Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Wiley, Neale
Master of Science - MSc
University of British Columbia
Magnetic resonance imaging (MRI) is a technique that allows non-invasive investigation of the properties of tissue through manipulation of the properties of the hyperfine splitting of the hydrogen atom in a magnetic field. Diffusion MRI is a variant of MRI that sensitises the imaging process to diffusion of water molecules in tissue through the use of strong gradient magnetic fields to dephase the signal of any protons undergoing diffusion, leading to a reduction in signal magnitude in that voxel. There is increasing use of iterative reconstruction algorithms of MRI data using the principles of compressed sensing (CS) that rely on the sparsity of information in the image domain to improve reconstructions of undersampled MRI data. Further refinements to the CS algorithm not only rely on the sparsity of single MRI images, but take advantage of low rank properties of some MRI data along a further dimension to permit even higher undersampling. Diffusion tensor imaging (DTI) acquires many sampled volumes of the same field of view encoded along different diffusion vectors. A reconstruction using low rank properties in the diffusion dimension could improve DTI image reconstructions. However, the random phase of diffusion imaging due to subject motion during the diffusion sensitising gradients of the acquisition destroys the low-rank nature of diffusion datasets. Phase correction of a randomly undersampled k-space is impossible, but the application of a phase correction factor to the sensitivity maps used in multicoil acquisitions can correct the phase error to restore low rank properties to the data. Alternatively, one can use a similar technique to perform a secondary reconstruction of diffusion tensor image data, as the magnitude images of a diffusion scan already discard all phase information, which includes the motion corruption. This secondary reconstruction efficiently denoises and simultaneously interpolates the images to a higher resolution by leveraging a projection into a low-rank subspace that shares information across different diffusion shots. This type of denoising performs on par with other contemporary denoising algorithms while reducing upsampling blurring through the use of zero-padded interpolation.
Subspace informed compressed sensing reconstructiontechniques for diffusion tensor imagingbyNeale WileyA THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2024© Neale Wiley, 2024The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Subspace informed compressed sensing reconstruction techniques for diffusion tensor imagingsubmitted by Neale Wiley in partial fulfilment of the requirements forthe degree of Master of Sciencein PhysicsExamining Committee:Shannon Kolind, Assistant Professor, Medicine (Neurology), UBCSupervisorAlex Mackay, Professor Emeritus, Physics and Astronomy, UBCSupervisory Committee MemberiiAbstractMagnetic resonance imaging (MRI) is a technique that allows non-invasive investigationof the properties of tissue through manipulation of the properties of the hyperfine splitting of thehydrogen atom in a magnetic field. Diffusion MRI is a variant of MRI that sensitises the imagingprocess to diffusion of water molecules in tissue through the use of strong gradient magneticfields to dephase the signal of any protons undergoing diffusion, leading to a reduction in signalmagnitude in that voxel.There is increasing use of iterative reconstruction algorithms of MRI data using theprinciples of compressed sensing (CS) that rely on the sparsity of information in the imagedomain to improve reconstructions of undersampled MRI data. Further refinements to the CSalgorithm not only rely on the sparsity of single MRI images, but take advantage of low rankproperties of some MRI data along a further dimension to permit even higher undersampling.Diffusion tensor imaging (DTI) acquires many sampled volumes of the same field of viewencoded along different diffusion vectors. A reconstruction using low rank properties in thediffusion dimension could improve DTI image reconstructions. However, the random phase ofdiffusion imaging due to subject motion during the diffusion sensitising gradients of theacquisition destroys the low-rank nature of diffusion datasets. Phase correction of a randomlyundersampled k-space is impossible, but the application of a phase correction factor to thesensitivity maps used in multicoil acquisitions can correct the phase error to restore low rankproperties to the data.Alternatively, one can use a similar technique to perform a secondary reconstruction ofdiffusion tensor image data, as the magnitude images of a diffusion scan already discard allphase information, which includes the motion corruption. This secondary reconstructionefficiently denoises and simultaneously interpolates the images to a higher resolution byiiileveraging a projection into a low-rank subspace that shares information across differentdiffusion shots. This type of denoising performs on par with other contemporary denoisingalgorithms while reducing upsampling blurring through the use of zero-padded interpolation.ivLay SummaryDiffusion tensor imaging is a magnetic resonance imaging (MRI) technique that allows us toinvestigate how water molecules move on a microscopic level in tissue, but it is a slow processthat must collect information about how water moves in dozens of different directions. There aremany ways to accelerate these scans by measuring less data and using computationalresources to fill in the missing information, allowing us to scan faster.The technique in the following thesis lays out a way to use the similarities in the data from eachdifferent diffusion direction to help reconstruct useful, high-quality images. This requiredsignificant error correction in the image reconstruction process to restore the similaritiesbetween different images. This technique was also applied to remove image noise and increasethe detail in previously acquired images by taking advantage of the shared information betweeneach image.vPrefaceThe work presented in this thesis was carried out at the UBC MRI Research Facility at theUniversity of British Columbia between Sept 2022 and April 2024. Participating volunteers in thisstudy were consented under the approval of the Clinical Ethics Research board at the Universityof British Columbia (A Registry of Magnetic Resonance Scans on Normal Volunteers,H20-03417).The idea for subspace based compressed sensing applied to DTI was inspired by my colleagueDr. Adam Dvorak and my supervisor, Dr. Shannon Kolind. The use of TSE for a suitablecandidate for this application was my own, and the coding of the sampling schemes, collectionof data and analysis of the acquired data was my work. Additionally, the application of thereconstruction to already acquired DTI data was mine from conception to finish, though myanalysis of FA, MD and tractography was greatly assisted by Sharada Balaji.viTable of ContentsAbstract......................................................................................................................................... iiiLay Summary.................................................................................................................................vPreface..........................................................................................................................................viTable of Contents......................................................................................................................... viiList of Tables................................................................................................................................. xiList of Figures...............................................................................................................................xiiList of Abbreviations.................................................................................................................... xivAcknowledgments....................................................................................................................... xvi1. Background information of MRI, compressed sensing and diffusion................................11.1 Magnetic resonance imaging.............................................................................................31.1.1 The physics of nuclear magnetic resonance.............................................................31.1.2 Refocusing pulses.....................................................................................................61.1.3 Gradient fields in MR for signal localization..............................................................71.1.4 Data readouts and k-space.....................................................................................101.1.5 Image reconstruction...............................................................................................111.1.6 The Nyquist limit......................................................................................................111.1.7 Parallel imaging...................................................................................................... 121.1.8 Noise in MRI........................................................................................................... 131.2 Compressed sensing....................................................................................................... 151.2.1 Basics of compressed sensing reconstruction algorithms...................................... 151.2.2 Least-squares minimization.................................................................................... 161.2.3 Undersampling........................................................................................................171.2.4 Subspace extension of compressed sensing..........................................................171.3 Structure of the brain....................................................................................................... 191.3.1 Structure of white matter and grey matter...............................................................191.3.2 Axonal bundles and tracts.......................................................................................201.4 Diffusion...........................................................................................................................211.4.1 Diffusion in grey matter and white matter............................................................... 22vii1.4.2 Use of diffusion in brain pathology and research....................................................231.5 Diffusion sensitisation of MR imaging..............................................................................241.5.1 Diffusion encoding gradients...................................................................................251.5.2 DIffusion encoding strength.................................................................................... 251.5.3 Diffusion encoding vectors......................................................................................271.5.4 Diffusion MRI artefacts............................................................................................281.5.5 Diffusion tensor imaging readout pulse sequences................................................ 291.5.6 Undersampling in 2D DTI scans............................................................................. 321.5.7 Uses of different DTI acquisition schemes..............................................................342. Applying subspace compressed sensing reconstructions to TSE-DTI........................... 352.1 Introduction...................................................................................................................... 352.1.1 Objectives............................................................................................................... 362.1.2 Motivation................................................................................................................362.1.3 Hypothesis.............................................................................................................. 372.2 Methods........................................................................................................................... 372.2.1 Sparsity in DTI........................................................................................................ 372.2.2 Randomly undersampled TSE scans......................................................................442.2.3 Complex DTI scans.................................................................................................462.2.4 Phase errors in complex DTI.................................................................................. 502.2.5 Phase navigation.................................................................................................... 512.2.6 Partial reconstruction phase correction...................................................................532.2.7 Application of subspace compressed sensing........................................................542.3 Results.............................................................................................................................552.3.1 Image reconstructions.............................................................................................562.3.2 SNR problems.........................................................................................................562.3.3 Possible uses of the subspace reconstructed SS-TSE DTI scan........................... 582.4 Conclusions..................................................................................................................... 593. Applying subspace compressed sensing reconstructions to already-reconstructedEPI-DTI........................................................................................................................................613.1 Introduction...................................................................................................................... 61viii3.1.1 Objectives..................................................................................................................... 613.1.2 Motivation................................................................................................................613.1.3 Hypothesis.............................................................................................................. 623.2 Methods........................................................................................................................... 623.2.1 EPI-DTI datasets.....................................................................................................623.2.2 Eddy and motion correction of EPI-DTI data.......................................................... 633.2.3 Subspace generation and size................................................................................633.2.4 Zero-padding to higher resolution...........................................................................643.2.5 Application of subspace compressed sensing........................................................643.3 Results.............................................................................................................................653.3.1 Image reconstructions.............................................................................................663.3.2 Voxel size vs. true resolution.................................................................................. 683.3.3 Denoising effect...................................................................................................... 693.3.4 Motion artefact compensation effect....................................................................... 703.3.5 Interpolation artefacts............................................................................................. 723.3.6 Calculated diffusion metrics: fractional anisotropy and mean diffusivity................. 733.3.7 Tractography...........................................................................................................773.4 Discussion....................................................................................................................... 783.4.1 Comparison to standard upscaling processes........................................................793.4.2 Absence of higher frequency data.......................................................................... 793.4.3 Removal of information not represented by basis vectors......................................803.4.4 Correction for motion-deteriorated volumes............................................................813.4.5 Regularization factor...............................................................................................833.4.6 Ground truth unavailability...................................................................................... 833.4.7 2D vs 3D Upscaling................................................................................................ 843.4.8 Rician noise characteristics and problems..............................................................853.5 Conclusions..................................................................................................................... 864. Conclusions and Discussion................................................................................................884.1 Applying subspace compressed sensing reconstructions to TSE-DTI............................ 88ix4.1.1 Impact of the results..................................................................................................... 884.1.2 Limitations...............................................................................................................894.1.3 Future work.............................................................................................................894.2 Applying subspace compressed sensing reconstructions to already reconstructed EPIDTI......................................................................................................................................... 904.2.1 Impact of the results................................................................................................904.2.2 Limitations...............................................................................................................914.2.3 Future work.............................................................................................................934.3 Concluding remarks.........................................................................................................95Bibliography...............................................................................................................................96xList of TablesTable 2.1 Comparison of subspace size and compression factors in different DTI scans ..........44xiList of FiguresFigure 1.1 Stejskal-Tanner DTI pulse sequence with EPI readout gradients ............................. 30Figure 1.2 EPI k-space filling order with an undersampling factor of 2 .......................................31Figure 1.3 Stejskal-Tanner DTI pulse sequence with SS-TSE readout gradients ...................... 32Figure 1.4 SS-TSE k-space filling order with an undersampling factor of 2 ............................... 33Figure 2.1 Information contained per principal component of a HARDI-60 DTI scan .................39Figure 2.2 First 12 principal subspace images created through projection of the HARDI-60 DTIdataset onto the basis spanned by the right singular vectors of the PCA decomposition withvaried window levels ...................................................................................................................40Figure 2.3 First 12 principal subspace images created through projection of the HARDI-60 DTIdataset onto the basis spanned by the right singular vectors of the PCA decomposition withequal window levels ....................................................................................................................41Figure 2.4 Information contained per principal component of a 128 direction HARDI DTI scan. 42Figure 2.5 A comparison of a projection onto the first 6 images of a PCA-generated subspacebasis and a similar projection onto the even real spherical harmonics .......................................43Figure 2.6 K-space phase sampling schemes used for a 4x undersampled SS-TSE HARDI-60DTI scan ..................................................................................................................................... 46Figure 2.7 Information contained per principal component of a HARDI-60 SS-TSE DTI scan ...47Figure 2.8 A magnitude-only subspace basis to reconstruct diffusion data failed to produceuseful images due to the complex phase errors in the diffusion image space ........................... 48Figure 2.9 Information contained per principal component of a complex HARDI-60 SS-TSE DTIscan ............................................................................................................................................ 49Figure 2.10 Low-frequency phase images of 9 different diffusion shots in a single slice of aSS-TSE diffusion scan ................................................................................................................50Figure 2.11 Reconstructions of a SS-TSE HARDI-60 DTI dataset with a standard compressedsensing reconstruction and with a phase-corrected subspace reconstruction ........................... 58Figure 3.1 Baseline b2000 diffusion images and the same images denoised and upscaled inplane by a factor of 1.98 via UPSIZER .......................................................................................66Figure 3.2 Cubic interpolation of a denoised image compared to an UPSIZER image at thesame resolution .......................................................................................................................... 67xiiFigure 3.3 UPSIZER denoised images reconstructed at original resolution of 2 x 2 x 2 mm, at aresolution of 1.24 x 1.24 x 2 mm, and at a resolution of 0.62 x 0.62 x 2 mm ............................. 69Figure 3.4 Comparison of two imaging sets denoised using lPCA, MPPCA, P2S and UPSIZERand their difference images from the original data ......................................................................71Figure 3.5 Original single b2000 diffusion image viewed coronally showing motion corruptionartefacts as black lines and UPSIZER-denoised images at the same resolution without motioncorruption ....................................................................................................................................72Figure 3.6 Blocking and line artefacts from an integer interpolation factor of 2 in each dimension.73Figure 3.7 FA and MD maps from original images and images denoised through variousdenoising algorithms ...................................................................................................................75Figure 3.8 Tractography of the optic radiation, seeded at the lateral geniculate nucleus from DTIdenoised using A. No denoising, B. P2S denoising, C. MPPCA denoising and D. UPSIZER ... 78Figure 3.9 UPSIZER reconstruction of a multishell DTI with 12 images with a b-value of 0s/mm2, 55 images with a b-value of 2000 s/mm2, 35 b-value of 1000 s/mm2 images and 24b-value of 500 s/mm2 images ..................................................................................................... 81Figure 3.10 Subplots of original images, magnitude, and complex (images denoised usingUPSIZER to compare CSF denoising in complex or magnitude spaces ....................................86xiiiList of AbbreviationsAMC-PROUD: Amsterdam UMC- PRospective Undersampling in Multiple DimensionsADC: Apparent Diffusion CoefficientAD: Axonal DiameterBART: Berkeley Advanced Reconstruction ToolboxBW: BandwidthCS: Compressed SensingCSF: Cerebrospinal FluidDTI: Diffusion Tensor ImagingDWI: Diffusion Weighted ImagingEPI: Echo Planar ImagingFSL: FMRIB Software LibraryFA: Fraction Anisotropy (or Flip Angle)FOV: Field of ViewGRAPPA: GeneRalized Autocalibrating Partial Parallel AcquisitionHARDI: High Angular Resolution Diffusion ImagingLPCA: Local Principal Component AnalysisMPPCA: Marchenko-Pastur Principal Component AnalysisMRI: Magnetic Resonance ImagingMD: Mean diffusivityPCA: Principal Component AnalysisP2S: Patch2SelfRF: RadiofrequencyRMS: Root Mean SquareSAR: Specific Absorption RateSENSE: Sensitivity EncodingxivSNR: Signal to Noise RatioSS-TSE: Single shot turbo spin echoTE: Time to echoTR: Time to repeatTSE: Turbo Spin EchoUPSIZER: UPsampling through Subspace Informed ZEro-Padded ReconstructionxvAcknowledgmentsThis work was performed on the traditional, ancestral, and unceded territory of thexwməθkwəy̓əm (Musqueam) People.In addition to the formal supervision by Dr. Shannon Kolind, I also received significant help onimage reconstruction from Dr. Adam Dvorak and diffusion tensor imaging from Sharada Balaji.I want to thank the MRI technologists at UBC MRI Research Centre for their enthusiasticassistance in performing some of the scanning required for this thesis.I also greatly appreciate my partner, Meghan Kerr, for cheering me up when my imagereconstructions were failing over and over again.xvi1. Background information of MRI, compressed sensing and diffusionIntroductionWith today's fast films, you can light the way your eye sees the scene. You can abuse thefilm and create subtleties in contrast with light and exposure, diffusion and filters. That'swhat makes it an art. Conrad Hall (1926-2003)While the above quote refers to the art of cinema, it seems almost equally applicable totoday’s engineering of magnetic resonance imaging and the advances of technology in allaspects of medical imaging. In MRI specifically, the variations in radiofrequency and gradientfield pulses and sequence timing can create medically relevant changes in image contrast thathave advanced our knowledge of biology and pathology significantly. These different contrastsallow further insight in terms of diagnosis and disease progression, as well as providing one ofthe best tools for in vivo investigations of living processes; in particular, the intricacies ofconnectivity in the mammalian brain. Since 1973, when the seminal magnetic resonanceimaging experiments were performed by Paul Lauterbur [1], MRI techniques and thetransformation processes of the sampled data from the scanner into human legible images ofanatomy and pathology have advanced dramatically, each further development helping tofurther our knowledge of anatomy, physiology and pathology.This thesis focuses primarily on the transformation of diffusion-encoded data we receivefrom the MRI system into high quality diffusion tensor images of the brain to aid in the navigationof the structure and geography of axons, the tiny nervous structures that enable that vastmajority of communication within and between the brain, spinal cord and all other organs of thebody.1Chapter 1 consists of an introduction to magnetic resonance imaging, MR imagereconstruction, advanced data reconstruction techniques, diffusion MRI and the diffusionencoding process.Chapter 2 details the application of low rank data reconstruction techniques to diffusiondata acquired with a turbo spin echo readout, the problems inherent in applying these todiffusion imaging and how they can and can’t be overcome.Chapter 3 takes the application of these concepts and applies it to conventionallyacquired and reconstructed diffusion data, demonstrating the ability of sparsity promotingtechniques to simultaneously denoise and increase the resolution of diffusion tensor imaging.Alternate techniques for denoising and interpolation are compared for image quality anddiffusion tractography estimated from the denoised images.Chapter 4 is a summary of the work, its implications for the field, and further concepts toexplore.21.1 Magnetic resonance imagingThis section provides an overview of the basic physics behind magnetic resonanceimaging and how it permits measurement of spatially localised information from a sample, aswell as techniques for image reconstruction from the sampled data. It also describes the basicsof diffusion weighting of images in MRI.1.1.1 The physics of nuclear magnetic resonanceThe core principle on which magnetic resonance imaging operates is the quantummechanical hyperfine splitting of the energy levels of an atom in the presence of a strongmagnetic field [15,16]. Fortunately, these details of quantum mechanics are not strongly relevantto the everyday operation of the magnetic resonance imaging technique. It is, in most cases,sufficient to note the intrinsic property known as “spin” of the hydrogen nucleus (or proton) canbe in two quantum states when subjected to a strong magnetic field: states aligned parallel orantiparallel to the magnetic field. The Zeeman effect [17] or hyperfine splitting of energy levelsrefers to the fact that a spin in a state parallel to the magnetic field has a slighted lower intrinsicenergy than one aligned antiparallel. Therefore a system of hydrogen atoms at thermalequilibrium inside a magnetic field will have fractionally more spins in the lower-energy parallelstate than the anti-parallel state, giving the system as a whole a net magnetic moment.This net magnetic moment is best characterised the fraction of spins in the parallel stateto the antiparallel state, given by the Boltzman distribution in equation [1]:3Where represents the number of spins aligned parallel with the magnetic field ,represents the number of spins aligned antiparallel, is the energy difference between theparallel and antiparallel states, k is the Boltzmann constant , and isthe temperature of the sample in Kelvin.The net magnetic moment then is proportional to the number of spins inherent in thesample multiplied by the ratio of to ; this equilibrium magnetisation is therefore both fieldstrength and proton density dependent and is conventionally denoted .Each proton spin in a magnetic field precesses around the field vector at the Larmorfrequency dependent on the strength of the magnetic field as given in equation [2]:where is the gyromagnetic ratio; for hydrogen this is . This leadsto a similar precession of the net magnetic moment around at the same rate.Spins can be flipped from parallel to anti-parallel states by using electromagnetic pulseswith energy corresponding to the difference between states, or, equivalently, radiation withfrequency given by equation [3]. This is the equivalent frequency as the Larmor frequency; theenergy gap between the hyperfine states is given in equation [4].where4For most human-scale MRI systems, is on the order of MHz, in the radiofrequency(RF) range of the electromagnetic spectrum; as such, this radiation is typically called an RFpulse. The net effect of an on-resonance RF pulse on the entire system of spins is to rotate thecombined magnetization vector of all the spins around an axis perpendicular to the magneticfield, with a rotation angle dependent on the total energy delivered by the RF radiation,generally called the flip angle. The flip angle is determined by the amplitude of the absorbedRF pulse ( ) as well as the duration it is applied (equation [5]).The magnetic moment then precesses at the Larmor frequency around the vector of the mainmagnetic field, conventionally denoted , and gradually returns to equilibrium state. This returnto equilibrium is characterised by the relaxation constants of the local environment, T1, T2* andT2, which represent thermal and magnetic interactions with the neighbouring spins. In areference frame rotating at , the net transverse magnetization, denoted , and thenet longitudinal magnetization, denoted , evolve according to equations [6] and [7]respectively.The spins can be further manipulated by additional RF pulses and the application of additional,location dependent magnetic fields conventionally called gradient fields , and .51.1.2 Refocusing pulsesAfter an initial RF excitation pulse, each spin in a voxel experiences local differences inits immediate magnetic environment, which causes it to precess at a different rate than itsneighbours. This has an effect called T2* decay of the transverse magnetization as the differentphase accumulation for each of the spins gradually causes increasing destructive interferencebetween their magnetic moments. However, the spins can be rephased and returned tocoherence through the application of an additional 180o RF pulse at a point in time. This RFpulse inverts spin orientation, causing the phase dispersion of the spins to reverse. At a timethe magnetization is back in coherence, and the T2* decay is undone. This is called a spinecho, and the time is hence called the “time to echo” or TE, and is conventionally referredto as TE/2.The net transverse magnetization still decays through dynamic magnetic fieldfluctuations arising from interactions with other spins in the local environment. As this is astochastic process, the dephasing caused by these fluctuations can not be fully refocused bythe 180o pulse and results in an irreversible exponential decay of the transverse magnetization.This is described by a decay constant T2, which is generally considerably longer than T2*. In asimilar equation to T2* in equation [6], the decay at TE of a spin echo experiment is given byequation [8].Alternatively, spins can be deliberately dephased through the application of deliberatespatial variations in the field, known as gradient fields, through the use of additionalelectromagnetic coils. After time TE/2, the gradient fields’ polarity is reversed, causing the spinsto rephase and the net magnetisation reforms in what is called a gradient echo. As the gradient6reversal process does not affect the dephasing due to static inhomogeneities, the decay at TEof a gradient echo experiment is dominated by T2*, and is given in equation [9].Both spin echoes and gradient echoes can be performed multiple times after an initialexcitation pulse, with repeated 180o RF pulses or gradient field applications forming a train ofechoes. Each successive echo will have decayed slightly more due to T2 or T2* dephasing,depending on the echo being a spin or gradient echo respectively. The magnetic moment willeventually decay enough that the signal amplitude is less than the inherent electrical noise ofthe system, and echoes will no longer be of sufficient magnitude to be detected. The echoformation process is usually halted before this point and the ensemble of spins is allowed torecover to its equilibrium state. The system of spins will recover through thermal fluctuations andinteractions between neighbouring spins to its equilibrium state with the net magnetic momentrealigned to the axis of the main magnetic field. This recovery process is determined by thechemical makeup of the local tissue that governs spin interactions and as such is tissuedependent; each spin within the molecules that make up tissue has a characteristic recoverytime denoted T1 as given in equation [6].After the system of spins has undergone sufficient T1 relaxation back to equilibrium forthe experiment’s purposes, the series of RF and gradient pulses can be repeated to sample thesystem again and record a new set of echoes.1.1.3 Gradient fields in MR for signal localizationGradient fields can also be used to localise spins within the system by causing them toprecess at spatially dependent frequencies. Gradients in modern MRI systems attempt toproduce spatially linear increases and decreases in the main magnetic field along , and7axes, or any combination thereof. These respective gradients are usually called , , and. These gradient fields can be used for an abundance of purposes in the MRI pulsesequence, with the most common purpose as a method of localising the source of signal in theMRI.They can be used to create frequency selective slices within a subject by applying agradient field along the axis being slice encoded during the initial excitation pulse(conventionally , though in this case is not necessarily aligned with ). This creates alinear frequency variation along that axis proportional to the linear variation created in .If the excitation RF pulse is tuned to a specific frequency , only the on-resonanceslab of the sample centred at such that experiences an excitation or spin flip.This slab has thickness proportional to the bandwidth of the excitation pulse, asdescribed in equation [11] . There are, however, caveats due to the slightly different resonantfrequencies of hydrogen in different molecules and imperfections in shape of the RF pulse,which can give rise to problems such as chemical shift artefacts [18] and excitations of sidelobes outside the selected slab [19].The slice dimension in space can be encoded sequentially by repeating these slice-selective RFpulses centred at different frequencies to encode the respective slices of the sample.When the echo is refocused at time TE, the transverse magnetization precesses aroundthe z axis and induces a current in one or more RF receiver coils around the sample, which is8recorded and digitised. During this readout, if an additional gradient field perpendicular to theslice select gradient is applied, the spins along each point in that gradient direction will haveindividual resonant frequencies that contribute to the collected signal. The receiver coils pick upthe linear sum of all these frequencies, and during image reconstruction each separatefrequency is decomposed from the signal through the Fourier transform, This allows themagnitude of each frequency to be used to localise the signal source’s position along theencoding axis. This is called frequency encoding of the signal, and this axis is denoted as thefrequency axis.Correspondingly, to further localise the signal within the excited slice, a phase encodinggradient can be applied perpendicular to the frequency encoding gradient between the 180orefocusing pulse and the echo . This introduces a linear phase difference between spins atdifferent points along the new axis. During successive excitations or echoes, the phaseencoding step is repeated with different gradient strengths . The accumulated phase ofeach phase step is given in equation [12], with representing the duration in which the phaseencoding gradient is applied.Multiple phase encoding steps can be applied consecutively through the use of sequentialrefocusing pulses, such as in turbo spin echo (TSE) or echo planar imaging (EPI), each with adifferent magnitude of phase encoding gradient applied. The phase evolution of each locationalong the phase axis will be recorded by the receiver coils, and, similar to frequency encoding,the phase encoding is used during image reconstruction to assign the signal to its position alongthe phase axis.9The signal can thus be localised within a 2D slice, with the source of the signal localisedalong one axis by its frequency and along the other by its phase. Alternatively, in 3D imaging,the slice selective gradient is skipped to excite the entire volume, and an additional phaseencoding step is applied along the slice direction to phase encode the slice dimension instead.1.1.4 Data readouts and k-spaceAs mentioned, at time TE the resulting transverse magnetization of the echo induces anelectric signal into one or more radiofrequency receiver coils. This received data is digitised andthen sorted by frequency and phase and stored as data in a multidimensional array calledk-space. K-space can be organised in 2D stacks of slices or 3D volumes, with one axisrepresenting the frequency dimension and the other(s) representing phase encoding of thesignal.Additionally, the imaging can be repeated with different signal characteristics such asdifferent RF pulses, sequence timings, or additional applications of the gradient field during theacquisition pulse sequence. These different repetitions can be placed into a fourth dimension ofk-space, allowing sampling of imaging parameters within one imaging session. Of particularinterest in this thesis is the application of a pair of large gradient pulses on either side of an 180orefocusing pulse to dephase spins that are undergoing diffusion, sensitising the experiment tothe microscopic movement of spins along the plane of the applied gradients. The fourthdimension of k-space can be filled with data collected with diffusion encoding gradients appliedon a large number of vectors in 3D space, to acquire information about the 3D diffusionproperties of spins on a microscopic level.101.1.5 Image reconstructionOnce k-space is filled after sufficient successive repetitions of the MR pulse sequence,the image acquisition is considered completed. The data is stored and transferred to areconstruction computer as k-space matrices, and it then undergoes an image reconstructionprocess to change it into human-viewable images of the sample.The most fundamental image reconstruction technique for MRI data is the Fouriertransform [20]. K-space is constructed as the 2D (or 3D) Fourier transform of the spatialdistribution of signal, so k-space can be inverse Fourier transformed to create images of tissueand anatomy.1.1.6 The Nyquist limitThe Nyquist limit is a fundamental concept in signal processing that provides guidelinesfor the proper sampling of continuous signals to avoid aliasing and to ensure accurate signalreconstruction. The theorem was independently developed by Harry Nyquist and ClaudeShannon in 1928 and 1949 respectively [21,22]. The Nyquist theorem states that in order toaccurately reconstruct a continuous signal, the sampling rate must be at least twice themaximum frequency component of the signal. Mathematically, if is the maximumfrequency in the signal, the sampling rate must be greater than . For acontinuous-time signal with receiver bandwidth limited to rad/s, the Nyquist sampling rate,or Nyquist limit, is given by . If the signal sampling rate is less than this criticalfrequency, aliasing can occur, leading to inaccurate reconstructions of the original signal.For sampling of the frequency encoded signal, analog-to-digital receivers of the electricalsignal have reached high enough sampling rates that the encoding easily surpasses the Nyquistlimit, and no aliasing is present in the data.11However, in the phase encoding directions, if sampling rates are beneath the Nyquistlimit, tissue outside the field of view (FOV) will have the same phase as tissue within the FOV,and hence both will be encoded into the image at the same location. This appears as an overlapof anatomy from outside the field of view into the region of interest and can obscure theinformation intended to be acquired. To ensure phase sampling above the Nyquist limit, the fieldof view can be extended to ensure all anatomy in the plane of the image phase direction has aunique phase. This necessitates a reduction of the phase resolution of the FOV or increasingthe number of phase encoding steps performed, which proportionately increases the totalscanning time to fill k-space. Alternatively, parallel imaging, the simultaneous sampling of thesignal with multiple receiver coils with different spatial sensitivities, can be used to reducealiasing from sampling beneath the Nyquist limit [23].1.1.7 Parallel imagingBy using an array of receiver coils that each have a known or measured spatially-varyingsensitivity to the acquired signal, the Nyquist theorem can be bypassed by using this additionalinformation about the spatial origin of the signal to “unwrap” the aliasing of signal caused bysampling below the Nyquist limit. This can be used to acquire fewer samples of the MRI data(also known as undersampling) by only acquiring every nth line in k-space. Practically, this hasthe effect of decreasing the FOV in the phase direction by a factor of n, which means tissueoutside the smaller field of view would be aliased into the image. When images arereconstructed, this extra information has to be unwrapped to its proper location through one ofvarious parallel imaging reconstruction algorithms. Common algorithms include SensitivityEncoding (SENSE) [24] and Generalised Autocalibrating Partial Parallel Acquisition (GRAPPA)[25].12SENSE and derivative algorithms use a measurement of receiver coils’ spatial sensitivitymaps to calculate an unfolding matrix through matrix inversion of the sensitivity maps. Theunfolding matrix, as its name implies, can undo the aliasing caused by undersampling of theimage by using the differences in signal intensity for each coil to separate aliased and unaliasedsignals. GRAPPA, on the other hand, performs an undersampled acquisition with a fullysampled central portion of k-space called an autocalibration region. GRAPPA then utilises theautocalibration data to create a 3D kernel that provides the necessary information to iterativelyestimate the unsampled points of k-space from neighbouring data. In both techniques, individualunaliased coil images are reconstructed and then combined to create the unwrapped full FOVimage.These techniques permit fewer phase encoding steps to produce unaliased images,effectively allowing a bypass of the Nyquist limit while maintaining the resolution of the scan,reducing acquisition time and/or removing aliased signal from outside the FOV. As such, theyhave been very valuable for accelerating the acquisitions of nearly every sort of MRI scan,helping patients tolerate scanning better and permitting more individual scans to be acquired inthe same timeframe.1.1.8 Noise in MRIImage noise in MRI refers to random fluctuations in the acquired signal that can degradeimage quality and affect diagnostic accuracy. MRI noise can originate from various sources,both intrinsic and extrinsic to the imaging process. Depending on the primary source of thenoise, noise in images can have a variety of features, but in complex MRI images noise istypically white. However, as most MRI images are viewed as magnitude images, the propertiesof the noise change, and the positive and negative fluctuations due to noise all become positivein low signal regions of the image. This is typically described as noise with pixel intensities that13follow a Rician distribution. The primary sources of noise in MR signals are electronic noisegenerated by the thermal fluctuations of electrons in the body being imaged and dielectriccoupling of the conductive tissue with the MR coils causing signal fluctuations. This noise isinherent within the signal and is difficult to prevent without cryogenic cooling of the receiver coilsand subject being imaged, a process to which in-vivo specimens generally object. [26–28]Additional noise can also be introduced through image undersampling through theparallel imaging acceleration techniques described in section 1.1.7, where estimations of themissing data through iterative reconstruction techniques are affected by noise in the sampleddata and can amplify these noise effects, usually with geometric patterns depending on thegeometry of the imaging coil receiver array [29].Denoising algorithms exist as post-processing techniques to improve the imagesignal-to-noise ratio (SNR) by estimating and removing noise from images [30,31]. Thesealgorithms can take advantage of the characteristic properties of the noise, self-similarities inimages [32,33], or redundant information in scans [34,35] to estimate the underlying signal andremove the noise. These algorithms can be tuned for performance, but the underlying variationin signal arising from the tissue itself can be sometimes hard to separate from image noise, andas such denoising algorithms can remove true detail from the underlying image.141.2 Compressed sensingCompressed sensing (CS), introduced to MRI data reconstruction by Lustig, Donoho andPauly in 2007 [5,6], is an approach to MR image recovery from undersampled k-space thatallows for even higher undersampling factors than conventional parallel imaging techniquessuch as SENSE or GRAPPA. It uses random or pseudo-random sampling of the phasedimension of the images and the image sparsity inherent to MR scans to reconstructartefact-free images from the undersampled data.1.2.1 Basics of compressed sensing reconstruction algorithmsThe foundation of compressed sensing lies in the assumption that the information inmost images can be sparsely represented in a certain domain. In MRI, this sparsity often refersto the fact that acquired images have a representation in a transform domain, such as a waveletdomain, that requires much less information to describe than the true images. This allows theformation of the image reconstruction problem as a regularised least-squares minimizationproblem as laid out in equation 13.In this equation, is the image to be reconstructed, is the acquired MRI data, is themeasurement matrix corresponding to the MRI acquisition, is a sparsity-promoting transform(e.g., the wavelet transform), and is a regularisation parameter that enforces a compromisebetween fidelity with the sampled data and the sparsity transform. Similarly to SENSE, CS alsouses maps of the individual coils’ spatial sensitivity as additional information to enhance thereconstruction by incorporating the sensitivity maps’ unfolding matrix as part of .151.2.2 Least-squares minimizationEquation 13 is solved through an iterative process that starts with an initial imageestimate, , that is transformed through into k-space to compare to the sampled data .Additionally, the factor , which represents a minimization parameter such as the -norm ortotal variation (TV) of in the wavelet domain, is calculated and added to the equation. Foreach iteration, a gradient descent algorithm iteratively updates the image estimate to attemptto minimise equation 13. At each iteration , the update is given by equation 14 in which isthe step size or learning rate and is the gradient of the objective function at thecurrent estimate .In this case, represents the imaging function in equation 13. Therefore, thegradient descent algorithm estimates through equation 15:and there are numerous computational methods to iterate this estimation swiftly, including theIterative Shrinkage-Thresholding Algorithm (ISTA), the Fast Iterative Shrinkage-ThresholdingAlgorithm (FISTA), Alternating Direction Method of Multipliers (ADMM) and others [36].The process is repeated for the new estimate for a specified number of iterations oruntil the difference between and is beneath a convergence limit . At this point,is deemed the reconstruction algorithm’s estimate of the image from the sampled data .161.2.3 UndersamplingThis regularised least squares minimization scheme permits high factors ofundersampling in the phase encoding dimension(s), meaning that only a small fraction of thek-space data is necessary to be acquired during each scan. It is typically able to equal orsurpass the maximum undersampling factors reliably reconstructed by SENSE or GRAPPA. Thealgorithm allows robust estimation of the underlying image data despite missing significantnumbers of data points in k-space through the regularisation process. If the data is sampledrandomly (or, as is usually done, pseudo-randomly), the artefacts generated by theundersampling are incoherent, as opposed to aliasing in the image created by periodicundersampling. Incoherent artefacts lead to large values in the regularisation term, so theiterative process then will converge on data that fills in the missing data points such that thereare the least undersampling artefacts, producing the best image estimate from the sampleddata.As an imaging reconstruction method, regularised least squares minimization performsvery well, provided the necessary conditions are met for the compressed sensing methodology[37–39].1.2.4 Subspace extension of compressed sensingIn some cases, the sparsity assumption can be extended to an additional dimension. If aset of measured images such as a T1 or T2 relaxation series of images can be representedsparsely along an additional transform domain, sparsity along that dimension can be used toinform the reconstruction process as well [7].17This is similar to the standard compressed reconstruction algorithm but contains asubspace basis projection operator in the least squares minimization scheme, where isthe size of the additional measured domain and is the size of the basis.If this transform is onto a sparse basis that represents the data in compactly, so that, the reconstruction algorithm reconstructs a potentially much smaller set of images thateffectively share information between each original imaging volume. This has the effect ofreducing the size of the minimization problem from volumes to basis images. The smallerminimization problem also shares imaging information between each of the volumes; eachvolume contributes its signal characteristics to all of the basis images. With k-space datameasured using a different pseudo-random pattern for each of the volumes, information abouteach measured phase step is shared to all of the basis images, resulting in less undersamplingartefacts and higher detail throughout. This means an optimal undersampling pattern shouldchange between each sampled volume but contain measurements of the maximum number ofdifferent phase steps across the imaging scheme to best represent the different frequenciesinherent in the sample.Through this new minimization scheme, the images reconstructed are not the sampledvolumes directly, but , the subspace images spanned by . To recover the measuredimaging volumes, a simple backprojection step through is performed, producing theimage .181.3 Structure of the brainThis section will provide an overview of the basic structure of nervous tissue in the brain,how it affects the diffusion properties relevant for this thesis, and why investigation of diffusion inthe brain is important. More information can be found in any basic biological or neurologicaltextbook.1.3.1 Structure of white matter and grey matterThe brain (and indeed the spinal cord as well) is broadly composed of two distinct tissuetypes: white matter and grey matter, as well as surrounding cerebrospinal fluid (CSF), awater-based fluid that supports, cushions and facilitates chemical exchange for the grey andwhite matter.Grey matter is primarily formed from neuron cell bodies and other brain cells, such asastrocytes. These cells are made of cell organelles floating in water-based cytoplasm, allcontained within a lipid bilayer cell wall. Gray matter cell bodies have roughly isotropic shapes,with dimensions in 3D space all on the same order of magnitude, and can generally beapproximated as spherical.White matter consists almost completely of the axons of neurons. These are long, thinextensions of the neurons that carry signals between different cells, enabling communicationbetween different parts of the brain, spinal cord and the rest of the body. Axons are composedof cytoplasm contained within a cell wall that is then wrapped in more layers of a phospholipidbilayer tissue called myelin that helps to accelerate conduction of electrical signals along theaxon. The shape of these axons is extremely elongated in one dimension while the other twoare similar in size, and can be approximated as long, thin cylinders.191.3.2 Axonal bundles and tractsAxons in white matter often are combined together in axonal bundles. These bundles arecollections of axons extending in a similar direction that create a further layer of structure withinthe white matter. Groups of axonal bundles are bound further yet into macroscopic axonal tractscomposed of large quantities of axon bundles that are aligned within a region. These tracts formconnections between different regions of the brain and body, connecting sensory inputs, motorand biochemical signals, and higher level brain functions. Studies of these tracts provideinformation about the structure and functionality of different regions of the nervous systems andhow sensory, motor and higher level brain signals are processed. For example, axonal tractsfrom the optic nerve can be identified and followed through the optic chiasm to the lateralgeniculate nucleus in the thalamus. From there, their paths can be tracked via the opticradiation into the primary visual cortex in the occipital lobe. Tractography studies, or studies thatanalyse brain tracts in different individuals, can then help identify abnormalities or variations inthese neural pathways, or predict how surgical resections of or through different regions of braintissue may affect neural processing, informing surgical planning for the benefit of patients. Asthese tracts also rapidly develop in neonates and children, tractography studies can also safelyand non-invasively provide information about natural brain growth processes and maturation.Additionally, neural pathways form and change throughout life as the physical underpinnings oflearning and memory, and their study can provide valuable insights for neuroscience,psychology and disease.201.4 DiffusionDiffusion is a term which describes the random motion of molecules or particles in afluid. In the context of neuroscience and medical imaging, diffusion properties of watermolecules are studied in different tissues, including CSF, grey matter and white matter in thebrain.Unconstrained or “free” water in CSF at thermal equilibrium is subject to a process calledself-diffusion, where molecules move and change directions randomly at a rate described by theStokes-Einstein equation:Where D is the diffusion coefficient in , is the Boltzmann coefficient, is the particleshape and fluid viscosity dependent mobility of the particles, and is the temperature of thewater in Kelvin. In free water, we can expect single molecules of water to move a root meansquare (RMS) distance in time as given in equation 18; the diffusion coefficient quantifieshow far and how fast water molecules spread on average over time.In the practical setting of an MRI scanner, this diffusion coefficient is commonlyestimated using a parameter called the apparent diffusion coefficient (ADC) in diffusionweighted imaging (DWI) studies, which averages the diffusion coefficient across multipledirections [2]. In water or other free fluids, the diffusion coefficient is isotropic, or the same in alldirections, so the ADC will reasonably represent the diffusion coefficient in any direction.211.4.1 Diffusion in grey matter and white matterIn grey matter, the self-diffusion of water is influenced by the cellular structure and thepresence of obstacles such as the cell membranes and organelles within the cell. The roughlyspherical nature of the cell permits diffusion mostly isotropically within the cell, although limitedin extent by the cell wall and internal structure. The actual, non-spherical nature of cells and thepresence of axons within grey matter usually lends a preference to one or more directions ofdiffusion within a voxel and molecules will diffuse preferentially along that direction, but thisanisotropy is generally low. If the time in equation 18 is short enough such that is muchsmaller than the cell diameter, diffusion in grey matter can be modelled as free diffusion. If islarger than or on the same order as the cell diameter, the cell wall will noticeably reduce theRMS distance a water molecule travels, and as a result the ADC will decrease. This allowsdiffusion scans to be sensitive not only to diffusion processes, but to cellular structuraldimensions on a microscopic level, as the time in which diffusion is allowed to occur can bevaried to quantify changes in ADC due to cell size [40].White matter in the brain primarily consists of the long, thin myelinated axons of neuronalcells, which are surrounded by glial cells and connect different regions of the brain. Within theaxons, the structure of the axon cell walls and surrounding myelin constrain the self-diffusion ofwater molecules along directions perpendicular to the axon, while still permitting diffusion alongthe axis of the axonal body. Again, if the time of measurement is short enough that issmaller than the axonal diameter, this would be similar to free diffusion. However, if the axondiameter is on the order of or smaller than during the measurement time in equation 18,diffusion will be strictly limited perpendicular to the axis of the axon and the only apparentdiffusion will be along the axonal body.22Diffusion tensor imaging (DTI) is a technique that characterises this anisotropic diffusionby measuring the diffusion of water in multiple directions to determine the primary directions ofdiffusion of water within imaging voxels [3]. The diffusion coefficient therefore has to berepresented by a tensor, (equation 19), as a single voxel can contain many axons with variedorientations and diffusion properties. This allows tractography, or the mapping of axonal tracts,by following the primary diffusion vectors through the voxels that represent the directions alongwhich water is free to flow and thus the axonal bodies. Additionally, parameters derived fromDTI such as fractional anisotropy (FA), mean diffusivity (MD) and axonal diameter (AD) [41]provide insights into the organisation and integrity of white matter tracts and can be used toidentify structural disorder and pathology in the brain.1.4.2 Use of diffusion in brain pathology and researchBoth anisotropic and isotropic diffusion in the brain can vary significantly across cell typeand brain structure. Normal tissue can have vastly different diffusion properties thanpathological tissue depending on cell type, cell damage, or changes in brain chemistry orenvironment; for example, changes in tissue ADC can help differentiate extracellular andcytotoxic oedema [42]. DTI can also help assess the cellular density and microstructuralchanges in brain tumours. High cellularity restricts water diffusion, and differences in diffusioncharacteristics can aid in tumour grading [43]. Probing the diffusion properties of the brain canlead to further information about pathological tissue on a microstructural level that mostanatomic imaging cannot resolve. Diffusion weighted imaging is of extra importance forevaluation of stroke [44], traumatic brain injury [45], brain tumours [46] and white matterpathologies [47].23Diffusion tensor imaging is further used for investigating connectivity within the brain,such as in psychiatric disorders, brain development and plasticity studies, degenerativediseases, or for surgical planning around important brain pathways, such as preserving the opticradiation during a temporal lobe resection [48]. DTI can visualise the structural connectivity ofthe brain through axonal tract mapping, or tractography [49]. Tractography techniques visualisethe orientation and integrity of fibre bundles, providing insights into neural pathways,development and connectivity patterns [50].The connectivity information provided by DTI is valuable in neuroscience, and isregularly used for studying brain development, especially in paediatric populations [51]. It helpscharacterise normal maturation processes, connectivity changes and identify abnormalities ordevelopmental disorders. Similarly, in adult and geriatric brains, it is important in assessingdegenerative changes due to normal ageing and diseases such as Alzheimer’s [52]. For mostconnectivity purposes, the highest imaging resolution possible and low imaging noise are crucialto track axonal bundles with the highest fidelity [53,54].1.5 Diffusion sensitisation of MR imagingOn a molecular level, diffusion is the random movement of molecules in space. The neteffect of diffusion over time in MRI, therefore, is the transposition of the hydrogen spins that areused to create the MRI signal. During the imaging process, spins will move a root mean squaredistance in space according to their diffusion tensor as previously described in equation19. We can build an estimate of this diffusion tensor by characterising the averagedisplacement of spins along multiple vectors within a voxel. The following sections detail theprocesses of measurement of this average displacement through diffusion tensor imaging.241.5.1 Diffusion encoding gradientsWhen applying a gradient magnetic field in MRI, each of the spins within a voxelexperience a different magnetic field and will precess at a different rate given bythe gyromagnetic ratio (equation 2). This causes spins within a voxel to gradually move out ofphase with each other, causing a reduction in the transverse magnetic moment throughdestructive interference. If the same gradient field is then applied with opposite polarity, thespins will experience a reversal of this dephasing effect and refocus, restoring the transversemagnetic field and hence the signal of the MRI. However, if the spins have a net displacement inthe gradient direction between the start and end of the application of the pair of gradient pulses,they will not experience exactly the same gradient field during the second gradient lobe, due tothe variation of the gradient field in space. Thus, the rephasing of the spins will be incompleteand the net transverse magnetization will still suffer from some destructive interference,producing a lower magnitude of signal recorded by the MRI.The classic diffusion experiment, therefore, as proposed by Stejskal and Tanner in 1965[55] has a pulse sequence (Fig. 1) that places large, equal gradient magnetic fields before andafter a 180o refocusing pulse. The 180o refocusing pulse has the effect of reversing thedephasing direction, so the gradients in the Stejskal-Tanner diffusion pulse sequence have thesame polarity. Any motion of the spins during these gradient pulses will cause spins toexperience a different magnetic field during the second gradient, leading to the desiredreduction in signal. In this way, the MRI signal can be sensitised to diffusion.1.5.2 DIffusion encoding strengthThe decrease in signal caused by the diffusion gradients is best presented as the ratio ofsignal with diffusion encoding to signal with no diffusion encoding [56]:25where is the signal intensity without the diffusion weighting, is the signal of thediffusion encoded scan, is the gyromagnetic ratio, is the strength of the diffusion gradient,is the duration of the diffusion gradient, is the time between the two gradients, and isthe diffusion coefficient as described in the Stokes-Einstein equation.For a given diffusion encoding, can be simplified to a single number,conventionally given the denomination , and the equation can be described asAs such, the diffusion sensitivity of a scan is denoted by in mm2/s , this combined parameterthat represents diffusion strength. It does not encapsulate all the information than can bepresent in a diffusion scan, as and will affect the diffusion signal characteristics in cells withdiameter comparable to the RMS distance travelled in that time, but is the conventional valuereported for diffusion images.describes the isotropic diffusion of free fluids, but the equation can be extended tofluid that has different diffusion coefficients along different vectors by replacing the diffusioncoefficient with the diffusion tensor . The solution to the equation becomes equation (X).261.5.3 Diffusion encoding vectorsAny combination of gradient fields has the net effect of a single gradient applied alongtheir vector sum in space, and there will be a 2D plane perpendicular to this vector along whichthere is no field gradient. For example, if the -gradient is applied during the diffusion pulse,there is no net gradient across the -plane, and spins that diffuse along the -plane will notexperience a location dependent change in magnetic field during the dephasing and rephasingdiffusion encoding gradient lobes. Thus, MRI diffusion scans are sensitive to diffusion along onlyone vector at a time. MRI acquisitions with different diffusion encoding vectors can be acquiredsequentially to characterise a 3D diffusion tensor . In the anisotropic case, the simplestdiffusion tensor is a 3x3 positive definite symmetric matrix, and requires minimum 6measurements along different vectors to characterise the 6 unknowns [56].The eigenvectors of represent the primary directions along which water in a voxel is diffusing,and their respective eigenvectors correspond to the diffusion coefficient along that vector.Voxels of brain tissue can contain multiple axonal fibres which extend in differentdirections, and fibres can cross, bend or merge, and hence the true diffusion tensor for a voxelis a complicated 3D tensor that is a representation of the diffusion of the sum of all fibres in thevoxel. Most DTI scans are therefore designed to resolve axon tracts that cross, bend orintersect, and acquire dozens or even hundreds of unique diffusion vector encoded samples[57]. Extraction of estimates of the true diffusion tensor from this data is an ongoing problem,and multiple techniques exist to characterise the complicated nature of a voxel’s diffusiondynamics [58–60], which remains out of the scope of this work.271.5.4 Diffusion MRI artefactsDiffusion scans are highly sensitive to a number of artefacts that can reduce imagequality. Motion artefacts are caused by subject motion or involuntary physiological motion (e.g.,pulsatory or respiratory motion), which introduces signal dropout, blurring, distortion, andmisalignment of diffusion images [61,62]. Images acquired with different diffusion gradients maytherefore not represent the same voxel of subject tissue, affecting the accuracy of diffusionmeasurements. Bulk motion artefacts are best prevented through subject immobilisation, thoughsome motion due to factors such as breathing and pulsation is unavoidable during scans.Displacement and rotation of different diffusion images can also be corrected through imagingvolume co-registration post acquisition using tools such as the FMRIB Software Library toolbox(FSL) [62,63].Eddy currents can be induced by the application of the diffusion gradient fields, whichproduces additional field effects during the imaging process. This can lead to distortions andgeometric inaccuracies in the images as well, as these additional field effects causeunaccounted phase and frequency changes in the receiver data. Proper gradient design andcompensation techniques are employed to mitigate eddy current artefacts, and eddy currentartefacts can be somewhat corrected for in post processing of imaging data [64–66].B0 inhomogeneities and will also introduce distortion into reconstructed images byproducing unaccounted phase and frequency encoding during the pulse sequence readout. Thiseffectively causes changes in the k-space readout trajectory, leading to misregistration of thesignal. [67,68]. The B0 inhomogeneities can be generated from multiple sources, including poorshimming, heating effects and magnetic susceptibility differences between tissues, particularlyat air-tissue interfaces or regions with ferromagnetic materials, such as metallic implants. Thesedistortions can again be corrected using post-processing methods [64,65].28Gibbs ringing, also known as truncation artefact, can occur during the reconstruction ofdiffusion images, leading to the appearance of oscillations around sharp tissue boundaries. Thisartefact can be minimised with acquiring a larger k-space matrix or through appropriatereconstruction algorithms that include Gibbs minimising imaging filters. [69]1.5.5 Diffusion tensor imaging readout pulse sequencesAfter sensitization to diffusion, the spins are refocused to produce echoes and then canbe read out by standard MRI readout pulse sequences. As a large number of diffusion vectorsmust be sampled to characterise , the total acquisition can be very long. Therefore, in theinterests of time, the vast majority of diffusion scans use an EPI read-out, which uses a series ofrapidly switching gradients to read out consecutive lines of k-space, with phase encodinggradient blips to increase the phase between each alternating gradient read-out lobe [70,71].K-space is filled from left to right, with the frequency readout polarity switching betweenconsecutive lines. A simple k-space filling scheme with an undersampling factor of 2 ispresented in Figure 1, with blue lines representing frequency-encoded readouts and red linesrepresenting the phase encoding blips that jump between phase steps.Other methods to read out the diffusion encoded signal are less widely used as they areslower and less SNR efficient. A single-shot turbo spin echo (SS-TSE) readout, asdemonstrated in figure 3, has the advantages over EPI that magnetic susceptibilities arerefocused and frequency readouts are all performed in the same direction, and images are notas distorted by b0 inhomogeneities.29Fig 1.1 Stejskal-Tanner DTI pulse sequence with EPI readout gradients. Frequency encoding is denoted in blue andphase encoding blips are denoted in bright red.30Fig 1.2 EPI k-space filling order with an undersampling factor of 2. Frequency readouts are denoted in blue andphase encoding blips are denoted in red.An SS-TSE readout is considerably slower than a comparable EPI readout and hasmarkedly less inherent SNR, due in part to increased T2 decay of the signal during the echotrain. Some of the SNR loss is mitigated by measuring the centre of k-space at the beginning ofthe readout train, which has the greatest signal, whereas an EPI readout must collect the centreof k-space at its midpoint. Unfortunately, this does not come close to compensating for the lowerSNR, especially in the later echoes that fill the outer lines of k-space, and the resulting datacollected can be extremely noisy, as will be seen in chapter 2.31Fig 1.3 Stejskal-Tanner DTI pulse sequence with SS-TSE readout gradients. Frequency encoding is denoted in blueand phase encoding steps are denoted in bright red.1.5.6 Undersampling in 2D DTI scansEPI DTI poses problems when attempting to randomly undersample. T2* decay alongthe echo train creates decreasing signal amplitudes for each successive echo. Since theencoding is performed sequentially across k-space, the signal amplitude and phase of each lineof k-space will vary and these differences, if not properly accounted for, can lead to moderate toaliasing and triple brain artefacts [72]. Undersampling in EPI currently works best with regularphase encoding intervals, where this decrease in signal amplitude can best be accounted for byapplying a signal modulation factor to each successive encoding [72]. Since compressedsensing requires a random undersampling pattern that causes incoherent aliasing artefacts, its32use for EPI scans adds an additional level of complexity to the acquisition, especially if theundersampling pattern changes between each shot. The reconstruction algorithm would have toapply the correct modulation factors between each encoding, not just once but for everydiffusion image acquired. This is further complicated by eddy currents, concomitant gradientfields and gradient non-linearities, which induce additional signal modulations and errors in thesignal encoding, compounding the problem.Fig 1.4 SS-TSE k-space filling order with an undersampling factor of 2. Frequency readouts are acquired in acentre-out order and are denoted in blue.SS-TSE DTI, on the other hand, can be sampled pseudo-randomly without issue[73].The centre-out sampling scheme reduces the effect of the signal modulation on the imagereconstruction. The RF refocusing pulse is also less affected by gradient imperfections, so33different phase encodings between each sample can be applied without the errors typical ofEPI. This allows compressed sensing to be used for TSE DTI without large error corrections tok-space. A SS-TSE diffusion k-space filling scheme with a pseudo-random undersampling factorof 2 is presented in figure 4, with blue lines representing frequency-encoded readouts acquiredin a centre-out order.1.5.7 Uses of different DTI acquisition schemesTo date, the vast majority of DTI scans are performed using an EPI readout train; thespeed and SNR advantages over SS-TSE DTI are too great to justify using SS-TSE for thereduction in distortion and artefacts alone. Furthermore, post processing algorithms have beendeveloped that counteract distortion effects for EPI, reducing the advantages of SS-TSE evenmore. The undersampling and distortion properties of SS-TSE, however, could mean it is perfectfor subspace reconstruction techniques as laid out in section 1.2.4.342. Applying subspace compressed sensing reconstructions to TSE-DTI2.1 IntroductionDiffusion tensor imaging could be a decent candidate for which to extend the principlesof joint sparsity and subspace reconstructions laid out in section 1.2.4. For this to be applied,however, the set of all diffusion tensor images needs a transform in which it is sparselyrepresentable to allow us to reduce the dimensionality of the reconstruction. Effectively, theremust exist a projection operator in which the significant information in a DTI scan isrepresented by a set of vectors smaller than the number of diffusion images. This requires theimages to be well aligned in image space, that is, an image voxel in each diffusion shot mustrepresent the same voxel of tissue across all images. To this end, it is necessary that distortiondue to eddy currents, gradient non-linearities, B0 inhomogeneities and phase encoding errorsbe kept to a minimum.An additional requirement is for random k-space sampling patterns, not only within eachslice of k-space, but a different sampling pattern for each diffusion vector to maximise theadvantages of the subspace reconstruction. If each diffusion shot has a different randomsampling pattern, each phase component of the imaging domain should be represented in oneor, more likely, multiple k-spaces, and will help provide information to remove undersamplingartefacts from unrepresented phase information when images are projected onto the subspacebasis [74].To this end, a SS-TSE readout train seems preferable to EPI since it lacks the severedistortion effects inherent in echo planar acquisitions, and does not pose the challenges withrandom undersampling characteristic of EPI. With a pseudo-random sampling scheme for anSS-TSE DTI acquisition and a sparse subspace representation of the diffusion information, we35should be able to use the subspace compressed sense technique to perform high fidelity,distortion free reconstructions of DTI scans.2.1.1 Objectives1. To identify and formulate a subspace that sparsely represents TSE DTI imaging.2. To build a TSE DTI acquisition that utilises a random undersampling pattern along boththe phase dimension and the diffusion dimension.3. To leverage compressed sensing techniques and the sparse representation of the DTIscan to reconstruct high fidelity, good SNR sets of diffusion images despite the poorinherent SNR of data samples and large undersampling factors.2.1.2 MotivationEcho planar imaging diffusion scans are able to make use of significant factors ofacceleration, including both linear undersampling within a slice and simultaneous multi-sliceacquisitions, but suffer from distortions and artefacts due to the EPI acquisition scheme. If aTSE DTI scan could be accelerated through undersampling to a duration similar to an EPI DTIscan with comparable or improved image quality, it might replace EPI as the standardacquisition for DTI scans. Utilising the principles of compressed sensing and sparsity couldallow TSE to be scanned in similar timeframes as EPI with minimal distortion and artefacts fromthe acquisition train. This could improve diffusion tensor imaging for many purposes, butespecially for analysis of white matter tractography in regions of the brain that historically sufferfrom strong distortion effects, such as the inferior frontal and temporal lobes, the brainstem andspinal cord. It might also better resolve small cranial nerves in these regions such as the optic orolfactory nerves, providing additional insight into the aetiology of pathologies such as opticneuritis or anosmia.362.1.3 HypothesisThere exists a subspace in which diffusion scans can be represented sparsely. Imagereconstruction in this subspace domain will allow us to acquire data with high undersamplingfactors and reconstruct good quality SS-TSE DTI acquisitions.2.2 MethodsInvestigations into the viability of applying the subspace compressed sensingreconstruction techniques began with determining whether the diffusion tensor imaging could berepresented sparsely through principal component analysis (PCA) of EPI DTI [75]. Once theexistence and nature of this subspace was established, a SS-TSE acquisition scheme witharbitrary choice of phase sampling patterns was developed and multiple phase samplingschemes were created and tested. Initial reconstructions using a subspace representation of thedata were attempted and abjectly failed due to the un-represented random phase information indiffusion scans [11]. Several phase correction schemes to remove this phase error weredeveloped and applied to achieve the final reconstruction algorithm.2.2.1 Sparsity in DTITo determine the existence of a sparse subspace that can represent diffusion data, anEPI-based High Angular Resolution Diffusion Imaging scan with 60 diffusion encoding directions(HARDI-60) was acquired. Distortion and motion correction was applied to the images throughFSL’s eddy and topup tools [64] by acquiring and including a reversed phase-encoding directionb0 scan [64–66]. These corrections were necessary to optimise sparsity by aligning the voxelsof each image, ensuring each voxel represented the same sample of tissue across all thediffusion images.37PCA was then applied along the diffusion dimension to reduce the dataset to obtain the rightsingular vectors of the decomposition, sorted by singular value. The singular value of each rightsingular vector is proportional to the weight of the vector, or, in other terms, the proportion oftotal imaging information represented by the vectors [76,77]. The vectors with the smallestsingular values represent the least coherent data across the diffusion volumes. Given that noiseis, by definition, the least coherent part of the images, the data represented by vectors withsmall singular values is the noise in the images. Thus, these vectors can be left out of thereconstruction with limited or no effect on the true underlying image information, and the size ofthe subspace spanned by the right singular vectors can be reduced to smaller than thedimensionality of the set of images.It can be seen in figure 5 that a corrected HARDI-60 with b-values of 0 and 1000 s/mm2had the vast majority of the coherent information stored in approximately 8-10 principalcomponent vectors. The vectors past the tenth had uniformly low singular values, andrepresented the noise information in the scan; therefore, excluding them from the reconstructionshould not affect the underlying diffusion information of the scan. Thus, the HARDI-60 scancould be reduced to a subspace size of 10, which is a sparsity transform with a dimensionalityreduction of 6.The right singular vectors in the principal component analysis comprised the subspaceprojection that compresses the DTI dataset; when truncated to the appropriate size, this was thesparsity transform required by the compressed sensing algorithm (equation 16); the datawas sparse, as expected.38Fig 2.1 Information contained per principal component of a HARDI-60 DTI scan. Zoomed-in region shows that thevast majority of information is concentrated within the first 10 principal components. Components past the 10thcontain almost entirely non-sparse noise-like data.It could further be seen that the vectors past the 10th contained noise components of thedata by projecting the diffusion data onto the full, untruncated basis spanned by the rightsingular vectors. This created subspace images (Figures 6 and 7) representing how much eachsingular vector was present in the data; visual inspection of the 10th+ subspace image showedRician noise and edge effects from slight misregistrations in the distortion correction. Figure 6has window levels set to show the structure of the subspace images, while figure 7 shows thesame images with consistent windowing between images to demonstrate the relative magnitude39of the information contained within each subspace image.Fig 2.2 First 12 principal subspace images created through projection of the HARDI-60 DTI dataset onto the basisspanned by the right singular vectors of the PCA decomposition. Image window and level are set to show structure,and vary between images. The 11th image and beyond are nearly entirely noise and edge effects from slight errors inimage distortion correction and registration.40Fig 2.3 First 12 principal subspace images created through projection of the HARDI-60 DTI dataset onto the basisspanned by the right singular vectors of the PCA decomposition. Image window and level are equal on every image,demonstrating relative image weighting in the decomposition. The vast majority of all coherent information in the scanis spanned by the first 7 singular vectors.Similar analysis was applied to a multi-shell diffusion scan with b-values of 0, 500, 1000and 2000 s/mm2and a total of 128 different diffusion vectors (HARDI-128). The HARDI-128scan had the vast majority of the coherent information contained within approximately 1641principal component vectors, with the increase in dimensionality compared to HARDI-60representing the different information represented by multiple b-values. This also demonstrateda sparse dataset with a potential dimensionality reduction of (128/16) = ~8, which suggestedmost diffusion scans could be compressed to a significantly smaller size through a sparsifyingtransform.Fig 2.4 Information contained per principal component of a 128 direction HARDI DTI scan. Zoomed-in region showsthat the vast majority of information is contained within the first 16 components. Components past the 16th containnon-sparse noise data.A comparison between a projection onto the PCA-generated subspace basis and asimilar projection onto the even- real spherical harmonics (Figure 9) demonstrated the physical42significance of these vectors; PCA reduced the images to a subspace that shared a lot offeatures to the even- real spherical harmonics. The plots did not correspond exactly; thesubspace more compactly represented the data than the spherical harmonics and,appropriately, a projection onto the spherical harmonics contained images with more non-noiseinformation than the PCA subspace. The PCA subspace was therefore a better candidate forthis reconstruction than the spherical harmonics as it represented the data more sparsely,especially in the higher order principle components, but it was reassuring that the subspacebasis of the diffusion images had physical meaning. It also suggested the use of the even- realspherical harmonics as the sparse subspace in the reconstruction, but given the sparser natureof the PCA-generated basis it remained the basis of choice.Fig 2.5 A comparison of a projection onto the first 6 images of a PCA-generated subspace basis (top) and a similarprojection onto the (image 1, bottom row) and the (images 2-6, bottom row) realspherical harmonics. Similarities are striking between the two sets of subspace images, suggesting that the subspacegenerated by PCA is a close approximation to the real spherical harmonics.This analysis of this dataset suggested the magnitude data of DTI scans were indeedsparse and could be represented by a much smaller number of component vectors, exactly asnecessitated by the subspace-augmented compressed sensing reconstruction technique underinvestigation. This held true for all DTI datasets tested; Table 1 shows approximate subspace43size for a variety of DTI scans and the dimensionality reduction it represented, though optimalsubspace size depended significantly on imaging parameters, SNR, and the accuracy ofregistration between volumes.Table 2.1: Comparison of subspace size and compression factors in DTI acquisitionsScan Shells Shell 0 Shell 1 Shell 2 Shell 3 TotalDirectionsSubspaceSizeReductionFactorb # Dir b # Dir b # Dir b # DirHARDI-601 0 1 1000 60 - - - - 60 10 6HARDI-601 0 1 1000 60 - - - - 60 9 6.7HARDI-983 0 10 2000 48 1000 24 500 16 98 16 6.1HARDI-1283 0 12 2000 56 1000 36 500 24 128 18 7.1HARDI-1283 0 12 2000 56 1000 36 500 24 128 16 8HARDI-1603 0 10 2000 150 - - - - 160 34 4.7HARDI-1931 0 1 3500 64 2000 64 1000 64 193 34 5.672.2.2 Randomly undersampled TSE scansThe EPI DTI data investigated presented the problem that it was necessary to performdistortion corrections on the dataset before it formed an optimally sparse subspace. PCA ofuncorrected data produced a subspace with a much larger number of significant components.Upon visual inspection of subspace images, the additional singular vectors contained spatialinformation about the distorted parts of anatomy, especially the frontal and occipital lobes andaround the air spaces in the skull, which had high susceptibility changes and corresponding44distortions. These regions were less sparse as they also included changes due to distortion aswell as anatomical and diffusion information.Distortion correction could not be immediately applied to EPI k-space; conventionalalgorithms required post-processing of the reconstructed imaging data to determine thenecessary distortion corrections and apply them in image space. Furthermore, the samplingschemes for EPI scans were required to be regular, not random, to apply the scanner-generatedcorrections to the readout data. TSE-DTI acquisitions were subject to less of these problems;they were subject to motion corruption but were much less affected by distortions fromsusceptibility and eddy current effects. If the subject held still (ie. there was no motion artefact inthe images), the magnitude data of TSE-DTI scans were sparsely representable withoutpost-processing to apply distortion correction.A modification to the standard Philips TSE DWI pulse sequence through the applicationof a 2-dimensional adaptation of the AMC-PROUD [78] patch to the scanner allowed arbitrarychoice of sampling scheme for each diffusion shot. This permitted the use of arbitrary samplingschemes for TSE-DTI scans, including different sampling schemes for each diffusion encodingvector acquired. Test acquisitions were made with pseudo-random sampling schemes similar todefault Philips compressed sensing sampling schemes (Figure 10, image 1), with increasingspace between k-space lines towards the edges of k-space, but with random adjustments toindividual lines of k-space applied to each shot to promote incoherence between differentdiffusion directions (Figure 10, image 2).A variety of other sampling schemes were tested as well, including tests of full samplingof the central lines of k-space (Figure 10, image 3) and Poisson distributions of k-spacesamples (Figure 10, image 4). These sampling schemes were compared to the above samplingscheme but demonstrated markedly worse performance in reconstructions. The acquisition of a45filled central k-space was an ineffective attempt to reconcile phase differences between diffusionshots (Figure 14) that did not have enough high-frequency k-space information to accuratelyreconstruct images. The Poisson sampling scheme presented difficulties in obtaining suitablesampling patterns for each shot; variabilities in the distribution of k-space acquisitions meantsome images had large regions of k-space with no data, hampering reconstruction. The variablepseudo-random sequence in the second image of figure 10 was subsequently used for itsoptimal performance in initial reconstructions.Fig 2.6 From left to right: 1. Pseudo-random CS phase sampling scheme with identical sampling for each diffusionshot, 2. Pseudo-random phase sampling that varied for each diffusion shot, 3. Poisson-distributed phase samplingscheme that varied for each diffusion shot, and 4. Poisson-distributed phase sampling with increased sampling of thecentre of k-space that varied for each diffusion shot. The x-axis represents each diffusion direction, and each pixelalong the y-dimension represents an unsampled (white) or sampled (black) phase step. All sampling schemesdemonstrated were used for a 4x undersampled SS-TSE HARDI-60 DTI scan.2.2.3 Complex DTI scansThe TSE-DTI scans were then reconstructed using the pics compressed sensingreconstruction algorithm with 1 regularisation provided by the Berkeley AdvancedReconstruction Toolbox (BART) [79] to see if they share the same sparse representation as EPI.46The reconstructions were extremely low SNR, but PCA of the magnitude images producedsimilar principal component vectors and eigenimages as distortion corrected EPI DTI.Fig 2.7 Information contained per principal component of a HARDI-60 SS-TSE DTI scan. Zoomed-in region showsthat the majority of significant information is contained within the first 10 components. Components past the 10thcontain non-sparse noise data. SNR was extremely low so noise components formed a significant fraction of theimaging information.Using the sparse subspace generated from PCA of the imaging data to perform thereconstruction of the data as discussed in section 1.2.4 failed to produce useful images, withwide, low signal artefacts obscuring the majority of the image detail, in different patterns foreach diffusion direction.47Fig 2.8 Using a magnitude-only subspace basis to reconstruct diffusion data failed to produce useful images due tothe complex phase errors in the diffusion image space. Three diffusion directions are shown with differentreconstruction errors due to the phase differences between the three scans.The explanation for these banding artefacts was due to the complex phase of DTIimages. All MRI reconstructions are complex-valued as necessitated by the Fourier transform,but the PCA of the image had been performed using magnitude images, and as such, the phaseinformation of the scans was not included within the subspace and was left out of the imagingreconstruction.When performing PCA on complex-valued diffusion information, there does not exist asubspace that represents the data smaller than the size data itself. This was demonstrated byplotting the singular values of the complex PCA (Figure 13), where the consistently high singularvalues of the principal components indicate that information from every singular vector wasnecessary to completely represent the data. The majority of the information was notconcentrated into a small number of components; that is, the data was not sparse. Uponinspection of the phase data of the scans, the reason was apparent: the DTI images containedrandom phase variations in each shot [11]. Figure 14 shows the phase of a single slice formultiple diffusion shots, demonstrating these random fluctuations in phase across the images.48Fig 2.9 Information contained per principal component of a complex HARDI-60 SS-TSE DTI scan. Information in thecomplex dataset is distributed more equally amongst every component as the random phase of each scan is notsparse.49Fig 2.10 Low-frequency phase maps of 9 different diffusion shots in a single slice of a SS-TSE diffusion scan. Phasefluctuates due to any physiological motion during diffusion encoding and is highly variable, especially aroundCSF-filled ventricles and blood vessels which exhibit constant flow effects that affect image phase.2.2.4 Phase errors in complex DTIAny subject motion during the diffusion gradients of a diffusion weighted MRI acquisitionwill add an additional phase error to the data that is not countered by the inverse diffusiongradient lobe. The phase error of a rigid body acquired at a position with linear velocityand acceleration during the diffusion encoding period is given by the integral of the magneticfield gradient along the direction of motion, and can be expressed as:50The velocity and acceleration terms are not refocused by the second diffusion encodinglobe in the Stejskal-Tanner pulse sequence, and cause phase errors in the resultantant data.Rotational motion will have similar effects, and non-rigid-body motion such as cardiac pulsatileeffects will induce even less predictable phase errors in the diffusion scan, resulting in the phasecorruptions visible in Fig 14.The unpredictable and time-dependent nature of involuntary translational and rotationalmovement of the subject’s head and pulsatile effects meant the phase corruptions of thediffusion data were due to highly inconsistent effects. This inconsistency gave rise to thevariability in phase error and was unmitigable through immobilisation. It also meant thatprospective motion compensation or cardiac gating would not likely fully remove the effects, aseven the MRI scanner table vibrations and their propagation through tissue would contribute tothese phase variations, as was noticed in phantom scans during MRI sequence testing.As such, incorporating retrospective phase corrections from the data itself into thereconstruction process seemed an appropriate avenue to restore the image sparsity and enablethe subspace reconstruction technique.2.2.5 Phase navigationThe phase errors due to motion in DTI are generally low frequency only as the speed ofsubject motion is slow (on the order of cm/s) compared to the speed of imaging, which reads outdecimetre sized FOVs in less than a millisecond. In real terms, this means the phase errors inthe diffusion data are contained in the low frequency portion of k-space only; one can estimatemotion-induced phase error via reconstruction of solely the central points of k-space. Therefore,reconstruction of an adequately sampled central portion of k-space can produce a good51estimate of this phase error. Following this, the image phase can be corrected by multiplicationof the complex images with the complex conjugate of the low-frequency phase error.Modifications to the sampling scheme (Figure 10) to sample enough of the centre ofk-space to self-correct the diffusion scans allowed accurate estimation of the phase error foreach shot. Applying this phase correction in image space was a straightforward voxel-wisemultiplication of the complex conjugate of the phase error. However, phase correction of theundersampled k-space prior to image reconstruction was not as straightforward, and this wasnecessary to restore the low rank behaviour of the diffusion scans prior to performing thesubspace reconstruction. Attempts to apply phase correction in k-space through the propertiesof the Fourier transform demonstrated the problem; multiplication by a phase correction factor inimage space was equivalent to a convolution in k-space with the Fourier transform of the phasecorrection. In a fully sampled k-space, this operation could have been performed, but with anundersampled k-space this convolution mixed the zero-filled lines of k-space with the sampleddata, blurring and irrevocably corrupting the sampled data. In mathematical terms, theundersampling operator did not commute with a phase correction operator , so the phasecorrection can only happen prior to an undersampling operator, not post.It was possible to phase correct the data at other points during the reconstructionprocess as well. The most effective technique was to add the phase error to the sensitivity mapsused in the reconstruction algorithm such that each shot had its own phase-correcting sensitivitymap. The error was then removed from the images along with the individual coil phase when thesensitivity operator was applied during the compressed sensing reconstruction process.When using the subspace compressed sensing algorithm, unfortunately, including thephase error in the coil sensitivity maps broke the back projection step of the iterative loop in theconjugate gradient methods used to update the imaging estimates. The inclusion of coil error52meant the sensitivity operator no longer commuted with the basis projection operator as theyboth acted on the diffusion dimension, which ruined the dimensionality that allowed the matrixmultiplications to predict the next estimate. From the original CS equation,represents the system matrix that acts on the estimate to compare to the measureddata . In this case, , with the undersampling operator, the sensitivity matrix thatrepresents the coils, the motion induced phase error and the Fourier transform operator.When performing conjugate gradient minimization algorithm, the next estimate is found using aback projection step to find the gradient of the objective function:whereThe introduction of the phase error term into , and its non-commutativity with preventedthe matrix multiplication of . The basis projection operator and the phase erroroperator’s fundamental incompatibility meant inclusion of phase corrections into thesubspace-informed compressed sensing reconstruction algorithm was not possible.2.2.6 Partial reconstruction phase correctionIt was still possible to incorporate the subspace reconstruction into the reconstructionalgorithm if the phase correction step occurred later in the reconstruction process. A standardcompressed sensing reconstruction, without the inclusion of the subspace, was used togenerate an initial estimate of the missing lines of k-space, after which phase correction could53be applied to the entirety of data either in k-space or image space through the steps outlined insection 2.2.5. This restored the data to a sparsely representable set of diffusion information, andallowed the use of the subspace-informed compressed sensing algorithm after projecting thedata back into k-space.Unfortunately, application of the phase correction through an initial compressed sensingestimate of the undersampled lines of k-space incurred a further SNR penalty to the images asthe initial, incomplete reconstruction was used to replace the true sampled data, and any errorsin that reconstruction propagated through the secondary reconstruction steps. The first step ofreconstruction used was just a few compressed sensing iterations, which increased the noisepenalty on the images but sped up the imaging process. Further refinements to the processincluded completing the initial reconstruction in its entirety, to reduce the noise penalty beforephase correction and Fourier transformation back into k-space. The optimal amount ofreconstruction before phase correction therefore seemed to be a completed compressedsensing reconstruction to remove any noise amplification passed to the secondaryreconstruction.2.2.7 Application of subspace compressed sensingAfter the initial reconstruction and phase correction of data as detailed in section 2.2.5,the diffusion data had its sparsity restored, and a sparse subspace reconstruction was valid.Furthermore, PCA of these initial phase-corrected reconstructions was used to generate thesparse set of basis vectors for use in the subspace reconstruction scheme without priorknowledge or assumptions of properties of the diffusion encoding of the images. PCA was ableto extract diffusion subspace components similar to the EPI investigation, despite the low SNRof the reconstruction (Figure 11). As an alternative, the spherical harmonics could have54been used as the subspace basis, with the downside of being a less sparse basis than the PCAcomponents.The complete reconstruction algorithm including phase correction is as follows:1. Estimate phase error images through a low-frequency only image reconstruction2. Perform initial compressed sensing reconstructions with phase-corrected sensitivitymaps (or perform initial compressed sensing reconstructions and apply phasecorrections to the resultant images)3. Perform PCA analysis of phase corrected images to generate a basis set of vectors thatsparsely represents the data4. Fourier transform phase corrected images back into k-space5. Perform subspace informed compressed sensing reconstruction with the basisgenerated in step 3 or the ... spherical harmonics.2.3 ResultsThere did not exist a subspace that sparsely represented diffusion data because ofimage phase corruptions unique to each shot caused by involuntary subject motion. However,with self-navigated phase corrections included into a two-stage image reconstruction scheme,sparsity was restored, and a subspace was identified that permitted a low-rank reconstruction.The low rank reconstruction dramatically reduced noise and improved image quality.Unfortunately, the undersampled TSE-DTI scans had such a low SNR in the acquired data thatmost high frequency information was beneath the noise floor of the acquisition and was absentfrom the reconstructed images.552.3.1 Image reconstructionsThe image reconstructions that use the two stage phase corrective subspacereconstruction algorithm achieved significantly better image quality than a compressed sensingreconstruction alone (Figure 15). Compared to the standard compressed sensingreconstruction, images had vastly better structural delineation of gyri and sulci, and both greyand white matter became differentiable above the noise. Diffusion information began to bevisible as well, with some contrast appearing between different tracts of white matter thatcorresponded to a comparable EPI diffusion scan, but was nowhere near the level of diffusioncontrast achievable with the EPI scan. On most of the images there remained SNR problemsinherent with the SS-TSE DTI acquisition that the subspace reconstruction was not able toovercome.2.3.2 SNR problemsThe inherent SNR of the acquired k-space lines from the SS-TSE DTI pulse sequencewas extremely low. To achieve similar spatial resolution to standard EPI sequences, imagevoxel sizes had to be approximately 2 mm in each plane, which produced an inherently smallsignal. Diffusion encoding further reduced the amplitude of the collected signal through thedephasing effects on spins undergoing diffusion. The time to perform the diffusion encodinggradient lobes also increased the TE of the acquisition, allowing further T2 decay of the signalbefore the echo formed. Furthermore, specific absorption rate (SAR) limitations on the scannernecessitated 120 degree refocusing pulses for the TSE readout train, which prevented fullrefocusing of the signal at each echo. Finally, T2 decay during the long TSE readout traindecreased the magnitude of the refocused echoes, especially at the outer edges of k-space,which were collected at the end of the echo train. The centre of k-space was acquired earliest inthe pulse sequence, and as such had sufficient signal for image reconstruction, but when the56edges of k-space were acquired, the magnitude of the sampled signal was beneath theelectrostatic noise floor created by thermal fluctuations in the MRI system. This translated intovery low SNR images with poor spatial detail, as all high-frequency information wasoverwhelmed by noise. When combined with an undersampling factor of 4 or higher, thecompressed sensing algorithm had very limited amounts of low-quality information to inform thereconstruction and the resultant images were unacceptable.The subspace compressed sensing reconstruction improved the SNR of the images andproduced a recognizable brain and anatomy. Unfortunately, high frequency information was stillabsent from the images, and while the images had some diffusion information present, theylacked nearly all the anatomical and physiological information provided by conventionalEPI-DTI. These reconstructions could have been improved primarily by decreasing the imageresolution to boost SNR, but as 2x2x2 mm voxel sizes are necessary for the SS-TSE DTI toperform equivalently to EPI DTI, that avenue was not explored.Other options to improve SNR were considered. The refocusing RF pulse train of theSS-TSE DTI scan was very energy intensive and deposited enough RF radiation in the samplethat the scanner quickly reached the SAR standards for head imaging set by [80] that exist tolimit RF tissue heating in the subject. To perform the entire imaging sequence beneath thesestandards, the flip angle of the refocusing pulses was set to 120o, and an undersampling factorof 4 was required to limit the total number of RF pulses per TR. The TR could have beenextended to lower the SAR and increase the flip angle with the consequence of increasing thescan duration, but as scan times were already approximately 10 minutes, an increase did notseem feasible. Increased SAR was an unfortunate consequence of the TSE sequence used inplace of EPI, and reduced the flexibility of the scanning parameters. It also would have beenunavailable to use in any subject with medical implants that place further limits on thedeliverable SAR.57Fig 2.11 Reconstructions of a SS-TSE HARDI-60 DTI dataset with a standard compressed sensing reconstruction(left) and with a phase-corrected subspace reconstruction (right). Vastly improved image quality is obvious in thesubspace reconstruction compared to the original reconstruction, but diffusion information and fine structural detailare missing.Additionally, the longer TR of the TSE-DTI scan (due to the increased time required toperform the readout train and to lower SAR) meant even with 4x pseudo-randomundersampling, the TSE-DTI scan remained 10 minutes for a 60 direction HARDI scan, whichwas considerably longer than a comparable EPI scan. Given that the primary reason to use thecompressed sensing sampling scheme was the reduction of scan time, this seemed too long fora technique that did not offer significant other advantages. For similar reasons of scanning time,increasing the SNR through increased sampling or signal averaging was also not an option.2.3.3 Possible uses of the subspace reconstructed SS-TSE DTI scanA SS-TSE DTI scan is therefore of limited use compared to an EPI scan, with the onlymajor advantage being the reduction in distortion effects from the readout train. However, thereare use cases where this is important, especially in imaging of tissues with high susceptibility58differences like an air/tissue boundary or near metallic implants in the skull or oral cavities, suchas some dental implants or braces. EPI scans are severely distorted to the point of illegibility insome of these cases, while a SS-TSE scan may still be usable. Notable areas that suffer fromsignificant distortion in EPI scans are the inferior frontal and temporal lobes, due thesusceptibility differences in the sinuses beneath these regions. Studies that particularly focus onbrain DTI in these areas may benefit from a TSE approach with the demonstrated imagereconstruction technique, but significant modifications would have to be made to improve theSNR of acquired data. In these infrequent use-cases, lower image resolution and slower scantimes would likely be necessary sacrifices for artefact-free TSE DTI scanning.2.4 ConclusionsA subspace-enhanced reconstruction could only be used for SS-TSE DTI scans after aphase-correcting initial reconstruction was performed due to the unique, non-sparse imagephase from subject motion in each diffusion shot. This limited the potential of the technique toenhance TSE DTI scans, and given their inherently lower SNR, high SAR and lengthyacquisition times, the inclusion of a subspace-based compressed sensing reconstruction toSS-TSE DTI added no improvement to image quality over an EPI DTI scan.The application of a subspace reconstruction did, however, markedly improve thereconstructed TSE-DTI image appearance and SNR, despite the extremely low quality of thesampled data. It could have potential for use for other TSE-based DTI applications if thesampled data’s SNR was improved by other means, and may be especially useful in regions oftissue prone to distortion effects.Additionally, given that the subspace reconstruction could be performed after an initialreconstruction and phase correction, it seemed this technique might be applicable to other setsof already-acquired and reconstructed EPI diffusion images to improve the SNR of the final59product. High b-value EPI DTI scans generally have similarly low SNR as well, so applying asubspace reconstruction after phase, motion and eddy current correction to reduce distortioneffects could considerably improve the quality and therefore utility of these scans. This led to theinvestigations outlined in the following chapter, exploring applicability of this reconstructionpipeline as a DTI denoising post processing tool.603. Applying subspace compressed sensing reconstructions to already-reconstructedEPI-DTI3.1 IntroductionThe lessons learned from reconstructions of randomly undersampled SS-TSE DTI,notably the fact that the subspace reconstruction can be applied after an initial reconstructionand phase correction of DTI data, suggested its use to further refine and denoise alreadyreconstructed EPI DTI scans. With the primary reconstruction of the images already completed,image artefacts such as distortion could be mitigated through standard correction algorithmssuch as FSL’s topup and eddy functions [61-63], and the sampling scheme used to collect theoriginal images was irrelevant. The images could simply be Fourier transformed back intok-space and re-reconstructed with the subspace-based compressed sensing algorithm to takeadvantage of the sparsity inherent in the dataset to improve the final image quality.3.1.1 ObjectivesTo investigate the application of subspace compressed sensing as a post-processingtechnique to improve the image quality of diffusion tensor images.3.1.2 MotivationThe use of subspace reconstruction could amplify the effective SNR of each image in aDTI scan through denoising based on the shared information between the individual diffusionvector images. This would allow more accurate estimation of the diffusion tensors within eachvoxel, with improved calculation of diffusion metrics through reduced propagation of errors dueto image noise. This should demonstrate better tracking of diffusion tracts and axonal bundles,especially when combined with an simultaneous interpolation process that takes advantage ofthe improved SNR to refine the image to a higher resolution. This could have the benefit of61improved tissue characterization on a finer scale, which could be used for better brain lesioncharacterization, preoperative surgical planning, and mapping of white matter tracts forneurological research. Indeed, improved image SNR and resolution at no cost of imageacquisition time could lead to superior patient and research outcomes in every use case fordiffusion tensor imaging.This would have the benefit of being implementable for a wide range of already-acquiredclinical and research datasets as nothing must be altered in the acquisition process of this datafor it to be viable for this post-processing. It could simply act as an intermediate step in thepost-processing of all these datasets to improve the quality of calculated results.3.1.3 HypothesisUsing the sparsity in the diffusion dimension to inform a compressed reconstruction, aset of any already acquired DTI images can be simultaneously denoised and, using theimproved SNR a sparse representation provides, interpolated to a higher resolution.Effectiveness of the denoising and upsampling factors will depend strongly on the number ofDTI directions, sparsity of the DTI dataset, and accurate artefact and distortion correction of thesource images.3.2 Methods3.2.1 EPI-DTI datasetsSample EPI-DTI datasets with a variety of diffusion parameters were sourced from animaging repository of healthy volunteer data. Additional sample datasets made available onlineby Stanford University, the “Stanford HARDI” dataset [81], and L'Université de Sherbrooke, the“Sherbrooke 3-shell” [82] dataset were also used as standardised datasets to assessperformance against other image denoising and interpolation methods.623.2.2 Eddy and motion correction of EPI-DTI dataFor each institutional dataset, individual diffusion images were motion and distortioncorrected using FSL’s eddy and topup functions [61-63]. These functions of FSL corrected forany patient motion between diffusion shots and reduced the distortions in EPI data sets fromsusceptibility effects and eddy currents. These corrections were necessary to align the voxels inimage space in order to optimise the sparsifying transforms. Given that these images weremagnitude-only MRI data, image phase was absent and hence correction was unnecessary,unlike the SS-TSE DTI dataset in chapter 2. The Stanford HARDI and Sherbrooke 3-shelldiffusion datasets were motion and distortion corrected prior to accessing them.3.2.3 Subspace generation and sizeOnce images were aligned in space via distortion correction, PCA was performed alongthe diffusion dimension of the dataset. The top principal components of the decomposition,formed a lower-dimensional subspace, which was used to reduce the dimensionality of thedataset in the reconstruction. The size of the subspace was set depending on the number ofsubspace vectors necessary to represent all the anatomical and diffusion information containedin the data, which varied depending on the characteristics of the dataset. On inspection of a plotof the eigenvalues of each data, cutoff points where the singular value dropped to a consistentlylow number represented a point where the remaining information contained in the furthersingular vectors contained only noise data. A projection onto the subspace verified thisassumption, as the subspace images did not contain patches of coherent data; they were Riciannoise-like in appearance. Figs. 5 and 8 in section 2.5.1 demonstrate the cutoff points for HARDI60 and 128 direction multishell scans respectively; as expected, the multishell scan had a largernumber of basis vectors before the cutoff point but both scans had a sparse basis.63This subspace cutoff and subsequent dimensionality reduction comprised the denoisingstep of the algorithm, where the noise-like components in the data were left out of thesubsequent secondary reconstruction of the images. Smaller and the correspondingdimensionality reduction reduced the perception of noise in the images, but it was necessary tokeep large enough that diffusion and structural information was not lost.Table 2.1 in section 2.2.1 gives typical for diffusion datasets; as can be seen, multishelldiffusion scans required higher to best capture the differences in information betweenb-values.3.2.4 Zero-padding to higher resolutionThe distortion-corrected DTI datasets were then 2D Fourier transformed into k-space.The matrix size for each slice of the 2D k-space was expanded by zero padding all four sides ofk-space, such that the original k-space filled the centre of the new matrix. For example, imageswith a 112 x 112 matrix size were zero padded to 180 x 180 for an in-plane combinedinterpolation factor of (180/112)2 = 2.58. Upon reconstruction, this zero-padding step providedinterpolation of the images, a technique which has demonstrated some of the best upscalingperformance for MR images by reducing blurring between voxels [83,84].3.2.5 Application of subspace compressed sensingThe newly zero-padded k-space was then reconstructed using subspace informedcompressed sense reconstruction, as detailed in section 1.4.6, using the subspace truncatedto size produced by principal component analysis of the original DTI data. This subspacereconstruction was performed with the pics tool using the Fast Iterative Shrinkage-ThresholdingAlgorithm (FISTA) provided by BART [79]. Images were reconstructed in 2D with 1-normregularisation and a regularisation factor of 0.003, and a subspace size determined empirically64for each dataset by inspection of PCA eigenvalue plots and subspace image features (section2.5.3).3.3 ResultsThe combined denoising and interpolation algorithm, hereafter called UPSIZER(UPsampling by Subspace Informed ZEro-padded Reconstruction), produced upscaled imageswith significant increases in diffusion image quality and higher resolution. Calculated metrics,including FA, MD and tractograms, had an associated higher resolution as well. Images andcalculated metrics were compared visually to baseline images and quantitatively to baselineimages zero-filled interpolated to the same resolution. Baseline images were also denoised byother state-of-the-art denoising algorithms, including Local Principal Component Analysis (lPCA), Marchenko-Pastur Principal Component Analysis (MPPCA), and Patch2Self (P2S) [33–35],which were then also zero-padded to interpolate to the same resolution. These denoiseddiffusion images and their associated metrics were also compared to UPSIZER to comparedenoising performance and characteristics.65Fig 3.1 Baseline b2000 diffusion images (top) and the same images denoised and upscaled in plane by a factor of1.98 via UPSIZER (bottom). Denoising performance was obvious to the naked eye and the level of detail in theimages was decidedly improved, with structural detail of the sulci and gyri in the brain becoming distinguishable andsmoother curves present in all regions of the brain due to the higher resolution.3.3.1 Image reconstructionsUPSIZER images had visibly higher SNR and image detail than baseline (Figure 16),and retained characteristic diffusion information in each image. More anatomical details wereresolvable in UPSIZER images than baseline scans. Brain gyri and sulci became easilydifferentiable after the UPSIZER algorithm was applied due to the higher resolution and reducedimage noise. The upscaled images were still subject to some blurring from the reconstructionprocess, where expected sharp boundaries at anatomical borders (ie., the edges of theventricles) were instead soft and the contrast change was spread over multiple pixels instead of66at a sharp line. This was due to the lack of high frequency information in the dataset;zero-padding the k-space matrices interpolated the images but did not replicate the trueinformation that would be necessary to add sharp edges and increase fine detail. For blurringcomparisons, denoised images were also upscaled using cubic interpolation, which calculatessub-voxels through a cubic fitting process. Blurring was reduced with the zero-paddedinterpolation inherent to UPSIZER; images in figure 17 showed a subtle increase in detail, mostvisible in the edges of brain gyri.Fig 3.2 Cubic interpolation of a denoised image (left) compared to an UPSIZER image at the same resolution. Thedifferences are subtle but image blurring is slightly reduced with the UPSIZER interpolation, most noticeably inregions with small details like the edges of lateral gyri or within the internal capsule. Arrows demonstrate some of themore visible changes.673.3.2 Voxel size vs. true resolutionThe increased matrix size of UPSIZER images lead to a smaller voxel size in thesecondary reconstructed images. However, the true image resolution was not accuratelyreflected by voxel size, but rather by smallest feature size in the images. Hence, whencomparing resolution of upscaled images it is insufficient to measure it by voxel size, as onecould interpolate to arbitrarily high resolution with no increase in anatomical information in theimages.Original images in the HARDI 128 dataset had 2 x 2 x 2 mm resolution, with a matrixsize of 112 x 112 x 74, which accurately represented both image voxel size and resolvablefeature size. UPSIZER images were upscaled to a size of 1.24 x 1.24 x 2 mm, with anassociated new matrix size of 180 x 180 x 74. This is an interpolation factor of 1.61 in each axialdimension, for a total in-plane factor of 2.58. At this scale, there was a definite increase to trueimage resolution as the upscaled image structure contained features and detail notdistinguishable in the original dataset. The smallest resolvable feature size remained slightlylarger than the voxel size; no single-voxel anatomical detail was visible due to the blurring of theimage from the interpolation process and lack of high-frequency k-space information. Upscalingto an even smaller voxel size was tested (Figure 18) and did not further improve the imagedetail; the increase in anatomical information was limited to approximately this scale. Thismaximum useful resolution increase was dependent on dataset properties, including baselineresolution, number of diffusion directions, and image SNR, but in-plane interpolation factors ofapproximately 2 or more were beneficial for resolution increases in all datasets tested.68Fig 3.3 UPSIZER denoised images reconstructed at original resolution of 2 x 2 x 2 mm (left), at a resolution of 1.24 x1.24 x 2 mm (middle) and at a resolution of 0.62 x 0.62 x 2 mm (right). The moderate increase of resolution in themiddle image shows increased detail and reduced partial voluming effect over the original resolution, especiallyaround the lateral gyri and sulci, but the further increment to image resolution in the right image did not add additionalinformation.3.3.3 Denoising effectThe sparsity constraints placed upon the data had a denoising effect on the images, asnoise in the data was non-sparse, and hence was not represented in the truncated basis set.This was a similar process to lPCA denoising or MPPCA denoising, which also took advantageof the low-rank nature of DTI data to remove noise components from the image.One main difference between the UPSIZER and the lPCA and MPPCA denoisingtechniques was that lPCA and MPPCA applied PCA on local patches of data to create sparsedatasets to filter out noise, while UPSIZER used a larger set of basis vectors that representedthe image globally. UPSIZER’s secondary zero-padded image reconstruction and inherentincrease in resolution also presented a difference, and the re-reconstruction had the potential to69reduce partial voluming effects and add more fine detail than the conventional interpolation usedto upscale the resolution of the lPCA and MPPCA denoised data.Results varied with number of diffusion directions, basis vectors and initial data SNR, butUPSIZER had a moderate denoising effect with performance in between the extreme imagesmoothing produced by lPCA and the weaker denoising of the MPPCA algorithm. UPSIZER’sdenoising effect produced images strikingly similar to the machine learning diffusion imagedenoising technique Patch2Self. P2S had an extremely strong resemblance in denoised imageappearance, including residual signal fluctuations. This suggested the residual signalfluctuations may actually be due to variations in tissue properties, rather than noisecontributions, as they are consistent results from two different denoising schemes.3.3.4 Motion artefact compensation effectDifference images from all denoised images and the original images showed noisepatterns with a Rician distribution, as was expected from magnitude MRI data. There wasminimal structure present in subtraction images that would indicate true diffusion signalcomponents left out of the basis set. However, on some diffusion slices where motion artefactintroduced a dropout in signal in certain regions of brain tissue, both UPSIZER and P2Sreplaced the signal dropout with an estimate of what the signal should have been. This can betruly appreciated with coronal or sagittal reformats of the problematic shots, as can be seen inFigure 20, where original images present artefactual dark slices that were smoothed byUPSIZER and P2S. MPPCA and lPCA denoising algorithms did not have this correction effectfor motion artefact.70Fig 3.4 Comparison of two imaging sets (top row) , the Stanford HARDI-? Images (left columns) and a HARDI128dataset (right columns), denoised using lPCA, MPPCA, P2S and UPSIZER and their difference images from theoriginal data. Original, lPCA, MPPCA and P2S images were also zero-padded to match the resolution increases inUPSIZER.71As long as the motion-induced signal dropout was limited to small areas on a singlediffusion shot, it was not well represented in the principle components used to form the basisvectors. This meant that when images were reconstructed to the subspace, the signal dropoutwas not contained within the truncated set of basis vectors and therefore did not contribute tothe data when it was projected back into image space, similar to the noise removal process.Fig 3.5 Original single b2000 diffusion image viewed coronally showing motion corruption artefacts as black lines (redarrows). UPSIZER-denoised images at the same resolution without motion corruption.3.3.5 Interpolation artefactsDuring testing, the upscaling process did occasionally introduce interpolation artefactsinto the resultant images, primarily with integer upscaling factors. This occurred as edges ofvoxels from the original images aligned with the edges of voxels in upscaled images, causingdistinct boundaries between sets of voxels, and creating blocky or linear structures in theresultant data. For example, this was apparent when a scan with an original matrix size of 112was upscaled to 224, a factor of 2 (Fig. 21). A zero-padded matrix with 256 voxels, with an72upscaling factor of 16/7 in each plane, had significantly reduced artefacts compared to 224. Theinterpolation artefacts were mitigated with stronger regularisation factors, to promote imagesmoothing, and using non-integer upscaling factors, to prevent the voxel edge alignment.Fig 3.6 Blocking and line artefacts from an integer interpolation factor of 2 in each dimension (left); horizontally andvertically straight lines of voxels were overemphasised and stood out visually (arrows). An interpolation factor of 5.22(right) showed much smoother curves and less blocking artefacts.3.3.6 Calculated diffusion metrics: fractional anisotropy and mean diffusivityFractional Anisotropy (FA) is a scalar value derived from DTI. FA provides informationabout the anisotropic nature of water diffusion, meaning it quantifies the directionality ororientation preference of the diffusion. FA is calculated from the eigenvalues of the diffusiontensor. The eigenvalues represent the rates of diffusion along each principal axis. The formulafor FA is typically expressed as:73Where are the primary eigenvalues of the 3x3 diffusion tensor. FA values range from 0to 1. A value of 0 indicates perfectly isotropic diffusion (equal diffusion in all directions, such asin free water), while a value of 1 indicates perfectly anisotropic diffusion (diffusion occurring inonly one direction, with complete restriction in the orthogonal plane).In the context of brain imaging, high FA values are often associated with well-organisedand coherent structures, such as the white matter tracts composed of myelinated axons. Thehighly anisotropic nature of diffusion in white matter is reflective of the long, thin cylindricalstructure of axonal fibres, where diffusion can occur along the long axis of the axon, but isrestricted by cell walls in the orthogonal plane.The FA values did not visually change significantly between denoised and undenoisedimages, with the only significant changes in appearance occurring in regions of very low FA,such as deep grey matter or CSF, where denoised images suffered less from noise-likeartefacts. The actual FA values, however, were higher in most white matter tracts of the brain.This was a known FA squeezing effect [85], where an FA bias was created by the positive noisefloor. The Rician noise contributing to the secondary and tertiary diffusion eigenvalues wasremoved by the denoising process. This increased the relative magnitude of the primaryeigenvalue of the voxel, and hence the FA. This effect was not significant in MPPCA or lPCAdenoising, but showed a similar pattern in P2S as UPSIZER. FA in grey matter and CSF hadsignificant percentage changes in most denoising schemes, but as FA was very low or zero inthese isotropic tissues, this was not overly concerning.74Fig 3.7 FA (left) and MD (right) maps from original images (top row) and images denoised through various denoisingalgorithms, with their percent difference from original images included. Denoising algorithms include lPCA (secondrow), MPPCA (third row), Patch2Self (fourth row) and UPSIZER (bottom row). Green arrows indicate increased MD insmall white matter tracts due to reduced partial voluming effects.75Uniquely to UPSIZER, there were notable increases in FA in thin white matter tractssuch as the internal capsule and optic radiata (Figure 22, arrows), where there appeared to beless partial voluming effects from neighbouring tissue due to UPSIZER’s incorporation of theinterpolation process into the denoising scheme. This demonstrated better separation of thesewhite matter tracts from surrounding voxels, and was a beneficial effect of the simultaneity of thedenoising and interpolation.Mean diffusivity (MD) is a scalar measurement derived from the eigenvalues of thediffusion tensor. It is the average of the three eigenvalues of the diffusion tensor and representsthe overall rate of diffusion within a voxel in units of . Mathematically, it is expressed as:MD is dependent on tissue microstructure. In voxels with high cellularity or othermicrostructural barriers, water diffusion is limited, leading to lower MD values. Conversely,regions with minimal obstruction to diffusion have high MD values. Even small tissuemicrostructural changes can have a greater impact on the MD than other properties such as T1or T2, meaning MD can be one of the most sensitive properties to tissue change, pathological orotherwise.Like FA, MD was not significantly altered by the upscaling process, and showed valueswithin a few percentage points of the original images. This was not unexpected, as MD, like FA,is a property that averages over all the diffusion scans, so changes in noise or signal intensityhad minimal effects. In each denoising scheme, there was an overall decrease in MD, as thedenoising processes removed noise that increased the diffusion eigenvalues of the original data.Also, similar to FA, a unique increase in MD from UPSIZER data was present in thin white76matter tracts such as the internal capsule and optic radiata. This also demonstrated reducedpartial voluming between these tracts and surrounding tissue.3.3.7 TractographyOne of the main uses of a diffusion tensor scan is identification of axonal tracts, which isone of the best ways to measure connectivity within the brain. Major and minor diffusiondirections in each voxel are calculated through constrained spherical deconvolution of thevoxel’s diffusion data [58,86], and are assumed to represent the extensions of axons within eachvoxel. Thus, if a diffusion direction points to another voxel which shares similar diffusionproperties, axons can be assumed to be extending through these neighbouring voxels, andhence tracked backwards and forwards from the seed points to their endpoints. Algorithms toidentify tracts are frequently probabilistic, to better identify smaller tracts or to separate tractsthat bend, cross or touch other tracts within a voxel. Noise and other errors like partial volumingeffects in the underlying data are propagated through the probabilistic tract generationalgorithms and this propagation of error can generate unrealistic tracts, affecting tractographyresults. DTI scans with higher matrix sizes and reduced noise increase the confidence incalculation of tractograms. Therefore, denoising and upscaling schemes are frequently usedprior to tractography calculations, and a method that combines denoising and upscaling toachieve the best performance possible should increase the relative robustness and accuracy oftractography.Tractograms were produced from original and denoised datasets using MRtrix [87],using constrained spherical deconvolution to calculate diffusion vectors in each voxel andprobabilistic tract generation from random seed points [88]. Seed points were generated in thelateral geniculate nucleus to isolate tracts belonging to the optic radiation [48,89], an importantbundle of white matter that plays a significant role in visual information processing. Performance77was compared at identifying Meyer’s loop,, a small bundle of fibres extending into the temporallobe that is important to preserve during temporal lobe resections. These optic radiationtractograms were generated using data interpolated to 0.875 x 0.875 x 0.875 mm in original,MPPCA, P2S and UPSIZER dataset for comparisons (fig X).Fig 3.8 Tractography of the optic radiation, seeded at the lateral geniculate nucleus from DTI denoised using A. Nodenoising, B. P2S denoising, C. MPPCA denoising and D. UPSIZER. Images are coloured by their normalised clustercoherence index [90]3.4 DiscussionThe application of a subspace-informed compressed sensing reconstruction toalready-acquired DTI images opens up avenues of using computationally intensive methods tosimultaneously denoise and interpolate datasets by using the redundancy of the data. It78leverages the fact that while it is difficult to achieve high resolution, high SNR and strongdiffusion weighting in one image, one can acquire large numbers of low resolution and low SNRscans. The algorithm uses the data‘s internal consistency to self-correct noise and errors in thedata and achieve better estimates of sub-voxels. However, there are limits to the power of thistechnique; it is constrained by the amount of redundancy within the data, and pitfalls exist whereartefacts or too-aggressive application of the algorithm could invalidate the accuracy of thedenoised data.3.4.1 Comparison to standard upscaling processesThere exist numerous techniques that are regularly used for interpolation of MR imagesto a higher resolution. The most common of these include cubic and sinc interpolation. Typicalpipelines for interpolation of DTI datasets use cubic interpolation, which produces reasonableresults in very fast calculation times. Zero-padding interpolation is slower and can introduceother effects, such as noise amplification, but in this case its major benefit is the improvement infeature resolution it offers; it helps reproduce the structure and detail of the underlying tissuethrough better representation of the information already contained in k-space. All data wasinterpolated using zero-padding interpolation, due to its performance and comparability toUPSIZER.3.4.2 Absence of higher frequency dataThere is a limit on the size of spatial information present in the original images and theircorresponding k-space data. No additional information was collected or created regardinghigher spatial frequencies, which would correspond to filling in a larger k-space matrix. Thereconstruction scheme attempted to estimate the zero-filled portion k-space as best as it canthrough its wavelet transforms, but this resulted mainly in image smoothing and slight blurring ofvoxels as the regularisation factor removed hard edges. The subspace projection had the79potential to estimate some small amounts of higher frequency information and to reduce partialvoluming effects, as can be seen in the FA and MD maps, but further investigation is required toquantify the increase in detail possible through this technique.3.4.3 Removal of information not represented by basis vectorsBasis vectors need to be chosen carefully to contain the entirety of real information- ifthe PCA decomposition is truncated too aggressively, diffusion differences begin to appear inimages, especially in areas of highly variable signal intensity like the corpus callosum. Furtherwork exists to construct an automated subspace size calculation that can use the totalinformation present in each basis vector plus noise characteristics of the basis images tocalculate an appropriate subspace size. This would mitigate the changes of choosing aninappropriately small subspace size and removing important information, while retaining thedenoising benefits of a high dimensionality reduction when performing the reconstruction.This appropriate subspace size also was more complicated when including multipleb-values in a multi-shell scan, as the number of necessary basis vectors to represent allimportant information changed depending on the number of b-values and the b-value itself. Asthe diffusion signal for high b-values depends additionally on axonal diameters of the tissue, itwas necessary to include an entire new set of information to represent AD within the basisvectors.It was possible to denoise and upscale different b-values separately, which preventedthe use of shared information between b-values but was more likely to retain b-value specificinformation. Separation of the b-values was particularly beneficial for the denoising in somecases with very low SNR scans, notably the Sherbrooke 3 Shell dataset, which contained a setof images with a b-value of 3000 mm2/s and very low SNR. Reconstruction of all b-valuessimultaneously shared the noise characteristics between each b-value, so while the b300080images improved, the other b-value images had suboptimal denoising performance whencompared to separate b-value reconstructions. Work remains to be done to optimise whetherreconstructions should be separated or combined; in general most cases are likely to benefitfrom denoising b-values separately, to limit the potential for loss of diffusion information and toavoid corrupting low b-value images with noise from high b-values.Original UPSIZER using all b-values UPSIZER using individual b-valuesFig 3.9 Reconstruction of a multishell DTI with 12 images with a b-value of 0 s/mm2 , 55 images with a b-value of2000 s/mm2, 35 b-value of 1000 s/mm2 images and 24 b-value of 500 s/mm2 images. Original data (left) shows a rawimage with b-value of 2000 s/mm2. UPSIZER denoised using all available b-value images (middle) show bestdenoising performance. UPSIZER denoised images using just the 2000 s/mm2 images (right) has reduced denoisingeffects but is less likely to remove 2000 s/mm2 specific diffusion information.3.4.4 Correction for motion-deteriorated volumesDuring any MRI acquisition, there is potential for subject motion which can causeartefacts on images. As individual slices of diffusion volumes are each fully acquired during oneTR (on the order of several seconds), the artefacts due to motion are usually misregistration ofdiffusion volumes across diffusion encoding directions as the subject changes position duringthe scan. However, motion during the diffusion gradients also affects the phase and can81deteriorate signal in the image if the motion prevents the refocusing effect of the second lobe ofthe diffusion gradient through a large enough net change in position. This gives the diffusionscan motion corruption artefacts where all or portions of the image are artefactually attenuated,appearing as low signal regions of tissue where signal would otherwise be expected. Oftenthese artefacts are most noticeable when volumes are reformatted to a sagittal or coronal view,appearing as dark lines where signal should reasonably be expected to be isointense to thevoxels above and below it. Some obvious examples are plotted on both slices and sagittalreformats in Fig. 20, with red arrows indicating motion attenuation of signal.The highly sampling diffusion space in the DTI scans permitted the backprojection stepof the subspace compressed sensing reconstruction to estimate the signal in these corruptedareas. When viewed in the subspace, the missing signal appears as an outlying noise peakalong the diffusion direction and is filtered out through the subspace fitting procedure. Imagesare reconstructed without this corruption, and the signal that best fits the subspace is filled ininstead. The incongruous lack of signal indicated by the red arrows in the left image of Fig. 20was estimated and corrected by UPSIZER in the right image in Fig 20 and the resultant imagesare much smoother.This signal dropout correction necessitated that a significant majority of the diffusionshots were uncorrupted in the same area, otherwise the subspace reconstruction wouldrepresent the signal dropout as an image feature and retain that information in the principlecomponents. However, if there was enough motion during the acquisition to corrupt a majority ofvolumes, the data is unlikely to be reliable enough to be usable at all.This estimation of corrupted data will not significantly affect calculated metrics such asFA and MD of the DTI scan, as the affected volumes will likely only represent a small number ofthe total volumes, assuming the motion is not too severe, but could certainly improve the82reliability of fitting procedures for higher-order spherical harmonics in deconvolution steps of theDTI data. Higher order components are more sensitive to rapid changes in signal in the diffusionspace, and thus motion corruption will affect them. As a result, the performance of tractographythat utilises high order deconvolutions should be improved in UPSIZER denoisedmotion-corrupted scans.3.4.5 Regularization factorThe value of regularisation is also important to consider- when was set too high in thecompressed sensing minimisation algorithm (equation 13) artefacts began to appear. They tookthe shape of lines or cubes of voxels with stark differences from their neighbours and gave ablocky or pixelated appearance to the images. This also dramatically affected FA and MD maps,causing artefacts appearing as striated and blocky voxels. When was set appropriately, thesevanished as the true images are better estimated more by the reconstruction process. Whenwas set too low, however, the benefits of the compressed sensing reconstruction were lost andthe whole process was equivalent to projection into the truncated subspace and back. Choice ofwas determined empirically and set to 0.003, but remained to be tested for optimalperformance characteristics as the optimal may change for different diffusion datasets.3.4.6 Ground truth unavailabilityIt would have been ideal to compare the denoised images to a ground truth diffusion scan;however, this was unacquirable for the DTI scans in questions, as all current DTI acquisitionshave noise and limited resolution and there is no other way to quantify diffusion in-vivo. It waspossible to downsample an image, add noise, then upscale and denoise it to see how well itreproduces the original scan. Unfortunately, due to the effects of original matrix size on theeffect of zero-filling [84], an attempt of this on already low-resolution images quickly introducedinterpolation artefacts at even small interpolation factors. Additionally, relying on the original83image as “ground truth” introduces other confounding effects as the original image wouldcontain factors that the denoising process would remove, with no method of determining if theoriginal or denoised images are more accurate.An indicator that the denoising process might be approaching the true diffusioncharacteristics of the tissue was the extreme similarity of the denoised scans to the Patch2Selfdenoised images. Two different denoising algorithms producing near-identical imagessuggested they were converging on a truth value. However, this effect could have been due tounderlying mathematical similarities in the two denoising techniques. Given the self-supervisedmachine learning nature of the Patch2Self algorithm, it was difficult to ascertain the similaritiesin the mathematical models used to provide the denoising effects.Further work could use 3D printed diffusion phantoms [91,92] as reference values fordenoising models, but at the time of writing these phantoms were unavailable for testing, andonly contained a small number of straight fibres, with no bending, crossing or touching fibrestructures. As the available complexity of these phantoms increases, they will provide better andmore accurate diffusion phantoms with known ground truths for testing MRI diffusion denoisingalgorithms and analysis schemes.3.4.7 2D vs 3D UpscalingMost upscaling of diffusion scans for tractography is done to achieve an isotropicresolution to remove any dependance on voxel shape during the probabilistic estimation ofdiffusion tracts. UPSIZER as presented can be upscaled in either 2D or 3D, depending on thenature of the Fourier transform back into k-space. Performance is similar between 2D and 3D,with proportional 2D or 3D combined interpolation factors available before artefacts are seen.The major difference is then computer memory use and processing time, where the84simultaneous compressed sense reconstruction of dozens if not hundreds of diffusion imagesgets significantly more memory intensive if done in 3D rather than slice by slice.Alternatively, standard cubic or zero-padding interpolation can be performed in the slicedirection after 2D upsampling, with similar performance and much lower memory requirements.In either way, images can be upscaled to an isotropic resolution as required by further imageprocessing pipelines. As noted in section 3.6.5, 2D UPSIZER interpolation had decreasedpartial voluming effects visible on MD and FA maps, and the 3D UPSIZER could have similaradvantages in the slice direction as well, but remains to be tested.3.4.8 Rician noise characteristics and problemsDue to the Rician characteristics of noise in magnitude MRI data [26,27], there wasbackground noise with a positive average signal in regions like the CSF-filled ventricles, whichshould have zero or near zero signal on even moderate b-value diffusion scans due to thecomparatively rapid diffusion in free water . Post denoising, this Rician noise had theunfortunate consequence of reconstructing images with the net positive average of the noise inthe ventricles. Other tissues of the brain, like grey or white matter, had positive signalmagnitudes greater than the noise. Therefore, these regions had both positive and negativenoise changes and were less affected by the Rician nature of the noise. If the MRI data wascomplex, the noise in zero or low-signal areas would have had a Gaussian distribution with anaverage magnitude of zero.The positive values of noise in the low-signal areas post reconstruction could be easilyremedied through saving complex diffusion images and phase correcting them as laid out insection 2. The complex noise, which should remain relatively unaffected by the low-frequencyphase correction procedure, would be Gaussian instead of Rician and would be correctlyremoved from zero- or low-signal regions of the image. Unfortunately, this negatively affects the85backwards compatibility of UPSIZER, as historically only the magnitude information of DTIscans was saved. A complex-valued DTI scan was processed in both complex and magnitudespaces with UPSIZER and results are demonstrated in figure 25. As can be seen, the complexUPSIZER reconstruction accurately reconstructs CSF as near-zero valued, while the magnitudereconstruction has a small positive value within the CSF.This concern only applied to regions of the image where image intensity was expected tobe zero or near zero; that is, CSF and empty air for diffusion images. Even CSF regularly hadnon-zero signal for low to moderate b-values, and was unaffected in those cases. Thedifferences arising from denoising complex and magnitude images were small in most non-CSFtissue; and can likely be ignored. Ideally, however, UPSIZER denoising should be performed ina phase-corrected complex domain.Fig 3.10 Subplots of original images (left), magnitude (middle) and complex (right) images denoised using UPSIZERto compare CSF denoising in complex or magnitude spaces. Difference images (far right) demonstrated significantreduction in image CSF magnitude when complex images were denoised, but other tissue remained relativelyunchanged.3.5 ConclusionsA secondary reconstruction of motion- and eddy-corrected diffusion tensor imaging data usingcompressed sensing with an additional low rank constraint provided a straightforward and fastmethod of reducing noise in the images, with the additional advantage of a simultaneous 2- or863- dimensional image increase in image resolution. The denoising and reconstruction propertieswere highly dependent on the choice of subspace size and regularisation parameters, which, ifincorrectly chosen, could remove important diffusion information, cause interpolation artefacts orhave suboptimal denoising performance. However, when chosen correctly, the denoising effectprovided an increase in SNR that was comparable to some of the most advanced denoisingtechniques available for DTI, (eg. lPCA, MPPCA, and P2S). The simultaneous interpolation ofthe images was at least equivalent to cubic interpolation and showed an increase in fine detailand reduced partial voluming in calculated metrics such as FA and MD beyond zero-paddedinterpolation alone. Tractography results were inconclusive; while denoised images showed animprovement in performance over original data, tractograms created from MPPCA and P2Sdenoised datasets showed similar or better detail for tracking of white matter pathways. Testingwould greatly benefit from a high-resolution diffusion phantom with a known ground truth to trulyquantify performance of UPSIZER and other denoising algorithms.874. Conclusions and Discussion4.1 Applying subspace compressed sensing reconstructions to TSE-DTIGiven the underlying phase problems with DTI, low-rank subspace reconstructions ofin-vivo data directly from k-space are currently impossible, as the phase errors due to motionduring acquisition are incoherent and cannot be represented sparsely. With perfectly motionless(ie. ex-vivo) subjects and a firmly stabilised scanning surface to prevent vibration, thesereconstructions may be possible, but this is of limited use as scanning time becomes vastly lessof an important factor under those conditions and hence undersampling may not be necessary.All subspace-informed reconstructions had to pass through an intermediatereconstruction and phase correction step, which dramatically limited the potential of thistechnique as errors in the intermediate step were propagated through the reconstruction.Additionally, given that the SS-TSE acquisition had inherently poor SNR, the resulting imageshad extremely limited spatial information and could represent only very basic diffusioninformation in the scan. Subspace reconstructed SS-TSE DTI scans were therefore not animprovement over the current standard EPI DTI scans, which remained much faster to acquire,with vastly higher SNR. A limited number of use cases may prefer a SS-TSE DTI scan over EPIDTI when image distortion reduction is a primary goal; however, SNR of the data would need tobe dramatically amplified for this style of imaging to be viable, even with the phase-correctedsubspace reconstruction.4.1.1 Impact of the resultsThe lengthy acquisition and poor image quality of the SS-TSE DTI image reconstructionscompared to conventional EPI DTI negated the advantages of the technique. The reconstructionwas hampered by the low SNR and intermediate reconstruction steps necessary to phase88correct the data, and likely will have limited impact .The phase-corrected reconstruction steps ofthis technique could alternatively be used for complex-valued denoising of EPI data such as insection 3.8.9, with the added steps of distortion and motion correction occurring after theintermediate phase-corrected reconstruction. This would improve the viability of PCA baseddenoising techniques by restoring the sparsity of the complex data.4.1.2 LimitationsThe biggest limitations of the technique were the phase errors inherent to DTI, extremelylow SNR of the chosen pulse sequence and length of the acquisition. The phase errorsprecluded the viability of sparse reconstruction methods directly from k-space, and theintermediate reconstruction and phase steps were at best a compromise for optimal SNR. LowSNR of the SS-TSE scans was avoidable but would have had a profoundly negative effect onthe resolution and length of the imaging process. The lengthy acquisition due to the long TSEecho train and the inherent SAR of the TSE echo generation minimised the advantages of theundersampling factors available from CS reconstruction and as such lent no significant timesavings to the acquisition, especially compared to conventional EPI based DTI.4.1.3 Future workAs techniques that minimise phase errors in DTI scans [93] become feasible andeffective for routine DTI scans, the application of a subspace reconstruction to a SS-TSE DTIscan might become more applicable. In the absence of the phase errors, the intermediatereconstruction and phase correction could be skipped, and the negative repercussions of thisstep negated. Phase-neutral scans would also lift the requirements for single shot techniquesfor diffusion scans, and permit sampling in a 3D k-space. 3D improves undersampling potentialby allowing acceleration in two phase dimensions, which further promotes the use ofcompressed sensing reconstruction schemes for reconstruction. This could mean these scans89have a reduced scan time compared to a 3D EPI DTI, with minimised distortion or susceptibilityeffects as an additional advantage. However, the underlying problem of the length of the TSEecho train and the extremely low SNR of the later echoes would still exist, and as long as thehigh frequency portion of k-space is overwhelmed by noise, spatial information will be missingfrom the data, reducing its potential value as an imaging technique despite the improvedreconstruction algorithms presented.4.2 Applying subspace compressed sensing reconstructions to already reconstructedEPI DTIThe application of the subspace reconstruction technique to distortion corrected EPI DTIscans demonstrates an entirely new method of image denoising in the post-processing steps ofa DTI dataset with performance on par with state-of-the-art techniques. The simplicity, robustmathematical basis and advantages of the simultaneous interpolation and noise removal lendweight to its candidacy as a denoising algorithm of choice.4.2.1 Impact of the resultsUPSIZER presents a novel denoising and image interpolation scheme using principalcomponent analysis, compressed sensing principles and zero-padding to achieve similar resultsto best-in-class algorithms. The benefits of UPSIZER include its simple and flexibly controllableparameters, the underlying physical basis of the subspace projection and the simultaneousupsampling to maximise the achievable resolution with less blurring and partial voluming. Theonly assumption UPSIZER makes about the DTI model is that the true diffusion information canbe represented sparsely, similar to other PCA-based denoising schemes. It has denoisingperformance similar to Patch2Self, but is founded on a relatively straightforward mathematicalbasis instead of a machine learning algorithm, and can therefore have its performance adjustedin flexible ways to suit individual datasets with varied SNR, b-values, matrix dimensions and90diffusion characteristics. Unlike MPPCA or lPCA, it also provides limited image signal motioncorruption compensation, and estimates appropriate signal replacements for regions of signalerroneously dephased by involuntary motion.Additionally, the increased resolution innately provided by UPSIZER in 2D or 3D couldimprove the accuracy of voxel-wise measurements on a finer scale, providing more informationabout connectivity and pathology for researchers and clinicians than un-denoised data or dataupsampled using conventional cubic or zero-padding upsampling algorithms. This mightimprove or enhance connectivity studies such as the Human Connectome Project, braindevelopment research scans of any number of different projects, surgical planning for tumourresection or anywhere diffusion tensor imaging is used and could benefit from finer parcellationof brain tissue.4.2.2 LimitationsThere are several drawbacks that need to be addressed with UPSIZER that primarily aredue to the potential for the loss of higher-order diffusion information in the truncation of theprincipal component basis. To this end, the biggest limitation of UPSIZER has to do with thecreation and selection of the subspace used in the reconstruction. While images with goodintra-image registration and distortion correction seem to form a principal component curve suchas Figure 5, which naturally falls to noise components very quickly, there is no guarantee thatthis basis actually represents the diffusion information. With poor registration due to significantmotion and distortion, the early principal components may contain more information about themisregistration than diffusion, and early truncation will remove diffusion data. Algorithms toestimate the ideal subspace sizes through analysis of the structure of the subspace images willneed to accommodate this type of error. Strict quality control of the image registration wouldalso minimise chances of these problems.91Additionally, the magnitude-only images of most saved datasets can lead toUPSIZER-introduced errors in low signal regions of the image. In complex images, white noisevalues in zero or low signal areas should follow a Gaussian curve around zero, but whenmagnitude data is saved, all the negative noise components are saved with positive values,leading to average noise values above zero. The first principal component from every dataset(and indeed the spherical harmonic) was equivalent to the average signal across allimages and as such, the noise in low signal areas averages to a positive, non-zero factor. Whenthe images are reconstructed with the basis, each image now gains this average value in alllow-signal areas, leading to errors in the estimates of the diffusion signal. This can skewcalculated values of the mean diffusivity, fractional anisotropy and diffusion eigenvalues.Fortunately, this is only likely to happen in air and freely diffusing tissue such as CSF, which arenot typically the focus of diffusion experiments, but for extremely high b-values or other very lowSNR scans it could create issues. The easiest way to rectify this positive noise bias is to savecomplex images, though that will introduce the necessity for phase corrections as laid out inchapter 2.A problem inherent with any diffusion tensor imaging denoising scheme is, so far, nonecan be tested against a known ground truth with complexity anywhere similar to the brain, andtheir true effectiveness at removing noise and retaining diffusion information can not be known.This limits the testing possible for all denoising algorithms, leading to preferential bias to imagesthat “look best” rather than represent the truth. While smooth, noise-free images appeal to theviewer aesthetically, the heterogeneity of brain tissue is such that diffusion scans in particularcould have inherent significant intervoxel variability that could mimic noise. Without a truth valueknown for at least one dataset, comparisons between denoising algorithms are generallyqualitative at best, and calculated values from denoised images are equally as prone to bias.924.2.3 Future workFuture work for UPSIZER should address some of its limitations and include testing ofthe algorithm on a wider scale. There are an incredible variety of diffusion scans that couldbenefit including scans with different diffusion weighting parameters, different scanner systems,and even different acquisition pulse sequences. The principles of the algorithm should remainsound as long as the data are sparsely representable, and thus should apply equivalently to andprovide denoising and upscaling for all manner of diffusion scans. Indeed, there is nothinginherent to UPSIZER that limits it to diffusion, and there exist a plethora of MRI scans thatcontain a fourth parametric dimension such as T1 and T2 mapping and multi flip angleexperiments that could be used to generate a subspace representation to use for denoising andinterpolation of the images in much the same way.For this particular application of UPSIZER, an algorithm specific to estimating a diffusionsubspace size for optimal denoising performance and retention of diffusion information shouldbe implemented and tested to automate this currently empirical determination step.Incorporating quality control steps to predict when subspace vectors no longer representspecifically diffusion data would also be beneficial; to this end, a large dataset of diffusion scanswould be useful to help identify features specific to diffusion information.Also as mentioned in the limitations of UPSIZER, a comparison to underlying groundtruth is necessary to truly compare the performance of different denoising algorithms and theireffectiveness at removing solely noise from the images. 3D printed diffusion phantoms [91,92]as a source of a ground truth data set is a promising avenue, but whether it can reachcomplexity levels close to neurological tissue remains to be seen. Nonetheless, tests of diffusiondenoising with a known underlying diffusion tensor are necessary to accurately comparealgorithms.93Another route of future research could be to investigate using the even- real sphericalharmonics as the subspace basis for reconstruction. This would guarantee a physicallymeaningful subspace, and remove the effects of problems that misregistration of images,artefacts, and distortion have on the PCA basis. The other benefit would be that the resultantsubspace images would be representative of the voxel-wise coefficients of each sphericalharmonic, the calculation of which is usually part of post-processing. However, the reducedsparsity of the spherical harmonic basis might reduce its ability to denoise and upscale theimages, especially as the subspace size quickly increases with the order of spherical harmonicsincluded. Instead of a subspace size of 8-20 from PCA, a subspace including only theharmonics already would contain 32 subspace components. For a HARDI-60 DTIscan, this is only a compression factor of ~2 rather than ~6, reducing the effectiveness of thereconstruction. Additionally, the image information’s dependency on b-value may also not berepresented sparsely with the use of the spherical harmonic basis and information may be lost,though that is by no means ensured with a PCA-basis either.Finally, the application of UPSIZER to diffusion datasets outside the brain would beextremely interesting, especially in even more resolution limited body parts such as the spinalcord or kidneys. These both pose even greater problems with subject motion and other tissuetypes within the imaging FOV, but on the other hand, the diffusion characteristics of theseorgans have much less complexity in their cellular organisation and as such may be even moresparsely representable than the brain. If the other confounding effects in these areas can becompensated for, UPSIZER may enhance diffusion imaging in these difficult-to-scan areas bypermitting fast, low-resolution acquisitions that are computationally upsized to usefulresolutions.944.3 Concluding remarksThe compressed sensing algorithm is one of the most powerful computational toolscreated for the analysis of sparse signals in the last few decades. With the recent advances inprocessing power cheaply available, the ease in which compressed sensing can extract reliableestimates of signals from noisy and poorly sampled data means it is tempting to apply it in everypossible situation. Therein lies a trap for an unwary scientist; not every signal is as sparse as itmay originally seem, and compressed sensing may not be the best route forward. Still, such isthe power of the algorithm that even when it may not work directly, a different approach mayproduce promising results.Such was the case with this body of work; a lack of in-depth knowledge of the subtletiesof diffusion imaging’s complex phase led me to believe it was ripe for the application ofcompressed sensing only to discover that diffusion data, as sampled by the MRI, does not forma sparse dataset along its diffusion dimension. After much labour to extract and correct thephase of these scans, the compressed sensing algorithm still proved powerful enough togenerate remarkably improved TSE-DTI images despite these original setbacks, and thisproved applicable to more conventional EPI diffusion scans via UPSIZER. 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Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial 4.0 International
Graduate
Kolind, Shannon
Subspace informed compressed sensing reconstruction techniques for diffusion tensor imaging
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