Spatially regularized compressed sensing for high angular resolution diffusion imaging

A majorize-minimize framework for Rician and non-central chi MR images

The statistics of many MR magnitude images are described by the non-central chi (NCC) family of probability distributions, which includes the Rician distribution as a special case. These distributions have complicated negative log-likelihoods that are nontrivial to optimize. This paper describes a novel majorize-minimize framework for NCC data that allows penalized maximum likelihood estimates to be obtained by solving a series of much simpler regularized least-squares surrogate problems. The proposed framework is general and can be useful in a range of applications. We illustrate the potential advantages of the framework with real and simulated data in two contexts: 1) MR image denoising and 2) diffusion profile estimation in high angular resolution diffusion MRI. The proposed approach is shown to yield improved results compared to methods that model the noise statistics inaccurately and faster computation relative to commonly-used nonlinear optimization techniques.

Leveraging EAP-Sparsity for Compressed Sensing of MS-HARDI in (k, q)-Space

Compressed Sensing (CS) for the acceleration of MR scans has been widely investigated in the past decade. Lately, considerable progress has been made in achieving similar speed ups in acquiring multi-shell high angular resolution diffusion imaging (MS-HARDI) scans. Existing approaches in this context were primarily concerned with sparse reconstruction of the diffusion MR signal S(q) in the q-space. More recently, methods have been developed to apply the compressed sensing framework to the 6-dimensional joint (k, q)-space, thereby exploiting the redundancy in this 6D space. To guarantee accurate reconstruction from partial MS-HARDI data, the key ingredients of compressed sensing that need to be brought together are: (1) the function to be reconstructed needs to have a sparse representation, and (2) the data for reconstruction ought to be acquired in the dual domain (i.e., incoherent sensing) and (3) the reconstruction process involves a (convex) optimization. In this paper, we present a novel approach that uses partial Fourier sensing in the 6D space of (k, q) for the reconstruction of P(x, r). The distinct feature of our approach is a sparsity model that leverages surfacelets in conjunction with total variation for the joint sparse representation of P(x, r). Thus, our method stands to benefit from the practical guarantees for accurate reconstruction from partial (k, q)-space data. Further, we demonstrate significant savings in acquisition time over diffusion spectral imaging (DSI) which is commonly used as the benchmark for comparisons in reported literature. To demonstrate the benefits of this approach,.we present several synthetic and real data examples.

Prospective acceleration of diffusion tensor imaging with compressed sensing using adaptive dictionaries

PURPOSE

Diffusion MRI requires acquisition of multiple diffusion-weighted images, resulting in long scan times. Here, we investigate combining compressed sensing and a fast imaging sequence to dramatically reduce acquisition times in cardiac diffusion MRI.

METHODS

Fully sampled and prospectively undersampled diffusion tensor imaging data were acquired in five rat hearts at acceleration factors of between two and six using a fast spin echo (FSE) sequence. Images were reconstructed using a compressed sensing framework, enforcing sparsity by means of decomposition by adaptive dictionaries. A tensor was fit to the reconstructed images and fiber tractography was performed.

RESULTS

Acceleration factors of up to six were achieved, with a modest increase in root mean square error of mean apparent diffusion coefficient (ADC), fractional anisotropy (FA), and helix angle. At an acceleration factor of six, mean values of ADC and FA were within 2.5% and 5% of the ground truth, respectively. Marginal differences were observed in the fiber tracts.

CONCLUSION

We developed a new k-space sampling strategy for acquiring prospectively undersampled diffusion-weighted data, and validated a novel compressed sensing reconstruction algorithm based on adaptive dictionaries. The k-space undersampling and FSE acquisition each reduced acquisition times by up to 6× and 8×, respectively, as compared to fully sampled spin echo imaging. Magn Reson Med 76:248-258, 2016. © 2015 Wiley Periodicals, Inc.

Validation of tractography: Comparison with manganese tracing

In this study, we used invasive tracing to evaluate white matter tractography methods based on ex vivo diffusion-weighted magnetic resonance imaging (dwMRI) data. A representative selection of tractography methods were compared to manganese tracing on a voxel-wise basis, and a more qualitative assessment examined whether, and to what extent, certain fiber tracts and gray matter targets were reached. While the voxel-wise agreement was very limited, qualitative assessment revealed that tractography is capable of finding the major fiber tracts, although there were some differences between the methods. However, false positive connections were very common and, in particular, we discovered that it is not possible to achieve high sensitivity (i.e., few false negatives) and high specificity (i.e., few false positives) at the same time. Closer inspection of the results led to the conclusion that these problems mainly originate from regions with complex fiber arrangements or high curvature and are not easily resolved by sophisticated local models alone. Instead, the crucial challenge in making tractography a truly useful and reliable tool in brain research and neurology lies in the acquisition of better data. In particular, the increase of spatial resolution, under preservation of the signal-to-noise-ratio, is key.

Segmentation of the Cerebellar Peduncles Using a Random Forest Classifier and a Multi-object Geometric Deformable Model: Application to Spinocerebellar Ataxia Type 6

The cerebellar peduncles, comprising the superior cerebellar peduncles (SCPs), the middle cerebellar peduncle (MCP), and the inferior cerebellar peduncles (ICPs), are white matter tracts that connect the cerebellum to other parts of the central nervous system. Methods for automatic segmentation and quantification of the cerebellar peduncles are needed for objectively and efficiently studying their structure and function. Diffusion tensor imaging (DTI) provides key information to support this goal, but it remains challenging because the tensors change dramatically in the decussation of the SCPs (dSCP), the region where the SCPs cross. This paper presents an automatic method for segmenting the cerebellar peduncles, including the dSCP. The method uses volumetric segmentation concepts based on extracted DTI features. The dSCP and noncrossing portions of the peduncles are modeled as separate objects, and are initially classified using a random forest classifier together with the DTI features. To obtain geometrically correct results, a multi-object geometric deformable model is used to refine the random forest classification. The method was evaluated using a leave-one-out cross-validation on five control subjects and four patients with spinocerebellar ataxia type 6 (SCA6). It was then used to evaluate group differences in the peduncles in a population of 32 controls and 11 SCA6 patients. In the SCA6 group, we have observed significant decreases in the volumes of the dSCP and the ICPs and significant increases in the mean diffusivity in the noncrossing SCPs, the MCP, and the ICPs. These results are consistent with a degeneration of the cerebellar peduncles in SCA6 patients.

Diffusion-MRI in neurodegenerative disorders

The ability to image the whole brain through ever more subtle and specific methods/contrasts has come to play a key role in understanding the basis of brain abnormalities in several diseases. In magnetic resonance imaging (MRI), "diffusion" (i.e. the random, thermally-induced displacements of water molecules over time) represents an extraordinarily sensitive contrast mechanism, and the exquisite structural detail it affords has proven useful in a vast number of clinical as well as research applications. Since diffusion-MRI is a truly quantitative imaging technique, the indices it provides can serve as potential imaging biomarkers which could allow early detection of pathological alterations as well as tracking and possibly predicting subtle changes in follow-up examinations and clinical trials. Accordingly, diffusion-MRI has proven useful in obtaining information to better understand the microstructural changes and neurophysiological mechanisms underlying various neurodegenerative disorders. In this review article, we summarize and explore the main applications, findings, perspectives as well as challenges and future research of diffusion-MRI in various neurodegenerative disorders including Alzheimer's disease, Parkinson's disease, amyotrophic lateral sclerosis, Huntington's disease and degenerative ataxias.

Recovery of CT stroke hypodensity--An adaptive variational approach

The present research was directed to effective image restoration with the extraction of ischemic edema signs. Computerized support of hyperacute stroke diagnosis based on routinely used computerized tomography (CT) scans was optimized to visualize the infarct extent more precisely. In particular, a beneficial support of time-limited appropriate decision of whether to treat the patient by thrombolysis is expected. Because of a limited accuracy in determining the area of core infarction, particularly in the early hours of symptoms' onset, a variational approach to sensed data recovery was applied. Proposed methodology adjusts fidelity norms and regularization priors integrated with simulated sensing procedures in a compressed sensing framework. Experimental study confirmed almost perfect recognition of ischemic stroke in a test set of over 500 CT scans.

Compressive sensing in medical imaging

The promise of compressive sensing, exploitation of compressibility to achieve high quality image reconstructions with less data, has attracted a great deal of attention in the medical imaging community. At the Compressed Sensing Incubator meeting held in April 2014 at OSA Headquarters in Washington, DC, presentations were given summarizing some of the research efforts ongoing in compressive sensing for x-ray computed tomography and magnetic resonance imaging systems. This article provides an expanded version of these presentations. Sparsity-exploiting reconstruction algorithms that have gained popularity in the medical imaging community are studied, and examples of clinical applications that could benefit from compressive sensing ideas are provided. The current and potential future impact of compressive sensing on the medical imaging field is discussed.

A 4D hyperspherical interpretation of q-space

3D q-space can be viewed as the surface of a 4D hypersphere. In this paper, we seek to develop a 4D hyperspherical interpretation of q-space by projecting it onto a hypersphere and subsequently modeling the q-space signal via 4D hyperspherical harmonics (HSH). Using this orthonormal basis, we derive several well-established q-space indices and numerically estimate the diffusion orientation distribution function (dODF). We also derive the integral transform describing the relationship between the diffusion signal and propagator on a hypersphere. Most importantly, we will demonstrate that for hybrid diffusion imaging (HYDI) acquisitions low order linear expansion of the HSH basis is sufficient to characterize diffusion in neural tissue. In fact, the HSH basis achieves comparable signal and better dODF reconstructions than other well-established methods, such as Bessel Fourier orientation reconstruction (BFOR), using fewer fitting parameters. All in all, this work provides a new way of looking at q-space.

A Compressed-Sensing Approach for Super-Resolution Reconstruction of Diffusion MRI

We present an innovative framework for reconstructing high-spatial-resolution diffusion magnetic resonance imaging (dMRI) from multiple low-resolution (LR) images. Our approach combines the twin concepts of compressed sensing (CS) and classical super-resolution to reduce acquisition time while increasing spatial resolution. We use subpixel-shifted LR images with down-sampled and non-overlapping diffusion directions to reduce acquisition time. The diffusion signal in the high resolution (HR) image is represented in a sparsifying basis of spherical ridgelets to model complex fiber orientations with reduced number of measurements. The HR image is obtained as the solution of a convex optimization problem which can be solved using the proposed algorithm based on the alternating direction method of multipliers (ADMM). We qualitatively and quantitatively evaluate the performance of our method on two sets of in-vivo human brain data and show its effectiveness in accurately recovering very high resolution diffusion images.

Accelerated Microstructure Imaging via Convex Optimization (AMICO) from diffusion MRI data

Microstructure imaging from diffusion magnetic resonance (MR) data represents an invaluable tool to study non-invasively the morphology of tissues and to provide a biological insight into their microstructural organization. In recent years, a variety of biophysical models have been proposed to associate particular patterns observed in the measured signal with specific microstructural properties of the neuronal tissue, such as axon diameter and fiber density. Despite very appealing results showing that the estimated microstructure indices agree very well with histological examinations, existing techniques require computationally very expensive non-linear procedures to fit the models to the data which, in practice, demand the use of powerful computer clusters for large-scale applications. In this work, we present a general framework for Accelerated Microstructure Imaging via Convex Optimization (AMICO) and show how to re-formulate this class of techniques as convenient linear systems which, then, can be efficiently solved using very fast algorithms. We demonstrate this linearization of the fitting problem for two specific models, i.e. ActiveAx and NODDI, providing a very attractive alternative for parameter estimation in those techniques; however, the AMICO framework is general and flexible enough to work also for the wider space of microstructure imaging methods. Results demonstrate that AMICO represents an effective means to accelerate the fit of existing techniques drastically (up to four orders of magnitude faster) while preserving accuracy and precision in the estimated model parameters (correlation above 0.9). We believe that the availability of such ultrafast algorithms will help to accelerate the spread of microstructure imaging to larger cohorts of patients and to study a wider spectrum of neurological disorders.

Fast iterative algorithm for the reconstruction of multishot non-cartesian diffusion data

PURPOSE

To accelerate the motion-compensated iterative reconstruction of multishot non-Cartesian diffusion data.

METHOD

The motion-compensated recovery of multishot non-Cartesian diffusion data is often performed using a modified iterative sensitivity-encoded algorithm. Specifically, the encoding matrix is replaced with a combination of nonuniform Fourier transforms and composite sensitivity functions, which account for the motion-induced phase errors. The main challenge with this scheme is the significantly increased computational complexity, which is directly related to the total number of composite sensitivity functions (number of shots × number of coils). The dimensionality of the composite sensitivity functions and hence the number of Fourier transforms within each iteration is reduced using a principal component analysis-based scheme. Using a Toeplitz-based conjugate gradient approach in combination with an augmented Lagrangian optimization scheme, a fast algorithm is proposed for the sparse recovery of diffusion data.

RESULTS

The proposed simplifications considerably reduce the computation time, especially in the recovery of diffusion data from under-sampled reconstructions using sparse optimization. By choosing appropriate number of basis functions to approximate the composite sensitivities, faster reconstruction (close to 9 times) with effective motion compensation is achieved.

CONCLUSION

The proposed enhancements can offer fast motion-compensated reconstruction of multishot diffusion data for arbitrary k-space trajectories.

Multi-shell diffusion signal recovery from sparse measurements

For accurate estimation of the ensemble average diffusion propagator (EAP), traditional multi-shell diffusion imaging (MSDI) approaches require acquisition of diffusion signals for a range of b-values. However, this makes the acquisition time too long for several types of patients, making it difficult to use in a clinical setting. In this work, we propose a new method for the reconstruction of diffusion signals in the entire q-space from highly undersampled sets of MSDI data, thus reducing the scan time significantly. In particular, to sparsely represent the diffusion signal over multiple q-shells, we propose a novel extension to the framework of spherical ridgelets by accurately modeling the monotonically decreasing radial component of the diffusion signal. Further, we enforce the reconstructed signal to have smooth spatial regularity in the brain, by minimizing the total variation (TV) norm. We combine these requirements into a novel cost function and derive an optimal solution using the Alternating Directions Method of Multipliers (ADMM) algorithm. We use a physical phantom data set with known fiber crossing angle of 45° to determine the optimal number of measurements (gradient directions and b-values) needed for accurate signal recovery. We compare our technique with a state-of-the-art sparse reconstruction method (i.e., the SHORE method of Cheng et al. (2010)) in terms of angular error in estimating the crossing angle, incorrect number of peaks detected, normalized mean squared error in signal recovery as well as error in estimating the return-to-origin probability (RTOP). Finally, we also demonstrate the behavior of the proposed technique on human in vivo data sets. Based on these experiments, we conclude that using the proposed algorithm, at least 60 measurements (spread over three b-value shells) are needed for proper recovery of MSDI data in the entire q-space.

Enhancement of the low resolution image quality using randomly sampled data for multi-slice MR imaging

Low resolution images are often acquired in in vivo MR applications involving in large field-of-view (FOV) and high speed imaging, such as, whole-body MRI screening and functional MRI applications. In this work, we investigate a multi-slice imaging strategy for acquiring low resolution images by using compressed sensing (CS) MRI to enhance the image quality without increasing the acquisition time. In this strategy, low resolution images of all the slices are acquired using multiple-slice imaging sequence. In addition, extra randomly sampled data in one center slice are acquired by using the CS strategy. These additional randomly sampled data are multiplied by the weighting functions generated from low resolution full k-space images of the two slices, and then interpolated into the k-space of other slices. In vivo MR images of human brain were employed to investigate the feasibility and the performance of the proposed method. Quantitative comparison between the conventional low resolution images and those from the proposed method was also performed to demonstrate the advantage of the method.

Sparse parallel transmission on randomly perturbed spiral k-space trajectory

Combination of parallel transmission and sparse pulse is able to shorten the excitation by using both the coil sensitivity and sparse k-space, showing improved fast excitation capability over the use of parallel transmission alone. However, to design an optimal k-space trajectory for sparse parallel transmission is a challenging task. In this work, a randomly perturbed sparse k-space trajectory is designed by modifying the path of a spiral trajectory along the sparse k-space data, and the sparse parallel transmission RF pulses are subsequently designed based on this optimal trajectory. This method combines the parallel transmission and sparse spiral k-space trajectory, potentially to further reduce the RF transmission time. Bloch simulation of 90° excitation by using a four channel coil array is performed to demonstrate its feasibility. Excitation performance of the sparse parallel transmission technique at different reduction factors of 1, 2, and 4 is evaluated. For comparison, parallel excitation using regular spiral trajectory is performed. The passband errors of the excitation profiles of each transmission are calculated for quantitative assessment of the proposed excitation method.

Parallel imaging and compressed sensing combined framework for accelerating high-resolution diffusion tensor imaging using inter-image correlation

PURPOSE

Increasing acquisition efficiency is always a challenge in high-resolution diffusion tensor imaging (DTI), which has low signal-to-noise ratio and is sensitive to reconstruction artifacts. In this study, a parallel imaging (PI) and compressed sensing (CS) combined framework is proposed, which features motion error correction, PI calibration, and sparsity model using inter-image correlation tailored for high-resolution DTI.

THEORY AND METHODS

The proposed method, named anisotropic sparsity SPIRiT, consists of three steps: (i) motion-induced phase error estimation, (ii) initial CS reconstruction and PI kernel calibration, and (iii) final reconstruction combining PI and CS. Inter-image correlation of diffusion-weighted images are used through anisotropic signals for improved sparsity. A specific implementation based on multishot variable density spiral DTI is used to demonstrate the method.

RESULTS

The proposed reconstruction method was compared with CG-SENSE, CS-based joint reconstruction, and PI and CS combined methods with L1 and joint sparsity regularization, in brain DTI experiments at acceleration factors of 3 to 5. Both qualitative and quantitative results demonstrated that the proposed method resulted in better preserved image quality and more accurate DTI parameters than other methods.

CONCLUSION

The proposed method can accelerate high-resolution DTI acquisition effectively by using the sharable information among different diffusion encoding directions.

Neuroimaging in repetitive brain trauma

Sports-related concussions are one of the major causes of mild traumatic brain injury. Although most patients recover completely within days to weeks, those who experience repetitive brain trauma (RBT) may be at risk for developing a condition known as chronic traumatic encephalopathy (CTE). While this condition is most commonly observed in athletes who experience repetitive concussive and/or subconcussive blows to the head, such as boxers, football players, or hockey players, CTE may also affect soldiers on active duty. Currently, the only means by which to diagnose CTE is by the presence of phosphorylated tau aggregations post-mortem. Non-invasive neuroimaging, however, may allow early diagnosis as well as improve our understanding of the underlying pathophysiology of RBT. The purpose of this article is to review advanced neuroimaging methods used to investigate RBT, including diffusion tensor imaging, magnetic resonance spectroscopy, functional magnetic resonance imaging, susceptibility weighted imaging, and positron emission tomography. While there is a considerable literature using these methods in brain injury in general, the focus of this review is on RBT and those subject populations currently known to be susceptible to RBT, namely athletes and soldiers. Further, while direct detection of CTE in vivo has not yet been achieved, all of the methods described in this review provide insight into RBT and will likely lead to a better characterization (diagnosis), in vivo, of CTE than measures of self-report.

Sparse regularization for fiber ODF reconstruction: from the suboptimality of ℓ2 and ℓ1 priors to ℓ0

Diffusion MRI is a well established imaging modality providing a powerful way to probe the structure of the white matter non-invasively. Despite its potential, the intrinsic long scan times of these sequences have hampered their use in clinical practice. For this reason, a large variety of methods have been recently proposed to shorten the acquisition times. Among them, spherical deconvolution approaches have gained a lot of interest for their ability to reliably recover the intra-voxel fiber configuration with a relatively small number of data samples. To overcome the intrinsic instabilities of deconvolution, these methods use regularization schemes generally based on the assumption that the fiber orientation distribution (FOD) to be recovered in each voxel is sparse. The well known Constrained Spherical Deconvolution (CSD) approach resorts to Tikhonov regularization, based on an ℓ(2)-norm prior, which promotes a weak version of sparsity. Also, in the last few years compressed sensing has been advocated to further accelerate the acquisitions and ℓ(1)-norm minimization is generally employed as a means to promote sparsity in the recovered FODs. In this paper, we provide evidence that the use of an ℓ(1)-norm prior to regularize this class of problems is somewhat inconsistent with the fact that the fiber compartments all sum up to unity. To overcome this ℓ(1) inconsistency while simultaneously exploiting sparsity more optimally than through an ℓ(2) prior, we reformulate the reconstruction problem as a constrained formulation between a data term and a sparsity prior consisting in an explicit bound on the ℓ(0)norm of the FOD, i.e. on the number of fibers. The method has been tested both on synthetic and real data. Experimental results show that the proposed ℓ(0) formulation significantly reduces modeling errors compared to the state-of-the-art ℓ(2) and ℓ(1) regularization approaches.

Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion MRI

Validation is arguably the bottleneck in the diffusion magnetic resonance imaging (MRI) community. This paper evaluates and compares 20 algorithms for recovering the local intra-voxel fiber structure from diffusion MRI data and is based on the results of the "HARDI reconstruction challenge" organized in the context of the "ISBI 2012" conference. Evaluated methods encompass a mixture of classical techniques well known in the literature such as diffusion tensor, Q-Ball and diffusion spectrum imaging, algorithms inspired by the recent theory of compressed sensing and also brand new approaches proposed for the first time at this contest. To quantitatively compare the methods under controlled conditions, two datasets with known ground-truth were synthetically generated and two main criteria were used to evaluate the quality of the reconstructions in every voxel: correct assessment of the number of fiber populations and angular accuracy in their orientation. This comparative study investigates the behavior of every algorithm with varying experimental conditions and highlights strengths and weaknesses of each approach. This information can be useful not only for enhancing current algorithms and develop the next generation of reconstruction methods, but also to assist physicians in the choice of the most adequate technique for their studies.

Comparison of sampling strategies and sparsifying transforms to improve compressed sensing diffusion spectrum imaging

PURPOSE

Diffusion Spectrum Imaging enables to reconstruct the ensemble average propagator (EAP) at the expense of having to acquire a large number of measurements. Compressive sensing offers an efficient way to decrease the required number of measurements. The purpose of this work is to perform a thorough experimental comparison of three sampling strategies and six sparsifying transforms to show their impact when applied to accelerate compressive sensing-diffusion spectrum imaging.

METHODS

We propose a novel sampling scheme that assures uniform angular and random radial q-space samples. We also compare and implement six discrete sparse representations of the EAP and thoroughly evaluate them on synthetic and real data using metrics from the full EAP, kurtosis, and orientation distribution function.

RESULTS

The discrete wavelet transform with Cohen-Daubechies-Feauveau 9/7 wavelets and uniform angular sampling in combination with random radial sampling showed to be better than other tested techniques to accurately reconstruct the EAP and its features.

CONCLUSION

It is important to jointly optimize the sampling scheme and the sparsifying transform to obtain accelerated compressive sensing-diffusion spectrum imaging. Experiments on synthetic and real human brain data show that one can robustly recover both radial and angular EAP features while undersampling the acquisition to 64 measurements (undersampling factor of 4).

Multidimensional MR spectroscopic imaging of prostate cancer in vivo

Prostate cancer (PCa) is the second most common type of cancer among men in the United States. A major limitation in the management of PCa is an inability to distinguish, early on, cancers that will progress and become life threatening. One-dimensional (1D) proton ((1)H) MRS of the prostate provides metabolic information such as levels of choline (Ch), creatine (Cr), citrate (Cit), and spermine (Spm) that can be used to detect and diagnose PCa. Ex vivo high-resolution magic angle spinning (HR-MAS) of PCa specimens has revealed detection of more metabolites such as myo-inositol (mI), glutamate (Glu), and glutamine (Gln). Due to the J-modulation and signal overlap, it is difficult to quantitate Spm and other resonances in the prostate clearly by single- and multivoxel-based 1D MR spectroscopy. This limitation can be minimized by adding at least one more spectral dimension by which resonances can be spread apart, thereby increasing the spectral dispersion. However, recording of multivoxel-based two-dimensional (2D) MRS such as J-resolved spectroscopy (JPRESS) and correlated spectroscopy (L-COSY) combined with 2D or three-dimensional (3D) magnetic resonance spectroscopic imaging (MRSI) using conventional phase-encoding can be prohibitively long to be included in a clinical protocol. To reduce the long acquisition time required for spatial encoding, the echo-planar spectroscopic imaging (EPSI) technique has been combined with correlated spectroscopy to give four-dimensional (4D) echo-planar correlated spectroscopic imaging (EP-COSI) as well as J-resolved spectroscopic imaging (EP-JRESI) and the multi-echo (ME) variants. Further acceleration can be achieved using non-uniform undersampling (NUS) and reconstruction using compressed sensing (CS). Earlier versions of 2D MRS, theory of 2D MRS, spectral apodization filters, newer developments and the potential role of multidimensional MRS in PCa detection and management will be reviewed here.

Estimation of the CSA-ODF using Bayesian compressed sensing of multi-shell HARDI

PURPOSE

Diffusion MRI provides important information about the brain white matter structures and has opened new avenues for neuroscience and translational research. However, acquisition time needed for advanced applications can still be a challenge in clinical settings. There is consequently a need to accelerate diffusion MRI acquisitions.

METHODS

A multi-task Bayesian compressive sensing (MT-BCS) framework is proposed to directly estimate the constant solid angle orientation distribution function (CSA-ODF) from under-sampled (i.e., accelerated image acquisition) multi-shell high angular resolution diffusion imaging (HARDI) datasets, and accurately recover HARDI data at higher resolution in q-space. The proposed MT-BCS approach exploits the spatial redundancy of the data by modeling the statistical relationships within groups (clusters) of diffusion signal. This framework also provides uncertainty estimates of the computed CSA-ODF and diffusion signal, directly computed from the compressive measurements. Experiments validating the proposed framework are performed using realistic multi-shell synthetic images and in vivo multi-shell high angular resolution HARDI datasets.

RESULTS

Results indicate a practical reduction in the number of required diffusion volumes (q-space samples) by at least a factor of four to estimate the CSA-ODF from multi-shell data.

CONCLUSION

This work presents, for the first time, a multi-task Bayesian compressive sensing approach to simultaneously estimate the full posterior of the CSA-ODF and diffusion-weighted volumes from multi-shell HARDI acquisitions. It demonstrates improvement of the quality of acquired datasets by means of CS de-noising, and accurate estimation of the CSA-ODF, as well as enables a reduction in the acquisition time by a factor of two to four, especially when "staggered" q-space sampling schemes are used. The proposed MT-BCS framework can naturally be combined with parallel MR imaging to further accelerate HARDI acquisitions.

PCLR: phase-constrained low-rank model for compressive diffusion-weighted MRI

PURPOSE

This work develops a compressive sensing approach for diffusion-weighted (DW) MRI.

THEORY AND METHODS

A phase-constrained low-rank (PCLR) approach was developed using the image coherence across the DW directions for efficient compressive DW MRI, while accounting for drastic phase changes across the DW directions, possibly as a result of eddy current, and rigid and nonrigid motions. In PCLR, a low-resolution phase estimation was used for removing phase inconsistency between DW directions. In our implementation, GRAPPA (generalized autocalibrating partial parallel acquisition) was incorporated for better phase estimation while allowing higher undersampling factor. An efficient and easy-to-implement image reconstruction algorithm, consisting mainly of partial Fourier update and singular value decomposition, was developed for solving PCLR.

RESULTS

The error measures based on diffusion-tensor-derived metrics and tractography indicated that PCLR, with its joint reconstruction of all DW images using the image coherence, outperformed the frame-independent reconstruction through zero-padding FFT. Furthermore, using GRAPPA for phase estimation, PCLR readily achieved a four-fold undersampling.

CONCLUSION

The PCLR is developed and demonstrated for compressive DW MRI. A four-fold reduction in k-space sampling could be readily achieved without substantial degradation of reconstructed images and diffusion tensor measures, making it possible to significantly reduce the data acquisition in DW MRI and/or improve spatial and angular resolutions.

Fast dictionary-based reconstruction for diffusion spectrum imaging

Diffusion spectrum imaging reveals detailed local diffusion properties at the expense of substantially long imaging times. It is possible to accelerate acquisition by undersampling in q-space, followed by image reconstruction that exploits prior knowledge on the diffusion probability density functions (pdfs). Previously proposed methods impose this prior in the form of sparsity under wavelet and total variation transforms, or under adaptive dictionaries that are trained on example datasets to maximize the sparsity of the representation. These compressed sensing (CS) methods require full-brain processing times on the order of hours using MATLAB running on a workstation. This work presents two dictionary-based reconstruction techniques that use analytical solutions, and are two orders of magnitude faster than the previously proposed dictionary-based CS approach. The first method generates a dictionary from the training data using principal component analysis (PCA), and performs the reconstruction in the PCA space. The second proposed method applies reconstruction using pseudoinverse with Tikhonov regularization with respect to a dictionary. This dictionary can either be obtained using the K-SVD algorithm, or it can simply be the training dataset of pdfs without any training. All of the proposed methods achieve reconstruction times on the order of seconds per imaging slice, and have reconstruction quality comparable to that of dictionary-based CS algorithm.

Pushing spatial and temporal resolution for functional and diffusion MRI in the Human Connectome Project

The Human Connectome Project (HCP) relies primarily on three complementary magnetic resonance (MR) methods. These are: 1) resting state functional MR imaging (rfMRI) which uses correlations in the temporal fluctuations in an fMRI time series to deduce 'functional connectivity'; 2) diffusion imaging (dMRI), which provides the input for tractography algorithms used for the reconstruction of the complex axonal fiber architecture; and 3) task based fMRI (tfMRI), which is employed to identify functional parcellation in the human brain in order to assist analyses of data obtained with the first two methods. We describe technical improvements and optimization of these methods as well as instrumental choices that impact speed of acquisition of fMRI and dMRI images at 3T, leading to whole brain coverage with 2 mm isotropic resolution in 0.7 s for fMRI, and 1.25 mm isotropic resolution dMRI data for tractography analysis with three-fold reduction in total dMRI data acquisition time. Ongoing technical developments and optimization for acquisition of similar data at 7 T magnetic field are also presented, targeting higher spatial resolution, enhanced specificity of functional imaging signals, mitigation of the inhomogeneous radio frequency (RF) fields, and reduced power deposition. Results demonstrate that overall, these approaches represent a significant advance in MR imaging of the human brain to investigate brain function and structure.

A connectome-based comparison of diffusion MRI schemes

Diffusion MRI has evolved towards an important clinical diagnostic and research tool. Though clinical routine is using mainly diffusion weighted and tensor imaging approaches, Q-ball imaging and diffusion spectrum imaging techniques have become more widely available. They are frequently used in research-oriented investigations in particular those aiming at measuring brain network connectivity. In this work, we aim at assessing the dependency of connectivity measurements on various diffusion encoding schemes in combination with appropriate data modeling. We process and compare the structural connection matrices computed from several diffusion encoding schemes, including diffusion tensor imaging, q-ball imaging and high angular resolution schemes, such as diffusion spectrum imaging with a publically available processing pipeline for data reconstruction, tracking and visualization of diffusion MR imaging. The results indicate that the high angular resolution schemes maximize the number of obtained connections when applying identical processing strategies to the different diffusion schemes. Compared to the conventional diffusion tensor imaging, the added connectivity is mainly found for pathways in the 50-100mm range, corresponding to neighboring association fibers and long-range associative, striatal and commissural fiber pathways. The analysis of the major associative fiber tracts of the brain reveals striking differences between the applied diffusion schemes. More complex data modeling techniques (beyond tensor model) are recommended 1) if the tracts of interest run through large fiber crossings such as the centrum semi-ovale, or 2) if non-dominant fiber populations, e.g. the neighboring association fibers are the subject of investigation. An important finding of the study is that since the ground truth sensitivity and specificity is not known, the comparability between results arising from different strategies in data reconstruction and/or tracking becomes implausible to understand.

Denoising and fast diffusion imaging with physically constrained sparse dictionary learning

Diffusion-weighted imaging (DWI) allows imaging the geometry of water diffusion in biological tissues. However, DW images are noisy at high b-values and acquisitions are slow when using a large number of measurements, such as in Diffusion Spectrum Imaging (DSI). This work aims to denoise DWI and reduce the number of required measurements, while maintaining data quality. To capture the structure of DWI data, we use sparse dictionary learning constrained by the physical properties of the signal: symmetry and positivity. The method learns a dictionary of diffusion profiles on all the DW images at the same time and then scales to full brain data. Its performance is investigated with simulations and two real DSI datasets. We obtain better signal estimates from noisy measurements than by applying mirror symmetry through the q-space origin, Gaussian denoising or state-of-the-art non-local means denoising. Using a high-resolution dictionary learnt on another subject, we show that we can reduce the number of images acquired while still generating high resolution DSI data. Using dictionary learning, one can denoise DW images effectively and perform faster acquisitions. Higher b-value acquisitions and DSI techniques are possible with approximately 40 measurements. This opens important perspectives for the connectomics community using DSI.

Optic radiation fiber tractography in glioma patients based on high angular resolution diffusion imaging with compressed sensing compared with diffusion tensor imaging - initial experience

OBJECTIVE

Up to now, fiber tractography in the clinical routine is mostly based on diffusion tensor imaging (DTI). However, there are known drawbacks in the resolution of crossing or kissing fibers and in the vicinity of a tumor or edema. These restrictions can be overcome by tractography based on High Angular Resolution Diffusion Imaging (HARDI) which in turn requires larger numbers of gradients resulting in longer acquisition times. Using compressed sensing (CS) techniques, HARDI signals can be obtained by using less non-collinear diffusion gradients, thus enabling the use of HARDI-based fiber tractography in the clinical routine.

METHODS

Eight patients with gliomas in the temporal lobe, in proximity to the optic radiation (OR), underwent 3T MRI including a diffusion-weighted dataset with 30 gradient directions. Fiber tractography of the OR using a deterministic streamline algorithm based on DTI was compared to tractography based on reconstructed diffusion signals using HARDI+CS.

RESULTS

HARDI+CS based tractography displayed the OR more conclusively compared to the DTI-based results in all eight cases. In particular, the potential of HARDI+CS-based tractography was observed for cases of high grade gliomas with significant peritumoral edema, larger tumor size or closer proximity of tumor and reconstructed fiber tract.

CONCLUSIONS

Overcoming the problem of long acquisition times, HARDI+CS seems to be a promising basis for fiber tractography of the OR in regions of disturbed diffusion, areas of high interest in glioma surgery.

Fiber tractography based on diffusion tensor imaging compared with high-angular-resolution diffusion imaging with compressed sensing: initial experience

BACKGROUND

The most frequently used method for fiber tractography based on diffusion tensor imaging (DTI) is associated with restrictions in the resolution of crossing or kissing fibers and in the vicinity of tumor or edema. Tractography based on high-angular-resolution diffusion imaging (HARDI) is capable of overcoming this restriction. With compressed sensing (CS) techniques, HARDI acquisitions with a smaller number of directional measurements can be used, thus enabling the use of HARDI-based fiber tractography in clinical practice.

OBJECTIVE

To investigate whether HARDI+CS-based fiber tractography improves the display of neuroanatomically complex pathways and in areas of disturbed diffusion properties.

METHODS

Six patients with gliomas in the vicinity of language-related areas underwent 3-T magnetic resonance imaging including a diffusion-weighted data set with 30 gradient directions. Additionally, functional magnetic resonance imaging for cortical language sites was obtained. Fiber tractography was performed with deterministic streamline algorithms based on DTI using 3 different software platforms. Additionally, tractography based on reconstructed diffusion signals using HARDI+CS was performed.

RESULTS

HARDI+CS-based tractography displayed more compact fiber bundles compared with the DTI-based results in all cases. In 3 cases, neuroanatomically plausible fiber bundles were displayed in the vicinity of tumor and peritumoral edema, which could not be traced on the basis of DTI. The curvature around the sylvian fissure was displayed properly in 6 cases and in only 2 cases with DTI-based tractography.

CONCLUSION

HARDI+CS seems to be a promising approach for fiber tractography in clinical practice for neuroanatomically complex fiber pathways and in areas of disturbed diffusion, overcoming the problem of long acquisition times.

Pushing the limits of in vivo diffusion MRI for the Human Connectome Project

Perhaps more than any other "-omics" endeavor, the accuracy and level of detail obtained from mapping the major connection pathways in the living human brain with diffusion MRI depend on the capabilities of the imaging technology used. The current tools are remarkable; allowing the formation of an "image" of the water diffusion probability distribution in regions of complex crossing fibers at each of half a million voxels in the brain. Nonetheless our ability to map the connection pathways is limited by the image sensitivity and resolution, and also the contrast and resolution in encoding of the diffusion probability distribution. The goal of our Human Connectome Project (HCP) is to address these limiting factors by re-engineering the scanner from the ground up to optimize the high b-value, high angular resolution diffusion imaging needed for sensitive and accurate mapping of the brain's structural connections. Our efforts were directed based on the relative contributions of each scanner component. The gradient subsection was a major focus since gradient amplitude is central to determining the diffusion contrast, the amount of T2 signal loss, and the blurring of the water PDF over the course of the diffusion time. By implementing a novel 4-port drive geometry and optimizing size and linearity for the brain, we demonstrate a whole-body sized scanner with G(max) = 300 mT/m on each axis capable of the sustained duty cycle needed for diffusion imaging. The system is capable of slewing the gradient at a rate of 200 T/m/s as needed for the EPI image encoding. In order to enhance the efficiency of the diffusion sequence we implemented a FOV shifting approach to Simultaneous MultiSlice (SMS) EPI capable of unaliasing 3 slices excited simultaneously with a modest g-factor penalty allowing us to diffusion encode whole brain volumes with low TR and TE. Finally we combine the multi-slice approach with a compressive sampling reconstruction to sufficiently undersample q-space to achieve a DSI scan in less than 5 min. To augment this accelerated imaging approach we developed a 64-channel, tight-fitting brain array coil and show its performance benefit compared to a commercial 32-channel coil at all locations in the brain for these accelerated acquisitions. The technical challenges of developing the over-all system are discussed as well as results from SNR comparisons, ODF metrics and fiber tracking comparisons. The ultra-high gradients yielded substantial and immediate gains in the sensitivity through reduction of TE and improved signal detection and increased efficiency of the DSI or HARDI acquisition, accuracy and resolution of diffusion tractography, as defined by identification of known structure and fiber crossing.

Reconstruction of white matter tracts via repeated deterministic streamline tracking--initial experience

Diffusion Tensor Imaging (DTI) and fiber tractography are established methods to reconstruct major white matter tracts in the human brain in-vivo. Particularly in the context of neurosurgical procedures, reliable information about the course of fiber bundles is important to minimize postoperative deficits while maximizing the tumor resection volume. Since routinely used deterministic streamline tractography approaches often underestimate the spatial extent of white matter tracts, a novel approach to improve fiber segmentation is presented here, considering clinical time constraints. Therefore, fiber tracking visualization is enhanced with statistical information from multiple tracking applications to determine uncertainty in reconstruction based on clinical DTI data. After initial deterministic fiber tracking and centerline calculation, new seed regions are generated along the result’s midline. Tracking is applied to all new seed regions afterwards, varying in number and applied offset. The number of fibers passing each voxel is computed to model different levels of fiber bundle membership. Experimental results using an artificial data set of an anatomical software phantom are presented, using the Dice Similarity Coefficient (DSC) as a measure of segmentation quality. Different parameter combinations were classified to be superior to others providing significantly improved results with DSCs of 81.02%±4.12%, 81.32%±4.22% and 80.99%±3.81% for different levels of added noise in comparison to the deterministic fiber tracking procedure using the two-ROI approach with average DSCs of 65.08%±5.31%, 64.73%±6.02% and 65.91%±6.42%. Whole brain tractography based on the seed volume generated by the calculated seeds delivers average DSCs of 67.12%±0.86%, 75.10%±0.28% and 72.91%±0.15%, original whole brain tractography delivers DSCs of 67.16%, 75.03% and 75.54%, using initial ROIs as combined include regions, which is clearly improved by the repeated fiber tractography method.

Fiber density estimation from single q-shell diffusion imaging by tensor divergence

Diffusion-weighted magnetic resonance imaging provides information about the nerve fiber bundle geometry of the human brain. While the inference of the underlying fiber bundle orientation only requires single q-shell measurements, the absolute determination of their volume fractions is much more challenging with respect to measurement techniques and analysis. Unfortunately, the usually employed multi-compartment models cannot be applied to single q-shell measurements, because the compartment's diffusivities cannot be resolved. This work proposes an equation for fiber orientation densities that can infer the absolute fraction up to a global factor. This equation, which is inspired by the classical mass preservation law in fluid dynamics, expresses the fiber conservation associated with the assumption that fibers do not terminate in white matter. Simulations on synthetic phantoms show that the approach is able to derive the densities correctly for various configurations. Experiments with a pseudo ground truth phantom show that even for complex, brain-like geometries the method is able to infer the densities correctly. In-vivo results with 81 healthy volunteers are plausible and consistent. A group analysis with respect to age and gender show significant differences, such that the proposed maps can be used as a quantitative measure for group and longitudinal analysis.

Accelerated MR diffusion tensor imaging using distributed compressed sensing

PURPOSE

Diffusion tensor imaging (DTI) is known to suffer from long acquisition time in the orders of several minutes or even hours. Therefore, a feasible way to accelerate DTI data acquisition is highly desirable. In this article, the feasibility and efficacy of distributed compressed sensing to fast DTI is investigated by exploiting the joint sparsity prior in diffusion-weighted images.

METHODS

Fully sampled DTI datasets were obtained from both simulated phantom and experimental heart sample, with diffusion gradient applied in six directions. The k-space data were undersampled retrospectively with acceleration factors from 2 to 6. Diffusion-weighted images were reconstructed by solving an l2-l1 norm minimization problem. Reconstruction performance with varied signal-to-noise ratio and acceleration factors were evaluated by root-mean-square error and maps of reconstructed DTI indices.

RESULTS

Superiority of distributed compressed sensing over basic compressed sensing was confirmed with simulation, and the reconstruction accuracy was influenced by signal-to-noise ratio and acceleration factors. Experimental results demonstrate that DTI indices including fractional anisotropy, mean diffusivities, and orientation of primary eigenvector can be obtained with high accuracy at acceleration factors up to 4.

CONCLUSION

Distributed compressed sensing is shown to be able to accelerate DTI and may be used to reduce DTI acquisition time practically.

Interpolated compressed sensing for 2D multiple slice fast MR imaging

Sparse MRI has been introduced to reduce the acquisition time and raw data size by undersampling the k-space data. However, the image quality, particularly the contrast to noise ratio (CNR), decreases with the undersampling rate. In this work, we proposed an interpolated Compressed Sensing (iCS) method to further enhance the imaging speed or reduce data size without significant sacrifice of image quality and CNR for multi-slice two-dimensional sparse MR imaging in humans. This method utilizes the k-space data of the neighboring slice in the multi-slice acquisition. The missing k-space data of a highly undersampled slice are estimated by using the raw data of its neighboring slice multiplied by a weighting function generated from low resolution full k-space reference images. In-vivo MR imaging in human feet has been used to investigate the feasibility and the performance of the proposed iCS method. The results show that by using the proposed iCS reconstruction method, the average image error can be reduced and the average CNR can be improved, compared with the conventional sparse MRI reconstruction at the same undersampling rate.

Sparse DSI: learning DSI structure for denoising and fast imaging

Diffusion spectrum imaging (DSI) from multiple diffusion-weighted images (DWI) allows to image the complex geometry of water diffusion in biological tissue. To capture the structure of DSI data, we propose to use sparse coding constrained by physical properties of the signal, namely symmetry and positivity, to learn a dictionary of diffusion profiles. Given this estimated model of the signal, we can extract better estimates of the signal from noisy measurements and also speed up acquisition by reducing the number of acquired DWI while giving access to high resolution DSI data. The method learns jointly for all the acquired DWI and scales to full brain data. Working with two sets of 515 DWI images acquired on two different subjects we show that using just half of the data (258 DWI) we can better predict the other 257 DWI than the classic symmetry procedure. The observation holds even if the diffusion profiles are estimated on a different subject dataset from an undersampled q-space of 40 measurements.

Diffusion propagator estimation from sparse measurements in a tractography framework

Estimation of the diffusion propagator from a sparse set of diffusion MRI (dMRI) measurements is a field of active research. Sparse reconstruction methods propose to reduce scan time and are particularly suitable for scanning un-coperative patients. Recent work on reconstructing the diffusion signal from very few measurements using compressed sensing based techniques has focussed on propagator (or signal) estimation at each voxel independently. However, the goal of many neuroscience studies is to use tractography to study the pathology in white matter fiber tracts. Thus, in this work, we propose a joint framework for robust estimation of the diffusion propagator from sparse measurements while simultaneously tracing the white matter tracts. We propose to use a novel multi-tensor model of diffusion which incorporates the biexponential radial decay of the signal. Our preliminary results on in-vivo data show that the proposed method produces consistent and reliable fiber tracts from very few gradient directions while simultaneously estimating the bi-exponential decay of the diffusion propagator.

Diffusion spectrum MRI using body-centered-cubic and half-sphere sampling schemes

The optimum sequence parameters of diffusion spectrum MRI (DSI) on clinical scanners were investigated previously. However, the scan time of approximately 30 min is still too long for patient studies. Additionally, relatively large sampling interval in the diffusion-encoding space may cause aliasing artifact in the probability density function when Fourier transform is undertaken, leading to estimation error in fiber orientations. Therefore, this study proposed a non-Cartesian sampling scheme, body-centered-cubic (BCC), to avoid the aliasing artifact as compared to the conventional Cartesian grid sampling scheme (GRID). Furthermore, the accuracy of DSI with the use of half-sphere sampling schemes, i.e. GRID102 and BCC91, was investigated by comparing to their full-sphere sampling schemes, GRID203 and BCC181, respectively. In results, smaller deviation angle and lower angular dispersion were obtained by using the BCC sampling scheme. The half-sphere sampling schemes yielded angular precision and accuracy comparable to the full-sphere sampling schemes. The optimum b(max) was approximately 4750 s/mm(2) for GRID and 4500 s/mm(2) for BCC. In conclusion, the BCC sampling scheme could be implemented as a useful alternative to the GRID sampling scheme. Combination of BCC and half-sphere sampling schemes, that is BCC91, may potentially reduce the scan time of DSI from 30 min to approximately 14 min while maintaining its precision and accuracy.

Model-based reconstruction of undersampled diffusion tensor k-space data

The practical utility of diffusion tensor imaging, especially for 3D high-resolution spin warp experiments of ex vivo specimens, has been hampered by long acquisition times. To accelerate the acquisition, a compressed sensing framework that uses a model-based formulation to reconstruct diffusion tensor fields from undersampled k-space data was presented and evaluated. Accuracies in brain specimen white matter fiber orientation, fractional anisotropy, and mean diffusivity mapping were compared with alternative methods achievable using the same scan time via reduced image resolution, fewer diffusion encoding directions, standard compressed sensing, or asymmetrical sampling reconstruction. The efficiency of the proposed approach was also compared with fully sampled cases across a range of the number of diffusion encoding directions. In general, the proposed approach was found to reduce the image blurring and noise and to provide more accurate fiber orientation, fractional anisotropy, and mean diffusivity measurements compared with the alternative methods. Moreover, depending on the degree of undersampling used and the diffusion tensor imaging parameter examined, the measurement accuracy of the proposed scheme was equivalent to fully sampled diffusion tensor imaging datasets that consist of 33-67% more encoding directions and require proportionally longer scan times. The findings show model-based compressed sensing to be promising for improving the resolution, accuracy, or scan time of diffusion tensor imaging.

White matter integrity, fiber count, and other fallacies: the do's and don'ts of diffusion MRI

Diffusion-weighted MRI (DW-MRI) has been increasingly used in imaging neuroscience over the last decade. An early form of this technique, diffusion tensor imaging (DTI) was rapidly implemented by major MRI scanner companies as a scanner selling point. Due to the ease of use of such implementations, and the plausibility of some of their results, DTI was leapt on by imaging neuroscientists who saw it as a powerful and unique new tool for exploring the structural connectivity of human brain. However, DTI is a rather approximate technique, and its results have frequently been given implausible interpretations that have escaped proper critique and have appeared misleadingly in journals of high reputation. In order to encourage the use of improved DW-MRI methods, which have a better chance of characterizing the actual fiber structure of white matter, and to warn against the misuse and misinterpretation of DTI, we review the physics of DW-MRI, indicate currently preferred methodology, and explain the limits of interpretation of its results. We conclude with a list of 'Do's and Don'ts' which define good practice in this expanding area of imaging neuroscience.

AN OVER-COMPLETE DICTIONARY BASED REGULARIZED RECONSTRUCTION OF A FIELD OF ENSEMBLE AVERAGE PROPAGATORS

In this paper we present a dictionary-based framework for the reconstruction of a field of ensemble average propagators (EAPs), given a high angular resolution diffusion MRI data set. Existing techniques often consider voxel-wise reconstruction of the EAP field thereby leading to a noisy reconstruction across the field. We present a dictionary learning framework for achieving a smooth EAP reconstruction across the field wherein, the dictionary atoms are learned from the data via an initial regression using adaptive spline kernels. The formulation involves a two stage optimization where the first stage involves optimizing for a sparse dictionary using a K-SVD based updating and the second stage involves a quadratic cost function optimization with a non-local means based regularization across the field. The novelty lies in a dictionary based reconstruction as well as an NLM-based regularization that helps preserving features in the reconstructed field. We document experimental results on synthetic data from crossing fibers and real optic chiasm data set that demonstrate the advantages of the proposed approach.