Non-Negative Spherical Deconvolution (NNSD) for estimation of fiber Orientation Distribution Function in single-/multi-shell diffusion MRI

Participants : Jian Cheng [University of North Carolina at Chapel Hill,USA] , Tianzi Jiang [LIAMA, China] , Shen Dinggang [University of North Carolina at Chapel Hill,USA] , Yap Pew-Thian [University of North Carolina at Chapel Hill,USA] , Rachid Deriche.

Spherical Deconvolution (SD) is commonly used for estimating fiber Orientation Distribution Functions (fODFs) from diffusion-weighted signals. Existing SD methods can be classified into two categories: 1) Continuous Representation based SD (CR-SD), where typically Spherical Harmonic (SH) representation is used for convenient analytical solutions, and 2) Discrete Representation based SD (DR-SD), where the signal profile is represented by a discrete set of basis functions uniformly oriented on the unit sphere. A feasible fODF should be non-negative and should integrate to unity throughout the unit sphere SS2. However, to our knowledge, most existing SH-based SD methods enforce non-negativity only on discretized points and not the whole continuum of SS2. Maximum Entropy SD (MESD) and Cartesian Tensor Fiber Orientation Distributions (CT-FOD) are the only SD methods that ensure non-negativity throughout the unit sphere. They are however computational intensive and are susceptible to errors caused by numerical spherical integration. Existing SD methods are also known to overestimate the number of fiber directions, especially in regions with low anisotropy. DR-SD introduces additional error in peak detection owing to the angular discretization of the unit sphere. This work proposes a SD framework, called Non-Negative SD (NNSD), to overcome all the limitations above. NNSD is significantly less susceptible to the false-positive peaks, uses SH representation for efficient analytical spherical deconvolution, and allows accurate peak detection throughout the whole unit sphere. We further show that NNSD and most existing SD methods can be extended to work on multi-shell data by introducing a three-dimensional fiber response function. We evaluated NNSD in comparison with Constrained SD (CSD), a quadratic programming variant of CSD, MESD, and an L1-norm regularized non-negative least-squares DR-SD. Experiments on synthetic and real single-/multi-shell data indicate that NNSD improves estimation performance in terms of mean difference of angles, peak detection consistency, and anisotropy contrast between isotropic and anisotropic regions.

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

Participants : Alessandro Daducci [LTS5, Ecole Polytech. Fed. de Lausanne (EPFL)] , Maxime Descoteaux [SCIL Lab., Sherbrooke University] , Michael Paquette [SCIL Lab., Sherbrooke University] , Sylvain Merlet, Emmanuel Caruyer, Rachid Deriche.

Validation is arguably the bottleneck in the diffusion magnetic resonance imaging (MRI) community. This work 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

Participants : Michael Paquette [SCIL Lab., Sherbrooke University] , Sylvain Merlet, Guillaume Gilbert [SCIL Lab., Sherbrooke University] , Maxime Descoteaux [SCIL Lab., Sherbrooke University] , Rachid Deriche.

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. In this work, 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. 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. In 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).

DSI 101: Better ODFs for Free

Participants : Michael Paquette [SCIL Lab., Sherbrooke University] , Sylvain Merlet, Maxime Descoteaux [SCIL Lab., Sherbrooke University] , Rachid Deriche.

Diffusion Spectrum Imaging (DSI) is a well established method to recover the Ensemble Average Propagator (EAP). The orientation distribution function (ODF) is computed from this discretized EAP and used for tractography. However, there are several important implementation considerations that are tossed aside in the literature and the publicly available softwares. We investigate all the real steps necessary to go from the DSI signal to the ODF and provide applicable recommendations that greatly improve the accuracy of the local orientation detected. These recommendations come ”free-of-charge” as they are applicable to all existing DSI data and do not require a significant increase in computation time.

Comparison between discrete and continuous propagator indices from Cartesian q-space DSI sampling

Participants : Mauro Zucchelli [Dpt of Computer Science, University of Verona] , Eleftherios Garyfallidis [SCIL Lab., Sherbrooke University] , Michael Paquette [SCIL Lab., Sherbrooke University] , Maxime Descoteaux [SCIL Lab., Sherbrooke University] , Gloria Menegaz [Dpt of Computer Science, University of Verona] , Sylvain Merlet.

DSI is often considered the state-of-the-art technique to analyze q-space measurements sampled from a Cartesian grid. The 3D fast Fourier transform is used to directly obtain a discrete version of the EAP (Ensemble Average Propagator). DSI was one of the first techniques used to infer complex fiber configurations as it allows resolving crossings. In principle, DSI also captures some radial information which,in theory, can be used to extract diffusion features of the EAP. However, a discrete propagator representation suffers from a limited frequency band, which makes infinite integration impossible. Hence, EAP derived indices 2,3 are problematic and quantitatively questionable, as one needs to artificially normalize and approximate the infinite integrals. Combined with the recent popularity of DSI in the Human Connectome Project, it is important to investigate the different angular and EAP indices that can be computed from these DSI datasets. In this work, we investigate alternatives to the discrete model-free approach of DSI and investigate the Simple Harmonic Oscillator based Reconstruction and Estimation 3 (SHORE) models based on the evaluation of (i) the orientation distri bution function (ODF) ; (ii) the return to the origin probability (RTOP) and (iii) the mean square displacement (MSD).

Odf Maxima Computation Using Hill Climbing Algorithm

Participants : Makhlouf Laouchedi [USTHB, Algeria] , Thinhinane Megherbi [USTHB, Algeria] , Linda Oulebsir-Boumghar [USTHB, Algeria] , Rachid Deriche.

Methods like Diffusion Spectrum Imaging (DSI), High Angular Resolution Diffusion Imaging (HARDI) and the High Order Tensor techniques have been proposed to reconstruct specific functions like the Orientation Distribution Function (ODF) whose maxima correspond to the directions of the multiple fibers. In this work, we are interested to extract all the crossing fibers characterized as the maxima of the Orientation Distribution Function (ODF). A Hill Climbing algorithm based approach has been developed and implemented to efficiently and accurately extract all the fibers. Promising experimental results obtained with synthetic and real data illustrate the potential of the technique.

Greedy NNLS: Fiber Orientation Distribution from Non-Negatively Constrained Sparse Recovery

Participants : Aurobrata Ghosh, Rachid Deriche.

In this work, we validated experimentally the merits of the Non-Negative Least Squares (NNLS) for the constrained sparse recovery of the Fiber Orientation Distribution (FOD) and compared it with classical l1-minimization. The FOD is a robust model for mapping crossing white matter fibers. However, its angular resolution depends on the spherical harmonic basis order, which can imply a large number of acquisitions. Further, it is necessary to compute the maxima of the FOD to derive the fiber directions. It is possible to kill the two proverbial birds with a single stone by using a non-negatively constrained sparse recovery FOD estimation with NNLS.

From our experiments, we confirmed results from literature to show that NNLS converges to highly sparse solutions which are correctly constrained, while l1-minimization is less sparse, contains negative solutions and is unstable with noisy data. Finally, we discussed the NLS algorithm and attributed the sparsity to its design, which mirrors the design of Orthogonal Matching Pursuit (OMP)

Crossing Fibers Detection with an Analytical High Order Tensor Decomposition

Participants : Thinhinane Megherbi [USTHB, Algeria] , Mouloud Kachouane [USTHB, Algeria] , Linda Oulebsir-Boumghar [USTHB, Algeria] , Rachid Deriche.

Diffusion magnetic resonance imaging (dMRI) is the only technique to probe in vivo and noninvasively the fiber structure of human brain white matter. Detecting the crossing of neuronal fibers remains an exciting challenge with an important impact in tractography. In this, we tackle this challenging problem and propose an original and efficient technique to extract all crossing fibers from diffusion signals. To this end, we start by estimating, from the dMRI signal, the so-called Cartesian tensor fiber orientation distribution (CT-FOD) function, whose maxima correspond exactly to the orientations of the fibers. The fourth order symmetric positive definite tensor that represents the CT-FOD is then analytically decomposed via the application of a new theoretical approach and this decomposition is used to accurately extract all the fibers orientations. Our proposed high order tensor decomposition based approach is minimal and allows recovering the whole crossing fibers without any a priori information on the total number of fibers. Various experiments performed on noisy synthetic data, on phantom diffusion, data and on human brain data validate our approach and clearly demonstrate that it is efficient, robust to noise and performs favorably in terms of angular resolution and accuracy when compared to some classical and state-of-the-art approaches.

Complete set of Invariants of a 4th order tensor: the 12 tasks of HARDI from Ternary Quartics

Participants : Théodore Papadopoulo, Auro Ghosh, Rachid Deriche.

In this work, we presented a simple and systematic method to compute a functionally complete set of invariants of a non-negative 3D 4th order tensor with respect to 3D rotations. Intuitively, this transforms the tensor's non-unique ternary quartic (TQ) decomposition (from Hilbert's theorem) to a unique canonical representation independent of orientation.

Invariants play a crucial role in diffusion MRI. In DTI (2nd order tensors), invariant scalars (FA, MD...)have been successfully used in clinical applications. But DTI has limiations and HARDI models (e.g. 4th order tensors) have been proposed instead. These, however, lack invariant features and computing them systematically is challenging.

The invariants we propose, can be computed from two simple reduction steps, which first reduce an orthogonal class and then a rotation transform class of equivalent representations from the TQ coefficients. The resulting invariants are, by construction, (1) functionally complete, (2) functionally irreducible (if desired), (3) computationally efficient and (4) reversible – or mappable to the TQ coefficients or shape. These were the novelties of our contribution in comparison to prior work.

Fiber Orientation Distribution from Non-Negative Sparse Recovery

Participants : Thinhinane Megherbi [USTHB, Algeria] , Auro Ghosh, Linda Oulebsir-Boumghar [USTHB, Algeria] , Rachid Deriche.

In this work, we tested our non-negatively constrained sparse recovery algorithm for estimating the FOD on single shell phantom data provided by the ISBI'2014 challenge. We used the NNLS algorithm to estimate high order FODs (24th order) from just 20, 30 and 60 gradient directions and for various b-values of 1000, 2000, and 3000.

From the results, which are yet to be published, but can be viewed in their preliminary form online, it is clear that amongst the single shell algorithms, ours was good at fitting the signal and estimating the number of compartments. It performed well even with as low as 20 gradient acquisitions. Its major shortcoming was in underestimating the crossing angle and this needs to be improved upon.

How to get more out of a clinically feasable 64 gradient dMRI acquisition: Multi-Shell versus Single-Shell

Participants : Rutger H.j Fick, Mario Zuccheli [Dpt of Computer Science, University of Verona] , Gabriel Girard [SCIL Lab., Sherbrooke University] , Maxime Descoteaux [SCIL Lab., Sherbrooke University] , Gloria Menegaz [Dpt of Computer Science, University of Verona] , Rachid Deriche.

For clinical applications the number of diffusion MRI (dMRI) samples that can be obtained is often limited by scanner time and patient comfort. For this reason one often uses short scanning protocols that acquire just 32 or 64 gradient directions using a single b-value to obtain diffusion measures such as the fractional anisotropy from Diffusion Tensor Imaging (DTI) or to estimate the white matter orientation using Constrained Spherical Deconvolution (CSD). Using 3D-SHORE and MAP-MRI, we show that by spreading the same number of dMRI samples over different b-shells (sampling angularly and radially) we can estimate not only the directionality of the white matter using the ODF, but also the radially dependent higher order diffusion measures that SHORE and MAP-MRI provide. This approach lends itself well for situations where acquisition time is limited, and is therefore particularly well suited for clinical applications.

This work has been submitted to ISMRM'2015.