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Acquisition- and modeling-independent resolution enhancement of brain diffusion-weighted magnetic resonance… Bajammal, Mohammad Salem 2016

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Acquisition- and Modeling-independentResolution Enhancement of Brain dwMRI VolumesbyMohammad Salem BajammalA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016© Mohammad Salem Bajammal 2016AbstractDiffusion-weighted magnetic resonance imaging (dwMRI) provides uniquecapabilities for non-invasive imaging of neural fiber pathways in the brain.dwMRI is an increasingly popular imaging method and has promising di-agnostic and surgical applications for Alzheimer’s disease, brain tumors,and epilepsy, to name a few.However, one limitation of dwMRI (specifically, the more common dif-fusion tensor imaging scheme, DTI) is that it suffers from a relatively lowresolution. This often leads to ambiguity in determining location and orien-tation of neural fibers, and therefore reduces the reliability of informationgained from dwMRI.Several approaches have been suggested to address this issue. One ap-proach is to have a finer sampling grid, as in diffusion spectrum imaging(DSI) and high-angular resolution imaging (HARDI). While this did resultin a resolution improvement, it has the side effects of lowering the qual-ity of image signal-to-noise ratio (SNR) or prolonging imaging time, whichhinders its use in routine clinical practice.Subsequently, an alternative approach has been proposed based on super-resolution methods, where multiple low resolution images are fused intoa higher resolution one. While this managed to improve resolution with-out reducing SNR, the multiple acquisitions required still resulted in a pro-longed imaging time.In this thesis, we propose a processing pipeline that uses a super res-iiAbstractolution approach based on dictionary learning for alleviating the dwMRIlow resolution problem. Unlike the majority of existing dwMRI resolutionenhancement approaches, our proposed framework does not require modi-fying the dwMRI acquisition. This makes it applicable to legacy data. More-over, this approach does not require using a specific diffusion model.Motivated by how functional connectivity (FC) reflects the underlyingstructural connectivity (SC), we use the Human Connectome Project andKirby multimodal dataset to quantitatively validate our results by investi-gating the consistency between SC and FC before and after super-resolvingthe data. Based on this scheme, we show that our method outperforms in-terpolation and the only existing single image super-resolution method fordMRI that is not dependent on a specific diffusion model. Qualitatively,we illustrate the improved resolution in diffusion images and illustrate therevealed details beyond what is achievable with the original data.iiiPrefaceThe work performed in this thesis has resulted in the following publications:• (In progress) Bajammal, M. and Ng, B. and Abugharbieh, R. ”HighResolution Diffusion MRI Data without Acquisition Modifications”This paper was based on a collaboration between Bajammal and BiSICLalumni Dr. Ng under the supervision and guidance of Prof. Abughar-bieh. Dr. Ng. contributed the ideas of: using online dictionary to en-able method scalability, using two different databases for buildingthe dictionary and testing to show generalizability, and using affinitypropagation to find prototype gradient volumes to reduce computa-tional load. Bajammal contributed the code implementations, gener-ated the results, and the idea of using a clustering approach on multi-shell data. The paper manuscript is still in its early stages and will beedited by all co-authors.• Bajammal, M. and Yoldemir, B. and Abugharbieh, R. ”Comparison ofStructural Connectivity Metrics for Multimodal Brain Image Analy-sis”, International Symposium on Biomedical Imaging (ISBI), Brooklyn-USA, Pages: 934–937, April 2015.This paper was based on a collaboration between Bajammal and BiSICLPhD candidate Yoldemir under the supervision and guidance of Prof.Abugharbieh. Yoldemir contributed the paper idea, preprocessing ofivPrefacethe data and parcellation, and implementation of the tractographymethod and three of the four anatomical connectivity metrics. Bajam-mal contributed the implementation of the fourth anatomical connec-tivity metric and the validation code as well as generated the results.In terms of paper writing, both students contributed equally. The pa-per was edited by the supervisor. Parts of this paper are included inChapter 4.• Yoldemir, B. and Bajammal, M. and Abugharbieh, R. ”Dictionary BasedSuper-Resolution for Diffusion MRI”, MICCAI Workshop on Com-putational Diffusion MRI (CDMRI), Cambridge-USA, Pages: 194–204,September 2014.This paper was based on a collaboration between Bajammal and BiSICLPhD candidate Yoldemir under the supervision and guidance of Prof.Abugharbieh. Bajammal contributed the algorithmic idea conception,implementation of the method and validation scheme, as well as gen-eration of the results. Yoldemir contributed the application idea con-ception, preprocessing of the data and parcellation, and the validationscheme. In terms of manuscript writing, Yoldemir contributed the ma-jority of the effort. The paper was edited by the supervisor. Parts of thispaper are included in Chapter 3.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation and Problem Statement . . . . . . . . . . . . . . 11.2 Magnetic Resonance Imaging of the Brain . . . . . . . . . . . 21.3 Diffusion Modeling . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Limitations and Alternative Imaging Schemes . . . . . . . . 121.4.1 Diffusion Spectrum Imaging . . . . . . . . . . . . . . 121.4.2 High Angular Resolution Diffusion Imaging . . . . . 141.5 Thesis Objectives and Proposed Approach . . . . . . . . . . 151.6 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . 171.7 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . 17viTable of Contents2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Shifting-based dwMRI Super-resolution . . . . . . . . . . . . 202.2.1 Translational Field of View Shifting . . . . . . . . . . 212.2.2 Orientational Field of View Shifting . . . . . . . . . . 222.3 Model-based dwMRI Super-resolution . . . . . . . . . . . . 242.3.1 Super-resolved Diffusion Tensor . . . . . . . . . . . . 252.3.2 Super-resolved Spherical Deconvolution . . . . . . . 263 Proposed Framework . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Processing Pipeline . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Diffusion Shell Clustering . . . . . . . . . . . . . . . . 303.2.2 Training Matrix Preconditioning . . . . . . . . . . . . 363.2.3 Joint-dictionary Learning . . . . . . . . . . . . . . . . 474 Validation Methodology . . . . . . . . . . . . . . . . . . . . . . . 574.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Functional Connectivity Estimation . . . . . . . . . . . . . . 584.3 Structural Connectivity Estimation . . . . . . . . . . . . . . . 595 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2 Quantitative Performance . . . . . . . . . . . . . . . . . . . . 675.3 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . 766 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Thesis Contributions and Future Work . . . . . . . . . . . . 83Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85viiList of Figures1.1 Basic elements of NMR experiment . . . . . . . . . . . . . . . 41.2 Spin alignment in external B0 field. . . . . . . . . . . . . . . . 62.1 Spatial and orientational field-of-view shifts . . . . . . . . . 233.1 An overview of the proposed processing pipeline. . . . . . . 293.2 Illustration of diffusion shells . . . . . . . . . . . . . . . . . . 323.3 Shell clustering correlation matrix. Volumetric patch dimen-sion: d = 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Shell clustering correlation matrix. Volumetric patch dimen-sion: d = 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 Shell clustering correlation matrix. Volumetric patch dimen-sion: d = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6 Effect of volumetric patch dimension on correlation matrixrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7 Sample diffusion clustering result. . . . . . . . . . . . . . . . 413.8 Comparison of condition number of sparse coding matrix. . 443.9 The proposed training matrix preconditioning process. . . . 453.10 Local structure detection using non-linear approximation . . 483.11 Result of non-linear thresholding . . . . . . . . . . . . . . . . 493.12 Binary thresholding for atom initialization . . . . . . . . . . 503.13 Coding matrix condition numbers after preconditioning . . 51viiiList of Figures3.14 Effect of dictionary optimization weight and size on recon-struction quality . . . . . . . . . . . . . . . . . . . . . . . . . . 553.15 Effect of dictionary atom size on reconstruction quality . . . 564.1 Comparison of four common SC metrics in terms of SC-FC(resting state fMRI) correlation for 38 subjects from the Hu-man Connectome Project. . . . . . . . . . . . . . . . . . . . . 634.2 Comparison of four common SC metrics in terms of SC-FC(task-fMRI) correlation for 38 subjects from the Human Con-nectome Project. The shaded bands represent the standarddeviation of SC-FC across 7 different tasks. . . . . . . . . . . 644.3 Parcels with highest connectivity to posterior cingulate cor-tex as obtained from: (a) Functional connectivity (FC), (b)normalized fiber count, and (c) average FA. Note how the ar-rangement of connected parcels obtained by normalized fibercount has a better resemblance to the parcels obtained by FC,compared to average FA. . . . . . . . . . . . . . . . . . . . . . 655.1 Relative reconstruction error η for HCP dataset [91] test sub-jects. The vertical bars show the standard deviation of erroracross diffusion directions at each shell. The vertical bars forthe b0 volume show the standard deviation of error acrossrepeated b0 acquisitions. . . . . . . . . . . . . . . . . . . . . . 685.2 Relative reconstruction error η for Kirby dataset [51] test sub-jects using dictionaries trained on HCP dataset [91]. The ver-tical bars show the standard deviation of error across diffu-sion directions. . . . . . . . . . . . . . . . . . . . . . . . . . . 71ixList of Figures5.3 Relative reconstruction error η for Kirby dataset [51] test sub-jects using dictionaries trained on the remaining subjects inthe same dataset. The vertical bars show the standard devia-tion of error across diffusion directions. . . . . . . . . . . . . 735.4 Relative reconstruction error η for Kirby dataset [51] test sub-jects, trained on the remaining subjects in the dataset. Theselection of test versus training subjects in this figure is theopposite of Figure 5.3. . . . . . . . . . . . . . . . . . . . . . . 745.5 Quantitative SC-FC performance of resolution enhancement 775.6 Quantitative SC-FC performance of resolution enhancement 785.7 Quantitative SC-FC performance of resolution enhancement 805.8 Qualitative comparison, raw diffusion images . . . . . . . . . 81xList of AcronymsAP Affinity propagationDSI Diffusion spectrum imagingDTI Diffusion tensor imagingDW-MRI Diffusion-weighted Magnetic Resonance ImagingEPI Echo planar imagingFA Fractional anisotropyFC Functional connectivityfMRI Functional magnetic resonance imagingFOV Field of viewGFA Generalized fractional anisotropyHARDI High angular resolution diffusion imagingHCP Human Connectome Project datasetk-space Space of spatial positionMRI Magnetic resonance imagingMSE Mean-squared ErrorNLM Non-local meansxiList of AcronymsNMR Nuclear magnetic resonanceODF Orientation distribution functionPGSE Pulsed Gradient Spin EchoPDF Probability density functionQBI Q-ball imagingq-space Space of spin displacementROI Region of interestSC Structural connectivitySNR Signal to noise ratioxiiAcknowledgementsI would like to express my gratitude and appreciation to my supervisor Prof.Rafeef Abugharbieh for her continuous support and guidance. Her helpand advice was key to my success.I would also like to thank all current and previous members of the Biomed-ical Signal and Image Computing Laboratory (BiSICL) at the University ofBritish Columbia for their companionship and support.I also wish to thank Dr. Pierrick Coupe´ from the Laboratoire Borde-lais de Recherche en Informatique, France, for running comparative exper-iments using their CLASR method.xiiiDedication... to my familyxivChapter 1Introduction1.1 Motivation and Problem StatementDiffusion-weighted magnetic resonance imaging (dwMRI) is an increas-ingly common approach of performing brain imaging, with many appli-cations in both research and clinical practice. For instance, dwMRI hasbeen utilized in the assessment and therapy planning of brain tumors [30,58], in the neurological analysis and modeling of schizophrenia [6, 23] andAlzheimer’s disease [56, 77], for the prognosis and treatment monitoringof multiple sclerosis [25, 40], the diagnosis and abnormalities detection intraumatic brain injuries [18, 53], as well as the assessment and planning ofsurgical interventions in epilepsy [33, 96], among many other applications.A major motivation behind the increasingly common utilization of dwMRIis that it provides powerful capabilities for non-invasive imaging of neuralstructures in the brain. The accurate estimation of these structures enablesa more precise understanding of the structural connectivity in the brain.However, the accuracy of estimating these neural structures is often ham-pered by the inherently low resolution of dwMRI. A single pixel (or voxel,for 3D data) can therefore contain many distinct fibers with differing orien-tations, especially in the commonly used diffusion tensor imaging scheme(DTI). At such locations, the orientation typically becomes ambiguous, whichleads to erroneous information about brain structure.Therefore, increasing the resolution of dwMRI data holds great promise11.2. Magnetic Resonance Imaging of the Braintowards more accurate delineation of fibers. Accordingly, there has been anumber of modified dwMRI imaging approaches aiming for increased res-olution, which will be explored in the next chapter. However, they tend tohave practical limitations such as reduced image quality and a long imagingtime. Such limitations motivate the search for another approach for increas-ing resolution. This work will present an alternative approach of achievingthis goal.1.2 Magnetic Resonance Imaging of the BrainDiffusion-weighted magnetic resonance imaging (dwMRI) is one of the sub-categories of magnetic resonance imaging (MRI). It is an imaging techniquethat uses water diffusion strength as a contrast in MRI images. As such,a research investigation involving dwMRI imaging would benefit from anoverview of the underlying principles of MRI imaging.MRI is an imaging method for creating an image of magnetic propertiesof the nuclei of objects being imaged. More specifically, MRI is based onthe physical phenomenon of nuclear magnetic resonance (NMR), which de-scribes the interaction of external electromagnetic radiation with nuclei ina magnetic field. It is this phenomenon that makes MRI imaging possible,via external electromagnetic radiation probing of the nuclei of an imagedobject.The interaction of a nucleus with external fields depends on the spin ofthe nucleus. The spin is a quantum mechanical measure of angular momen-tum. The value of a quantum mechanical spin depends on the number ofprotons and neutrons in the nucleus. Accordingly, each atom and isotope(atom variants having different number of neutrons) has a particular spinvalue.The quantum mechanical spin of a nucleus is a major determinant of21.2. Magnetic Resonance Imaging of the Brainwhether or not a material composed of that nucleus can be imaged usingMRI. A nucleus that has a spin value of zero is not affected by magneticfields and can not be imaged in MRI. This is illustrated in Figure 1.1. Anucleus has a zero spin when the number of protons and neutrons are botheven numbers.In order to be able to detect a nucleus using MRI, it should have a non-zero integer or half-integer quantum mechanical spin, which is the case forodd values of number of protons or neutrons. Fortunately, a large number ofbiological tissues are composed of materials whose spin values are integeror half-integer. In practice, almost all medical MRI imaging is based on thehydrogen nucleus because of the large proportion of water in body tissuesand the fact that hydrogen’s spin value is half-integer.Another property of the nucleus that affects its interaction with externalfields is its magnetic moment µ. Like the quantum spin, the magnetic momentalso depends on the number of protons and neutrons in the nucleus. Assuch, each atom or isotope has a magnetic moment value. For the commonlyused hydrogen nucleus, the magnetic moment is µ = 2.79N-m/T. The valueof the moment indicates the amount of torque a nucleus will experiencewhen a force is exerted on it by an external field.The magnetic moment is not a scalar value, but rather a vector quan-tity. The orientation of the vector is aligned with the axis of rotation of thenucleus. Most objects of medical interest are relatively large and macro-scale, containing billions of nuclei. Therefore, it is more practical to definethe net magnetization vector as the combination of the individual magneticmoments in the nuclei.In a tissue in its normal state (i.e. without any external excitations), theindividual magnetic moments are randomly distributed due to the randomlocations and orientations of nuclei. Therefore, the net magnetization vectoris practically zero. On the other hand, when a tissue is placed in an exter-31.2. Magnetic Resonance Imaging of the Brain =   Nucleus with        integer       spin numberObject beingimaged =   Nucleus with        half-integer       spin numberExternalElectromagneticRadiationDetectedMRI SignalNone =   Nucleus with        zero       spin numberStatic magnetic eldFigure 1.1: An illustration of the basic elements of an NMR experiment, andthe required properties of the imaged object.41.2. Magnetic Resonance Imaging of the Brainnal magnetic field, a much more interesting behavior arises. The magneticmoments of the nuclei start to align with the external magnetic field. Thisis illustrated in Figure 1.2.However, the alignment of a given nucleus occurs in one of two oppo-site directions: alignment that is parallel to the external field, and anotherthat is anti-parallel to the field. That is, the magnetic moment can be eitherone of two opposite vectors. The number of nuclei which are oriented ineach of the two opposite directions is not equal. The parallel direction is ina lower energy state compared to the anti-parallel direction. As such, thereare slightly more nuclei assuming the parallel direction. Fortunately, eventhis slight difference results in a detectable net magnetic moment, especiallyconsidering the fact that the scale of interest in imaged tissues contains hun-dreds of billions of nuclei.When the magnetic moments of the nuclei begin to align with the exter-nal field, a rotational force is at work during this alignment. Due to angularmomentum, this rotational force causes the magnetic moment to resonateback and forth around the new external field axis. The angular frequencyof this periodic rotation is given by Larmor’s equation, one of the cornerstoneequations in magnetic resonance imagingω = γB0 (1.1)where ω is the Larmor frequency: the angular frequency of resonance (alsoreferred to as frequency of precession) in units of radians, B0 is the strengthof the external magnetic field in units of Tesla (T), and γ is a material prop-erty constant referred to as the gyromagnetic ratio. For hydrogen, γ = 42.57MHz/T.As long as the external B0 field is on, the precession continues with-out dampening as there are no frictional forces. Accordingly, the externalfield B0 results in a steady state condition where the magnetic moments51.2. Magnetic Resonance Imaging of the BrainObject under external static magnetic eld B0Object without any external excitationssomeanti-parallelnucleiNetmagnetizationvector = 0Netmagnetizationvector not zero =     Magnetic moment of an individual nucleusFigure 1.2: Alignment of magnetic moments under an external static mag-netic filed B0. Parallel nuclei are more common and favorable due to lowerenergy state compared to anti-parallel.61.2. Magnetic Resonance Imaging of the Brainare of known precession frequency. Even though the individual momentsare steadily rotating about the z-axis (the axis of B0), the net magnetizationvector has a steady orientation parallel to the z-axis, and we writeMz =M0.However, because the individual precessing moments are not in-phase, thetransverse component Mx,y is zero.Therefore, the net magnetization vector M when B0 is applied is givenby the steady time-invariant expressionM =M0zˆ (1.2)However, in order to induce a signal in a detector coil, a time-varying mag-netic field is necessary (based on Faraday’s induction). Accordingly, theindividual magnetic moments need to be perturbed and the resultant per-turbed net magnetization creates a signal in the detector.The perturbation of the magnetic moments is achieved through an ad-ditional external field that is time-varying, referred to as the B1 or RF pulse.Because the magnetic moments are precessing at a frequency of ω, the ap-plied B1 pulse also needs to have a frequency of ω in order to magneticallycouple (i.e. to resonate) with the precessing moments. The direction of theB1 pulse is perpendicular to B0. Assuming that B0 is oriented along the z-axis and B1 along the x-axis, the result of applying the pulse is to tip theorientation of the net magnetization from the longitudinal z-axis towardsthe transverse x,y-plane. The net change in orientation is referred to as thetip angle.The tip angle depends on the magnitude and duration of the RF pulse.Longer and more intense pulses yield larger tip angles. The intensity is typ-ically much smaller than the staticB0, because magnetic resonance couplingallows for cumulative perturbation, and therefore a small pulse applied fora given amount of time can accumulate and yield sufficient perturbation.Under this process of tipping the net magnetization vector, if a detector71.2. Magnetic Resonance Imaging of the Braincoil is positioned such that its axis lies in the transverse (x,y) plane, an ACvoltage signal oscillating at the Larmor frequency is generated in the coilbased on Faraday’s induction. This detected signal is the MR signal. Thisdetection provides an external readout of the changes in net magnetizationoccurring within tissue.However, the detected MR signal corresponds to all precessing nuclei.In other words, the detected signal is a combination of all magnetic mo-ments from all nuclei regardless of their location. So far, the process doesnot specify the location from which the detected signal is generated. Thisissue must be addressed because, after all, the association of an MR signalwith a location is a necessary information in order to form an image.The main approach of establishing spatial information in an MR signalis the use of magnetic field gradients. Such field gradients have spatiallyvarying intensities within the region of interest in the imaged object. Thegradient is typically linear with relatively small intensity variation in unitsof milliTesla per meter.Applying a gradient to a region of interest results in having a linearlyvarying B0 within the region. The result of a linearly varying B0 is a lin-early varying precession frequency ω. That is, each location is now associ-ated with a particular frequency. This is in contrast to the initial case with-out gradients, where all locations had the same frequency and therefore nospatial information was available.However, it is important to note that, so far, the applied gradient onlyencodes spatial information along one direction only. That is, if we are toimagine the region of interest to be an image slice, the applied gradientso far only provides information about which image column the signal iscoming from. Therefore, an additional gradient is required in order to fullyencode the spatial information in an image slice.While the first spatial direction was determined using frequency-encoding,81.2. Magnetic Resonance Imaging of the Braininformation about the second direction is defined through phase-encodingdirection. This is exactly similar to frequency-encoding, in the sense that itis also based on a spatially varying field. However, it is applied in a seconddirection that is perpendicular to the frequency-encoding direction. The re-sult of applying the phase-encoding gradient is that the precession phasesare now linearly varying along the new direction. The effect of the gradientis to encode the phase information in the detected signal, hence the name“phase-encoding gradient”.With the frequency-encoding direction dividing the image into columnsand the phase-encoding direction dividing the image into rows, the spa-tial unit at the intersection of both gradients is referred to as a voxel. Thex-dimension of the voxel is specified from the frequency information andits y-dimension is specified from the phase information. The intensity ofthe voxel is determined through Fourier transform. After the applicationof both gradients, the detected MR signal in the coil is composed of dif-ferent frequencies at different phases. An image is then formed by takingthe Fourier transform of the detected MR signal, which result in phase andmagnitude information of the various frequency components in the signal.So far, the applied sequence of gradients can provide information aboutan image slice. However, no aspect of the process determines which sliceshould be imaged within a region of interest. This part of the imaging pro-cess, which enables the acquisition of different slices that can span a 3Dvolume, is referred to as the slice-select direction encoding.A common method of encoding the slice-select direction involves a twostep process that applies an RF pulse on top of a linear gradient. First, alinear gradient is created along the slice-select direction (i.e. the directionperpendicular to the slices). Then, instead of applying an RF pulse that res-onates with the precessing moments produced by B0 as before, the RF pulseis designed to have frequency components that correspond to only a small91.3. Diffusion Modelingrange that lies within the applied gradient. The center frequency of the RFpulse and the bandwidth of the frequency range it contains determine theposition and thickness of the selected slice, respectively.Therefore, the creation of an MRI volume involves the application of allthree gradient directions. The sequence in which the gradients and pulsesare applied is referred to as the pulse sequence, and the process and stepsthat have been described in this section represent the main parts of a pulsesequence.1.3 Diffusion ModelingThe purpose of diffusion modeling is to fit the DW-MRI dataset into a modelthat enables extraction of various useful information about the microstruc-ture of the brain. The diffusivity information in a DW-MRI dataset is mea-sured using the pulse sequence described in the previous section.One of the earliest and most common approaches of modeling diffusioninformation is diffusion tensor imaging (DTI) [9]. DTI uses at least six dif-fusion weighted volumes to create a tensor that describes the diffusion ofwater within each voxel.Accordingly, the cornerstone element in DTI is the diffusion tensor. Thiscan be expressed in a matrix form as followsD =Dxx Dxy DxzDyx Dyy DyzDzx Dzy Dzz (1.3)which, due to the symmetry of diffusion (i.e. equality between both sidesthe diagonal; Dxy = Dyx, etc), can be fully determined using only six of thequantities in the matrix. The diagonal elements represent the diffusivities ofthe three axes. The off-diagonal elements are the correlation (or covariance)between any two axes.101.3. Diffusion ModelingThus, D represents a diffusion covariance matrix. The eigenvectors ofDrepresent the axes of an ellipsoid representing diffusion in the voxel. Theeigenvalues (λ1, λ2, λ3) ofD represent the three diffusivities along the threeaxes of the ellipsoid. The diffusion tensor orientation is taken to be the ori-entation of the principal eigenvector, which is the eigenvector that corre-sponds to the largest eigenvalue.Another important property of the diffusion tensor that is based on eigen-values is the fractional anisotropy (FA) [19], defined as followsFA =√32((λ1 −ADC)2 + (λ2 −ADC)2 + (λ3 −ADC)2λ21 +λ22 +λ23)(1.4)where ADC is the apparent diffusion coefficient (also referred to as mean dif-fusivity), and is proportial to the tensor trace: ADC = (Dxx +Dyy +Dzz)/3,which is equivalent to the average of eigenvalues. ADC is also sometimesused to refer to each direction separately, in which case the ADCs corre-spond to the eigenvalues.FA is a normalized quantitative indicator of the degree of anisotropy in avoxel. A near-zero FA value indicates an isotropic diffusion within the voxel.That is, the diffusivities along all three orthogonal axes are similar. Thisprovides an indication that water is diffusing equally along all directions,and therefore the local tissue structure is unrestrictive and non-directional.On the other hand, FA values closer to one indicate a highly anisotropicdiffusion. This is a diffusion that is very large along one direction only,and very small in the other two directions. As such, high FA results froma voxel which has a local tissue structure that is restricting water diffusionalong one direction.111.4. Limitations and Alternative Imaging Schemes1.4 Limitations and Alternative Imaging SchemesWhile DTI offers useful insight into brain structure through the use of ten-sors, it has a number of limitations. The accurate construction of fiber tractsis important in gaining insights into brain function since fiber tracts actas the infrastructure enabling communication between brain regions [39].However, accuracy of the reconstructed fiber tracts is often hampered bythe often low resolution of DTI data. A voxel can thus comprise several dis-tinct fiber bundles with differing orientations, leading to partial volume effect[4]. At such locations, diffusion information typically becomes ambiguous,and tractography is often falsely terminated.Fortunately, the dwMRI acquisition process has a multitude of param-eters that affect the quality of the final volume data. These include factorssuch as magnetic field strength, voxel size, diffusion orientations, amongother factors. This has resulted in a significant body of research devotedto optimizing such factors with the aim of achieving more accurate fiberestimation.The following sections will review some of the key alternative imagingschemes (mainly Diffusion Spectrum Imaging - DSI, and High Angular Res-olution Imaging - HARDI) that are addressing the limitations of DTI.1.4.1 Diffusion Spectrum ImagingOne approach that has been proposed to capture more detailed diffusioninformation is diffusion spectrum imaging (DSI) [93]. DSI acquires full dif-fusion information in a substantial region of the q-space, which is the 3Dspace representing the extent and orientation of spin diffusions along thethree axes of motion (x,y,z). This is in contrast with the more common DTIimaging scheme, which does not aim to fully sample the q-space.Accordingly, in order to sample a large region of the q-space, the im-121.4. Limitations and Alternative Imaging Schemesage acquisition in DSI is different from DTI. A major difference between thetwo techniques is that DSI adds additional gradients during imaging. Thatis, in addition to the typical slice selection, frequency-encoding, and phase-encoding gradients, three new gradients are added to fully encode diffusionin a 3D Cartesian grid comprising many diffusion magnitudes and orienta-tions. In other words, DSI is essentially a fully 6D imaging modality, whichsamples both the k-space (spatial position sampling) and q-space (spin dis-placement sampling) simultaneously.As a result of the additional three dimensions of q-space, DSI providesan explicit representation of diffusion. In other words, DSI does not requirediffusion modeling. This is because at each voxel, the q-space volume hasbeen shown [93] to be a direct representation (through a Fourier transform)of the spin-displacement PDF (probability density function) at each voxel.This is in contrast with DTI, which can not be performed without modelingdiffusion using tensors.Experimental studies comparing DSI relative to DTI have shown im-proved detection of fiber crossings in human and rat brains [55, 94]. In anumber of ROIs, the fibers reconstructed from DSI have shown better re-semblance to known anatomical pathways in the brain.Despite the improvement in fiber crossing detection, DSI has a signifi-cant drawback. The sampling of q-space requires a significant amount ofdata, with typically a 3D grid of 500 q-values used in scanning, represent-ing DW gradients over a multitude of orientations and magnitudes. Thisresults in a much longer acquisition time. While a typical DTI acquisitiontakes about 3-5 minutes, DSI acquisition time is usually around 40 minutes[16, 57], and is typically performed with a smaller number of repetitionscompared to DTI in order to reduce scan time, and therefore the SNR isgenerally lower. Accordingly, this is a significant drawback that preventsDSI from being a practical imaging scheme.131.4. Limitations and Alternative Imaging Schemes1.4.2 High Angular Resolution Diffusion ImagingIn an effort to reduce the significant burden of sampling a 3D q-space grid inDSI, an alternative approach that uses only angular samples was proposed[5, 27, 88]. This approach, which came to be known as high angular resolutiondiffusion-weighted imaging (HARDI), only samples a spherical subset of thediffusion space, which is referred to as a diffusion shell. That is, the acqui-sition is simplified to a single (or, sometimes, a few [1, 21]) diffusion shellsinstead of fully sampling the entire 3D diffusion space as in DSI.By restricting the acquisition to a high-resolution spherical shell insteadof a full Cartesian grid, HARDI simplifies the acquisition while maintainingsome of the information provided by DSI. This simplification is further sup-ported by the fact that the typical subsequent processing pipeline is basedon tracking the locally estimated orientations. Accordingly, HARDI’s sim-plification strategy of focusing more on angular diffusion data relative toradial data can be argued to be an efficient strategy.The choice of the magnitude(s) of shells is a trade off between the ben-efits of HARDI (few shells and orientations) and DSI (full Cartesian grid).However, it is known that higher magnitudes and more orientations yieldbetter performance [74].However, because full information on the q-space is no longer available,HARDI requires a diffusion modeling step, which was not required in thecase of DSI. Several modeling approaches have been proposed in literature.However, one of the most common approaches of reconstructing diffusionODF from HARDI data is Q-ball Imaging (QBI) [87]. An attractive featureof QBI is that it is a model-free method. In QBI, the measured sphericaldiffusion signal is directly used to reconstruct the diffusion ODF. This isperformed through the Funk-Radon transform, which extends the planarRadon transform to spherical tomographic reconstruction. QBI has been141.5. Thesis Objectives and Proposed Approachshown to yield better reconstruction of fiber crossings compared to DTI [87,89].Despite the improved reconstruction afforded by QBI, the technique hassome drawbacks. QBI has been shown to require high diffusion-weightingfactors (b-values) in order to be able to resolve fiber crossings [48, 86], withsuggested values of about b = 3000 − 4000smm−2. In comparison, typicalDTI values are around b ∼ 1000smm−2. Due to the higher b-values in QBIand the denser gradient directions sampling, the resultant SNR can be verylow [62]. Furthermore, the higher number of gradient orientations (∼ 60to few hundreds) results in a significant acquisition time of around ∼ 30minutes to a few hours [48], which hampers practical clinical use.1.5 Thesis Objectives and Proposed ApproachThe aim of this thesis is to propose an approach for increasing the spa-tial resolution of brain dwMRI data. Increasing the resolution will helpin enabling more detailed extraction of information from dwMRI data, andtherefore help improve the estimation of brain fiber structures. In this sec-tion, we describe the main objectives that this work is aiming to address.Recently, a powerful approach of enhancing the resolution of images us-ing dictionary learning has been shown to yield good performance in nat-ural images [46, 97]. Dictionary learning is a process in which a signal orimage is represented using as few learned basis functions as possible. Thedetails of this process are discussed at length in Chapter 3.In the dictionary learning based super-resolution process, two dictio-naries are created: a high-resolution dictionary and a corresponding low-resolution dictionary. Through these joint dictionaries, a high-resolutionimage can be generated from a new unseen low-resolution image.In a similar fashion, we propose a processing pipeline in this thesis that151.5. Thesis Objectives and Proposed Approachis built on top of the same joint-dictionary learning approach and extend itto multi-shell dwMRI data. We chose to create a pipeline that adopts thisapproach in order to be able to create a resolution enhancement of dwMRIwithout resorting to acquisition modifications. The details of the proposedprocessing pipeline are described in Chapter 3.An objective of the proposed approach is to not require modifications tothe acquisition scheme. This stems from the fact that most of the proposedalternatives to DTI require acquisition methods which require a very longimaging time to obtain acceptable image quality, hampering its utilizationin routine clinical use as explained earlier in section 1.4.To this end, we aim to propose an approach that should not require mod-ifications to the dwMRI acquisition process. This ensures that the proposedprocessing pipeline maintains its applicability in a wide variety of clinicaland research settings. This also allows the proposed approach to be usedon legacy data in pre-existing clinical or research data sets. These objectivescan not be attained using other approaches that are based on acquisitionmodificationsAnother objective of the proposed approach is to be independent fromthe choice of diffusion modeling. In other words, the proposed approach isnot attempting to change or improve the diffusion model. This ensures thatthe proposed approach can be used with whatever diffusion model used invarious acquisitions. It is also not a tractography method, but rather can beused as an input to tractography. This makes it suitable for use with anytractography method preferred by the end user.Furthermore, as a result of the preceding features, the proposed ap-proach should be modular and flexible. In other words, it can be used withany other method that aims to enhance structural information. For instance,it can be used as an additional step after other approaches that use acquisi-tion modifications.161.6. Thesis Organization1.6 Thesis OrganizationThe rest of this thesis is organized as follows. In Chapter 2, we examine keyrelated literature aiming to achieve similar objectives of increasing dwMRIdata resolution. We categorize them into main categories and explore theadvantages and disadvantages of each category.In Chapter 3, the proposed framework is described. We examine in de-tails the rationale and design of each block in the proposed framework, anddetermine the various parameters of the methods in order to yield goodperformance. We begin by describing the preconditioning steps employedto improve the condition of the coding matrix, after which we determine theparameters of the preconditioning process, followed by determining the pa-rameters of dictionary learning.In Chapter 4, we describe the proposed validation methodology. Chap-ter 5 presents quantitative and qualitative results examining the performanceof the proposed framework, and also examines the dependency of perfor-mance on variations in datasets, as well as comparisons against other meth-ods.1.7 Thesis ContributionsThe following is a brief overview of the contributions of this thesis. Thiswill be explained in detail in the following chapters.We propose a dwMRI data processing pipeline (built on a dictionarylearning approach) that enhances the resolution of dwMRI after the datahas been acquired. The pipeline does not require modifications to the dwMRIacquisition process, and therefore is more practical in clinical conditionswhere simple acquisition methods are typically used.Due to the absence of acquisition requirements, the pipeline can be read-171.7. Thesis Contributionsily used with existing legacy databases of dwMRI data. This can be usefulwhen trying to utilize or reuse datasets which has been acquired with olderacquisition technologies. Absence of acquisition requirements also makesit usable with different imaging schemes, such as DTI or HARDI (single,and multiple-shell). Some of the existing super-resolution approaches arespecific to HARDI or DTI.We also note that the proposed pipeline does not require repeated dwMRIacquisitions, unlike classical shifting-based super-resolution methods. An-other contribution is that the pipeline performs a resolution enhancementdirectly on raw dwMRI data, resulting in a model-free pipeline.This lack of required modeling is useful because it does not restrict theend user to a particular model, which gives the freedom of using differ-ent diffusion models in different research or clinical situations, depend-ing on the specific research conditions or objectives. Finally, we note thatthe model-independence also allows for cumulative enhancements, in whichany other diffusion-specific or tractography-specific enhancements can bereadily applied on top of the pipeline.18Chapter 2Related Literature2.1 OverviewObtaining structural information about the brain in a non-invasive fashionis one of the major applications of brain dwMRI, among other applicationssuch as using diffusion anisotropy to diagnose brain infarction [95] as wellas brain development and aging [72].This structural information takes the form of a set of connected fibersegments (referred to as tracts) that are reconstructed from dwMRI. Thefiber tracts in turn give raise to a network of inter-connected spatial regionsin the brain.The cornerstone of obtaining structural reconstruction is to perform atractography on the dwMRI data. In its most common form, tractographygenerates a set of fiber tracts by connecting the largest eigenvectors of thediffusion tensor at each voxel, starting from multiple seed points. This ap-proach is referred to as deterministic streamline tractography. Many other trac-tography approaches have been proposed, such as methods based on globally-consistent reconstructions as well as probabilistic reconstructions.The accurate construction of fiber tracts is important in gaining insightsinto brain function since fiber tracts act as the infrastructure enabling com-munication between brain regions [39]. However, accuracy of the recon-structed fiber tracts is often hampered by the inherently low resolution ofdwMRI data. Currently achievable spatial dwMRI resolution is around 2192.2. Shifting-based dwMRI Super-resolutionmm3, while the actual neuronal fiber diameter is on the order of 1 µm. Avoxel can thus comprise several distinct fiber bundles with differing orien-tations, leading to partial volume effect [4]. At such locations, diffusion in-formation typically becomes ambiguous, and tractography is often falselyterminated.Several methods have been explored in literature to address the partial-volume effect. The common goal of these methods is to reduce the ambigu-ity present in the imaged voxels. While there is a wide variety of approachesand strategies proposed in literature, we grouped them into two major cat-egories: shifting-based dwMRI super-resolution, and model-based dwMRIsuper-resolution.At this point it is important to note that the enhancement of structuralreconstruction can also be achieved via improvements in tractography al-gorithms. While research in this field is very active, the goal of the currentinvestigation is to explore structural enhancement approaches that are in-dependent of tractography. The rationale is that such approaches are bene-ficial because any structural enhancements they offer may then be accumu-lated in addition to improvements in tractography algorithms.2.2 Shifting-based dwMRI Super-resolutionOne of the earlier attempts to improve resolution of dwMRI was throughthe use of multiple shifted acquisition, in which the imaging process ac-quires multiple volumes, each representing a slightly shifted region of phys-ical space. This is followed by a subsequent super-resolution reconstructionmethod [10, 41]. The main difference between this approach and more tradi-tional Q-space imaging enhancements is that the latter aims to resolve moredetails through a modified imaging acquisition, while super-resolution en-tails a post-imaging processing approach.202.2. Shifting-based dwMRI Super-resolutionAs implied by the name, super-resolution is a method of increasing theresolution of an image. The central concept of super-resolution is the re-construction of high-resolution details from multiple lower-resolution im-age acquisitions. The main benefit of super-resolution is providing a higherresolution image without requiring modifications to the imaging hardwareor optics. This can be beneficial in situations where such hardware modifi-cations may be difficult or impractical.The rationale for super-resolution is based on an information fusion pro-cess. The super-resolution process assumes that, if multiple low-resolutionimages of a scene are available, and if such images were related to each otherby relatively small shifts, then a higher resolution image can be obtained byfusing the information from the set of shifted low resolution images into asingle high resolution image [90].2.2.1 Translational Field of View ShiftingOne of the first of investigations in translational-shifting super-resolution ofdwMRI was by Peled and Yeshurun [71]. In this work, super-resolution re-construction was directly applied to multiple acquisitions of dwMRI data.Specifically, each slice was acquired eight times, with each acquisition at asubvoxel spatial shift relative to the first acquisition. This was achieved bychanging the FOV (field of view) in the frequency-encoding and the phase-encoding directions (i.e. within the in-plane directions in a multi-slice ac-quisition).However, this approach has been proven later to have a fundamentalproblem [75]. This is because spatial sub-voxel shifts in the FOV in the in-plane directions simply correspond to linear phase modulations in the k-space. This means that, the multiple acquisitions correspond to the samek-space points, and no new points or new information is acquired. Anyobservable improvement may therefore be attributed to increased SNR be-212.2. Shifting-based dwMRI Super-resolutioncause of the use of more averaging acquisitions, which is already a standardpractice in most acquisitions.Accordingly, subsequent research efforts have focused on super-resolvingthe through-plane (i.e. the slice-select) direction of the acquisition. Oneapproach [32] performs sub-voxel shifts in the FOV along the slice-selectdirection.2.2.2 Orientational Field of View ShiftingAn alternative approach to translational sub-voxel shifting has been pro-posed based on orientational FOV shifting [76, 79]. For each low resolutionacquisition, the FOV was rotated around the frequency-encoding direction,resulting in a series of rotations (a total of six equidistant orientations wereacquired, 30o apart).The orientational FOV shifting approach has been shown to yield betterresolution enhancement relative to the sub-voxel spatial shift [80] for thesame number of low-resolution acquisitions. The methods adopting thisapproach are motivated by the rationale that orientational shifting wouldarguably result in a more efficient sampling of the k-space for the same num-ber of shifting. This has not been explored quantitatively in literature. Thedifference between the two FOV shifting approaches is illustrated in Figure2.1.Despite the reported resolution enhancements using shift-based super-resolution, the process has a number of limitations. First of all, it is very im-portant to note that the vast majority of super-resolution literature in MRIhave focused on standard (non-diffusion weighted) images. Its use in DTI(or other Q-space based modalities for that matter) received little attention.In most super-resolution studies, a total of around 6-8 FOV shifts were ac-quired (whether spatial or rotational), which were based on empirical ex-amination and no theoretical limit has been investigated. Strictly speaking,222.2. Shifting-based dwMRI Super-resolutionFigure 2.1: Illustration of the spatial sub-voxel FOV shifting super-resolution in comparison to the orientational shifting super-resolution.232.3. Model-based dwMRI Super-resolutionthese shifted acquisitions are in-addition to the few additional repeated ac-quisitions performed to average the signal and boost the SNR. Accordingly,around 10 acquisitions can be expected to be performed for each gradientdirection in a DTI acquisition. And with the recommended optimal num-ber of gradients being around 30 [43], this results in the requirement of hav-ing around 300 acquisitions for super-resolution. Recently, one of the firstDTI studies [73] used only 12 gradients (total acquisition time of around 9minutes) with no repetitions, which is much lower than the recommendednumber of gradients. More research needs to be done in investigating theuse of shift-based super-resolution for DTI.Another limitation of shifting-based super-resolution approaches is thatit is not possible to super-resolve information within the between-planes di-rection, thus limiting any possible resolution enhancement to the two otherdimensions only, due to limitations imposed by the encoding scheme [75].2.3 Model-based dwMRI Super-resolutionAn alternative class of super-resolution approaches may be identified as in-volving the construction of more complex models of dwMRI data. Specifi-cally, methods belonging to this approach focus on increasing the resolveddetails in various diffusion models. This then enables exploring the recon-structed neural structures at a greater resolution.This approach is different from the methods described in the previoussection in the sense that it does not require multiple shifted acquisitions.However, due to the often mutual dependency between model and acqui-sition, some of these methods may also include certain acquisition require-ments. This section presents a review of key literature adopting this model-based super-resolution approach.242.3. Model-based dwMRI Super-resolution2.3.1 Super-resolved Diffusion TensorA model-based approach aiming to super-resolve diffusion tensors has beenproposed [35, 66]. This approach essentially proposes an alternative diffu-sion tensor reconstruction method such that the diffusion tensors are con-structed at a higher resolution grid.The basic framework is based on expressing the diffusion signal in termsof a diffusion tensor at a higher resolution. This is in contrast with the stan-dard method of diffusion tensor construction, where the diffusion signal isexpressed in terms of a diffusion tensor at the same resolution (i.e. samegrid as dwMRI data). This can be expressed in a generic form as followsSLR(xLRi)= f(DHR(xHRj))(2.1)where SLR is the original (low-resolution) diffusion signal, xLRi representsthe original (low-resolution) grid, f indicates a function of DHR, which isthe diffusion tensor at the higher resolution grid xHRj . Further details onequation parameters can be found in [35].Accordingly, a relation is established between the low (original) resolu-tion dwMRI data and high-resolution tensors. These tensors are then con-structed using an inverse problem approach. An energy function is there-fore created, and then minimized to generate the tensors.While better tensor resolution has been demonstrated [35], a major lim-itation of this approach is that it is inherently restricted to using diffusiontensors as the model, which has been shown to be suboptimal in modelingdiffusion in complex neural fiber structures [8]. Therefore, this approachcan not be used with other diffusion models such as ODF (orientationaldensity function), for instance. Its applicability is limited due to the require-ment of a tensor model.252.3. Model-based dwMRI Super-resolution2.3.2 Super-resolved Spherical DeconvolutionAnother modeling method has been proposed which aims to construct ahigh resolution orientational probability density function (ODF) based on aspherical deconvolution of the diffusion-weighted data [84, 85]. As impliedby the name, the ODF assigns a probability for the existence of a fiber atvarious orientations on a sphere.In this model, the diffusion-weighted signals (as measured on a diffu-sion shell sphere) are modeled as the convolution of a certain transfer func-tion with an ODFS (θ,φ) = F (θ,φ) ∗R (θ) (2.2)where θ,φ are the elevation and azimuthal angles in a sphere and S (θ,φ) isthe measured diffusion signal at a voxel at the orientation (θ,φ). F (θ,φ) isthe desired unknown ODF, and R (θ) is the transfer function. The transferfunction would typically be determined from regions in the data where theODF is anatomically known to be a single coherent orientation of fibers.Therefore, given a diffusion-weighted signal S (θ,φ) and the transferfunction R (θ), a deconvolution process would then enable the estimation ofthe ODF F (θ,φ). Since the functions involved in the convolution are spher-ical, the deconvolution is also performed over a sphere [38].The super-resolution approach used by these methods is based on per-forming a spherical deconvolution to estimate the ODF at resolutions higherthan what is present in the measured data S (θ,φ) [84]. This is achieved byintroducing non-negativity constraints on the ODF, which stems from theanatomical impossibility of having negative fiber density. This has the dualbenefit of reducing background noise which may cause the ODF to takenegative values, and subsequently allows a finer estimation of the ODF dueto the reduced background noise.While this approach has demonstrated higher resolution in the gener-262.3. Model-based dwMRI Super-resolutionated ODFs [84], it is specifically restricted to choosing ODFs as a model.This makes it unusable with the wide variety of other modeling methods.Furthermore, the proposed method requires the use of HARDI data. Assuch, this super-resolution approach can not be used with the more com-mon DTI data, which is also often the type of data of many legacy dwMRIdatabases.27Chapter 3Proposed Framework3.1 OverviewThe proposed framework for enhancing the resolution of dwMRI differsfrom the existing shifting-based and model-based dwMRI super-resolutionapproaches discussed in Chapter 2. In the proposed framework, there is norequirement for a modified acquisition sequence, nor a choice of a particulardiffusion model. The proposed framework makes no assumptions aboutacquisition or modeling.In this chapter, we will describe the framework in detail. The frame-work has two components: a processing pipeline and a validation method-ology. The processing pipeline is described in section 3.2, and the validationmethodology is described in Chapter 4.3.2 Processing PipelineRecently, a powerful approach of super-resolving data using a dictionarylearning approach has been shown to yield good performance in super-resolving natural images [46, 97]. Dictionary learning is a process in which asignal or image is represented using as few basis functions as possible. Thekey concept of dictionary learning is that the basis functions are learnedfrom the data, rather than being constructed from a generating function.In the process of super-resolution via dictionary learning, two dictionar-ies are typically created: a high-resolution dictionary and a corresponding283.2. Processing PipelineFigure 3.1: An overview of the proposed processing pipeline.293.2. Processing Pipelinelow-resolution dictionary. Through these joint dictionaries, a high-resolutionimage can be generated from a new, previously unseen, low-resolution im-age.In a similar fashion, we propose a processing pipeline in this thesis thatis built on top of the same joint-dictionary learning approach and extend itto multi-shell dwMRI data. We chose to create a pipeline that adopts thisapproach in order to be able to create a resolution enhancement of dwMRIwithout resorting to acquisition modifications, which was the case in thealternative super-resolution methods described earlier in Chapter 2.The proposed processing pipeline is shown in Figure 3.1. The processstarts with selecting a dwMRI training dataset. The data is then clusteredinto a single diffusion shell. We recall from Chapter 1 that a diffusion shellis simply a spherical subset of the diffusion space instead of the entire 3Ddiffusion space. This will be discussed in more details in section 3.2.1.The clustering is then followed by a preconditioning process that aimsto improve the condition number of a coding matrix. Finally, the processterminates with a dictionary learning step applied on the dwMRI data.3.2.1 Diffusion Shell ClusteringIn general, dwMRI volumes may be described as residing on diffusion shells.Each diffusion shell describes a certain strength of diffusion weighting. Sub-sequently, each acquired diffusion direction represents a sample on the cor-responding diffusion shell. At the limit, as more directions are acquiredin each shell and as inter-shell gaps are reduced, the acquired data set ap-proaches that of a DSI acquisition. Figure 3.2 shows an illustration of theshell structure in a typical dwMRI dataset.Historically, most acquisitions were typically limited to single shells.This was due to practical limitations in the acquisition hardware and cor-responding software pipeline algorithms. Increasingly, this is being gradu-303.2. Processing Pipelineally replaced by multi-shell acquisitions. This form of acquisition allows forhigher angular contrast between the various diffusion directions [13, 17, 45]and also enables the use of richer diffusion modeling [2, 22, 47].However, the benefits afforded by multiple-shells are offset by the moreelaborate sampling scheme and the resulting substantial increase in the sizeof the dataset. While the design of multi-shell acquisitions is still an activefield of research, some studies [12, 81] have provided suggestions that anoptimal acquisition can be achieved using 3 shells and around 160 - 280total measurements (across all shells).Such multi-shell acquisitions posit a number of difficulties for the pro-posed resolution enhancement framework. First, there is a practicality issueof loading and learning on the large number of volumes in a multi-shell ac-quisition, as can be noted by observing the large number of dots in Figure3.2. For instance, a typical acquisition following the aforementioned scheme(for example, a typical subject data in the Human Connectome Project –HCP) requires around ∼ 8 GB of memory space, not including any othersoftware or system resources requirements. While this is gradually becom-ing a non-issue on some modern computers with ample resources, it maystill present a practical limitation on processing multi-shell acquisitions fora large majority of users (which was the case on the machines used to con-duct this research).Another, more fundamental, issue of training on multi-shell acquisitionsis the increased ill-conditioning of the training matrix. This is due to finerangular sampling in typical multi-shell acquisitions which increases theodds of having highly-similar patches. The resulting ill-conditioned matrixprevents the creation of a joint hi-res/low-res dictionary, which is necessaryfor the proposed resolution enhancement framework, as will be describedin more details in section 3.2.2.We propose to address these issues through a clustering approach. More313.2. Processing Pipeline2y diffusion axis10-1-221x diffusion axis0-1-212-20-1z diffusion axisFigure 3.2: Illustration of the diffusion shells structure in a typical dwMRIdata. Each shell represents a certain strength of diffusion weighting. Eachdot represents a 3D diffusion volume.323.2. Processing Pipelinespecifically, we propose to cluster the measurements across all acquiredshells into a set of representative dwMRI volumes. By design, this proposeddiffusion shell clustering should be able to condense a multi-shell acquisitioninto a relatively smaller and heterogeneous single shell that would still cap-ture the bulk of information present in the multi-shell dataset. While sucha clustering approach, by design, reduces the amount of information, weadopt this approach to reduce the computational load and the ill-condition-ing of the training matrix as explained earlier.There are a number of existing clustering methods that are widely usedto cluster many types of data. Common methods include K-means cluster-ing, K-medoids, and Gaussian mixture models, to name a few. However,most of these clustering methods require identifying the number of clus-ters before hand. Identifying the correct number of clusters remains a chal-lenge, and may be different for different acquisitions. This is especially truefor multi-shell data, where there can be great variability between one acqui-sition and another.At this point, we recall that one of the main objectives of this work isto propose a framework that is independent of the acquisition scheme. Assuch, we aim for the processing pipeline to be able to handle older and com-mon DTI data, as well as various forms of the richer and more recent multi-shell HARDI data.In order to achieve this goal, we propose to utilize affinity propagation(AP) [28] to perform diffusion shell clustering. In AP, the dataset is iter-atively analyzed to generate a set of exemplars: data points that are mostrepresentative of their respective clusters. The affinity propagation processbegins with a similarity matrix, which indicates how well a certain pointserves as an exemplar to other points. The exemplars then minimize thepairwise error or distance between themselves and potential cluster mem-bers. We refer the reader to [28] for more details.333.2. Processing PipelineMost importantly, the final number of clusters is determined automat-ically. In addition, there is no initial set of clusters. Instead, all points arecandidate exemplars. All points are represented as nodes on a network, andmessages are passed between nodes depending on the degree of similaritybetween the pair, denoted by s(i, j). The iterative propagation of such affin-ity messages throughout the network results in a final set of clusters andexemplars. Briefly, the iterations initialize and subsequently update twomatrices: responsibility matrix r(i, j) indicates suitability of j to be an ex-emplar for i, and availability matrix a(i, j) indicates how suitable it is for ito choose j as an exemplar. We refer the reader to [28] for algorithm details.We now show how we will utilize the AP method to implement diffu-sion shell clustering. We begin by constructing a feature vector φm for eachdwMRI volume (regardless of overall shell structure) to be the standard de-viation of DW volumetric patches, as follows:pl,n,m = vec(Pi,j,kVm) (3.1)φm =√√1NN∑i=1(pi,1,m − pi,1,m)2, . . . ,√√1NN∑i=1(pi,n,m − pi,n,m)2 (3.2)where P is defined as an operator extracting isotropic volumetric patchescentered at the voxel (i, j,k),Vm is them-th volume in the acquired data, andpl,n,m is the vectorization of the n-th volumetric patch of them-th volume atthe l-th index.We then build a DW measurements correlation matrixΨ from the fea-ture vectors, as follows:Σi,j = cov(φi ,φj ) (3.3)Ψ = diag(Σ)− 12 Σdiag(Σ)− 12 (3.4)343.2. Processing Pipelineafter which we utilizeΨ to provide similarity measurements for use duringaffinity propagation:r(i, j)|t=t0 = Ψi,j −maxj∗,j{Ψi,j∗}(3.5)which is then propagated across the nodes of the network as explained ear-lier in this section. All of the resultant cluster exemplars (regardless of theircount, i.e. no thresholding is performed) are then used for the remainderof the learning process.We recall that the goal of using a clustering approach was to provide areduction in the computational load and to reduce the occurrence of highlysimilar atoms. The training matrix preconditioning step of section 3.2.2 per-forms the rest of this reduction, and hence we opted for a simpler and easierto adjust feature vector. Accordingly, our feature vector φm has a single pa-rameter: the isotropic patch dimension, d. The choice of this parameter’svalue is expected to change the correlation matrixΨ. As such, we need todetermine an optimal value of d.In our case, the optimality condition would be a matrix Ψ that is asdiscriminative as possible between different diffusion clusters, while stillmaintaining high correlation within the same cluster. Accordingly, an opti-mal value of d should try to maintain as much dispersion as possible in thecorrelation values inΨ. In Figures 3.3 to 3.5 we show examples of the result-ingΨ for different values of d. These example figures were generated froma randomly selected subject in the HCP dataset (the dataset is described insection 5.1).The results of the effect of d on Ψ are summarized in Figure 3.6. Thefigure shows the range of the resultantΨ for various values of d for 10 ran-dom subjects from the HCP data. We note that the range is monotonicallydecreasing, and that its maximum occurs at a value of d = 3. This gives thelargest discriminative power for our chosen feature vector. Accordingly, the353.2. Processing Pipelinevalue of d = 3 will be used for the remainder of this work. An example ofthe result of clustering is shown in Figure 3.7. Note how the correlation ma-trix after clustering clearly shows a clustered grouping of diffusion indices.Each block or group of diffusion indices can now be represented using anyindex within that cluster.3.2.2 Training Matrix PreconditioningDictionary learning methods are typically highly non convex [3, 59]. Strictlyspeaking, the optimizations involved in these methods are generally com-binatorial in nature:minα||α||0 subjectto y =Dα (3.6)for a full-rank D ∈ Rn×m (n < m). This has been shown to be an NP-hardproblem [65]. An approach that has been used to address this issue is toperform a relaxation from an l0 to an l1 norm, which results in a convexifi-cation of the problem that yields a result that approximates the true sparsecoding vector [15]. However, when the sparse coding is combined with adictionary update (which is a required combination in dictionary learningmethods [3, 59]),minD,{αi }Mi=1M∑i=1||yi −Dαi ||22 subjectto ||αi ||1 ≤ k (3.7)the problem becomes a nested optimization, in which the first optimizesthe sparsity of each coding vector αi for a given D, and the second opti-mizes over D. The nested nature of the optimization has naturally resultedin algorithms that solve the problem by also alternating between two opti-mizations [3, 24]. Most of the variations between the algorithms lie in thechoice of heuristic for each of the two steps.However, due to their high non-convexity, such approaches will oftenfall into local minima or saddle points [3]. In fact, even when assuming a363.2. Processing Pipeline *   (d = 19)gradient index10 20 30 40 50 60 70 80 90gradient index1020304050607080900.960.9650.970.9750.980.9850.990.9951Figure 3.3: An illustration of the resultant correlation matrix Ψ obtainedwith a feature-vector volumetric patch dimension d = 19.373.2. Processing Pipeline *   (d = 11)gradient index10 20 30 40 50 60 70 80 90gradient index1020304050607080900.940.950.960.970.980.991Figure 3.4: An illustration of the resultant correlation matrix Ψ obtainedwith a feature-vector volumetric patch dimension d = 11.383.2. Processing Pipeline *   (d = 3)gradient index10 20 30 40 50 60 70 80 90gradient index102030405060708090 0.880.90.920.940.960.981Figure 3.5: An illustration of the resultant correlation matrix Ψ obtainedwith a feature-vector volumetric patch dimension d = 3.393.2. Processing PipelineFeature vector volumetric patch dimension, d2 4 6 8 10 12 14 16 18 20range of * 3.6: The effect of the feature-vector volumetric patch dimension, d,on the range of the resultant correlation matrixΨ. Larger values are better.Vertical bars show standard deviation across 10 HCP volumes.403.2. Processing Pipelinebefore clusteringdiffusion gradient indexdiffusion gradient index0.880.90.920.940.960.981after clusteringdiffusion gradient indexdiffusion gradient index0.880.90.920.940.960.981Figure 3.7: An example result of diffusion shell clustering. The upper figureshows the Ψ matrix before clustering, and bottom figure shows the matrixafter clustering. Note how the bottom figure clusters the data, thereforeprovides information on which gradient indices are relatively identical andtherefore replaceable by any index within a given cluster. 413.2. Processing Pipelineperfect sparse coding step, only a convergence to a local minimum is guaran-teed. This guarantee does not hold when approximate sparse coding meth-ods are used, which is often the case due to the combinatorial nature ofsolving an exact sparse coding as explained earlier.As a result, the choice of training matrix initialization will have an effecton the convergence of the training. Different initializations will convergeto different local minima. In previous works utilizing dictionary learningmethods in image processing, the training matrix was typically initializedto include all patches from the supplied training set [54, 69, 78]. The re-sults obtained from this initialization approach provided acceptable perfor-mance in different applications, such as face recognition, image denoising,and remote sensing.However, this approach presents a challenge when used with dwMRIdata. Volumetric patches constructed from dwMRI data have a greater de-gree of similarity compared to natural images. This can be attributed to acouple of reasons. First, most existing applications work with 2D data whiledwMRI is a 4D dataset. The increased dimensionality allows for greateroverlap between patches, hence increasing possibility of similarity betweenpatches. This increased similarity results in a higher chance of generatingmulticollinear atoms when a coding matrix is constructed. In addition, nat-ural training images often represent a much larger variety of choices forpatches compared to the more monotonous dwMRI data. Furthermore, es-pecially in the increasingly more common HARDI imaging schemes, theangular resolution of acquisition is very high. At a given physical location,this results in very similar volumetric patches across acquisition orienta-tions, which further increases the chances of having multicollinear atoms.To illustrate the aforementioned points, we show an example compar-ing the condition number of a dictionary coding matrix from both natu-ral images and dwMRI data. This is illustrated in Figure 3.8. We observe423.2. Processing Pipelinethat, for the same dictionary size, dwMRI data yield significantly more ill-conditioned coding matrices compared to natural images. As such, thismakes it impossible to use the dictionary learning framework to performsuper-resolution since the coding matrix becomes singular, preventing in-version operations. This is explained in section 3.2.3 in more detail.However, we propose to solve the problem using a preconditioning ap-proach. We perform an alternative construction of the training matrix inorder to improve the condition number. In this alternative construction, weseek to populate the training matrix with initial atoms that have reducedoverlap and thus reduced collinearity. We propose to achieve this by popu-lating the initial atoms along salient 3D structures in a dwMRI volume. Ourmotivation for following this approach is that including more salient struc-tures as initial training atoms would be expected to reduce overlap com-pared to allowing smooth, non-structured, patches into the training matrix.An overview of the proposed preconditioning approach is shown in Figure3.9.We propose to detect the salient local structures using a shearlet decompo-sition [50]. This method decomposes a volume into a set of coefficients thatrepresent 3D surface-like discontinuities at various locations, scales, andorientations. Each coefficient is associated with a particular combination ofscaling, shearing, and translation of a generating function. By capturing thespatial structure of high dimensional discontinuities (instead of capturing a1D discontinuity in each direction), this system provides a better represen-tation of high dimensional structures.The shearlet system has been shown to yield optimally sparse represen-tations compared to similar frameworks, such as curvelets and contourlets[34, 49]. In this work, we use the implementation provided by the authors[50]. As for the filter parameters, we used the parameters that has beenshown in a previous study [70] to yield good reconstruction in MRI, which433.2. Processing PipelineDictionary size100 200 300 400 500 600 700 800 900Condition number of sparse coding matrix100105101010151020natural imagesdwMRI dataFigure 3.8: An example comparison of the condition number of the codingmatrix in both natural and dwMRI data. The coding matrix of dwMRI dataexhibit orders of magnitude higher condition number compared to natu-ral images, which presents a challenge in using the matrix in the necessaryinversion operations.443.2. Processing PipelinedwMRI training volumeShearlet non-linear thresholdingTraining matrix atomsFigure 3.9: The proposed training matrix preconditioning process.453.2. Processing Pipelinewere diamond flat filters, with 4 scalings and 4 shear levels in the first twoscales and 8 shear levels in remaining two scales.We are now at a stage of being able to decompose the dwMRI volumeinto a multi-scale geometrical representation system. What we need to dois to determine a method for extracting the salient local structures in orderto populate the initial training matrix.We perform this task by a non-linear thresholding of the decomposedshearlet coefficients. Using this approach, by selecting only the highest Mcoefficients, we extract the most 3D-surface-like structures in the dwMRIvolume. We also note that since the thresholding is done non-linearly, theselectedM coefficients out of all N coefficients have no specific scale or ori-entation, and therefore the structures that are extracted are not at a prede-termined location, scale, or orientation.Accordingly, we run a non-linear thresholding experiment to determinethe most suitable threshold for detecting structures. The result is shown inFigure 3.10. We begin by observing that, as expected, the curve starts witha large error due to the absence of most coefficients. Then there is a largedrop in error, after which the approximation gradually plateaus.We emphasize that our goal is not to faithfully reconstruct the volume,and hence large M/N is not desirable. Instead, we are looking for only themost geometrically salient structures. A good point to look for structureswould be at the onset when we barely include any coefficients but the ap-proximation error drops sharply. This indicates that, at this point, the vol-ume has now gained strong salient structures. In Figure 3.10, this occursaround the value 0.03. As such, this will be our choice for non-linearlythresholding the coefficients. Figure 3.11 shows an example of the resultobtained from using this thresholding value.Finally, we binary threshold the reconstructed volume at different val-ues. We then assign all resulting voxels of this operation as center points of463.2. Processing Pipelinethe atoms to be initialized into the training matrix. The dictionary is thenlearned on the training matrix, and the condition number of the resultingcoding matrix is plotted. The result of this experiment is shown in Figure3.12. We can clearly observe a minimum point around 2%, which is thenmonotonically increasing afterwards. As such, we choose the thresholdingvalue of 2%.Using the above parameters, we now examine the end result of the pre-conditioning process by observing the new condition numbers. This is shownin Figure 3.13. The figure confirms that we have achieved a low conditionnumber for the coding matrix.3.2.3 Joint-dictionary LearningIn this section, we describe details pertaining to construction of the joint-dictionaries, and the process of using the dictionaries to super-resolve data.We begin by generating a training set using the approach proposed in sec-tion 3.2.2. Let the training set be denoted by Ω, and defined as followsΩ = { (ρiL,ρiH ) | ρL ∈Rd ,ρH ∈R8d} (3.8)where ρL indicates a low-resolution patch, ρH indicates a high-resolutionpatch, i is the patch index within the training set, and d is the dimensionof the low-resolution patch. Next, the low-resolution dictionary is createdusing the same minimization as in [46, 97]:DL, {αi} = argminDL, {αi }∑i||ρiL −DLαi ||22 + λ||αi ||1 (3.9)where DL ∈Rn×m, with m atoms of size n, and αi ∈Rm is the sparse codingvector of the i-th patch, and λ is an optimization weight controlling thesparsity of the coding vector.The existing dictionary-based super-resolution methods [46, 97] typi-cally use the K-SVD (K-singular value decomposition) algorithm [3] to im-473.2. Processing PipelineSparsity (M/N, percent)0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09M-term approximation error, ||f - f M|| 22101102103104105106Figure 3.10: The M-term approximation of a shearlet decomposition of adwMRI data. Note the onset of a drop in approximation error occurringaround 0.03, which gradually plateaus afterwards. This indicates that, atthis point, the volume has now gained strong salient structures. This willtherefore be the M/N ratio used to detect structures. Vertical bars showstandard deviation across 10 random HCP volumes.483.2. Processing PipelineFigure 3.11: Example result of non-linearly thresholding the shearlet coeffi-cients. The threshold is set to the point 0.03 in Figure 3.10. The intensitiesin the upper figure reflect diffusion signal, while intensities in the bottomfigure reflects salient shearlet locations.493.2. Processing PipelineReconstructed volume thresholding, percent1 2 3 4 5 6 7 8 9 10Condition number, normalized10-210-1100Figure 3.12: The condition number of the resulting coding matrix for vari-ous levels of binary thresholding of the shearlet reconstructed volume. Ver-tical bars show standard deviation across 10 random HCP volumes.503.2. Processing PipelineDictionary size100 200 300 400 500 600 700 800Condition number of coding matrix77.588.599.510Figure 3.13: The condition numbers of dictionary coding matrix after thetraining matrix preconditioning step. The condition numbers are low at avarious dictionary sizes.513.2. Processing Pipelineplement the joint-dictionary construction. While K-SVD has demonstratedgood performance, it posits a challenge for implementing the super resolu-tion method for dwMRI data.This is due to the batch-based nature of the algorithm, which requiresaccess to the entire training set. For dwMRI data (especially the increas-ingly common multi-shell acquisitions), the training set can be in the rangeof tens of millions of samples, which represents a computational burden ifrequired to be loaded at once. Furthermore, it has been shown that batch-based methods such as K-SVD can not effectively handle large training sets[60].At this point, we recall that one of the objectives of this thesis is to pro-vide an acquisition-independent framework. With the above limitations ofK-SVD regarding large training sets (which hinders the use of newer acqui-sitions like multi-shell), it is no longer a good choice for implementing thejoint-dictionary dwMRI super-resolution.In contrast, in this work we propose to implement the joint-dictionarysuper resolution process via an online learning approach using the SPAMS(SPArse Modeling Software) algorithm [60]. Using this algorithm, the train-ing set is only minimally loaded (one or a few training samples at a time),which makes the learning online. Our choice of this implementation stemsfrom the fact that dwMRI data is orders of magnitude larger in size com-pared to natural images that have been the target of existing dictionary-based super-resolution methods [46, 97]. As such, we use an online learningapproach in order to gradually construct the dictionaries from a sequence ofportions of the dwMRI training set.After computing the sparse coding vector αi in (3.9), the high-resolutiondictionary DH is constructed as in [46, 97]DH = argminDH∑i||ρiH −DHαi ||22 (3.10)523.2. Processing Pipelinewhich can be solved as in [97] using pseudo-inversion:DH = PHA+ = PHAT (AAT )−1 (3.11)where PH is a matrix of the patch set ρiH and A is a matrix of the codingcoefficient vectors αi .For dwMRI data, this is presents a problem due to the spatial and an-gular overlap between patches as described in sections 3.2.1 and 3.2.2, lead-ing to multicollinearity and ill-conditioning of A which hinders the stabil-ity of the inversion in (3.11). The approach we propose to solve this is-sue is described in section 3.2.2, which preforms a preconditioning on thedwMRI training matrix in order to improve the condition number of theresultant coding matrix A. This enables the successful computation of thehigh-resolution dictionary DH for dwMRI data.After the two dictionariesDL andDH are constructed, the training phaseis concluded. At this stage, a new un-seen dwMRI dataset can now be super-resolved. We now describe the super-resolution process utilizing the twolearned dictionaries, which follows the same approach as [46, 97].First, the input volume is converted into a patch set that matches the sizeof low resolution dictionary DL. Next, each patch in the set is sparse codedagainst the low resolution dictionary. After that, the resultant coding vectoris multiplied against the high resolution dictionary, which finally generatesthe high resolution patch set. The patch set is then finally assembled backinto a high-resolution volume, averaging overlapping areas.We now turn to the problem of determining the set of dictionary learn-ing parameters for super-resolving dwMRI. We start by examining the ef-fect of the dictionary learning optimization weight λ on the reconstructionquality. That is, the reconstruction of a given volume from its downsampledversion. The quality of this reconstruction is shown in Figure 3.14.From the figure, we make a number of observations. First, we note that533.2. Processing Pipelinedecreasing the weight λ increases the reconstruction quality. This plateausat around λ = 10−3. We also note that, for a given value of λ, larger dic-tionaries result in better reconstruction. This holds until around 700 - 1100atoms, where performance plateaus. This behavior is reasonable since alarger dictionary of atoms allows for learning more example atoms. Ac-cordingly, we choose λ = 10−3 with a dictionary of 900 atoms as parametersfor the rest of this paper. The effect of the choice of images used in train-ing, and whether performance depends on the type of image used, will beexplored in great detail in Chapter 5.Next, we determine the atom size parameter. In this analysis, we ex-amine the reconstruction quality at different isotropic length dimensionsof the atom. The performance is shown in Figure 3.15. The figure showsthat better performance is achieved with smaller atom sizes, and that thebest atom size is 3 voxels (isotropic). This performance is reasonable since atraining matrix with smaller atoms would provide more consistency to thedictionary learning method.543.2. Processing PipelineDictionary optimization weight, 610-4 10-3 10-2 10-1MSE, normalized0.750.80.850.90.95150 atoms100 atoms300 atoms500 atoms700 atoms900 atoms1100 atomsFigure 3.14: The effect of dictionary optimization weight λ, at different dic-tionary sizes, on reconstruction performance. Note the plateau occurringaround 700 - 1100 atoms.553.2. Processing PipelineAtom size, isotropic3456789MSE, normalized0. 3.15: The effect of dictionary atom size (isotropic) on reconstruc-tion performance. Vertical bars show standard deviation across 10 HCPvolumes.56Chapter 4Validation Methodology4.1 OverviewIncreasing the resolution of a dwMRI volume beyond what is available inthe original data presents a challenge in terms of validation. This is becauseof the lack of ground truth information at the higher resolution. dwMRI re-mains the only technique for obtaining non-invasive in vivo informationabout fiber tracts in the brain. As such, we have no other high resolu-tion source of information to directly validate the generated high resolutiondwMRI volumes against.In order to address this issue, we propose to use a validation approachthat builds on the existence of a strong dependence between structure andfunction in the brain [39, 52]. Accordingly, brain regions that have strongstructural connections also have a strong functional connection [92, 99].Motivated by this relationship between how functional connectivity (FC)reflects the underlying structural connectivity (SC), we quantitatively vali-date our results by investigating the consistency between SC and FC beforeand after super-resolving the data. In other words, a greater SC-FC consis-tency indicates a better reconstruction quality, compared to a low SC-FC.This validation approach is beneficial for a number of reasons. First,fMRI data is typically readily included with many dwMRI acquisitions, re-flecting the same subject at the same point of time. This has the benefitof having an independent modality (fMRI, in this case) used to validate an-574.2. Functional Connectivity Estimationother modality (dwMRI) of the same subject. Furthermore, fMRI is typicallynot resolution limited. In fact, fMRI voxels are rarely meaningful individ-ually without being parcellated into a larger group. For these reasons, im-provements in dwMRI reconstructions would translate into greater SC-FCconsistency.In the next sections, we start by describing the process of obtaining FCfrom fMRI data. We then proceed to estimate SC from dwMRI. After thetwo estimates are obtained, we finally calculate the SC-FC correlation as aquantitative metric for dwMRI resolution enhancement.4.2 Functional Connectivity EstimationFunctional information on the brain is estimated from functional-MRI data(fMRI). For each voxel, the intensity of the data represents the level of neu-ral activity, as measured through oxygen-level changes (referred to as theblood-oxygen-level dependent contrast, BOLD). In addition, due to the dy-namic nature of brain function, the data is continuously recorded for a pe-riod of time. The result is a functional time course per voxel.We begin the estimation of functional connectivity by parcellating thebrain voxels into regions. Parcellation is commonly performed on fMRI databecause functional information is rarely specific to a certain voxel, but ratheroccur synchronously with a larger group of voxels.We parcellate the brain into 200 regions using Ward clustering [63], whichhas been shown to perform better than other fMRI parcellation approaches[82]. The number of parcellation regions was set based on the results of aprevious study [83] which recommended the use of 200 or more parcels.Accordingly, let Z be a t × d matrix of fMRI time courses, where t isthe number of time points and d is the number of brain regions. We then584.3. Structural Connectivity Estimationestimate the functional connectivity (FC) using Pearson’s correlation:FC = ZTZ/(t − 1) (4.1)which, in a comparative study with other common FC metrics, has beenshown to yield better performance [26].4.3 Structural Connectivity EstimationWhile the neuroimaging research community has largely settled on Pear-son’s correlation for FC connectivity measurement [26], investigations ofstructural connectivity (SC) metrics have received less attention. An orig-inal contribution of this thesis includes presenting a comparison of struc-tural connectivity metrics assessed from the perspective of the largely ac-cepted inherent relationship between brain structure and function [7].Quantifying structural connectivity in the brain is most commonly basedon quantifying one or more aspects of the streamlines reconstructed usingdeterministic tractography, though computationally expensive approachesbased on probabilistic tractography techniques were also explored [11]. Thechoice of which streamline property to measure and of how to map it intoa structural connectivity metric are key aspects affecting the structural con-nectivity estimates.Arguably the most common SC metric is the number of reconstructedstreamlines between pairs of brain regions, commonly referred to as fibercount. A variant of this approach involves a normalization of the fiber countby the total volume of the region pairs they connect to account for the vari-able size of the brain regions [36]. Besides metrics based on fiber count, useof the total length of reconstructed streamlines has also been suggested asa measure of structural connectivity [61] aiming to correct for the fact thatlonger tracts have larger accumulated error, leading to lower fiber counts.594.3. Structural Connectivity EstimationAnother metric, the average fractional anisotropy (FA) along streamlinesconnecting regions, has also been proposed as a proxy for structural con-nectivity strength [14].It is important to acknowledge that all of the aforementioned SC met-rics are confounded by several factors, limiting their interpretability. First,tractography can only delineate bundles of fibers in the brain, and not in-dividual fibers. The term fiber count can thus be misleading. Indeed, us-ing the term streamline count has been recently proposed as an alternative[44]. Nonetheless, we use the term fiber count for easier interpretation andto conform to the jargon used in existing literature, with the understand-ing that it is the streamlines that are actually being counted. Moreover, wenote that the number of fibers is dependent on the number of seeds usedfor tracking the fibers, the tractography method used, and several featuresof the pathway such as curvature, length and width [44]. Additionally, wehighlight that FA not only depends on the reliability of local diffusion in-formation, but also on a large number of modulating factors such as axonalordering, axonal density, amount of myelination, and increase in extracel-lular or intracellular water [44]. Such confounding factors did not impedethe adoption of a variety of SC metrics, driven by a practical need for quan-tifying the degree of connection between brain regions.We propose that reconciling the presence of confounding factors withthe practical need for connectivity estimation calls for a detailed analysis,in a quantitative comparison, to determine which SC metric has the high-est potential of being of practical use in multimodal brain image analysisefforts. To this end, we compare four commonly used SC metrics in termsof their impact on the relationship between estimates of SC and FC. On 38subjects from the Human Connectome Project (HCP) database [91] (whichis described in detail in the datasets section – section 5.1), we show that re-gion volume-normalized fiber count best correlates with FC. We also show604.3. Structural Connectivity Estimationthat total fiber length has the least bias towards distance between brain re-gions. We further demonstrate that these results hold across seven differenttasks and resting-state data.We now describe the calculation of the four SC metrics. Let rki,j be the kthreconstructed fiber between a pair of structurally connected regions Pi andPj . We consider four widely used structural connectivity measures in thiswork: fiber count (fi,j), fiber count normalized by the total volume of theconnected regions (Ni,j), total length of fibers connecting region pairs (Li,j),and average FA along the fibers. For each subject, we compute the Pearson’scorrelation between the FC and SC estimates to quantify the SC-FC rela-tionship for each SC metric. More formally, the metrics can be expressed asfollows:Ni,j =fi,jV (Pi) +V (Pj )(4.2)Li,j =∑kl(rki,j ) (4.3)whereV (·) is the volume of the corresponding region, and l(rki,j ) is the lengthof rki,j .Prior to the computation of SC metrics, we reconstruct the fibers viaglobal tractography on constant solid angle orientation distribution func-tion (ODF) using MITK [68]. Global tractography was chosen over the morecommon streamline tractography since it was recently shown to facilitatehigher SC-FC consistency [98]. In global tractography, short fiber segmentsare connected together to generate the set of fiber tracts that best explainsthe measured dMRI data. As such, at regions with unreliable local diffusioninformation, the geometry of the surrounding fibers drives the tracking pro-cess to prevent premature termination of fibers as is commonly observed instreamline tractography.The results of the SC-FC correlation for resting-state and task fMRI are614.3. Structural Connectivity Estimationshown in Figures 4.1 and 4.2, respectively. As observed from these figures,average FA has lower correlation with FC compared to the rest of the exam-ined SC metrics (fiber count, volume-normalized fiber count, and total fiberlength). This is true for both resting-state and task fMRI. We speculate thatthe reason for the observed low average FA correlation can be attributed tothe large number of factors affecting local diffusion anisotropy [44].Figure 4.1 also shows that the volume-normalized fiber count has thehighest correlation with FC compared to the rest of the examined SC metricsfor both resting-state and task fMRI. The pairwise differences between SC-FC correlation assessed using normalized fiber count and other SC metricswere found to be statistically significant at p < 0.001 based on the Wilcoxonsigned rank test. Our results thus imply that the compensation (due to nor-malized fiber count) for the differences in number of fibers due to the vari-able size of brain regions yields better depiction of structural networks.We also note that the relatively consistent SC-FC correlation levels acrossa variety of tasks and resting-state data support the notion that SC forms thebackbone of the brain connectivity around which functional reorganizationoccurs to respond to different tasks. A diverse repertoire of functional brainconnectivity patterns can thus arise constrained by the same structural sub-strate.Figure 4.3 shows a qualitative comparison between functional and struc-tural connectivity patterns. We averaged subject-specific connectivity ma-trices to compute group results. Specifically, the top 10 parcels having strongestconnectivity to posterior cingulate cortex (PCC) are overlaid onto the brainusing these group-level connectivity matrices. PCC was selected as the seedas it is known to be a structural and functional hub facilitating efficient com-munication in the brain [37]. This figure shows that SC patterns estimatedusing normalized fiber count resemble FC patterns more than those esti-mated using average FA.624.3. Structural Connectivity Estimation5 10 15 20 25 30 350.−FC Correlation  fiber countnormalized fiber countaverage FAtotal fiber lengthFigure 4.1: Comparison of four common SC metrics in terms of SC-FC (rest-ing state fMRI) correlation for 38 subjects from the Human ConnectomeProject.634.3. Structural Connectivity Estimation5 10 15 20 25 30 3500.−FC Correlation  fiber countnormalized fiber countaverage FAtotal fiber lengthFigure 4.2: Comparison of four common SC metrics in terms of SC-FC (task-fMRI) correlation for 38 subjects from the Human Connectome Project. Theshaded bands represent the standard deviation of SC-FC across 7 differenttasks.644.3. Structural Connectivity EstimationFigure 4.3: Parcels with highest connectivity to posterior cingulate cortex asobtained from: (a) Functional connectivity (FC), (b) normalized fiber count,and (c) average FA. Note how the arrangement of connected parcels ob-tained by normalized fiber count has a better resemblance to the parcelsobtained by FC, compared to average FA.65Chapter 5Results5.1 DatasetsTwo different publicly available datasets were used for the experiments inthis chapter. The first dataset consists of dwMRI data from 38 subjects (17males and 21 females, ages ranging from 22 to 35 years) from the HumanConnectome Project (HCP) Q2-13 dataset [91]. This release of the datasethas 40 subjects for which dwMRI data was available. We excluded twosubjects from the dataset (subjects #209733 and #528446) as per HCP’s rec-ommendation, due to reported structural brain abnormalities. The dwMRIdata had a voxel size of 1.25 mm (isotropic), 3 diffusion shells (at b = 1000,2000 and 3000 s/mm2) and a total of 288 gradient indecies. The HCP dwMRIdata used in this chapter includes the suggested minimal preprocessingpipeline already applied by the HCP team, including corrections for EPIdistortion, eddy current, gradient nonlinearity and motion artifacts [31].Further details on the dwMRI acquisition can be found in [91].The second set of data is the Kirby 21 dataset [51]. This dataset com-prises scans of 21 subjects (11 males and 10 females, 32 ± 9.4 years old). Weused two modalities from this dataset: rs-fMRI data and dwMRI data. Thers-fMRI data consisted of a 7-minute acquisition with a TR of 2 s and a voxelsize of 3 mm (isotropic). We then preprocessed the data for motion correc-tion and bandpass filtered from 0.01 and 0.1 Hz using an in-house MATLABcode. We then divide the brain into 150 parcels using Ward clustering [64]665.2. Quantitative Performanceapplied on the voxel time courses. The dwMRI data consisted of 32 diffusiongradients with a b-value of 700 s/mm2, in addition to a single b0 image. Thevoxel size was 0.83×0.83×2.2 mm3. However, since anisotropic voxels werepreviously shown to be suboptimal for further processing of dwMRI datain the sense that fiber branching is less detectable [67], we resampled thedata to 2 mm isotropic voxels prior to any subsequent analysis and process-ing. Finally, we then warped the functionally derived group parcellationmap to the b0 volume of each subject using FSL [42] in order to facilitatecomparisons of structural and functional metrics.5.2 Quantitative PerformanceWe perform quantitative assessments in three categories of tests: tests onthe HCP data, tests on the Kirby data, and tests that combine both datasets(i.e. training on one and testing on the other). In order to quantify the qual-ity of the constructed volumes at higher resolutions, we performed twosets of experiments. The first experiment measures the similarity betweena ground truth volume and a reconstructed volume from its downsam-pled version. The quality is quantified using a reconstruction error metricη =MSRSR/MSEInterp., which are theMSE from the super-resolved recon-struction and spline interpolation, respectively.The second experiment aims to quantify the quality of the constructedvolumes at a higher resolution than that of the ground truth. The quantifica-tion metric of this experiment follows the SC-FC correlation measurementsdescribed earlier in Chapter 4.Multi-shell resultsFigure 5.1 shows the reconstruction error η for a number of HCP test sub-jects. These results were generated from training the dictionaries on 30 ran-675.2. Quantitative Performance1 2 3 4 5 6 7 800. subjectRelative reconstruction error,  η  interpolation3rd diffusion shell2nd diffusion shell1st diffusion shellb0 volumeStudent Version of MATLABFigure 5.1: Relative reconstruction error η for HCP dataset [91] test subjects.The vertical bars show the standard deviation of error across diffusion di-rections at each shell. The vertical bars for the b0 volume show the standarddeviation of error across repeated b0 acquisitions.685.2. Quantitative Performancedom subjects from the HCP data, and then testing the reconstruction qual-ity on the remaining 8 subjects from the same dataset. We observe thatthe figure shows η values in the range of ∼ 0.4 for b0 volumes, indicatingan improvement of around 60%. We also observe a highly consistent recon-struction quality of b0 volumes as indicated by the small standard deviationof error.The figure also shows the reconstruction quality of the diffusion shells.We observe that the quality of diffusion shell reconstruction is lower thanthat of the b0 volumes. This performance is not unexpected and we attributeit to the lower SNR of the diffusion shell acquisitions compared to the b0volumes and also the lower number of repetitions compared to b0 volumes.This is further supported by the observation that, within the diffusion shellresults, higher shells show lower reconstruction quality compared to lowershells, recalling that higher shells have lower SNR compared to lower shells.For the same reasons, we also observe that the consistency of reconstructionquality for diffusion shells is generally lower than that of the b0 volumes.Next, the results of Figure 5.2 use the same dictionary trained for Figure5.1, but examines the generalization ability of the dictionary by testing iton unseen data from the Kirby dataset. That is, the results of this figureare based on training the dictionary on the HCP dataset and testing on theKirby dataset, thereby assessing generalization performance of the learneddictionary to a different acquisition type.Accordingly, we make a number of observations on Figure 5.2. First, weobserve that the range of reconstruction quality values are relatively withinthe same range as that of Figure 5.1. This shows a good generalization per-formance of the learned dictionaries. However, we still note that there is aslight reduction in the reconstruction quality of b0 volumes between Fig-ures 5.1 and 5.2. We emphasize that this reduction is only partially relatedto the generalizability of the dictionary. The reason for this is that the HCP695.2. Quantitative Performanceb0 volumes were acquired 18 times and averaged in order to improve SNR,while Kirby b0 volumes were only acquired once. Hence, the reconstruc-tion quality of HCP b0 volumes is expectedly better than Kirby b0 volumes,regardless of generalizability of the dictionary. Nonetheless, the quality ofKirby b0 volumes constructed from the dictionary is still relatively similarto that of HCP b0 volumes.In addition, we also note that the gap between the b0 and diffusion vol-umes is smaller in Figure 5.2 compared to Figure 5.1. We suggest that thismay be attributed to the fact the difference between b0 and diffusion vol-umes is smaller for Kirby data compared to HCP. For Kirby data, this is 700s/mm2 while for HCP data the difference is 1000 s/mm2 and higher.Single-shell resultsNext, we examine the effect of changing training data on the quality of re-constructions. This is shown in Figure 5.3. As in the previous figure, thisexperiment also uses the Kirby dataset for test subjects. However, the train-ing data is changed from Kirby to HCP. The previous experiment used HCPdata for training and Kirby data for testing, while this experiment in Figure5.3 uses Kirby data for both training and testing. The difference betweenthe two experiments is the choice of training dataset.The training data is constructed from a randomly selected list of 11 sub-jects from the Kirby dataset. The testing dataset is then chosen to be theremaining 10 subjects in the dataset. The results are shown in Figure 5.3.We make a number of observations regarding this experiment. First,we note that the overall range of reconstruction error is relatively similarto that of Figure 5.2. This shows that, for the same test dataset, the perfor-mance is not highly sensitive to training data. Nonetheless, given that thetraining and test subjects now belong to the same dataset, we do observe animprovement in the consistency of reconstruction. More specifically, while705.2. Quantitative Performance5 10 15 2000. subjectRelative reconstruction error,  η  interpolationdiffusion volumesb0 volumeStudent Version of MATLABFigure 5.2: Relative reconstruction error η for Kirby dataset [51] test subjectsusing dictionaries trained on HCP dataset [91]. The vertical bars show thestandard deviation of error across diffusion directions.715.2. Quantitative Performancethe reconstruction error ranged between 0.5 to 0.6 in Figure 5.2, the errorrange in Figure 5.3 is more consistent at around 0.5. We also note that thegap between b0 and diffusion volumes is now considerably smaller. Thismay be attributed to the fact that the dictionary is now learning from a sin-gle diffusion shell compared to learning from a cluster of shells in the HCPdata. Therefore, there is now expectedly less ambiguity in reconstructingdiffusion volumes, which helps improve reconstruction quality. This also isexpected to result in more consistent reconstruction quality across diffusionvolumes for the same shell, which is indeed the case in Figure 5.3.Next, we perform another experiment on the sensitivity to the choice oftraining data. In this experiment, we reverse the roles of the training andtesting data used in the previous experiment. The test data of Figure 5.3are now used as training data, and the training data of that figure are nowused as test data. The result of this experiment is shown in Figure 5.4. Weobserve that the results of both experiments are clearly very similar. Theaforementioned observations about the range of error and reconstructionconsistency in the previous experiment also clearly hold in this experiment.This shows that, for a given dataset, the results do not exhibit a sensitivityto the choice of training subjects.SC-FC correlation We then proceed to the next set of experiments wherewe compare SC-FC correlations in order to assess the improvement in ex-tending the resolution beyond ground truth.To the best of our knowledge, the only previous work that tackled theproblem of super-resolving dMRI data from a single acquisition indepen-dent of the diffusion model was by Coupe et al [20] (referred to as CLASR– collaborative and locally adaptive super-resolution). Specifically, the au-thors showed that super-resolving b0 image using a locally adaptive patch-based strategy, and using this high-resolution b0 image to drive the recon-725.2. Quantitative Performance1 2 3 4 5 6 7 8 9 1000. subjectRelative reconstruction error,  η  interpolationb0 volumediffusion volumesStudent Version of MATLABFigure 5.3: Relative reconstruction error η for Kirby dataset [51] test subjectsusing dictionaries trained on the remaining subjects in the same dataset.The vertical bars show the standard deviation of error across diffusion di-rections.735.2. Quantitative Performance11 12 13 14 15 16 17 18 19 20 2100. subjectRelative reconstruction error,  η  interpolationb0 volumediffusion volumesStudent Version of MATLABFigure 5.4: Relative reconstruction error η for Kirby dataset [51] test sub-jects, trained on the remaining subjects in the dataset. The selection of testversus training subjects in this figure is the opposite of Figure 5.3.745.2. Quantitative Performancestruction of diffusion images, outperforms interpolation methods. To thebest of our knowledge, CLASR is the only existing super-resolution methoddeveloped for dwMRI that is independent of acquisition and the diffusionmodel employed, which are objectives of this thesis.To quantify the improvement, we analyzed the consistency between mea-sures of intra-subject SC and FC. We estimated SC using the fiber counts be-tween brain region pairs, and FC using Pearson’s correlation between par-cel time courses. We chose to employ deterministic streamline tractographywith the diffusion tensor model, which is by far the most popular tractogra-phy approach to date. However, we highlight that our super-resolution ap-proach can be used with any diffusion model and any tractography method.Tractography was carried out using Dipy [29], with 750,000 seed points forall examined volumes.For each subject, SC and FC are vectors of size d(d−1)/2 comprising thecorresponding connectivity estimates between each region pair, where d isthe number of brain regions. We then calculated Pearson’s correlation be-tween intra-subject SC and FC to quantify the consistency between the twoconnectivity estimates. Using this correlation measure, we compared theproposed super-resolution approach with trilinear and spline interpolationin addition to a CLASR.Figure 5.5 shows the SC-FC correlation for each subject tested. Takingthe average SC-FC correlation across the group when using the original dataas a baseline, the improvement was 5.7% with spline interpolation, 13.6%with CLASR, and 27.1% with our proposed method. On the other hand,there was a 6.3% decrease in the correlation when trilinear interpolationwas used. The difference in the performance of our method and every othermethod tested was found to be statistically significant at p < 0.01 based onthe Wilcoxon signed-rank test, showing its potential for enhanced structuralconnectivity assessment. Our results thus suggest that low spatial resolu-755.3. Qualitative Resultstion of dMRI data can partially account for the low SC-FC correlation, andstatistically significant improvements can be achieved using super-resolveddwMRI data.To investigate why trilinear interpolation resulted in a lower SC-FC cor-relation compared to the original data, we calculated the number of tractsreconstructed with each method. The local intra-parcel connections wereexcluded since they have no effect on SC-FC correlation. Figure 5.6 showsthe number of inter-parcel tracts averaged across the group along with thecorresponding standard deviations. As observed from this figure, perform-ing tractography on volumes upsampled with trilinear interpolation resultedin a lower number of tracts compared to the original volumes, even thoughthe same number of seed points were used to initiate tracking for all of themethods we compared. We speculate that the reason of this phenomenonis the additional partial volume effects introduced by the blurring of thedata during trilinear interpolation, which hamper the tractography quality.Spline interpolation, however, is known to cause less blurring comparedto trilinear interpolation, and our results suggest that upsampling dMRIdata using spline interpolation can be beneficial for tractography. The over-all trend of inter-parcel tract counts closely resembles to that of the SC-FCcorrelation, with our proposed method outperforming all other methodstested. This shows that dictionary based super-resolution is a viable post-processing solution for dwMRI that can help in mapping the white matterbrain architecture more accurately.5.3 Qualitative ResultsWe now present a qualitative comparison between the fiber tracts recon-structed from the original (2 mm) and super resolved (1 mm) dwMRI data.We employ the same tractography approach as in the SC-FC comparisons765.3. Qualitative Results1 2 3 4 5 6 7 8 9 IDSC−FC Correlation  Original Trilinear Spline CLASR [8] Our methodStudent Version of MATLABFigure 5.5: SC-FC correlation for 10 subjects with SC estimated from thedata at its original resolution (2 mm isotropic), and high-resolution data(1 mm isotropic) obtained using trilinear interpolation, spline interpola-tion, CLASR, and the proposed method. Our method outperforms allother methods tested for eight of the subjects, and performs comparableto CLASR for two subjects (subjects 4 and 10).775.3. Qualitative ResultsOriginal Trilinear Spline CLASR [8] Our method4.555.566.57x 104Inter−parcel tract countStudent Version of MATLABFigure 5.6: Number of inter-parcel tracts reconstructed from the dataat its original resolution (2 mm isotropic), and high-resolution data (1mm isotropic) obtained using trilinear interpolation, spline interpolation,CLASR and the proposed method. Intra-parcel tracts are not included heresince they do not contribute to SC-FC correlation. We emphasize that trac-tography is initiated with the same number of seeds for each method.785.3. Qualitative Results(deterministic streamline tractography using the diffusion tensor model with750,000 seed points). We generated the tract-density maps by calculatingthe total number of fiber tracts present in each voxel. Figure 5.7 (a),(c) and(b),(d) show sample tract-density maps with the original and super-resolveddMRI data, respectively. As observed from these figures, the tract-densitymaps generated from the super-resolution data clearly show more spatialinformation. Figure 5.7 (e),(f) and (g),(h) show the corticospinal tracts ex-tracted using a region of interest (ROI) placed on the brain stem for tworepresentative subjects. It can be observed that fiber tracts reconstructedfrom the super-resolution data can capture the fan-shape configuration ofthe corticospinal tract more fully.Next, we present a qualitative comparison between the raw dwMRI im-ages obtained from original (2mm) and super resolved (1 mm) data. Fig-ure 5.8 shows an example comparison. As observed from the figure, theproposed method has better resemblance to ground truth. The structuralfeatures in the image are also more pronounced, while appearing blurredfor spline interpolation. This provides an example of the type of detailsobtained by super-resolving the dwMRI data.795.3. Qualitative ResultsOriginal	   Super-­‐resolved	  (a)	   (b)	  (c)	   (d)	  (e)	   (f)	  (g)	   (h)	  Figure 5.7: Qualitative comparison between the tract-density maps and fiber tracts re-constructed from the original (left) and super-resolved (right) dwMRI data. Original datafrom the Kirby set has 2 mm isotropic resolution which is super-resolved to 1 mm isotropicresolution. Each row corresponds to a different test subject. Tract-density maps of super-resolved data ((b) and (d)) show markedly improved spatial detail compared to those oforiginal data ((a) and (c)). Corticospinal tracts reconstructed from super-resolved data ((f)and (h)) can capture the fan-shape configuration more accurately than those generated fromoriginal data ((e) and (g)) 805.3. Qualitative ResultsFigure 5.8: Qualitative of comparison of raw diffusion images from the HCPdataset [91]. Note the closer resemblance between the proposed methodand ground truth.81Chapter 6Conclusions6.1 DiscussionLow spatial resolution is a known limitation of dwMRI, which often hindersthe performance of subsequent analysis and determination of structural in-formation. We proposed the use of a simple yet effective super-resolutionprocessing pipeline on dwMRI to capture a more accurate portrayal of thewhite matter architecture. This approach does not require multiple dwMRIacquisitions and is applicable to legacy data. Quantitatively, we demon-strated that SC-FC consistency can be markedly increased with the use ofour approach in estimating SC. We also qualitatively illustrated that the gainin spatial resolution remarkably improves the fiber tracts and tract-densitymaps generated. Taken collectively, our results suggest a super-resolutionbased framework holds great promise in enhancing the spatial resolutionin dwMRI, without requiring additional scans or any modifications of theacquisition protocol.We also presented a closer investigation of the validation strategy andpresented a comparison of four SC metrics computed from tractographyresults with respect to their relationship to FC. Among the metrics consid-ered, we showed that volume normalized fiber count has the highest cor-relation with FC for both resting-state and task data. On the other hand,our results showed that average FA has the lowest correlation with FC. Wespeculate that the reason of this low correlation is the non-specificity of FA,826.2. Thesis Contributions and Future Workwith several inadvertent factors (such as axonal density, axonal ordering,and amount of myelination) modulating it along with the reliability of lo-cal diffusion information. In addition, we also demonstrated that total fiberlength metric reduces the fiber length bias associated with shorter fibers.Our results therefore suggest that average FA may not be the best metric toquantify SC, and that the choice among other SC metrics warrants specialattention depending on the question being addressed and the scale of theproblem (e.g. whole-brain or local regional analysis).6.2 Thesis Contributions and Future WorkWe now describe the major contributions of this thesis. We proposed adwMRI data processing pipeline (built on a dictionary learning approach)that enhances the resolution of dwMRI after the data has been acquired.The pipeline does not require modifications to the dwMRI acquisition pro-cess, and therefore is more practical in clinical conditions where simple ac-quisition methods are typically usedDue to the absence of acquisition requirements, the pipeline can be read-ily used with existing legacy databases of dwMRI data. This can be usefulwhen trying to utilize or reuse datasets which has been acquired with olderacquisition technologies. Absence of acquisition requirements also makesit usable with common types of dwMRI, such as DTI or HARDI (single,and multiple-shell). Some of the existing super-resolution approaches arespecific to HARDI.We also note that the proposed pipeline does not require repeated dwMRIacquisitions, unlike classical shifting-based super-resolution methods. An-other contribution is that the pipeline performs a resolution enhancementdirectly on raw dwMRI data, resulting in a model-free pipeline.This lack of required modeling is useful because it does not restrict the836.2. Thesis Contributions and Future Workend user to a particular model, which gives the freedom of using differ-ent diffusion models in different research or clinical situations, depend-ing on the specific research conditions or objectives. Finally, we note thatthe model-independence also allows for cumulative enhancements, in whichany other diffusion-specific or tractography-specific enhancements can bereadily applied on top of the pipeline.While we demonstrated the benefits of the described pipeline, it is im-portant to acknowledge that the performance of the proposed method in-herently depends on the training dataset, as in any machine learning methodthat involves training or prior information. The age span of the subjects weused in our experiments was 23-61, showing that the method can generalizeto a large range of ages. However, how well abnormalities such as tumorand edema can be modeled with dictionary learning is currently unclearand warrants further research in future work.84Bibliography[1] Iman Aganj, Christophe Lenglet, Guillermo Sapiro, Essa Yacoub,Kamil Ugurbil, and Noam Harel. Multiple q-shell ODF reconstructionin q-ball imaging. 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