system. HDR content is now widespread through the introduction of HDR camera settings in which a series of pictures is taken with different exposure times and are automatically blended together to produce one HDR image. Compared to standard images, HDR images can show more detail in the dark and in the bright areas of the picture [3]. Due to their limited contrast, ordinary (non-HDR) displays need to reduce the dynamic range of HDR images by a process called tone-mapping [4]. Because of this compression of dynamic range, most tone-mapped images appear quite unlike the original real scene.  Currently, HDR displays, built only in very small numbers for professional use, are the only displays that can show HDR images and videos with realistic luminance values. HDR displays use an array of ultra-bright individually tunable light emitting diodes (LEDs) to backlight a common liquid crystal display (LCD). This combination can show images with very bright 2 highlights while maintaining very low luminance values in dark areas. The dynamic range of such displays is comparable to the capabilities of the human eye.  Another important factor of new displays is power consumption. The number of consumer electronics items in households in developed countries has seen a drastic rise over the course of the last ten to twenty years [5], and with it, the total power consumption of this sector has been rising. A study from 1999 shows that consumer electronics in U.S. homes account for more than 10% of residential power consumption. This is comparable to the amount of power consumed by lighting in these households [6]. The study also finds that televisions consume more than a quarter of this 10%, leading the list of single product power consumption. From 2000 to 2009 the average number of TV sets in U.S. homes increased by roughly 18% to 2.9 per household [7]. Bearing in mind these facts, it is apparent that displays with improved power efficiency could significantly reduce total power consumption. In LCD screens, 90% of the power is used in the backlight [8]. Thus, improving the backlight's efficiency is the most effective way to save energy. The methods to improve efficiency include global backlight dimming [8], where the whole screen becomes dimmer for darker scenes, and local dimming of LEDs in an array [3], [9], [10], [11] and [12], where backlight LEDs are dimmed only in localized dark image regions. This is the method that HDR displays are able to employ to reduce power consumption without sacrificing image quality [2].  Current HDR displays have limitations associated with displaying images with spatially uniform luminance levels. If the light from each LED is mainly localized to its immediate vicinity, (which is desirable for local dimming), often an objectionable periodic luminance pattern is apparent, arising from the underlying pattern of the LEDs themselves. The problem is 3 caused by the LED?s point spread function (PSF): the luminance pattern created on a screen by the light output of a single LED. In this dissertation, a novel PSF for HDR displays is presented, which solves the problem of uniformity while maintaining high local contrast.  Another improvement is aimed towards higher local contrast and less light absorption through the front LCD, through use of electronically switchable RP-LCDs: liquid crystal devices sandwiched between two sheets of reflective polarizers. These devices can either transmit or reflect light on its way from the LEDs towards the screen. With several RP-LCDs within the region of the PSF region of one LED, the RP-LCDs can reflect the light in dark areas of a given image. This light can be recycled to increase the brightness of the bright areas, hence the local contrast is enhanced and the LED can be dimmed to save energy. In standard HDR displays this light would have been absorbed by the front LCD, which increases the power consumption. This dissertation has the following structure: in Chapter 2 background information on the human visual system, the HDR display and competing display technologies are discussed. A novel PSF which solved the intrinsic trade-off in current HDR displays between the uniformity and contrast is introduced in Chapter 3. A practical way to produce a similar PSF is shown in Chapter 4. In Chapter 5, the HDR backlight with RP-LCDs is described. Chapter 6 gives an overview of the power consumptions of various backlights including the ones developed in Chapters 4 and 5. Conclusions of this research are discussed in Chapter 7. 4 Chapter  2: Background  This chapter introduces the necessary background for the dissertation. It covers aspects of the human visual system with its strengths and weaknesses, as well as giving an overview of the work that has been done on the high dynamic range (HDR) display and selected additional display technologies. 2.1 Aspects of the Human Visual System The human visual system is a complex system carrying out multiple tasks, such as reception of light, identification of objects and patterns, and determining distances to and between objects. Humans are able to see very faint stars in a dark night and very bright highlights under direct sunlight. However, there are also a few limitations to our visual system, which one can take advantage of. This section focuses on the capabilities and limitations of our visual system, which play a role in HDR displays. 2.1.1 Dynamic Range The human eye has two different kinds of photoreceptors, called the rods and the cones. The rods can cover a luminance range from 10-6 cd/m2 to approximately 101 cd/m2 and the cones? range is from 10-3 cd/m2 to approximately 108 cd/m2. Depending on the light environment, the two kinds of photoreceptors either work individually or together. Rods are more sensitive to light, which is why they are responsible for night vision. Unlike rods, cones contain one of three types of pigments (red, green, blue), which enables us to perceive colour. Since we have about 5 20 times more cones than rods and the response time to stimuli is faster for cones, we are able to see much finer detail and more rapid changes when the cones are in use.1 A complex adaptation system is in use to maintain sensitivity over this very wide range of luminance. One component of this system is the changing retinal sensitivity, which decreases rapidly with increasing luminance. This effect decreases the instantaneous receptor range by about 3 to 4 orders of magnitude. In combination with other adaptation methods such as pupil changes and pigment depletion, the near-instantaneous receptor range increases to 5 to 6 orders of magnitude 2. It is the goal of HDR displays to be able to show this kind of range in luminance. 2.1.2 Local Contrast We are able to distinguish a wide range of luminance values across one scene, but have very limited local contrast. For example, we see bloom and flare lines around bright objects, caused by optical imperfections within our eyes: scattering in the cornea, lens and retina; and diffraction in outer areas of the lens. Due to these effects, the maximum perceivable contrast at a high contrast boundary is approximately 150:1 [13]. Boundaries with higher contrast than this are perceived as blurry and differences in relative magnitudes cannot be discerned. The results of the scattering and diffraction effects can be combined into a so-called veiling luminance or veiling glare [14], which limits the perceived contrast in small regions and depends                                                   1 A good review about the three luminance ranges and colour perception can be found in [56]. 2 More information on the visual adaptation is found in [57]. 6 on the angle of separation between the bright and dark area, the luminance and the solid angle of the source. The effect of veiling luminance is used to hide imperfections of the HDR display: Due to the blur of the backlight, caused by the relatively low resolution of the LEDs compared to the resolution of the front LCD, sharp very high contrast boundaries (larger than the contrast ratio of the front LCD) are not possible, since some of the light needed for the high-luminance area will spread into the dark area. However, no observer will notice this if the light leakage is smaller than the veiling luminance created by the high-luminance area (see Figure 1).  Figure 1: Comparison of display and eye introduced blur at contrast boundaries of 1 cd/m2 to 10,000 cd/m2, 10,000 cd/m2 to 5,000 cd/m2 and 5,000 cd/m2 to 1 cd/m2. Data from [3]. 2.1.3 Contrast Sensitivity Another critical ability of the human visual system is to distinguish an object from its background. This ability depends on the differences in luminance and in colour, with the 1101001000100000 5 10 15 20 25 30Log Luminance (cd/m2) Horizontal position on screen (mm) Desired ImagePerceived ImageFinal HDR Image7 luminance having a larger effect than colour [15]. Since relative, rather than absolute, difference in luminance is most important, a variable called contrast modulation is used, which is defined in [16] as:                             (2.1) where L+ and L- stand for the maximum and the minimum luminance, respectively. The higher the contrast modulation, the easier it is to distinguish an object from its background. Often used is also the reciprocal of the smallest visible contrast modulation, this is called contrast sensitivity. To determine the smallest visible contrast modulation, most studies use sinusoidal test patterns with different contrast modulations and a range of frequencies, since a strong frequency dependence has been found. Commonly, the sinusoidal test patterns are rotated by a random angle and the observer has to specify the angle. Based on the correctness of the angle, it is determined if the pattern was visible or not.3 It is common to perform a Fourier analysis of a given image to determine if a pattern is visible to the human eye or not. The contrast modulation of the main frequencies in the image can be compared to the contrast sensitivity function, which is the contrast sensitivity as a function of angular frequency [17]. With this method, the visibility of a pattern in an image can be determined.                                                   3 More information on how these measurements are taken can be found in [15]. 8  Figure 2: Example of typical contrast sensitivity function. Measured data from [18], fit curve from [19]. The following characteristics are common in contrast sensitivity functions in a log-log plot: the maximum sensitivity is reached between 2-5 cycles/degree, a nearly linear decrease towards low frequencies and no sensitivity for frequencies higher than 50 cycles per degree. An example of a contrast sensitivity function can be found in Figure 2. In this graph, patterns with lower contrast sensitivity than given by the fit curve would be visible to the human eye; patterns with higher contrast sensitivities cannot be perceived. With common viewing distances of three to five meters and LED spacings of the backlight of around 1 cm, the range from 0.1 to 1 cycle per degree is most relevant for the research in this dissertation. 9 2.2 Dynamic Range and Static Contrast in Displays Dynamic range and contrast ratio are important specifications for the capabilities of displays. Unfortunately, improper use of these terms for advertisement has caused considerable confusion; for this reason their scientific definitions are presented here. Dynamic range or dynamic contrast ratio is the ratio of the highest luminance the display can produce under any device setting to the lowest luminance it can produce under any device setting. The Static contrast ratio (or simply called the contrast ratio and abbreviated CR) is the maximum ratio of simultaneous luminance values on the screen. To increase the dynamic contrast ratio, LCD displays often have the ability to dim their uniform brightness backlight globally for dark scenes in a video and increase the backlight output for bright scenes. However, a large dynamic contrast does not give an advantage within one single frame of a video or in still images. In this case the CR limits the image quality. Hence, the CR is considered the more useful metric for display specifications than the dynamic contrast [20]. Nowadays, common LCD displays have a contrast ratio of 1000-2000:1, with a peak luminance of about 300 cd/m2. The limiting factors of the contrast ratio in an LCD display are the liquid crystals themselves. The liquid crystal material, sandwiched between two absorptive polarizers, works as a light modulator, permitting variable degrees of light transmission as the degree of alignment of the molecules is adjusted by an applied electric field. But even in the lowest transmission state there is still some light leakage, so that the minimum luminance is non-zero. This is what limits the contrast ratio of LCD displays. 10 2.3 High Dynamic Range Display As mentioned in the previous section, LCDs have a finite contrast ratio. To increase the contrast ratio of an LCD display, a second light modulator is needed [3]. Since contrast ratios are multiplicative, the following relation holds: Two modulators with CRs of c1:1 and c2:1 in series will theoretically produce contrast ratio of (     )  . In the early stage of HDR displays, two prototypes were built [21], one with a projector as a backlight and one with an LED array as backlight for a common LCD display. For these prototypes, contrast ratios well beyond 50,000:1 were measured, with a peak luminance of 8,500 cd/m2 [21]. These HDR displays could show sparkling bright highlights, while still maintaining a very dark black. The newest HDR displays are currently developed by Dolby [22]. The technique of using a backlight with spatially varying intensity to produce a high dynamic range image is known as local dimming.  This approach has the additional advantage of reducing the power consumption compared to the power required by a uniform backlight, due to the locally reduced power density associated with the local dimming. The average power level compared to a screen with uniform brightness backlight has been determined to be 25-30% [2]. User studies have shown that HDR images are strongly preferred over low dynamic range images. The same study showed that higher peak luminance and higher contrast improve the 11 perceived image quality [23]4. The discussed benefits are the reason for choosing the HDR technique as a starting point of this dissertation. Currently, HDR displays can produce one of the highest image qualities of displays on the commercial market. But further improvements are needed to produce images that are perceived as perfectly realistic to humans. 2.3.1 Perceptually Accurate Dynamic Range Another important factor in displays is the range of luminance values over which the luminance can be controlled reasonably accurately, which is known as the perceptually accurate dynamic range. Any display will have some non-zero error in luminance value, arising from inaccuracy of the display?s electro-optic response, or the digitization error and/or electronic noise in the control signal.  At lower luminance levels, the fractional size of this error becomes larger, such that below a certain threshold luminance value, the error is perceptible, which of course is undesirable.  Normally, the perceptually accurate dynamic range will be the display?s peak luminance divided by that threshold value. Common LCD displays use a non-linear 6-bit or 8-bit modulator, which typically means that an individual LCD pixel can be actuated by a signal determined by a 6-bit binary number (0 to 63 in base 10) or an 8-bit binary number (0 to 255 in base 10).  Often the output luminance is approximately proportional to the square of the corresponding control number, which makes the luminance step size smaller at lower luminance values, which in turn helps to extend the                                                   4 A comprehensive overview of the HDR display technology can be found in [2] 12 perceptually accurate dynamic range.  An HDR display has a wider range of luminance values and therefore should have an even greater perceptually accurate dynamic range.  For this reason, the control signal for an HDR display should be at least 14-bit which enables the range from 0.05cd/m2 to 4,000cd/m2 to be displayed with perceptual accuracy [2]. In HDR displays, an even higher bit depth can be obtained by using an 8-bit controller for the LEDs in the backlight and another 8-bit controller for the LCD pixels in the screen, which results in an effective ?bit-depth? of 16. 2.3.2 Current Problems of HDR Displays High contrast HDR displays exist at the cost of uniformity. This is the intrinsic trade-off in HDR displays. Standard HDR displays have the problem that in spatially uniform images a pattern is perceptible, caused by periodic placement of the LEDs. More light diffusion could solve that problem, but this decreases the maximum contrast ratio. The problem is the point spread function (PSF), which is the luminance pattern created on the screen by one LED: generally it has a peak value directly in front of the LED in question and it decreases monotonically and asymptotically toward zero with distance from the peak.  Previously used PSFs have all been problematic.  If they are broad enough to combine to provide uniform luminance, it is then impossible to produce high contrast local dimming.  Conversely, PSFs that are sufficiently narrow to enable high contrast local dimming cannot blend to produce uniform illumination. Generally the compromises selected in HDR displays have been problematic.  13 2.4 State of the Art of Displays 2.4.1 Liquid Crystal Displays Most flat-screens used in laptops or as TV/computer screen are LCDs. These displays are non-emissive, which means that they do not emit light, but rather modulate light coming from a backlight. This light modulation can be performed by the combination of a first absorptive polarizer, a liquid crystal cell and a second absorptive polarizer, which is rotated by 90? with respect to the first. The liquid crystal cell can be electrically controlled to rotate the polarization of the light by a given degree. One absorptive polarizer absorbs theoretically half the light of randomly polarized light, the other half is transmitted. The intensity of the polarized beam transmitted through the second polarizer is given by the equation                         (2.2) where the angle ? is the relative rotation angle between the two polarizers [24]. If the liquid crystal cell rotates the linear polarization coming through the first absorptive polarizer by 90?, then the maximum amount of light can go through the second absorptive polarizer: this is called the white state. If the liquid crystal cell does not rotate the incoming linear polarization at all, the minimum amount of light goes through: this is called the black state. With any rotation between 0? and 90? intermediate grey levels are yielded. A liquid crystal (LC) material with the described behaviour is called twisted-nematic. There are other LC materials with slightly different cell setups, which might increase the contrast or the viewing angle, including In-Plane-Switching (IPS) and Multiple Vertical Alignment (MVA) [25]. 14 To make a display show colours, red, green or blue colour filters are added to every single liquid crystal cell to form a sub-pixel. The three of them combined form one LCD pixel. The white backlight is usually uniform and is created by light sources located at the side of the active area shining into light guides, which distribute the light more or less uniformly (shown in Figure 3). Inexpensive versions of LCDs usually come with cold-cathode fluorescent lamps as backlight sources; higher quality panels use LEDs.  Figure 3: Example of an acrylic light guide (not to scale) with an LED light source which can illuminate a LCD screen (not shown). The light is contained in the light guide by a specular reflector on one side and total internal reflection at the top side, if the angle of incidence larger than the critical angle (around 42? for acrylic). A way to save energy in LCD displays is backlight recycling; the use of one sheet of reflective polarizer (RP) within the display can improve the efficiency by around 60% [26]. Reflective polarizers transmit one direction of linear polarization and reflect light that is polarized perpendicularly. In LCD displays without the RP, 50% of the backlight is absorbed by the first absorptive polarizer of the LCD, but the RP can reflect this light, which then can be reused. Nowadays, all standard LCD displays employ one layer of RP to increase the efficiency. Two other ways to reduce power consumption are a) the previously mentioned global backlight 15 dimming in dark scenes and b) the decrease of the maximum brightness of displays; this however degrades image quality substantially, especially in bright viewing environments [23]. 2.4.2 Low Dynamic Range Direct-LED LCD Displays In the last 2-3 years, low cost LCD displays with direct-LED technology have been introduced by major display manufacturers. With only tens of LEDs and strong light diffusion, uniform brightness backlights can be produced, which are cheaper than edge-lit backlights, since no special light guide is necessary to distribute the light evenly. However, this strong light diffusion diminishes any local contrast, so that only global backlight dimming is feasible with these backlights.  2.4.3 Field Emission Backlit LCD Displays In recent years, LCD backlights using carbon nanotube field emission have been developed by Samsung [27] and other display manufacturers. In these backlights carbon nanotubes are used as electrodes, which emit electrons when voltages of around 15kV are applied. As in cathode ray tube displays, the electrons penetrate different phosphors, which emit red, blue or green light. Contrasts of the same order of magnitude as for the early stage HDR prototypes were found for these displays. However, this technology suffers from high power consumption and emission of strong radio-frequency electromagnetic radiation. 2.4.4 Plasma Displays A plasma display is an emissive display where the sub-pixels are made out of phosphors, which emit red, green or blue light. In this case phosphor is not being excited by electrons as in 16 cathode ray tubes, but rather by ultraviolet photons. Each sub-pixel consists of a small separated chamber filled with neon (Ne) and xenon (Xe). Ignited by a transistor the gas can form a plasma, which emits the ultraviolet photons required to excite the phosphor at the walls. The front of the cell is clear so that the visible light emitted by the phosphor can be seen by the viewer. Plasma displays can usually show higher contrasts than common LCD displays, since each pixel can either create or not create light; there is not light leakage in dark areas of a shown image. However, they suffer from high power consumption due to poor luminous efficacy of only 2-2.5 lm/W [28], compared to luminous efficacies of LEDs with up to 150 lm/W [29]. 2.4.5 Organic Light Emitting Diode Displays Organic Light Emitting Diodes (OLEDs) are organic compounds sandwiched between two electrodes, which emit light in response to electric current. In OLED displays each sub-pixel consists of an OLED (either red, green or blue), which produces emitted light within this sub-pixel. Since no backlight is needed for OLED displays, higher efficiencies and contrasts are theoretically achievable with this technology.  At the moment, the OLED industry struggles with low efficiency and short lifespans of blue OLEDs [30]. The different time-scales of degradation of the different colours can cause colour shift. To always accurately reproduce the correct colours, OLED displays need to be colour calibrated throughout their lifetime. This has been the major handicap preventing OLED displays from becoming the new display standard. 17 2.4.6 Recent Developments in High Dynamic Range Technology  Due to relatively high cost, HDR displays are currently produced only in small numbers, mostly for video-editing purposes in the motion picture industry. While first prototypes of HDR displays had LED spacings of about 25 mm, newer versions have spacings of only a few millimeters. However, even with these small spacings, the intrinsic problem of the trade-off between the uniformity and local contrast persists. The issue of the uniformity of HDR backlights has been recognized and studied by industry and academia [31], [32], [33]. Usually a certain degree of non-uniformity is accepted to increase local contrast, but no one has been able to take advantage of the full contrast potential of HDR, due to the inevitable high degree of non-uniformity. The traditional method is to reduce the artifacts of the backlight pattern with the front LCD. This however, can be computationally intense; especially in the case of wide PSFs, for which a larger number of LEDs needs to be taken into account for every single pixel on the front LCD. In [31], an algorithm is described that reduces a commonly found contrast of 2.5% to 1%. However, it is highly questionable if this algorithm can be used to correct in real-time, especially, since display manufacturers try to decrease the refresh rates below 5 ms to improve the image quality of video games. There have been attempts to solve the trade-off between the uniformity and contrast by changing the point spread function. In [32], 2-dimensional super-Gaussians are used as point spread functions, which are optimized with the goal of uniformity. The reported contrast on the ideally uniform screen was 18.9%. To improve the uniformity, some of the excess light in bright areas was absorbed by the front LCD. After two iterations the non-uniformity was reduced to 18 1.4%. In [33], a shape of a 3-dimensional refractive optical device (i.e. similar to a lens) is calculated, which creates a fairly uniform brightness PSF in a hexagonal shape with a root mean square (RMS) error of 3.2% within this hexagon. In these last two cases, the PSFs have no or minimal overlap with the surrounding PSFs, which makes real-time corrections feasible. However, when there is no overlap between PSFs, luminance gradients across several PSFs result in ?staircase? luminance values provided by the backlight. Artifacts caused by abrupt changes in the backlight are difficult to compensate with the front LCD. 19 Chapter  3: Point Spread Function that Creates a Uniform Backlight and Ideal Gradients  In this chapter, the concept of point spread functions (PSF) is introduced. The intrinsic PSF of LEDs is discussed and issues of current HDR displays caused by this PSF are considered. A new PSF is proposed and compared to the standard PSF. An experimental setup that can create this new PSF is described and measurement data are shown. In HDR displays, backlight LEDs are located directly behind the LCD, which enables local dimming, the reason why such a wide range of luminance values can be observed on the screen. However, when displaying an image that has spatially uniform luminance levels, a pattern caused by the periodic placement of the LEDs or by shadows of separation walls is perceptible within the image. The shape of the PSF is the cause of the non-uniformity on the screen. A PSF describes the spatial luminance distribution on a screen coming from a point source. If only one LED was illuminated in a standard HDR display, a circular patch of light with a bright centre would be visible on the screen. This is the PSF of one LED. Without taking into account back reflections from diffusers or reflective polarizers, the rotationally invariant PSF of an LED can be described by the following equation:  ( )        ( ) ( ?     )             (3.1) In this equation L(x) is the luminance at a given position, L0 is the maximum luminance at position x = 0, a stands for the fixed distance between the LED and the screen, and x denotes the position on the screen in one dimension. One cosine factor comes from the fact that LEDs can be 20 considered to be Lambertian emitters [34]. The angular radiation pattern of a Lambertian emitter can be found in Figure 4. The second cosine in Eq. 3.1 is needed because light hits the screen at angle  . The light falloff over large distances is governed by the inverse-square law, which results in a very slow decrease towards zero (see Figure 8 on page 25).  Figure 4: A cross section of the radiation from Lambertian surface, which is directly proportional to the cosine of the angle between the observer's position and the surface normal. The number of photons per second directed into any wedge is proportional to the area of the wedge [35]. In HDR displays, the addition of diffuser sheets and reflective polarizers substantially increases the width of a PSF. This decreases the local contrast since dark areas of the screen are still illuminated by distant bright LEDs. Another issue with this slow decrease toward zero becomes critical when the non-uniformities of the backlight are corrected by the front LCD, which is described in [9] and [31]. In this case, the backlight luminance for every pixel of the front screen needs to be calculated in real-time. Considering that a screen with a current standard resolution (1080p or 1080i) contains 2.1 megapixels, this is a difficult computational challenge to carry out at video rate.  A PSF that slowly decreases toward zero increases the number of 21 LEDs that need to be taken into account in this calculation. For a standard HDR screen, the calculation involves thousands of LEDs, which is not practical. To summarize, the intrinsic trade-off between contrast and uniformity creates a design conflict when selecting the width of the PSF: Too narrow and the light doesn?t blend; too broad and the local contrast diminishes, since it is not possible to illuminate the region in front of just one LED.  3.1 Novel Point Spread Function in 1D and 2D This section describes the design challenge in mathematical terms and describes the PSF that solves this problem. In 1D, the constraints are that the LED spacing interval is 1 spatial unit and the average screen luminance when all LEDs are on is 1 luminance unit. The ideal PSF should have the following characteristics: ?  ( )     when the absolute value of x exceeds a specified cut-off (|x| > C) ?  ( )   (  ) ? The cut-off should be as small as possible, while maintaining overlap and ?smoothness? as described below. In addition, the desirable characteristics are as follows: ? If the LEDs are uniformly illuminated, the luminance on the screen is perfectly uniform: ( ( )   ), ? If the LEDs are illuminated with a linear gradient ( ( )      ), the luminance on the screen is a linear gradient,  22 ? If the LEDs are illuminated with a quadratic gradient ( ( )          ), the luminance on the screen is a quadratic gradient. These goals cannot be achieved with a cut-off of C = 0.5 unit spacings, since there would be no overlap between the PSFs. The next larger cut-off is C = 1.0 unit spacing, which can be realised with a triangle function (see Figure 5). If this shape would be used as a one-dimensional PSF a constant luminance and a linear gradient could be yielded. For this, an LED spacing of 1 unit spacing is necessary and the output of each is either constant or tuned to be linear increasing from one LED to the next.  Figure 5: A triangle PSF with cut-off of 1 and integrated luminance of 1. If the output from one LED to the next is increased quadratically, small dis-continuities can be observed at the locations where two PSF meet (i.e. at every LED position). The resulting luminance pattern is a piece-wise linear curve describing a quadratic increase, but it is not a smooth quadratic gradient (slightly visible in Figure 6). The human eye is trained to detect 00.250.50.751-2 -1 0 1 2Luminance Position on screen (unit spacing) 23 changes in luminance, and therefore the spatial derivative of a given luminance curve is meaningful. The derivative of the luminance curve yielded by a quadratic increase in LED output from one LED to the next is shown in Figure 7. The dis-continuities in the first derivative at the position of integer values of unit spacings could be apparent to the human eye.  Figure 6: 1D blending of seven LEDs (located at position 1-7) with a triangle point spread function shown in Figure 5. A uniform luminance level (blue) can be achieved as well as smooth linear (red) gradient. When the output of each LED is increased quadratically (green) small dis-continuities in the slope are found at the position of each LED (e. g. at 1 unit spacing). 0204060801000 1 2 3 4 5 6 7 8Luminance (a. u.) Position on screen (unit spacing) 24  Figure 7: Spatial derivative of luminance when output of each LED is increased quadratically from one LED to the next. Discontinuities are observed at every LED position (i.e. at integer unit spacings) It has been shown that the triangle PSF does not fulfill the requirements that were set for a PSF, since it cannot produce smooth quadratic luminance gradients. However, a PSF that has the shape of a quadratic B-spline (see Figure 8) with a cut-off of C = 1.5 unit spacings satisfies all the constraints ? for  | |     :       ( )         ? for      | |     :  ( )     (| |     )  ? for  | |     :    ( )               (3.2) -50-40-30-20-10010200 1 2 3 4 5 6 7 8d Luminance / dx Position on screen (unit spacings) 25  Figure 8: Comparison of proposed PSF (solid) and Lambertian PSF (dotted) with the same full-width at half-maximum and integral. Splines are commonly used as fitting tools in signal and image processing5. A continuous representation of a discrete signal can be yielded by fitting the data with splines in one or more dimensions. Two examples are (a) increasing the resolution of an image through interpolation or (b) curve fitting to discrete data points (Figure 9).  The quadratic B-spline consists of three piecewise connected polynomials, which are smoothly connected to have the same slope and the same curvature at the connection points (i.e. C2 continuous). If the quadratic B-spline is used for fitting, the resulting curve is continuously differentiable. When this spline is used as PSF, the resulting luminance on the screen can be                                                   5 More detailed information on splines can be found in [58]. 00.250.50.751-6 -4 -2 0 2 4 6Luminance Position on screen (unit spacings) 26 calculated through a convolution of this spline with Dirac delta functions at the locations of the LEDs [31]. Each Dirac delta function is multiplied by a value representing this LED?s flux output. The graphical representation of this method is shown in Figure 10, where a linear increase in LED outputs is calculated in one dimension.   Figure 9: Example of fitting discrete data points (circles) with splines to yield a continuous curve, for interpolation or differentiation. The key value of this approach is that the curve at any point is entirely determined by a few closest points, it passes through those points exactly, and the curve is continuous in both first and second derivatives. 00.510 5 10 15 20 2527  Figure 10: Creation of linear gradient through the convolution of the quadratic B-spline with linear increasing LED outputs at the LED locations (integer unit spacings). The colour curves are the scaled PSFs of seven LEDs and the black curve is the linear gradient yielded by the summation over the contributing LEDs. In the same way as demonstrated in Figure 10, a linear arrangement of LEDs with the PSF described in Eq. 3.2 can also create a constant luminance level and a quadratic gradient (Figure 11). In both figures imperfect edge effects can be seen in the areas of the first and the last unit spacing introduced by the spread and the overlaps of the PSFs. This effect only plays a role, when an LED is not emitting any light or on the edge of an LED backlight, when there is no neighbouring LED. All overlapping PSFs have this effect, but it is reduced by using the quadratic B-spline instead of the original Lambertian PSF of an LED, as can be seen in Figure 16 on page 32. 01234567-1 0 1 2 3 4 5 6 7 8 9Luminance (a. u.) Position on screen (unit spacings) 28  Figure 11: 1D blending of seven LEDs with the new point spread function. A constant luminance level (blue) can be achieved as well as perfectly linear (red) and quadratic (green) gradients.  Figure 12: Spatial derivative of luminance when the new PSF is used and the output of each LED is increased quadratically from one LED to the next. The gradient is perfectly smooth, unlike in Figure 7. 0255075100-1 0 1 2 3 4 5 6 7 8 9Luminance (a. u.) Position on screen  (unit spacings) -60-50-40-30-20-1001020-1 0 1 2 3 4 5 6 7 8 9d Luminance / dx Position on screen (unit spacings) 29 The next task was to find a 2D version of this conceptual approach. A rotationally invariant PSF with the shape of the quadratic B-spline would have the following form:  (   )   ((     )   )                 (3.3) On a rectangular grid, a rotationally invariant version of the quadratic B-spline can create constant luminance and gradients only in certain areas of the screen: along lines in x or y-direction of the grid that are directly on top of LEDs. In the case of constant LED outputs across all LEDs, the areas in between the position of four LEDs would have a luminance lower than the surrounding areas.  Another way to set up a 2D function is to create it as a multiplicative separable function that can be described by the following equation:  (   )   ( ) ( )         (3.4) Two different visualizations of the 2D PSF are found in Figure 13 and in Figure 14. As with the rotationally invariant case, this approach can also create a constant luminance along lines in x and y-direction on top of LED positions. However, with this function the areas between LEDs have the same constant luminance. In Figure 14 it can be seen that this function is not rotationally invariant, especially the orange and yellow parts in that figure show an increased light level in the four ?corners?, which increases the light level in areas surrounded by four LEDs. Hence, with this 2D PSF and an array of LEDs located on a square grid with a spacing of 1, a completely uniform brightness screen is obtained. In addition, perfectly linear and quadratic gradients can be shown in any direction, not only along the x or y-axis of the square grid. 30  Figure 13: 2D point spread function in 3D. The area has a size of 3 ? 3 unit spacings.  Figure 14: 2D point spread function is not rotationally invariant. The area has a size of 3 ? 3 unit spacings. 31 3.2 Comparison to Lambertian Point Spread Function To show the benefits of the new PSF with the shape of a quadratic B-spline, it is compared to a Lambertian PSF, which is the PSF of a standard LED (Equation 3.1). The width of the Lambertian PSF is adjustable since it depends on the screen distance at which the PSF is measured. The chosen value was set so that the full-width half-maximum is the same for the two functions.  Figure 8 on page 25 shows the two functions in one dimension. The quadratic B-spline is very localized with a cut-off at 1.5 unit spacings. In contrast, the Lambertian PSF has non-zero contributions at distances larger than 6 unit spacings.  Figure 15 shows the luminance on a screen created by a linear arrangement of 7 LEDs, which are controlled to have either constant or linearly increasing output. The black lines correspond to the new PSF and the red lines to the Lambertian PSF. It can be seen that the new PSF creates a perfect uniform brightness screen and a linear gradient. On the other hand, the Lambertian has an overlying oscillation pattern and is unable to provide a uniform screen or linear gradient.  Figure 16 compares the light falloff at the end of a linear arrangement of 10 LEDs from 90% to 10%. The new PSF falls off within 1.1 unit spacings, the Lambertian PSF spreads out further and needs 3.1 unit spacings to decrease from 90% to 10%. This shows that the new PSF provides both uniformity and high local contrast and solves the intrinsic trade-off between uniformity and contrast commonly found in HDR displays: wide Lambertian PSFs can be used to create a uniform screen, but contrast will be substantially reduced, at the same time, narrow Lambertian PSFs can create high contrast, but only at the cost of poor uniformity. This new solution is a break-through in HDR display technology. 32  Figure 15: Output of seven LEDs with new PSF (black) and Lambertian PSF (red). Goal is a uniform brightness screen and a linear gradient.  Figure 16: Comparison of luminance fall-off from 90% to 10% between new PSF (1.1 unit spacings) and Lambertian (3.1 unit spacings). 00.20.40.60.81-2 0 2 4 6 8 10Luminance (a. u.) Position on screen (unit spacings) 00.20.40.60.817 8 9 10 11 12 13 14 15Luminance (a. u.) Position on screen (unit spacings) Lambertiannew PSF90% 10% 33 Figure 17 compares the new PSF (left) and the Lambertian PSF (right) in two dimensions. Every subfigure was created by a simulation of 10 ? 10 LEDs. In the top subfigures, all LEDs have the same output. The new PSF creates a perfectly uniform brightness screen, but in the case of the Lambertian PSF every single LED is clearly visible. In the centre subfigures LED output has a linear increase from the left bottom corner at an arbitrary angle of 28? from the horizontal plane. The bottom subfigures show a quadratic increase in output at the same angle. The gradients produced by the new PSF are perfectly linear and quadratic, respectively.   34  Figure 17: Simulation of 10 ? 10 LED arrays with the new PSF (left) and Lambertian PSF (right): Uniform brightness screen (top), linear gradient at 28? (centre) and quadratic gradient at 28? (bottom). 35 3.3 Experimental Implementation of Backlight with Novel PSF The next step was to build an experimental implementation of an HDR backlight with the new PSF. In the first approximation of the PSF, a distance between the LEDs and the screen was chosen, which, together with the diffuser screen, produced a shape similar to the desired PSF. This, however, did not produce the sharp cut-off necessary for the new PSF. An optical barrier was used to achieve the cut-off. To get a close match between desired and actual PSF, a customized transmission filter with a certain 4-fold symmetric distribution was used. In this experimental setup an absorptive filter and walls were used to decrease reflections and light leakage. Reflective materials would be favourable from an efficiency point of view, but at this point the experimental setup was built to prove the benefits of the new PSF under real conditions. In the end, the screen should appear to be uniform to the human eye and able to show smooth linear and quadratic gradients.  Figure 18: Schematic of experimental setup that created the new PSF. Due to edge effects (see Figure 11 on page 28), there was one LED spacing of non-active area around the active screen. The active area was chosen to be 2 ? 3 units with a unit spacing of 10 cm. The whole experimental setup consisted of five columns of four LEDs electrically connected in series. Hence, each column was individually dimmable with a potentiometer to create gradients across the screen. 36 The LEDs emitted a phosphor-based white spectrum. These LEDs are known for a colour difference between light emitted at normal angle and wider angles [36]. The emission spectrum of the LEDs was measured at different angles: 0?, 45? and 90?. This colour shift created a colour pattern on the screen with a more bluish light directly on top of LEDs and a more yellowish light between LEDs. The colour difference between the light emitted at normal angle and at 70? was determined with a spectroradiometer  [37] to be ?x = 0.036 and ?y = 0.05 in the CIE XYZ colour space. To combine the difference in one number the following equation was used: ?    ??    ?          (3.4) The total colour difference ?xy for the LEDs was 0.062. To hide this colour pattern a randomization element on the LED was necessary to mix the colours. A diffuser cap (shown in Figure 20) was attached on every LED. With the diffuser, the colour difference was reduced to ?xy = 0.004. Due to the fact that any point on the screen was illuminated by light from multiple LEDs that was emitted at various emission angles, the colour difference on the screen was further reduced and made imperceptible. The diffuser had the following specifications: it was a cylindrical (diameter: 10 mm, height: 3.2 mm) piece of diffuse acrylic with a centre hole (diameter: 2.4 mm, depth: 1.8 mm) for the LED lens. A highly specular reflective film was added between the LED base and the diffuser cap, to prevent back scattering of the light from the diffuser cap onto the absorbing LED base.  37  Figure 19: Emission spectra of LED for different angles: 0? (blue), 45? (red) and 90? (green). The curves were normalized to have the same area. It shows that for larger angle the spectrum shifts to larger wavelengths, which causes a colour shift.  Figure 20: Photograph of diffuser cap (white) with a radius of 10 mm placed on top of an LED. 0123400 450 500 550 600 650 700 750 800Luminance (a.u.) Wavelength (nm) 0 deg45 deg90 deg38  Figure 21: Sketch of the backlight structure. The hexagonal holes are there to position the LEDs, the baffles are visible as well as the grid structure to attach the filters. The screen (not shown) sits ontop of the side walls.  Figure 22: Photograph of backlight structure with 20 LEDs separated by black baffles. The LED spacing was 10 cm. 39 The setup shown in Figure 21 and Figure 22 was built in a way that the structure of walls and filters could be removed and replaced by a structure without walls and filters, but with the same thickness to emulate a standard HDR backlight. The dimensions of the structure were the following: height of walls (from the top of the diffuser cap to filter plane): 24 mm; distance from the diffuser cap to screen: 76 mm. The screen for this application needed to have a high transmission for high efficiency, but also needed to be diffuse enough to randomize the incoming light, thus preventing parallax effects. If the diffuser would not be diffuse enough, viewers at different angles would not see the same pattern on the screen. A diffuse acrylic with a transmission of 53% was used.  The filters had a size of 94 mm ? 94 mm and were made out of a transparency film with a black pattern printed by a laser printer. Since the maximum black level of this printer was not absorptive enough, two of the same transparencies were used in series. An iterative process was established to find the correct filter pattern. The PSF of one LED was measured off the screen with a calibrated Charge-Coupled Device (CCD) camera [38]. Since there was a 1 to 1 spatial correspondence between the filter and the PSF on the screen, any deviation between the measured PSF and the desired PSF could be accounted for by changing the transmission through the filter at specific spots. To change the grey scale value of the filter, the non-linear printer response as well as the decrease of transmission for larger angles from normal had to be taken into account. Both effects were measured and parameterized to use them in an automated optimization procedure. It analyzed a photograph of the illumination on the screen created by a filter, determined the regions of either too much or too little light and calculated the pattern of a new filter to compensate. 40  It was estimated that overall about 20% of the light produced by LEDs was transmitted through the diffuser screen. 92% of the light emitted by the LED is transmitted through the diffuser cap; 56% of the light was absorbed by the walls and filters and 47% of that light was reflected by the diffuser screen and was absorbed by the opaque wall structure or the filters. 3.4 Measurements  3.4.1 Measurement Procedure All measurements were taken with a CCD camera [38], which was calibrated to yield luminance values. For this calibration, a highly collimated light source was placed centric behind an acrylic diffuser, which created a slowly varying luminance pattern on the diffuser. A mask out of black cardboard with accurately cut holes was put in front of the diffuser. A luminance meter [39] was used to measure the luminance value of each hole. In a next step, a picture (shown in Figure 23), of this masked diffuser was taken in the manual mode of the CCD camera at a certain exposure time (1/13 s). The aperture size was not changed over the course of this research project. All pictures were taken from a tripod at a distance of 1.5 m and with a resolution of 3264 ? 2448 pixels.  The obtained image had the file format ?.orf?, which is a Raw image format used in Olympus cameras. Raw image formats do not undergo any automatic processing by the camera and contain the image information rendered directly by the camera sensor. This is important to guarantee genuine and reproducible results. When these files are opened in photo editing software, image specifications like colour temperature have to be entered and image processing tools are available (changing contrast, saturation, gamma factor, etc.). For reproducibility, a 41 colour temperature of 4000 Kelvin was assigned to every picture in this research project and all image processing tools were deactivated. After opening the images in photo editing software, the 8-bit per colour channel image was transformed with a standard algorithm in an 8-bit grey scale image, since we were only interested in luminance and not in colour information. All images were saved in the ?.tif? format without compression.  Figure 23: A slowly varying luminance pattern on a diffuser masked by black cardboard with holes. The calibration image was imported into a MATLAB procedure, which created a 3264 ? 2448 matrix, with grey scale values between 0 ? 255 as entries. Since each of the holes in the cardboard shown in Figure 23 contained hundreds of pixels, an average grey scale value was determined in MATLAB. These values were compared to the luminance values measured with the luminance meter. A log-log graph was created with MATLAB grey scale values on the x-axis and the product of luminance and exposure time on the y-axis (see Figure 24). A power function 42 was fitted to the data, which was used in a MATLAB procedure to determine luminance values from images taken with the CCD camera.  Figure 24: Luminance calibration of CCD camera: MATLAB grey scale value (0-255) vs product of luminance and exposure time shown on a log-log plot. A power function (formula given in graph) was fitted to the data, which was used in a MATLAB procedure to determine luminance values from image data. In this dissertation, the data acquired with the described method is usually shown in line scans through the image, which means that only one row of luminance data is shown against the position on the screen where it was measured off. This has the advantage that luminance values can be read off the figure. The line scan is a truthful representation of the measured data. In contrast, an image like Figure 25 went through automated image processing of the screen it is y = 0.0052x1.5384 R? = 0.9989 0.010.11101001 10 100 1000Luminance x exposure time (cd/m? s) MATLAB grey scale value 43 shown on or the printer it was printed with. In the line scan graphs, the position on the screen is usually given in unit spacings, from one LED to the next.  Figure 25: Example picture of a PSF taken with the CCD camera. The red line indicates the location of the line scan shown in Figure 26.  Figure 26: Line scan through a picture of a large PSF (see Figure 25) taken with the CCD camera. 00.020.040.060.080.1-20 -15 -10 -5 0 5 10 15 20Luminance (cd/m?) Position on screen (unit spacings) 44 3.4.2 Measurement Results The data measured off the backlight screen with the CCD camera is shown as horizontal line scans going right through the centre of three PSFs. Four different backlight scenarios were studied: uniform brightness screen, linear and quadratic gradients as well as a step function. All scenarios were measured with the two different point spread functions: the new PSF with walls and filter; and the Lambertian PSF, to represent a standard HDR display without walls or filters.  Figure 27: Measured line scan across uniform brightness screen with the new PSF (red) and the Lambertian PSF (blue). First derivative of the data is shown on the right. The respective goals are shown in black. Figure 27 shows measurements across the screen with all LEDs at the same output level. Due to the fact that more light is absorbed when filters and walls are in place, the graph of the new PSF shows a lower average luminance. With the Lambertian PSF, there are high luminance bumps on top of the locations of the LEDs. In contrast, the higher luminance areas with the new PSF are between the LEDs and are created by light leakage: Through multiple reflections, a small amount of light can leave the area of the corresponding PSF. This amount has its 45 maximum right outside of the PSF, which is located centred between two LEDs. When both measurements are normalized, the amplitude of the oscillation is 4.5 times lower for the new PSF than for the Lambertian PSF. The root mean square (RMS) error is 1.3% for the new PSF compared to 4.5% for the Lambertian PSF. The graph on the right side of Figure 27 shows the first derivative of the data shown on the left side. The first derivative is of interest, because human vision is especially sensitive to changes in luminance. The substantial improvement through the new PSF is most visible here.   Figure 28: Measured line scan of linear gradient with the new PSF (red) and the Lambertian PSF (blue). First derivative of the data is shown on the right. The respective goals are shown in black. Figure 28 shows luminance measurements of a screen where LED output decreased linearly with distance. Figure 29 shows the screen with a quadratic decrease in LED output. Both graphs more or less show the same behaviour as seen at the uniform brightness screen: At positions where the Lambertian PSF gives luminance bumps, the new PSF exhibits luminance dips. Improvement can be seen best in the first derivative, with markedly reduced variation. The RMS errors have been determined for all gradients. The use of the new PSF improved the linear 46 gradient from 4.8% for the Lambertian PSF to 2.0%. In the case of the quadratic gradient the improvement reduced the RMS error from 5.0% for the Lambertian to 2.1% for the new PSF. The last measurement of this experimental backlight had the following setup: the two columns of LEDs on the left were illuminated and the other LEDs were switched off. The light falloff was measured, which can be seen in Figure 30. The graph shows three curves: the new PSF (red), the Lambertian PSF (blue) and a setup without the filters, but with walls: a Lambertian PSF with cut-off (green).  Figure 29: Measured line scan of quadratic gradient with the new PSF (red) and the Lambertian PSF (blue). First derivative of the data is shown on the right. The respective goals are shown in black. The Lambertian PSF has a relatively quick falloff, but exhibits a long tail, which is apparent on the right graph of Figure 30, since the data is plotted on a logarithmic y-axis. The new PSF drops smoothly from high to low light levels. As expected, the Lambertian PSF with cut-off exhibits the same behaviour in the low luminance region, since this is caused by the walls. However, since the walls are abrupt baffles, they create undesirable artifacts like the highly non-47 uniform step visible in the right graph of Figure 30 at about 3 unit spacings. It is clear that a filter is needed to hide the abrupt cut-off effect of the walls.  Figure 30: Line scans across screen with two LEDs on (at the screen location of 0.5 and 1.5 unit spacings): Lambertian PSF (blue), the new PSF (red) and a Lambertian PSF with walls (green). In the graph on the right, luminance is plotted on a logarithmic axis. As described in Section 2.1.3, the contrast sensitivity function can be used to determine if a pattern in an image is visible or invisible to the human eye. This method was used on the line scans of a uniform brightness screen shown in Figure 27 on page 44. With a Fourier transformation the spatial frequencies contributing to the measured oscillation were found and their respective amplitudes were determined. These frequencies were then binned into bands with exponentially increasing bandwidths and the corresponding contrast sensitivity was plotted against the centre frequency of each frequency band.  The results are shown in Figure 31 together with experimental data of a contrast sensitivity function published in [18]. When a data point is lower than the contrast sensitivity function, this 48 pattern is visible. It can be seen that for the Lambertian PSF a pattern is visible with a spatial frequency of 0.26 cycles/degree, which corresponds to the unit spacing in this experimental setup (10 cm LED spacing at a viewing distance of 1.5 m). Figure 31 demonstrates that the new PSF does not show any pattern, and thus, is perceived as uniform by the human eye. As expected, the contrast sensitivities of the two setups was very similar for spatial frequencies smaller and larger than 0.26 cycles/degree, since the focus of this project was to reduce the visible pattern corresponding to the frequency of the LED spacings.  Figure 31: Curve that approximately follows the measured data of the contrast sensitivity function (CSF) (black) from [14]: Data points higher than the black curve indicate that the corresponding patterns are invisible to the human eye. With filters (circles) the deviation on the screen at 0.26 cycles/degree is invisible. 49 3.5  Summary and Conclusions A new PSF for HDR displays was proposed and its benefits over a Lambertian PSF representing a PSF commonly found in HDR displays were shown. Simulations in 1D and 2D showed that the new PSF can create uniform brightness screens, perfectly linear and quadratic gradients and a higher local contrast. It was discussed how the new PSF could be produced and an experimental setup of a backlight was developed. The experimental setup showed the improved behaviour. However, the RMS errors between 1.3% and 2.1% show that it was hard to create a PSF that matched the shape of the quadratic B-spline exactly, even in a highly absorptive environment. Since reflective materials tend to cause more light leakage and reflections, it would be even harder to create a good match in a highly reflective setup. On the other hand, less absorption and thus higher efficiency is crucial for a possible application in HDR displays. 50 Chapter  4: Establishing the Feasibility of an Efficient Backlight with a Special Point Spread Function  A backlight needs to be efficient to be useful in a display application. However, in Section 3.5 it was concluded that it would be hard to realize an efficient backlight that created a PSF with the shape of a quadratic B-spline. In this chapter, the search for a practical and efficient backlight is described, which uses a PSF that shows similar characteristics to the mathematically perfect PSF described in Section 3.1, but is more amenable to being created by practical and efficient optical structures.  This chapter describes every stage of the research project: Starting from an initial idea of the backlight model, a simplified computer simulation was developed that tested the main idea, which then was tested experimentally. The experimental setup was then step-wise changed to make it more energy efficient. In the end, an efficient backlight design with the correct characteristics was found. This design was built and measurements were taken.  4.1 Backlight Model A theoretical model was established, with the goal of being more practical and efficient than the absorptive setup discussed in Section 3.3. A possible embodiment is shown in Figure 32. Instead of using a point source such as an LED, every backlight module in this model consisted of a light emitting plane (LEP) parallel to the screen with a certain pre-determined luminance distribution across the LEP. The angular light emission was supposed to be that of a Lambertian emitter. Opaque baffles were placed between these modules and the height of the baffles was set 51 to half the distance between the LEP and the screen. In this system, baffles provide high contrast, since they limit the spread of light. When baffles are combined with a point source, cut-off artifacts are produced. However, in combination with the LEP, smooth PSFs could be produced. Through luminance changes on the LEP, very different PSFs could be created on screen.  Figure 32: First idea of an efficient backlight setup that might be able to create a suitable PSF. The idea was to experimentally realize this model by using an LED within a reflective cavity with a partially transmitting filter out of a diffuse material on top (see Figure 32). The light transmission of the filter should vary spatially, and in the low transmission regions, most of the light should be reflected back into the reflective cavity of the LED. The transmitted light should be diffuse, so that the filter could act as the LEP. This can be realized with a bulk diffuser as filter, for which the transmission is based on the thickness. 4.2 Simulation Setup To show the viability of this model, the following simulation was created (see Figure 33). The LEP and the screen were discretized into two matrices: E(x,y) described the emitted luminance pattern of the LEP and I(x,y) described the received intensity pattern on the screen. 52 E(x,y) covered the area of 1 ? 1 unit spacing, while I(x,y) covered the area of 3 ? 3 unit spacings - this was enough since the PSF was truncated by opaque baffles, which covered all four sides of the emitting surface with a height of half the screen distance.   Figure 33: Schematic side view of simulation setup. E(x,y) described the pixelated light emission surface with the width of one unit spacing. I(x,y) described the pixelated intensity pattern at distance d. The width of I(x,y) was three unit spacings. Opaque baffles surrounded E(x,y) with the height of d/2. After defining the locations of the pixels on the emission plane and the screen corresponding to the matrix entries of E(x,y) and I(x,y), rays were traced from every pixel of the emission plane to every pixel on the screen. To calculate the delivered flux by every ray, Equation 3.1 was used, which takes into account the inverse square law and the two cosine factors of a Lambertian emitter and of incident light on a plane. The rays which hit the baffles were absorbed. With this simulation the PSF on the screen could be calculated for a given luminance distribution on the LEP. An optimization procedure was set up to find the most suitable PSF. In a first approximation, an E(x,y) with dimensions 10 ? 10 was chosen (see Figure 34). Due to the 4-fold symmetry along the x and y- axes and along diagonals, the problem has 15 variables. The distance between the LEP and the screen variable was also included, which added one more parameter to the problem. 53  Figure 34: Example of an E(x,y) with dimensions 10 ? 10. The 15 independent variables are indicated in red. The error function of the optimization procedure was based on the characteristics that were found for the quadratic B-spline: the uniformity and ability to show linear and quadratic gradients. In every optimization step, E(x,y) was determined for a given I(x,y) and this PSF was used to calculate the output of a larger backlight array in three different cases: (i) all LEDs having the same flux output, (ii) their output increased linearly in one direction, and (iii), their output increased quadratically in one direction. The relative RMS errors for these three cases (?uni, ?lin and ?quad for uniform brightness, linear and quadratic gradients, respectively) were determined and combined into the following error function:           (                  )  ?              (4.1) The 16 variables were optimized by a MATLAB Pattern Search [40] procedure to minimize this error function F. On a standard desktop PC, the optimization converged within a few minutes to a value of F = 1.5% at a screen distance of 0.68 unit spacings. The individual RMS errors (0.5%, 0.7% and 1.2% for uniform, linear and quadratic) were smaller than the RMS 54 errors measured on the experimental setup with the quadratic B-spline PSF by a factor of 2-3 (see Section 3.4). The MATLAB code, which calculates F, can be found in Appendix A.  To make sure a sufficiently high resolution was used, the matrix dimensions of E(x,y) and I(x,y) were increased to 20 ? 20 and 60 ? 60, respectively. This increased the size of the simulation by a factor of 16 and the number of variables from 16 to 56. The optimization time increased by two orders of magnitude. The final value of the error function was not substantially lower: F = 1.4%.  4.3 Experimental Verification of a Practical PSF 4.3.1 Absorptive Filter The first step was to physically emulate the system described above. For the light emitting plane with different luminance values a uniform brightness light box was used, and a transparency on which a 10 ? 10 grey scale matrix was printed was added (see Figure 35). The uniform brightness backlight consisted of an array of closely packed LEDs enclosed by mirrored walls and an acrylic diffuser on top. To increase the contrast of the absorptive filter, two printed transparencies were used in series. The transmission through two identical transparency films printed with different grey scale values had been measured beforehand. A calibration based on this measurement made it possible to relate the luminance goal of every pixel to a grey scale value printed in the 10 ? 10 grey scale matrix. The transparencies were surrounded by black baffles. A layer of neutral 55 density filter was added on top of the transparencies to reduce the amount of back reflections from the diffuser screen, which had not been simulated.  The dimensions of the experimental setup were the following: the filter was 10 cm ? 10 cm, the screen was 30 cm ? 30 cm, the baffles had a height of 4 cm, which is half the spacing between the screen and the absorptive filter. A line scan was taken across the screen of the setup. Good agreement was found between the measurement and the simulated PSF (see Figure 36).  Figure 35: Experimental setup of backlight module with absorptive filter (10 cm ? 10 cm) on top of uniform brightness backlight. The baffles have a height of 4 cm. This experimental setup closely matched the simulation.  While this agreement is encouraging, it does not represent fulfillment of the overall design objective, because this arrangement differs in important ways from a potentially practical design. For example, back reflections from the screen were mostly absorbed by the neutral density filter and the transparencies, but in an actual application such absorption could be considered unacceptably wasteful. For this reason it is necessary for the spatially variable reflective filter to be reflective from both sides and the resultant series of inter-reflections would modify the PSF, thus requiring a compensatory design change. 56  Figure 36: Luminance measurement on screen with absorptive filter (green) and simulation result (red). To introduce inter-reflections between the diffuser screen and the reflective filter, a more sophisticated filter design was created: a grey-level matrix was printed on a sheet of thick paper, instead of printing it on transparency film. The sheet of paper was placed on the uniform brightness backlight, with the printed pattern facing the backlight. That positioned the white paper side toward the screen, creating the necessary inter-reflections, while the printed pattern created the variable transmission. To increase the contrast of the variable transmission filter, a printed transparency was added underneath the sheet of paper. 00.020.040.060.080.1-20 -15 -10 -5 0 5 10 15 20Luminance (a. u.) Position on screen (cm) MeasurementSimulation57  Figure 37: A 10 ? 10 matrix with horizontal, vertical and diagonal symmetry has 15 independent variables (shown in Figure 34). Based on these variables, 15 filters were produced with positions of high transmission pixels (white) and zero transmission pixels (black). The ray-tracing simulation did not take reflections into account; this was the reason the first step had been to work with an absorptive filter and opaque baffles. However, the newly introduced inter-reflections changed the system substantially. For example, in the first setup with only absorptive surfaces, areas on the filter with no transmission gave no contribution to the PSF on the screen; with inter-reflections, the whole filter is lit to some extent and all areas of the filter contribute (to different degrees) to the PSF. Due to these back reflections, the simulation was no longer accurate enough to calculate the PSF coming from a given luminance pattern on the LEP. However, taking inter-reflections into account in the simulation would have been difficult. 58 A new way to find a suitable filter pattern had to be developed. As mentioned, the LEP had 15 variables and the goal was to determine the influence of every single variable in our system. Hence, 15 independent filters were created (see Figure 37), each corresponding to one variable under the given symmetry. The filters were printed with black ink onto thick sheets of paper (230 g/m2). With the additional transparency (with the same print pattern) in series, no light transmission was measureable in the dark areas of the filters, the only light came from the areas shown in white in Figure 37. By measuring the PSF of one such filter, the luminance pattern created by this individual variable was determined. The individual PSFs, or called sub-patterns, of all 15 filters were measured. How different the individual contribution were, can be seen in Figure 38, where line scans through five selected PSFs are shown. Each of them showed some degree of a cut-off artifact from the opaque baffles, but the idea was that the artifacts could be eliminated when multiple variables contribute to the total PSF. The method, which was used to find the best PSF and the associated filter pattern, was based on the fact that any PSF (LPSF) could be seen as a linear combination of the 15 sub-patterns (L1 ? L15) with different linear combination coefficients (ci):       ?                      (4.2)  For example, giving the case of no filter and just the uniform backlight, the resulting PSF would be a linear combination out of all 15 sub-patterns, and each of them with a linear combination coefficient of 1. In the case of a completely black filter, all coefficients would be zero.    59  Figure 38: Line scans of five examples of sub-patterns measured with filters shown in Figure 37. An optimization procedure in MATLAB was used to find the linear combination of these 15 measured sub-patterns that gave the best PSF by changing the linear combination coefficients in the range of 0-1, since 1 corresponded to the maximum transmission. In each iteration, the calculated PSF was used to simulate a larger backlight system with either a constant LED output, or linearly/quadratically increasing outputs from one LED to the next. The results were analysed based on the RMS errors and the before mentioned error function (Equation 4.1).  Once the optimization found the minimum of the error function, the linear combination coefficients had to be translated into grey scale values and printed on paper. For this calibration, a variety of grey scale patterns were printed on a sheet of transparency film and on a sheet of paper. The two sheets were put on the uniform backlight in the same way they were supposed to 0501001502002500 0.05 0.1 0.15 0.2 0.25 0.3Luminance (a. u.) Position on screen (m) 60 be used in the experiment: First the transparency, then the paper with its white side facing up. The normalized luminance values for different grey values can be found in Figure 39.  Figure 39: Luminance gradient measured off a grey scale pattern on a film of transparency and a sheet of paper in series.  In this case the normalized luminance on the y-axis of Figure 39 corresponds to the linear combination coefficients. Since it is the goal to find the grey scale value for a given linear combination coefficient, the axes of the graph were flipped and a new polynomial was fitted. This sixth order polynomial was used for the calibration.  00.10.20.30.40.50.60.70.80.910 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Normalized Luminance Grey scale 61  Figure 40: Grey scale of a printed pattern versus the luminance measured off the pattern. A sixth order polynomial is fitted to it, which is used in a MATLAB procedure for calibration. With this new filter in place, the PSF on the screen was measured and compared to the PSF that was calculated through the optimization. Good agreement was found (see Figure 41); the only discrepancy is in the low-light areas caused by a small offset in the measurements of the sub-patterns. This caused the calculated luminance to be slightly higher than the measured luminance. So far, the experimental setup consisted of only one backlight module. However, if backlight modules were to have an unaccounted effect on neighbouring modules, this could cause a problem. To make sure that something like light leakage does not change the PSF dramatically, a larger setup with multiple modules was created. The size of modules was decreased to 25 mm ? y = -15.796x6 + 59.041x5 - 86.987x4 + 64.023x3 - 24.976x2 + 5.676x + 0.018 00.10.20.30.40.50.60.70.80.910 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Grey scale Normalized luminance 62 25 mm and the pattern of the paper filter was printed multiple times on letter size paper (see Figure 42).  Figure 41: Line scan of luminance measurement on screen with an absorptive filter with varying transmission (circles) and simulation result (solid line). The uniform brightness screen showed a contrast modulation (defined in Section 2.1.3) of only 0.8% and an RMS error of 0.7% (see Figure 43). Two transparencies with a 5 ? 6 matrix were printed, with linearly and quadratically increasing transmission from left to right to create gradients (see Figure 44). RMS errors of the measured luminance gradients compared to perfect gradients were determined to be 3% (linear) and 2% (quadratic), respectively.  00.20.40.60.811.21.40 0.05 0.1 0.15 0.2 0.25 0.3Luminance (a. u.) Position on screen (m) calculatedmeasured63  Figure 42: Setup with 5 ? 6 filters (25 mm ? 25 mm each) separated by black baffles. Filters are printed on the side facing the backlight and the white side of the paper is facing up (left). When the backlight is on, the filter patterns can be seen (right).  Figure 43: Measurement data (black) on the screen created by the setup shown in Figure 42. A cosine function (red) with amplitude of 0.8% was fitted to the measurement data. The RMS error was determined to be 0.7%. 0.980.9911.011.020 1 2 3 4Luminance (a. u.) Position on screen (unit spacings) 64  Figure 44: Measurement results of linear (blue) and quadratic (red) gradients along with fitted polynomials of first and second order (equations are shown in graph). The RMS errors were 3% and 2% for the linear and the quadratic gradient, respectively. 4.3.2 Reflective Filter After starting with a mostly absorptive experimental setup, introducing some inter-reflections and establishing a new method to find the best PSF for a system, the next step was to increase efficiency through a new filter: A filter was needed that can spatially vary the transmission through reflection rather than absorption. The idea was to produce such a filter by using a variable thickness bulk diffuser, since thicker regions of the bulk diffuser reflect more light than thinner regions. To do this, a stack of 1 mm thick high-transmission diffuser sheets were used (100 mm ? 100 mm) and cut holes (10 mm ? 10 mm) into the sheets for pixels with high transmission values. The individual sheets with holes were drawn with computer-aided design (CAD) software and cut with a water jet cutter. y = 1.79x2 + 2.49x + 1.37 y = 11.23x + 5.01 010203040500 1 2 3 4Luminance (a. u.) Position on screen (cm) 65 The stack contained 10 sheets of Matte Lexan (T=80%) and 2 sheets of standard printer paper (T=20%) were added to achieve a transmission of almost zero in certain areas. On top of the stack there was one sheet of acrylic diffuser without holes. The filter was on top of a uniform backlight and surrounded by black baffles (see Figure 45).   Figure 45: Schematics of the experimental setup with reflective filter out of stacked diffuser sheets on top of uniform brightness backlight. In this example, the transmission decreases from the centre towards the sides. The width of the setup was 10 cm, the baffles were 4 cm high and each diffuser film has a thickness of 1 mm. The luminance goal of the filter was determined with the same method as described before: measuring the 15 contributing sub-patterns. A first iteration of the filter was created by stacking diffuse layers with holes at specific locations and measured the transmission at every pixel. Due to cross-talk from one pixel to the next, an iterative process was necessary: Beginning with the highest transmissions, the luminance was measured and diffuser sheets added or removed. The iteration was finished once the luminance value at each pixel was at least as high as its goal luminance, but not high enough to add an additional diffuser sheet.  66 Figure 46 shows a line scan across the top of the reflective filter (red) and the goal luminance (black). An important difference was the smoothness of the measured luminance compared to the goal. That was mainly caused by the top diffuser, in which light from a bright pixel could scatter into a low transmission pixel. Thus, high contrast boundaries could not be created with this filter.  Figure 46: Luminance across the reflective filter: goal (black) and measurement before adding absorptive transparency (red). It is not possible to accurately match the luminance of every pixel to its goal luminance, since the transmission is changed in steps of a few percentage points, when a diffuser sheet is added or removed. These luminance steps are decreasing with number of sheets in the stack and are largest for the first one added; it reduces the transmission by 20% percentage points, from 100% to 80%.  00.20.40.60.811.20 1 2 3 4 5 6 7 8 9 10Relative luminance Position on filter (cm) goalmeasurement67 Since the luminance across the reflective filter varies within each pixel region (see Figure 46), it is difficult to compare the luminance of a pixel to its goal luminance. As a simplification, it was assumed that the flux emitted by a pixel is more meaningful than the luminance distribution within each pixel. Hence, a calibrated photograph of the filter was taken and the flux was determined by integrating over the area of each pixel. To match the flux goals of each pixel accurately, the amount of excess light was determined and a sheet of transparency was printed with a calibrated 10 ? 10 pixel pattern of black ink to absorb the small amounts of extra light at every pixel position. A measurement of the resultant PSF on the screen was taken and compared to the MATLAB simulation of the linear combination of the sub-patterns (see Figure 47). Only a small deviation between the measurement and the simulated PSF was found. This is probably caused by the earlier discussed limitations that high contrast boundaries are not possible with this reflective filter and the non-constant luminance values across a single pixel (see Figure 46). However, with the measured PSF, a uniform brightness backlight was simulated and the resulting contrast modulation on screen was sufficiently small (2.4%) to show that this method of creating a reflective filter can work in principle. 68  Figure 47: Line scan across screen for setup with reflective filter (circles) and theoretical PSF (red curve). 4.3.3 Single LED Backlight A uniform brightness backlight for the reflective filter is not a feasible option for an HDR display since the light sources are commonly LEDs. This is why the next step was to create a backlight for the reflective filter that consisted of one LED inside a reflective cavity. With sufficient distance between the LED and the reflective filter, the reflective filter could have been lit nearly uniformly. However, thin displays are generally favoured over thicker ones and it was expected here that the non-uniformity created by the smaller distance between LED and reflective filter could be taken into account when developing the filter.  For the experimental verification (shown in Figure 48), the module size was reduced from 100 mm ? 100 mm to 25 mm ? 25 mm, since this is a more reasonable size-scale for actual HDR backlights. The height of the reflective cavity was set to 3.25 mm and the inside was covered 0204060801001201400 0.05 0.1 0.15 0.2 0.25 0.3Luminance (a.u.) Position on screen (m) SimulationMeasurement69 with a highly reflective film. Enhanced Specular Reflector (ESR) film [1], was used for this, a dielectric multilayer film with a reflectivity of about 98.5% across the whole visible spectrum. The LED was centred in the cavity, with all but the light emitting part covered in ESR. An acrylic diffuser (t=2.75mm, T=53%) was placed on top of the cavity.  Figure 48: Experimental setup with LED in recycle cavity underneath an acrylic diffuser, which is surrounded by retro-reflective baffles. To reduce the absorption in the system even further, the opaque baffles were exchanged for reflective baffles. However, light that is specular or diffusely reflected by the baffles can reach areas of the screen that would not be possibly reached with opaque baffles. To reduce this light leakage, retro-reflective baffles were used. Instead of absorbing the light that hits the baffles, or reflecting it onto the screen causing light leakage, the light is reflected back onto the reflective filter and can be recycled (see Figure 49). Because the retro-reflectors do not work perfectly for 70 all angles and also permit some light to be transmitted, a diffuse reflective backing out of paper and ESR were added to increase the reflectance of the baffles. In that way, the light that is transmitted by the retro-reflectors was still reflected. Due to this non-perfect behaviour, the retro-reflective baffles did not prevent light leakage to the same extent as the opaque baffles did. However, since only a small fraction of the light that is reflected by the non-perfect retro-reflectors is reflected diffusely, less light leakage is introduced compared to diffuse or specular reflective baffles. It was expected that it would be possible to account for the slight increase of light spread due to the imperfections of the retro-reflectors.  Figure 49: Schematics of different baffle materials: a) opaque, b) specular reflective, c) diffuse reflective and d) retro-reflective. In the previous steps, the 15 individual filters (see Figure 37 on page 57), which were used to find the sub-patterns were absorptive. For the next step, these filter patterns were cut into ESR using a CO2 laser cutter. The highly reflective film reflected all the light hitting it back into the 71 LED cavity, while the holes permitted full light transmission. The idea was to cut a halftone pattern to realize the variable transmission.  Figure 50: Picture of the experimental setup: retroreflective baffles on an acrylic diffuser, which is lit from underneath by an LED in a reflective cavity, but is masked by a filter with four holes (see Figure 37, top centre filter) to measure the sub-pattern of this filter. The 15 contributing sub-patterns were measured with the single LED backlight (see Figure 50) and with the MATLAB optimization procedure it was found, to our surprise that the centre pixels, by themselves, produced a nearly ideal PSF. That meant that in this particular case a continuously variable transmission filter was not required. This was unexpected, since light coming from a small contained area would produce cut-off artifacts on the screen due to the baffles. That was the reason why our initial hypothesis was that we would need multiple light emitting areas, so that the artifacts cancel out. However, in the experimental setup, the light did 72 not come from a small contained area: the light exiting the reflective cavity through the acrylic diffuser continuously spread out over a larger area. Two more beneficial effects were due to reflections in the optical setup: (1) the retroreflective baffles didn?t retro-reflect perfectly, which caused some light to be reflected to areas of the filter where no light had previously emitted; and (2), reflections off the diffuser screen also caused some light to be reflected in these formerly dark areas of the filter. All three effects contributed to the lucky coincidence that a localized source at the centre produced a less localized light output that had the desired properties. This truly was lucky, because if an even more localized output distribution were required, this might have required a thinner diffuser (possible sacrificing Lambertian angular emission) and/or less reflective baffles, which would have reduced efficiency. Therefore we were especially pleased to find that this was a case where the desired pattern turned out also to be one of the most practical arrangements to fabricate. The only modification necessary was to reduce the size of the LED cavity to the size of the 4 centre pixels (5 mm ? 5 mm). Figure 51 shows a drawing of the new experimental setup and Figure 52 shows a photograph of the reflective cavity in a piece of acrylic. 73  Figure 51: Experimental setup with LED in recycle cavity with a reduced size underneath an acrylic diffuser, which is surrounded by retro-reflective baffles. The bottom side of the diffuser around the reflective cavity is covered with ESR to reflect light that would exit the acrylic diffuser otherwise.  Figure 52: Reflective cavity (5mm ? 5mm, red arrow) cut into a piece of acrylic and covered with ESR next to a Canadian penny with a diameter of 19 mm. 74 4.3.4 Mechanical Description of Efficient Multi-LED Backlight  Given the result from the previous section that a suitable PSF can be created with a small reflective cavity and an acrylic diffuser on top, an efficient multi-LED backlight was built consisting of 3 ? 5 LEDs with the individual modules 25 mm ? 25 mm. Each row of 3 LEDs (in series) was dimmable to create gradients along the long axis. The thickness of the whole backlight structure was 34 mm, including the LED backplane and the screen, and thus, was larger than the module width by a factor of 1.35. The baffles were 10 mm high, which is also the distance between the end of the baffles and the screen. Figure 53 and Figure 54 show the optical paths in the structure and explain the optical components. The dimensions were the same as shown in Figure 51. A picture of the backlight without the screen is shown in Figure 55.  Figure 53: Diagram of optical paths Part 1: The photons are emitted by the LED; the photons are reflected and directional randomized in the reflective cavity; the photons are scattered in the acrylic diffuser. 75  Figure 54: Diagram of optical paths Part 2:  The photons transmitted by the acrylic diffuser can hit the retro-reflective baffles and are reflected back; the other photons hit the acrylic diffuser screen; these photons are scattered by the diffuser and are either reflected or transmitted.  Figure 55: Picture of efficient multi-LED backlight without screen and with all LEDs switched on. 76 4.4 Measurements and Simulations Using a calibrated CCD camera, the luminance distribution was measured in several cases.  With all LEDs of the backlight emitting the same flux a contrast modulation of 0.8% and relative RMS error of 0.6% was measured for a line scan along the longer axis (see Figure 56). The contrast modulation was determined by fitting a cosine curve to the data, with the wavelength of one unit spacing. According to the contrast sensitivity function, this oscillation pattern is barely visible to the human eye.   Figure 56: Variation on the uniform brightness screen of the backlight. The measured data (black) has an RMS error of 0.6%. A cosine function (red) was fitted to the data to determine the contrast modulation perceived by a human eye: 0.8%. Figure 57 shows selected data points from the line scan measurements of the linear gradient and the quadratic gradient. The actual measurement resolution was about 100 times finer. The 0.980.9911.011.020 1 2 3Luminance (a. u.) Position on screen (unit spacings) DataCosine fit77 RMS errors of the complete line scans of the gradients were determined to be 1.6% (linear) and 1.8% (quadratic).  The light falloff, a measure of the local contrast, is shown in Figure 58. For this measurement only one column of LEDs was switched on. A contrast of 5:1 was found at the location of the next LED and a contrast of 33:1 at the distance of 2 unit spacings. The exponential character of this falloff can be seen by the linearity of the data when plotted on a logarithmic scale (see Figure 58).  Figure 57: Data points extracted from line scans across the screen of backlight, showing the linear gradient (circles) and the quadratic gradient (crosses). The black lines are fits using first (linear) and second (quadratic) order polynomials, respectively. The RMS errors between the complete line scans and the fits are 1.6% (linear) and 1.8% (quadratic). 00.20.40.60.810 0.5 1 1.5 2 2.5 3Luminance (a. u.) Position on screen (unit spacings) 78  Figure 58: Line scan demonstrating a decrease of light level with increasing distance to LED (dotted, red). The active LED is positioned at ?0?. The solid black line is plotted on a logarithmic scale shown on the axis on the right-hand side. The linearity of the solid curve suggests an exponential drop of the light level.  Another way to show the local contrast is by measuring the flux transfer from one backlight module to its neighbouring modules (sideways and diagonal). The flux transfer with a) opaque and b) retro-reflective baffles was determined, by taking a picture of the backlight with only one LED switched on. From this picture the flux that is received at neighbouring LED modules can be determined (see Figure 59). With this information, the two different light falloffs in a larger screen were simulated using an iterative process: At the first step of the simulation, the flux of the centre LED module was set to one, all other LED modules had no flux. In the second step, flux transferred to the eight neighbouring LED modules, with values according to our measurements. The flux of the centre LED was set to one again and the flux transfer to the 0.00010.0010.010.1100.20.40.60.810 1 2 3 4 5Luminance (a. u.) Luminance (a. u.) Position on screen (unit spacings) linear y axislogarithmic y axis79 neighbours and from the neighbours to their neighbours calculated. This process was continued until a stable flux distribution was found.  The two iteratively simulated transfer flux functions were compared to the theoretical falloff of an LED, which is placed on a perfectly reflective Lambertian diffuser with a screen with Lambertian reflectance of 53% added at the same distance as in the experimental setup of the backlight (20 mm). This situation emulates the backlight without baffles. The results in Figure 60 show that the best contrast can be achieved with black baffles, but retro-reflective baffles still represent a substantial improvement over the case without baffles.     Figure 59: False colour image of a photograph of the experimental setup with 3 ? 5 LED modules and retro-reflective baffles. The flux in each square was determined and normalized to the centre module. 6.7% of the light spread to each sideways adjacent module, while 3.1% spread to each diagonal module. 80 Finally, the efficiencies of the backlights had to be determined. They were measured to be 61% with retro-reflective baffles and 25% with opaque baffles. For this, the flux output of the LEDs and the flux output of the screen of the backlight was determined. These efficiencies were verified using ray-tracing software [41]. For this verification, computer-aided design (CAD) models of the two setups were created. By slightly varying start values of reasonable surface properties, efficiencies were found that agreed with the measured efficiencies within 0.3 percentage points. The simulated surface properties were the following: ? the LED dye had a diffuse reflectance of 50% and absorbed 50%, ? the acrylic diffuser had a transmission of 53%, reflection of 46% and 1% is absorbed, ? the retro-reflectors reflected 96%, while 4% were absorbed, ? the black baffles reflected 12% and absorbed 88%, ? ESR had a bulk reflectance of 96% and 4% were absorbed. The bulk reflectance of ESR was around 98.5%. The observed 96% reflectance could be explained with 250 ?m gaps along the edges within the cavity.   81  Figure 60: Based on measurements of the small backlight, the flux transfer from one LED module to the next was determined, and a larger display was simulated. The given scenario was that one LED at position '0' was switched on and the amount of flux, which traveled to a distant module is shown in the graph. The ?triangle? data points show the flux transfer for a backlight with opaque baffles, the ?cross? data points are for the same backlight with retro-reflective baffles, and the ?circle? data shows the ray tracing simulation results for a backlight without baffles. For future improvements of the efficiency it is important to know where the flux was absorbed, this was another reason why the ray tracing simulation was done (see Figure 51 on page 73 for diagram of the setup). In the case of the retro-reflective baffles approximately 19% of the light was absorbed when rays hit the walls of the recycle cavity and 12% were absorbed when light within the recycle cavity was reflected back onto the LED dye. The retro-reflectors and the acrylic diffuser absorbed 5% and 3%, respectively. In the case of the opaque baffles, the 1.E-151.E-131.E-111.E-091.E-071.E-051.E-031.E-01 0 2 4 6 8 10 12Flux (a. u.) Position on screen (unit spacings) no bafflesretroreflective bafflesopaque baffles82 baffles had the major contribution to the absorption in the system: 53%. Recycle cavity and LED absorbed 13% and 8% respectively, while the acrylic diffuser absorbed only 2%. In summary, we found that the newly developed backlight is efficient, can show a uniform screen and smooth gradients. The rapid light falloff increases local contrast and the feasibility of image corrections by the front LCD. The backlight is relatively thin for direct-LED display and the method is scalable to higher or lower LED resolutions. 83 Chapter  5: Reflective Polarizer Liquid Crystal Devices in an HDR System  So far, we have modified the PSF to improve the contrast and uniformity of HDR displays. This chapter introduces a second approach to increase contrast. These two different approaches are compared in Chapter 6. The general idea of the project discussed in this chapter was to use an additional reflective light modulator between LEDs and the front LCD. The light modulator consisted of a liquid crystal cell sandwiched between two dielectric reflective polarizers. These reflective polarizer LCDs (RP-LCDs) were electrically driven to have a transmission and a reflection state (Figure 61) and could also have intermediate reflectivity values.   Figure 61: Transmission and reflection state of RP-LCDs for orthogonal linear polarizations 1 and 2. Polarization 1 is either transmitted or reflected, whereas Polarization 2 is always reflected. Usually, a recycle cavity is used that randomizes the polarization of the reflected light sufficiently, so that all light can be transmitted in the transmission state. 84 In this project, RP-LCDs are introduced as an intermediate sized light modulator, with a higher resolution than the wide-spaced LEDs, but lower resolution than the high resolution LCD screen (see Figure 62). In that way, one LED shines on an array of several RP-LCDs. We expected (1) an increase in the contrast, since it was another light modulator in series with the LED backlight and the front LCD and (2) a drop in the energy consumption for certain images. In high local contrast images, which exhibit bright highlights in dark regions, the LEDs of standard HDR displays cannot be sufficiently dimmed, because the light is needed to show the bright highlight. The light hitting the dark areas is absorbed by the front LCD, but the contrast is limited by the contrast of the LCD. With an array of multiple RP-LCDs per LED, the light reaching the dark regions is reduced, which should increase the contrast, and the reflected light should be recycled to make the highlight brighter, which can be seen in Figure 62. This recycle effect enables the dimming of the LEDs, which saves energy.  Figure 62: Theoretical working principle of RP-LCDs as additional light modulators. In this chapter the above theoretical description is tested experimentally and algorithms, which determine the flux output of a backlight with RP-LCDs, are described in this chapter. 85 These will be used in Chapter 6 to calculate the energy consumption of a backlight using the described backlight method. 5.1 Dielectric Mirrors as Reflective Polarizers This section explains the theory of dielectric mirrors and how they can be modified to work as reflective polarizers. It is commonly known that a properly designed stack of thin layers of dielectric material can yield ultra-high reflectivity (R ? 99.999%) for a narrow wavelength range (see [42] and [43]). This effect is based on constructive interference from all the interfaces within the stack. For this to happen at normal incidence for a wavelength   , the non-equal refractive indices (n1 and n2) and layer thicknesses (d1 and d2) need to have the following requirements:                            (5.1) The part of the wave that is reflected off the higher index material n1 will then be in phase with the part of the wave that is transmitted through this first interface and reflected off the next interface, because the optical path difference ?  is an integer multiple of   :       ?                               (5.2) The term     comes from a phase shift introduced by the reflection off the lower index material n2. A stack of layers will not only reflect the light with wavelength   , but a band of wavelengths around the centre wavelength   . The width of this band is linearly proportional to the refractive index difference: ?       .           (5.3) 86 This effect is more complicated if the light is entering the stack at an angle; then the stack must be treated as two reflectors, one for each plane of polarization. Any stack consisting of isotropic materials has a Brewster?s angle and a surrounding angle band over which most of the light is transmitted and not reflected. This means that these materials are not useful as omnidirectional mirrors. However, [44] and [45] showed that birefringent materials can be used to create reflectors with nearly constant reflectance for almost all incident angles. Birefringent materials have different indices of refraction along their main axes. For an omnidirectional reflector, the two in-plane axes x and y of the two materials can be equal:                                      (5.4)  To eliminate the Brewster?s angle, the out-of-plane axes of the two materials need to be equal:                   (5.5) To yield high reflection over a large range of wavelengths (e.g. the whole visible spectrum), individual stacks with different centre wavelengths and overlapping reflection bands can be combined. In this process it is important to realize that reflection bands generally shift to shorter wavelengths for increasing angle from normal, because the effective phase thickness of the layers decreases. That is the reason that for normal angles even the near-infrared needs to be reflected to ensure reflectance of the whole visible spectrum for large angles. As mentioned previously, the commercially available product ESR [1] uses the described methods and exhibits a reflectance of approximately 98.5% across all visible wavelengths. 87 A reflective polarizer (RP) is an optical component that transmits one polarization and reflects the orthogonal polarization. To create an RP with the discussed dielectric multilayer method the following birefringent requirements need to be met:         ,          (5.6) this ensures the omnidirectional character,         ,            (5.7) this introduces the high reflectance for light polarized along the x-axis of the stack,         ,              (5.8) this eliminates the reflective interfaces along the y-axis of the stack and ensures high transmission for light polarized along this axis.6 Another way to create an RP is by using a nano-wire grid. This method is described in [46]. Since dielectric RP film is commercially available [47], this was used in the experiments described in this chapter. 5.2 Scale Model of Backlight Module with Uniform Brightness Backlight A model of one backlight module was designed and built, consisting of a uniform brightness backlight and nine RP-LCDs. It was a larger-than-life model, with the RP-LCDs being of size 10                                                   6 More information on dielectric reflective polarizers can be found in [26] 88 cm ? 10 cm and with a spacing of 10 cm between the backlight and the RP-LCDs. The side walls of the setup were covered with ESR. The backlight consisted of a very bright (up to 10,000 cd/m2) uniform brightness light box exceeding the 30 cm ? 30 cm size of the setup. It was created by more than a thousand closely packed LEDs with a spacing of 10 mm and a diffuser screen. A highly diffuse reflective polyolefin fiber based paper (R ? 95%) [48] was placed on top of this light box. This was for two reasons: (1) the reflectivity of the backlight was known to be around 95% and (2) it randomized the polarization of incoming light, which was measured as described below. The second reason is especially important, since the RP-LCDs introduced polarization dependence into the system: One direction of polarization is always reflected by the first RP. If this reflected light was not able to change its polarization, it would never be transmitted. However, the reflected light was reflected by the diffuser film and the new polarization of the light was random.  The randomization of linear polarized light through reflection off the backlight diffuser film was proven with a separate experiment. Light from an incandescent light was linearly polarized and shone onto a piece of the film from an angle of about 5? off normal. An illuminance meter [49] with a rotating linear polarizer in front was also located at approximately 5? off normal. Were the linear polarization preserved, a large difference between the maximum and the minimum amount of light transmitted through the rotating linear polarizer would be expected for different angular positions of the second linear polarizer. However, a difference of only a few percent was measured, which is sufficiently small to assume polarization randomization. 89 5.2.1 Measurements The reflectance, transmission and absorption values of RP-LCDs in transmission (see Figure 63) and reflection mode (see Figure 64) was measured. These values were measured with a spectroradiometer [37] within an integrating sphere, which produced uniform illumination from all angles, in the following way. The spectroradiometer was used to measure the luminance of the RP-LCD with an angle of 40? between the normal of the RP-LCD and the spectroradiometer. Two luminance measurements were taken, where behind the sample there was placed a white standard (R = 98.5%) and a light absorber (R = 5%), respectively. In two additional measurements the luminance was measured for just the white standard and the absorber, without the RP-LCD. From these four measurements the reflectance, the transmission and the absorption of an RP-LCD in one state (either reflection or transmission state) can be determined. However, the values of reflectance, transmission and absorption of the RP-LCDs depend on the incident angle. 40? was used since it is close to 45? (the half-angle for flux coming from a Lambertian light source).  The RP-LCDs used in this experiment exhibit a colour change when switching between reflection and transmission mode: at normal incidence under transmission mode the light colour is almost neutral (white), but the transmitted light in reflection mode has a purple appearance; with increasing angle the neutral appearance in transmission mode slowly turns yellowish and the purple under reflection mode turns blue and later into cyan. The contrast (transmitted flux in transmission mode divided by transmitted flux in reflectance mode) is the highest (3.6:1) for normal incidence. 90  Figure 63: Reflectance (R), transmission (T) and absorption (A) of the RP-LCD in the transmission mode.  Figure 64: Reflectance (R), transmission (T) and absorption (A) of the RP-LCD in the reflectance mode. To be able to compare measurements to single wavelength simulations in ray-tracing software [41], the following measurements were done at 560nm, since it showed the largest 00.20.40.60.81400 450 500 550 600 650 700 750 800Wavelength (nm) R - transT - transA -trans00.20.40.60.81400 450 500 550 600 650 700 750 800Wavelength (nm) R - reflT - reflA - refl91 contrast between the transmission and the reflection mode. On top of the RP-LCDs was a nearly Lambertian diffuser sheet, so that luminance measurements could be taken. A neutral density filter between the RP-LCDs and the diffuser ensured that back reflections from the diffuser did not interfere with the measurements. With all nine RP-LCDs of the backlight module in the transmission mode, the luminance of every RP-LCD was measured at 560nm. A reflectance pattern was created by switching some RP-LCDs into the reflective mode. Subsequently, the luminance of the RP-LCDs, which stayed in the transmission mode, was measured. The gain factor, which is the factor by which the luminance of the RP-LCDs in the transmission mode increased due to the recycled light from the reflective RP-LCDs, was determined. Seven different reflectance patterns were analyzed (see Figure 65). For example, the centre RP-LCD stayed in the transmission mode while all others were switched to reflection mode (see pattern A in Figure 65). For this pattern, the luminance of the centre RP-LCD was 1.49 times brighter when all the other RP-LCDs were in reflection instead of transmission mode (see Table 1).   Figure 65: Seven reflectance patterns with RP-LCDs. Grey RP-LCDs are in the reflective state. Gain factors were determined for the white RP-LCDs in the transmission state.    92 Table 1: Measurement and simulation results for seven different reflectance patterns (see Figure 65). Reflective pattern Measured gain factor (?0.008) Simulated gain factor (?0.005) A 1.487 1.482 B 1.464 1.472 C 1.446 1.453 D (centre) 1.405 1.398 D (side) 1.373 1.378 E (centre) 1.411 1.402 E (corner) 1.362 1.368 F 1.186 1.192 G 1.207 1.198 5.2.2 Verification with Ray Tracing Software A 3D model of the experimental setup was created in a ray-tracing program [41]. Since the polarization effects of an RP cannot be correctly modeled in ray tracing software, the RP-LCDs were modeled as partly specular reflective surfaces. This simplification works as long as the backlight randomizes the polarization of the reflected light, which has been discussed earlier. That this is true can be seen in the following example: A perfect reflective polarizer with 100% transmission for one linear polarization and 0% transmission for the orthogonal polarization transmits 50% of randomly polarized light, and thus, could be modelled as 50% reflective. 93 In the ray tracing program, the experiment was modeled using the reflection, transmission and absorption (RTA) values measured at 40? for 560nm as starting point. With these values the same seven RP-LCD patterns (see Figure 65) which had been measured, were simulated. The RTA values for transmission and reflection mode were modified until simulation results of the seven reflectance patterns matched the measurement results of these patterns (see Table 1). Since the RTA values of the RP-LCDs depend on the angular distribution, a perfect match between the RTA measurement in the integrating sphere and the RTA values, which explain the measurement results of the whole setup, was not expected. However, the RTA values agree within ?0.6 percentage points (see Table 2). Table 2: Measured and simulated reflectance, transmission and absorption values of RP-LCDs at 560nm.  Transmission mode Measured          Simulated Reflection mode Measured          Simulated Reflection 67.2% 66.1% 82.5% 82.8% Transmission 26.1% 27.0% 9.2% 8.5% Absorption 6.6% 6.9% 8.2% 8.7% For the ray tracing simulation the following properties were used: The backlight had a Lambertian angular emission spectrum and a diffuse reflectivity of 95%, walls covered with ESR were modeled as 98% specular reflective. 5.2.3 Numerical Solution for Flux The next step was to generalize the results that were found. A method was established that determined the flux through every RP-LCD for a given reflectance pattern. To determine the 94 emitted flux pattern, it is important to find the total output of the system, which decreased with increasing reflective area. The total efficiency was normalized by the efficiency for the case where all RP-LCDs are in the transmission state. This fractional efficiency can be found in Figure 66. In the graph, 0% reflective area represents the case where all RP-LCDs are switched to the transmission mode and 100% means all RP-LCDs are switched to the reflection mode. A 2nd order polynomial was fitted for interpolation, which is needed to model the system and calculate the emitted flux, as described in the next paragraph. The contrast of a transmissive compared to a reflective RP-LCDs was about 3.1:1 for the combination of RPs and LCDs that were used in this project. With this information it was possible to numerically solve for the gain factors for the reflective and the transmissive RP-LCDs, if the constraints were used that (1) the sum of all gain factors divided by the number of RP-LCDs had to give the efficiency for the given reflective area; and (2) the contrast between the transmissive and reflective RP-LCDs was 3.1:1. This method also worked with intermediate reflective states, if this was taken into account in the calculation of the reflective area and the contrast was reduced. 95  Figure 66: Fractional efficiency of RP-LCD backlight module with 2nd order polynomial fit, which is needed for modelling the system. The total efficiency of the backlight (light output of module over light emitted by source) depends on the reflective area. It starts at 64% when the entire area is transmissive and then drops with increasing reflective area to 32-37% when 90-100% of the area is reflective. However, this high reflectance range is where this backlight technology has the biggest advantage over current HDR systems, since the LEDs can be dimmed due to the high recycle effect. 5.3 Setup with Single LED as Backlight A uniform brightness light source for these RP-LCDs is not very likely to be found in an actual application; a reasonable light source instead is an LED. Hence the uniform brightness backlight discussed in the previous section was replaced by an LED in the centre of the 30 cm ? 30 cm box. y = -0.000021x2 - 0.002752x + 0.996911 00.20.40.60.810 10 20 30 40 50 60 70 80 90 100Fractional efficiency Reflective area (%) 96 Since the rest of the setup was up-scaled by roughly a factor of 10, a 10 times larger LED was emulated: The dye area of the original LED had a diameter of 2.5 mm with a reflectance of about 50%. A pattern with a diameter of 25 mm was printed on a piece of paper that resulted in a reflectance of 50%. The grey piece of paper had a hole for the actual LED in the centre. The rest of the actual LED was covered with a square white piece of diffuser (see Figure 67).  Figure 67: Picture of 3 ? 3 RP-LCD array with up-scaled LED. With this setup, 10 different reflectance patterns were created and the flux of every RP-LCD determined for these patterns. One measurement was done without the RP-LCDs to determine the flux coming directly from the LED. This part of the flux is not constant across the 10 cm ? 10 cm RP-LCDs (see Figure 68). Due to this fact, for the measurement, the flux was integrated over an area of 9 cm ? 9 cm using an illuminance meter [49] within an integrating device, which had a 9 cm ? 9 cm aperture covered with a neutral density filter and a diffuser. The walls were covered with highly reflective material. The measurement setup is shown in Figure 69. 97 Since the measurement results of the module with a uniform brightness backlight were verified with ray tracing, a second verification process for the backlight module with one LED was not necessary. Therefore, the measurements did not need to be made for individual wavelengths, but were done with an illuminance meter, which weights the wavelengths according to the luminosity function to correlate with human brightness perception.  Figure 68: Picture of the screen of the backlight module with nine RP-LCDs and one LED.  98  Figure 69: Measurement setup for backlight module with one LED. 5.3.1 Numerical Solution for Flux Output An algorithm was established which calculated the flux at each of the nine RP-LCD positions based on the reflectance pattern and the reflection, transmission and absorption values of the RP-LCDs in the two different modes. The algorithm took into account that the flux pattern of the light coming directly from the LED is not uniform; however, it was assumed that all the recycled light in the box was spatially uniform. The recycled light was calculated based on the reflectance and the absorption values within the cavity and a geometric series was formed. The distribution of this recycled light was based on the relative transmission values of the RP-LCDs. The algorithm was given a 3 ? 3 luminance goal matrix and then the procedure optimized the 99 reflectance values of the nine RP-LCDs to minimize the RMS error between the luminance goal and the calculated luminance values. The MATLAB code can be found in Appendix B. The reflection, transmission and absorption values of the RP-LCDs were measured in an integrating sphere with a luminance meter. As mentioned earlier, the angular distribution of the light plays an important role in the RTA properties and can be different in the integrating sphere and the backlight module. This is the reason why the measured data in Table 3 are only approximate values. To verify these values, the algorithm described above was used in an optimization procedure in which the RTA values of the RP-LCD were variable. The goal of the optimization was to minimize the deviation between the measured and simulated flux values. Convergence was found for simulated values that differed only by 2 percentage points from the measured values (see Table 3). Since these values depend on the incident angle, perfect agreement was not expected and the observed difference is sufficiently low. The agreement between the simulated and the measured flux values was within 5%, which was sufficient to use the algorithm in Chapter 6 to calculate the energy efficiency of this backlight method when showing actual HDR images.     100 Table 3: Measured and simulated photometric RTA values for RP-LCDs in a backlight module with one LED.  Transmission mode Reflection mode measured simulated measured simulated Reflection 67.5% 69.8% 78.7% 77.4% Transmission 27.7% 26.3% 15.1% 14.6% Absorption 4.8% 3.6% 6.2% 8.0% 101 Chapter  6: Simulation of Power Consumptions of Various Backlights  In the previous chapters, novel HDR backlights with important benefits in uniformity and contrast were described. This chapter contains a theoretical study on the power consumption of these backlights. In HDR displays, power consumption depends critically on the content of the screen. Full brightness is only needed in bright regions of the image, while LEDs can be dimmed in darker regions of the shown image. In this chapter, the theoretical power consumptions of three different HDR backlights are determined, while showing 20 randomly chosen HDR images:  ? A backlight using 3 ? 3 RP-LCDs per LED module, as discussed in Chapter 5 ? A backlight having the measured PSF discussed in Chapter 4 ? A backlight commonly found in current HDR displays where all LEDs are placed in one large reflective cavity For easier comparison of the above backlights, the simulated power consumption is normalized to the power consumption of a common LCD display with uniform brightness backlight. The relative power consumption is determined by the comparison of the average LED flux output. It is assumed that the luminous efficacy (which is output of luminous flux per input of electrical power) of all display light sources is the same and the efficiency loss of the light distributing optics (e.g. light guides) in a common LCD display is neglected. 102 6.1 HDR Image Preparation In HDR displays, the front LCD shows the high resolution details in colour, while the backlight produces a blurred low-frequency grey scale luminance pattern. Since only the backlight contributes significantly to the power consumption of HDR displays, the creation of the low frequency luminance patterns had to be studied. Therefore, preparation of the 20 HDR images was required in order to use the images in the backlight simulations. The HDR image preparation procedure described in [50] was followed. The first step was to change the colour images to grey scale. Therefore, only the brightest RGB sub-pixel, i.e. the maximum of the three RGB values of a given pixel was used. This is the standard procedure in HDR image processing, since the LEDs need to produce at least as much light as is required to produce the brightest colour channel. In this project, each of the three HDR displays was simulated with several different numbers of LEDs. However, the number of LEDs had to be a multiple of 16 ? 9, the common aspect ratio for displays. It is common to reduce the resolution of a given HDR image to determine luminance goals in certain areas of the image. An image can be split into equally sized squares, where each square contains a large and equal number of pixels. Each square can be though to be lit by one LED. To reduce the resolution of the HDR images from 1920 ? 1080 to a smaller multiple of 16 ? 9 to match the number of simulated LEDs in the backlight, a ?down-sampling? process was used.  For the down-sampling, a new image matrix was created with reduced dimensions. The average luminance of all pixels within each square area was determined and written into the new down-sampled matrix cell of the corresponding LED as luminance goal in this area (see Figure 103 70). The down-sampled image was normalized by the maximum matrix entry, so that the maximum matrix entry was one. Then the square root of each matrix entry was taken. This was necessary to equally divide the dynamic range of the 16 bit display: The front LCD is usually driven by an 8 bit controller and so are the LEDs of the backlight [50]. A series of down-sampled images, in grey scale and with the square root taken, can be found in Appendix C.  Figure 70: Schematic of the used down-sampling method. A 6 ? 4 matrix (left) with indices from a to x is reduced in size to a 3 ? 2 matrix (right). The entries of the new down-sampled matrix are determined by averaging over the values (va - vx) in the corresponding 2 ? 2 areas of the original matrix.  6.2 Power Consumption of Standard LCD with Uniform Brightness Backlight The standard LCD display does not use any HDR technology; since the backlight cannot be locally adjusted, all luminance variations have to be made by the front LCD. This simplifies the calculation of the power consumption considerably. Since this type of LCD display is the most common display, the power consumption of all HDR displays studied in this project are normalized and compared to the power consumption of the standard LCD display. In a standard LCD display, the whole backlight has constant luminance. For the power consumption comparison, this uniform backlight needs to produce enough light to match the 104 luminance of the brightest LED of an HDR display showing the same image. Since the images were normalized so that the brightest luminance was equal to 1, the average backlight luminance of the uniform backlight was also 1 for every image. This was defined as normalized power consumption of 1. 6.3 Power Consumption of HDR Display with Reflective Polarizer LCDs This investigation was based on the backlight module with RP-LCDs described in Chapter 5. One module consisted of one LED and 3 ? 3 RP-LCDs. All modules were independent from the neighbouring modules, since there was no gap between the walls and the screen. The modules were placed on a grid consisting of squares with one LED in each square. LED resolutions of 16 ? 9, 32 ? 18, 64 ? 36, 128 ? 72 were simulated. Since there were 3 ? 3 RP-LCDs per LED, images for this study were down-sampled to match a resolution 9 times finer than the LED resolution (e.g. 48 ? 27, 96 ? 54, etc.). In this way, every single RP-LCD had a luminance goal. For the simulation, the algorithm described in Section 5.3.1, was used for each LED module individually: transmission values of all nine RP-LCDs and the LED output were adjusted to closely match the nine respective luminance goals given by the goal matrix. In MATLAB, the optimization method fmincon [51], [52] was used to minimize root mean square (RMS) error between the simulated and goal luminance values. The code, which calculated the RMS error of the nine RP-LCDs can be found in Appendix B.  To be able to compare average LED values of this system to a standard LCD display with uniform brightness screen, the LED output had to be calibrated. An LED value of 1 was given to the LED output, which created an average luminance of 1 across the 9 pixels in a system without 105 the RP-LCDs in place. This scenario is comparable to the standard LCD with uniform brightness backlight. When the RP-LCDs were in place, the average total flux transmitted was only around 56% of the light produced by the LEDs. However, this efficiency loss was built into the described algorithm; it increases the LED output accordingly, so that the goal luminance is matched.  Figure 71: Average relative power consumption of backlight with RP-LCDs and different numbers of LEDs. Numbers correspond to the LED resolution. For every picture, the average LED value was determined. This corresponded to the relative power consumption, where 1 meant the same power consumption as a standard LCD display. As shown in Figure 71, even with an array of only 16 ? 9 LEDs, this backlight used less than 50% of the power a standard LCD screen consumes. As expected, the power consumption dropped with increasing number of LEDs since the areas corresponding to each LED module decreased. The data in Figure 71 seemed to be converging, so the next goal was to determine the minimum power consumption (the value it converges toward with increasing numbers of LEDs). 00.10.20.30.40.50.6100 1000 10000Relative power consumption Number of LEDs 16 ? 9 32 ? 18 64 ? 36 128 ? 72 106 The average luminance of the 20 normalized full resolution HDR images was 0.196 and the RP-LCD backlight system had an average efficiency of about 56%. This meant that the minimum power consumption was 0.196/0.56 = 0.35. This corresponded to a system where every single LCD pixel has one RP-LCD, for which there would be 640 ? 360 LEDs (1920/3 ? 1080/3). This showed the diminishing return of increased numbers of LEDs: This system had 25 times the number of LEDs the system with 128 ? 72 LEDs had, while the power consumption was only 1.8 percentage points lower.  Figure 72: Average RMS error of  RP-LCD backlights with different numbers of LEDs. For all images the fractional RMS error between the goal image matrix and the simulated luminance matrix values was determined and the average was taken over all images. This is a measure of image quality and display capability. The difference between the goal luminance and the simulated luminance has to be corrected by the front LCD. As expected, RMS error decreases with increasing number of LEDs. 00.050.10.150.20.25100 1000 10000Fractional RMS error Number of LEDs 107 6.3.1 Matching Luminance Values of Highlights The previously described optimization was based on the minimization of RMS error. Another option is to provide always at least the amount of light needed and never less. In this case highlights (small very bright areas) can be shown accurately, but dark areas might appear washed out. However, high peak luminance values are preferred by viewers [23].  Figure 73:  Line scan through one picture. The luminance values of the original picture are shown in black, the simulated luminance values are shown in red (RMS optimization) and green (maximum luminance optimization). To achieve the new requirement that the simulated luminance should never be below the target luminance, a new error function was defined for the optimization:           ?    ,          (6.1) 00.10.20.30.40.50.60.70.80.91Relative Luminance GoalRMSmax Luminance108 where errrms is the RMS error and di is the difference between one simulated pixel and the pixel in the goal image; however, if the simulated luminance is higher than the goal,    is defined to be zero. With this function highlights are weighted stronger.  Figure 74: Average relative power consumption of backlight with RP-LCDs and different numbers of LEDs, which always produces enough light to show highlights (diamonds) compared to a backlight with minimal RMS error (crosses). The effect of this optimization is demonstrated in Figure 73 by line scans through the original picture and the simulated luminance matrices, showing the entries of only one row of each matrix. Most of the high luminance values of the original image are now matched. Since with this error function most backlight modules provide a slightly larger amount of light than when using the RMS method, power consumption increased compared to the previous simulation (see Figure 74). In the previous simulation the optimization goal was to minimize the RMS error. This explains why the RMS error increased with the new error function (see Figure 75). In summary, if the high luminance values are matched by the backlight, power consumption increases by approximately 8% and RMS error increases by nearly a factor of 2. 00.10.20.30.40.50.6100 1000 10000Relative power consumption Number of LEDs max LuminanceRMS109  Figure 75: Average RMS error of RP-LCD backlights with different numbers of LEDs, which always produces enough light to show highlights compared to backlight (diamonds), compared to a backlight with minimal RMS error (crosses). 6.4 Power Consumption of HDR Displays with Different Point Spread Functions In HDR displays, the shape of the PSFs has an effect on local contrast and power consumption. The faster the light falloff, the higher is the local light control and thus the match between the goal backlight luminance and the calculated luminance. In this section two different PSFs will be compared and the power consumption and RMS error corresponding to them will be determined. 6.4.1 Deconvolution of HDR Images A high resolution image of the backlight luminance can be calculated by convoluting the shape of the PSF of the system with the positions and driving values of all LEDs (see Section 3.1 and [31]). In this section the reverse is done: an image with a certain PSF is deconvoluted to find 00.10.20.30.4100 1000 10000Fractional RMS error Number of LEDs max LuminanceRMS110 the LED values. Deconvolution is often used to de-blur images with an unknown PSF (e.g. astronomical pictures blurred by the PSFs of optical elements in telescopes [53]). A standard deconvolution method, the Lucy-Richardson algorithm [54], [55] was used, which is based on maximizing the likelihood that deconvoluted LED values are an instance of HDR images under Poisson statistics. The HDR images that were used for the deconvolution were down-sampled as described in 6.1, such that every matrix entry corresponds to one LED. 6.4.1.1 Modified PSF Measured off Experimental Backlight The first PSF used for this study was the modified PSF described in Chapter 4, where the PSF was measured off the experimental backlight. Since the HDR image is down-sampled, it is useful to also use a down-sampled PSF down-sampled, so that there is one PSF value for each LED module. The flux transfer model described in Chapter 4 was applied to determine the PSF. The PSF in one dimension can be found in Figure 60 on page 81. The 2D PSF used in this study can be found in Figure 76. The flux of this PSF is normalized, so that the integrated flux equals one. The PSF is symmetric in x and y direction and was represented in a matrix of dimension 13 ? 13, shown in Figure 76. This dimension was a compromise between accuracy and computational speed, since smaller dimensions favor computational speed, but do not take the light into account that is outside of the respective area (13 ? 13 LED modules), which decreases the accuracy of the simulation.    The centre matrix entry reads 0.375, meaning that 37.5% of the flux emitted by the centre LED is emitted into the area on the screen right above the backlight module. The four adjacent areas correspond to four neighboring backlight modules and each receives 9.5% of the flux emitted by the centre LED. The colour shown in Figure 76 is based on a colour gradient from red 111 (largest value) over yellow to green (smallest value). The rapid light falloff can be seen, which provides a high contrast. In Chapter 4, the efficiency of these backlight modules was determined: 61% of the created light is emitted by a top diffuser.  Figure 76: Normalized and down-sampled modified point spread function measured off the experimental backlight. Every cell represents one LED module and the number in each cell represents the flux in this LED module coming from the central LED. 6.4.1.2 Simulation of Standard PSF of HDR Backlight  For the second PSF of this study a common HDR setup was simulated with ray-tracing software [41]. In this backlight, the PSF of the LEDs is not changed by any optical structure. The LEDs shine on a diffuser and the backplane is highly reflective. To find the PSF of one such LED in this system, LED was modeled as a surface with a radius of 1.25 mm with an absorption of 50% and diffuse reflection of 50%. The screen was at a distance of 20 mm from the LED and had a transmission of 53% and reflection of 47%, which corresponded to the screen distance and specifications used in the experimental setup of the backlight. The backplane of the LED module was made 98% specular reflective, emulating the reflector film ESR commonly used for this purpose. The schematics of the simulation can be found in Figure 77. 112  Figure 77: Schematics of the simulation setup to determine the PSF of an HDR display that consists only of LEDs and a diffuser screen. ESR is used as a reflective backplane. The distance between the ESR plane and screen was 20 mm. After the ray-tracing, the average of the flux within each backlight module area was computed to create down-sampled versions of the PSF (see Figure 78). In this case, only 15.5% of the flux emitted by the centre LED stays in the corresponding area on the screen, compared to 37.5% in the case of the modified PSF in Figure 76. The colour gradients in Figure 76 and Figure 78 are the same: the maximum value is 0.375 and minimum is 3.8E-08. This visualisation helps to demonstrate that the light falloff of the modified PSF is much quicker and hence the contrast is higher than of the standard PSF, which is much broader. To determine the efficiency of this backlight module, 100% specular reflective side walls were introduced. This ensures that no light is lost through the sides. With this change, it was simulated that 91.5% of the light produced by the LED is transmitted through the screen. 113  Figure 78: Normalized and down-sampled point spread function simulated for a standard HDR setup. Every cell represents one LED module and the number in each cell represents the flux in this LED module coming from a central LED. 6.4.1.3 Results for Different Point Spread Functions The deconvolution method that was used kept the average luminance of the image constant, which meant that the average LED value did not depend on the shape of the PSF. Therefore, the average LED value in an image, which corresponded to power consumption, was the same for the modified PSF and the standard PSF. The relative power consumption of the 20 images was determined for different numbers of LEDs: from uniform brightness and 32 ? 18 to 1920 ? 1080, where the latter one represented the theoretical minimum power consumption, since every LCD pixel corresponded to one LED and can be dimmed. Figure 79 shows the theoretical relative power consumption of these 20 images for a display with 100% efficiency. For every backlight with a certain number of LEDs, the power consumptions of the 20 images were averaged and the backlight efficiencies (61% for the modified PSF and 91.5% for the standard PSF) taken into account (see Figure 80). It shows that an increase in the number of LEDs decreases power consumption and that the backlight with modified PSF needs about 1.5 times more power than a backlight with a standard PSF. 114  Figure 79: Relative power consumption for 20 different HDR images for a uniform brightness backlight and a set of different backlights with different numbers of LEDs: 32 ? 18, 48 ? 27, 96 ? 54, 192 ? 108, 384 ? 216, 1920 ? 1080 (with efficiency 1). Every data series has 20 data points, which correspond to the 20 images.  Figure 80: Average power consumption of backlights showing 20 HDR images: backlight with modified PSF (diamonds) and standard PSF (squares) with different numbers of LEDs (label). 00.20.40.60.810 0.1 0.2 0.3 0.4 0.5 0.6 0.7Relative power consumption Average luminance of image uniform32 ? 1848 ? 2796 ? 54192 ? 108384 ? 2161920x108000.10.20.30.40.50.6100 1000 10000 100000 1000000Relative power consumption Number of LEDs Modified PSFStandard PSF32 ? 18  48 ? 27  96 ? 54  192 ? 108  384 ? 216  1920 ? 1080 115  Figure 81: Average RMS error of backlights having either modified PSF (diamonds) or standard PSF (squares) with different numbers of LEDs. The fractional RMS errors were determined for the two kinds of displays (see Figure 81). As expected, both backlights exhibit smaller errors with more LEDs. Due to a higher degree of localization, the RMS error of the modified PSF is smaller by almost a factor of two compared to the RMS error of the standard PSF.  6.4.2 Matching Luminance Values of Highlights As discussed in 6.3.1, the capability of showing luminance realistic highlights is an important feature of HDR displays. However, through the deconvolution process some of the highlights were not provided with enough light, especially in the case of the standard PSF (see Figure 82). 00.10.20.30.4100 1000 10000 100000Fractional RMS error Number of LEDs Modified PSFStandard PSF116  Figure 82: Line scan through one image. Shown are the luminance values of the original picture (black), and line scans through images created by a convolution of the deconvoluted image with modified PSF (red) and with standard PSF (green). The LED values needed to be slightly increased to match the goal luminance values, however, since one point on the screen depends on multiple LEDs, there is a risk of overshooting the luminance values. This is why an iterative solution was used: First, in one image, the maximum difference between goal luminance values and the simulated backlight luminance is located. Second, the respective LED value was increased by half of the difference. This was repeated until all luminance goals were matched by the image. Since in the case of the modified PSF only 38% of the flux stayed in the centre area (see Figure 76), in one iteration step, the luminance value of this area is only increased by 19% of the absolute difference. Adjacent LED module areas will be increased by around 5% of the absolute 00.10.20.30.40.50.60.70.80.911 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47Relative Luminance Pixel of down-sampled image GoalModified PSFStandard PSF117 difference. This produces minimal unnecessary overshooting. After updating the simulated backlight with the new LED value, the location of the next maximum difference was determined and this LED value increased accordingly.  Figure 83: Line scan through one picture created with matched luminance goal. Shown are the luminance values of the original picture (black), the luminance values after the iterative increase of the LED values with modified PSF (red) and with standard PSF (green). Figure 83 shows line scans across one image to illustrate the effect of the increased luminance values for highlights. After the iteration process the luminance values are matched by both PSF types. In the case of the standard PSF, it is clearly visible that light from these highlights leaks into dimmer regions, diminishing the local contrast. 00.10.20.30.40.50.60.70.80.911 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47Relative luminance Pixel of a down-sampled image GoalModified PSFStandard PSF118  Figure 84: Average power consumption of backlights showing 20 HDR images: backlight with modified PSF (diamonds) and standard PSF (squares) with different numbers of LEDs. The relative power consumptions yielded with basic deconvolution are shown as black crosses. For both PSFs, the increased LED values also increases the power consumption, but due to wider PSF, the backlight with the standard PSF needs more additional energy than the backlight with the modified PSF (see Figure 84). In the case of the new PSF, the matching of the high luminance values decreased the respective RMS errors; in case of the standard PSF it increased (see Figure 85). 00.10.20.30.40.5100 1000 10000 100000Relative power consumption Number of LEDs Modified PSF (max lum)Standard PSF (max lum)Modified PSF (decon)Standard PSF (decon)119  Figure 85: Average RMS error of backlights matching the maximum luminance values of the goal images, having either modified PSF (diamonds) or standard PSF (squares) with different numbers of LEDs. The RMS errors yielded with basic deconvolution are shown as black crosses. 6.5 Summary and Conclusions All here discussed HDR backlight systems consume less power than standard LCD displays by a factor between 2 and 5, when showing image content with an averaged luminance of 20% of maximal white. The averaged luminance of these randomly chosen HDR images was a few percentage points lower than for standard NTSC TV content [2]. It was shown that more localized PSFs have a smaller RMS error and can produce luminance realistic highlights with less power, as long as the LED module efficiency is the same. The power consumption and the RMS errors of the three backlights, able to show bright highlights, are summarized in Figure 86 00.10.20.30.4100 1000 10000 100000Fractional RMS error Number  of LEDs Modified PSF (max lum)Standard PSF (max lum)Modified PSF (decon)Standard PSF (decon)120 and Figure 87, respectively. With the actual LED module efficiencies, an HDR display using the technology from the experimental backlight discussed in Section 4.3.4 uses about 11 percentage points more power than a standard HDR display, no matter how many LEDs are used.  Figure 86: Relative power consumption of three different HDR backlights, which provide enough light for highlights. The HDR display with RP-LCDs did not show a benefit over the experimental backlight: the power consumption was similar to the backlight with modified PSFs, but the RMS error was higher. However, the RMS error was smaller than the error of a standard HDR. The main problem was the low efficiency, which is mainly caused by the relatively low transmission and substantial absorption of the RP-LCDs. Limited contrast was the main reason for the higher RMS error: the RP-LCDs had relatively low transmission in transmission state and relatively high transmission in the reflection state. 00.10.20.30.40.50.6100 1000 10000 100000Relative Power consumption Number of of LEDs Modified PSFStandard PSFRP-LCD121  Figure 87: RMS errors of three different HDR backlights, which provide enough light for highlights. For the future, it is expected that the average luminance of HDR content will not change substantially. However, small and bright highlights will probably play a much more important role, since it has been shown that high peak luminance values make images to be perceived as more interesting and realistic [23]. This is when smaller PSFs such as modified PSF or displays with RP-LCDs are most efficient and have the largest benefit over standard HDR displays. In the future, HDR displays with high local contrast will be needed. The required contrast could be provided by the modified PSF and the RP-LCD backlight. 00.10.20.30.4100 1000 10000 100000Fractional RMS error Number of LEDs Modified PSFStandard PSFRP-LCDs122 Chapter  7: Conclusion  The goal of this research was to improve contrast and uniformity of HDR displays. Nowadays, HDR displays have the highest image quality, but suffer from a trade-off between contrast and uniformity. A new point spread function was developed that combines uniformity and local contrast and can create smooth gradients. This enables display manufacturers to exploit the full potential of HDR displays: a contrast comparable to the capabilities of the human eye, while still being able to show a uniform screen.  A backlight using a new PSF that was optimized for uniformity and smooth gradients was developed. It has been shown that a new backlight configuration producing a new PSF that is quite efficient, uniform and can show smooth gradients. Small visual imperfections can be easily removed computationally by compensation through the front LCD. Due to the rapid light falloff the image processing necessary for HDR displays based on such a backlight would be significantly reduced. The new design is sufficiently thin for a direct-LED backlight and is suitable for large scale use. The fact that the backlight modules are easily scalable makes them applicable for high-end displays with unit spacings of only a few millimeters as well as for low-cost displays with only a few dozens of LEDs. In the latter case, a uniform brightness screen can be achieved with a limited number of LEDs without losing the capability of local dimming and smooth gradients. The PSF of the developed backlight falls off to zero much quicker than the PSFs in standard HDR displays. This increases the local contrast and the capability to match a certain luminance goal within the area of each LED module for a given image on the screen. So far, this method 123 needs marginally more power than a standard HDR display, since the backlight is less efficient; however, it has been shown that highlights can be shown more efficiently with more localized PSFs. Since future HDR content is expected to exhibit more highlights, displays with the newly developed highly localized PSF will become more efficient in comparison to displays with conventional less localized PSFs. A second kind of HDR backlight with RP-LCDs as additional light modulators between LEDs and LCD screen were studied. 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Unser, "Splines - A Perfect Fit for Signal and Image Processing," IEEE Signal Processing Magazine, pp. 22-38, 1999.    131 Appendices  Appendix A    MATLAB code for ray tracing simulation function F = fitness1(x)   hs = x(16) %distance between source and screen in unit spacings   %% Defining source  source_res_x = 10;                  %Defines how many point sources are in one direction within the unit cell source_res_y = source_res_x; spacing = 1/source_res_x;                 %Point source spacing in unit cells   source = zeros(source_res_y, source_res_x); source_x = zeros(source_res_y, source_res_x);       %gives x positions of point sources source_y = zeros(source_res_y, source_res_x);       %gives y positions of point sources   m = 0; n = 1; for i = 1:source_res_x/2     %Creates first half quadrant out of input values      for j = 1:source_res_x/2-m         source((i-1)*(source_res_x)+m +j) = x(n);         n= n+1;      end      m=m+1; end   for i = 1:source_res_y/2      for j = 1:source_res_x/2         source(i,j) = source(j,i);      end end   for i = 1:source_res_y/2     for j = 1:source_res_x/2         source(source_res_y - (i-1),j) = source(i,j);         source(i,source_res_x - (j-1)) = source(i,j);             source(source_res_y - (i-1),source_res_x - (j-1)) = source(i,j);      end end   for i = 1:source_res_y %Determines the x and y positions of the point sources     for j = 1:source_res_x         source_x(i,j) = (-0.5 + spacing/2) + spacing*(j-1); 132         source_y(i,j) = (0.5 - spacing/2) - spacing*(i-1);     end end %% Defining screen screen_res_x = 3*source_res_x + 1;                 %Defines how many detector areas are in one direction within the PSF area screen_res_y = screen_res_x;   screen = zeros(screen_res_y, screen_res_x); screen_x = zeros(screen_res_y, screen_res_x);       %gives x positions of screen points screen_y = zeros(screen_res_y, screen_res_x);       %gives y positions of screen points   for i = 1:screen_res_y     for j = 1:screen_res_x         screen_x(i,j) = -1.5 + spacing*(j-1);         screen_y(i,j) =  1.5 - spacing*(i-1);     end end     %% Do ray-tracing for i = 1:source_res_y     for j = 1:source_res_x         for n = 1:screen_res_y             for m = 1:screen_res_x                 if source(i,j) > 0;                     dy = (screen_y(n,m) - source_y(i,j));   %distance in y direction                      dx = (screen_x(n,m) - source_x(i,j));   %distance in x direction                     r_plane2 = (dy^2 + dx^2);                    %square of distance of points in one plane                     r = sqrt(r_plane2 + hs^2);                 %length of r vector                     alpha = acos(hs/r);  %angle between Normal and r vector                     y_12 = abs(source_y(i,j) + dy/2);   %y component of the centre point between source and screen                     x_12 = abs(source_x(i,j) + dx/2);   %x component of the centre point between source and screen                     if y_12 <= 0.5                      %Makes sure that centre point is within unit cell: abs(0.5)                         if x_12 <= 0.5                  %Makes sure that centre point is within unit cell: abs(0.5)                             screen(n,m) = screen(n,m) + source(i,j) * cos(alpha)^2 * (hs/r)^2;                         end                     end                 end             end         end     end end  %% Stitching nine PSFs together sc_uni = zeros(screen_res_y + source_res_y*2,screen_res_x + source_res_x*2); 133 sc_lin = zeros(screen_res_y + source_res_y*2,screen_res_x + source_res_x*2); sc_quad = zeros(screen_res_y + source_res_y*2,screen_res_x + source_res_x*2);   for i = 1:3     for j = 1:3         for n = 1:screen_res_y             for m = 1:screen_res_x                 sc_uni((i-1)*source_res_y + n,(j-1)*source_res_x + m) = screen(n,m) + sc_uni((i-1)*source_res_y + n,(j-1)*source_res_x + m);                 sc_lin((i-1)*source_res_y + n,(j-1)*source_res_x + m) = j*screen(n,m) + sc_lin((i-1)*source_res_y + n,(j-1)*source_res_x + m);                 sc_quad((i-1)*source_res_y + n,(j-1)*source_res_x + m) = j^2 * screen(n,m) + sc_quad((i-1)*source_res_y + n,(j-1)*source_res_x + m);             end         end     end end  %cuts out area of one unit cell on screen ssc_uni = sc_uni(screen_res_y - source_res_y:screen_res_y, screen_res_x - source_res_x:screen_res_x);        ssc_lin = sc_lin(screen_res_y - source_res_y:screen_res_y, screen_res_x - source_res_x:screen_res_x); ssc_quad = sc_quad(screen_res_y - source_res_y:screen_res_y, screen_res_x - source_res_x:screen_res_x);   %% Analysis mean = 0; dev_uni = 0; dev_lin = 0; dev_quad = 0; mean_lin = zeros(source_res_y+1,1); mean_quad = zeros(source_res_y+1,1); xs = zeros(source_res_y+1,1); fit_lin = zeros(source_res_y+1,1); fit_quad = zeros(source_res_y+1,1); min = 1000; max = 0;    for i = 1:source_res_y+1     for j = 1:source_res_x+1         mean = ssc_uni(i,j) + mean;         mean_lin(i,1) = ssc_lin(j,i) + mean_lin(i,1);       %Takes mean over each column         mean_quad(i,1) = ssc_quad(j,i) + mean_quad(i,1);       %Takes mean over each column         if ssc_uni(i,j) > max             max = ssc_uni(i,j);         end         if ssc_uni(i,j) < min             min = ssc_uni(i,j);         end              end     xs(i,1)=(i-1)*(1/source_res_y) -0.5; end 134   mean; mean = mean / ((source_res_y+1) * (source_res_x+1));    %Normalizes mean mean_lin = mean_lin/(source_res_y+1);    %Normalizes mean of each column mean_quad = mean_quad/(source_res_y+1);  %Normalizes mean of each column   po_l = polyfit(xs,mean_lin,1); %Fits a linear gradient  po_q = polyfit(xs,mean_quad,2);  %Fits a quadratic gradient  for i = 1:source_res_y+1     fit_lin(i,1) = po_l(2) + po_l(1)*xs(i,1);     fit_quad(i,1) = po_q(3) + po_q(2)*xs(i,1) + po_q(1)*xs(i,1)^2; end   for i = 1:source_res_y+1     for j = 1:source_res_x+1         dev_uni = (ssc_uni(i,j)-mean)^2/mean^2 + dev_uni;         dev_lin = (ssc_lin(j,i) - fit_lin(i,1))^2/((fit_lin(i,1))^2) + dev_lin;         dev_quad = (ssc_quad(j,i) - fit_quad(i,1))^2/((fit_quad(i,1))^2) + dev_quad;     end end   dev_uni = sqrt(1/((source_res_y+1) * (source_res_x+1))*dev_uni);     %fractional standard deviation dev_lin = sqrt(1/((source_res_y+1) * (source_res_x+1)) * dev_lin);                        %fractional standard deviation dev_quad = sqrt(1/((source_res_y+1) * (source_res_x+1)) * dev_quad);                      %fractional standard deviation   F=sqrt(dev_uni^2 + dev_lin^2 + dev_quad^2);    135 Appendix B    MATLAB code simulating HDR module with nine RP-LCDs  function [F,I,t, LED] = optim_1LED(x, I_goal)  t_min = 0.146; %Minimum transmission through RP-LCD t_max = 0.263; %Maximum transmission through RP-LCD   r_backlight = 0.95; %reflectivity of backlight const = x(10);   t = zeros(3,3); %3x3 matrix with the transmission values of each RP-LCD r = zeros(3,3); %3x3 matrix with the reflection values of each RP-LCD  abs = zeros(3,3); %3x3 matrix with the absorption values of each RP-LCD I = zeros(3,3); %3x3 matrix for the calculated flux values of each RP-LCD I_0 = zeros(3,3); %3x3 matrix with the flux output of in the case without RP-LCDs   norm_factor = 16.606/9; %Normalization factor to normalize average flux to 1.   I_0(1,1) = 0.9322/norm_factor; I_0(1,3) = 0.9322/norm_factor; I_0(3,1) = 0.9322/norm_factor; I_0(3,3) = 0.9322/norm_factor; I_0(1,2) = 1.72498/norm_factor; I_0(2,1) = 1.72498/norm_factor; I_0(2,3) = 1.72498/norm_factor; I_0(3,2) = 1.72498/norm_factor; I_0(2,2) = 5.97771/norm_factor;   t_avg = 0; for i=1:3     for j = 1:3         t(i,j) = x((i-1)*3+j);    %imports variables         t_avg = t_avg + t(i,j)/9; %determined average transmission     end end   total_r = 0;        %total amount of reflected light r_avg = 0;          %average reflectivity   for i=1:3     for j = 1:3         abs(i,j) = 0.08 - (t(i,j)-t_min)/(t_max - t_min)*0.036; %interpolates the absorption linearly         r(i,j) = 1-t(i,j) - abs(i,j); %Calculates reflection based on T and A         r_avg = r_avg + r(i,j)/9;   %Averages reflectivity of all RP-LCDs 136         total_r = total_r + r(i,j) * I_0(i,j); %Determines total reflected light     end end   r_avg = r_avg * r_backlight;     %takes reflection off backlight into account total_r = total_r * r_backlight; %takes reflection off backlight into account dev = 0;   for i=1:3     for j = 1:3         I(i,j) = const * t(i,j)*(I_0(i,j) + total_r/(1-r_avg)*t(i,j)/(t_avg*9)); %geometric series to calculate flux output         if I_goal(i,j) >0             dev = dev + (I(i,j)-I_goal(i,j))^2;         else             dev = dev + I(i,j)^2;         end     end end                137 Appendix C     Figure 88: Grey scale HDR image with full resolution: 1920 ? 1080. It was taken by Ryan Whitehead. 138  Figure 89: Down-sampled image: 96 ? 54. The square root of every pixel value is taken. 139  Figure 90: Down-sampled image: 16 ? 9. The square root of every pixel value is taken.  

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