Spherical Polar Fourier EAP and ODF Reconstruction via Compressed Sensing in Diffusion MRI

International audienceIn diffusion magnetic resonance imaging (dMRI), the Ensemble Average Propagator (EAP), also known as the propagator, describes completely the water molecule diffusion in the brain white matter without any prior knowledge about the tissue shape. In this paper, we describe a new and efficient method to accurately reconstruct the EAP in terms of the Spherical Polar Fourier (SPF) basis from very few diffusion weighted magnetic resonance images (DW-MRI). This approach nicely exploits the duality between SPF and a closely related basis in which one can respectively represent the EAP and the diffusion signal using the same coefficients, and efficiently combines it to the recent acquisition and reconstruction technique called Compressed Sensing (CS). Our work provides an efficient analytical solution to estimate, from few measurements, the diffusion propagator at any radius. We also provide a new analytical solution to extract an important feature characterising the tissue microstructure: the Orientation Distribution Function (ODF). We illustrate and prove the effectiveness of our method in reconstructing the propagator and the ODF on both noisy multiple q-shell synthetic and phantom data

Compressive Sensing Ensemble Average Propagator Estimation via L1 Spherical Polar Fourier Imaging

International audienceIn diffusion MRI (dMRI) domain, many High Angular Resolution Diffusion Imaging (HARDI) methods were proposed to estimate Ensemble Average Propagator (EAP) and Orientation Distribution Function (ODF). They normally need many samples, which limits their applications. Some Compressive Sensing (CS) based methods were proposed to estimate ODF in Q-Ball Imaging (QBI) from limited samples. However EAP estimation is much more difficult than ODF in QBI. Recently Spherical Polar Fourier Imaging (SPFI) was proposed to represent diffusion signal using Spherical Polar Fourier (SPF) basis without specific assumption on diffusion signals and analytically obtain EAP and ODF via the Fourier dual SPF (dSPF) basis from arbitrarily sampled signal. Normally the coefficients of SPF basis are estimated via Least Square with weighted L2 norm regularization (L2-SPFI). However, L2-SPFI needs a truncated basis to avoid overfitting, which brings some estimation errors. By considering the Fourier relationship between EAP and signal and the Fourier basis pair provided in SPFI, we propose a novel EAP estimation method, named L1-SPFI, to estimate EAP from limited samples using CS technique, and favorably compare it to the classical L2-SPFI method. L1-SPFI estimates the coefficients in SPFI using least square with weighted L1 norm regularization. The weights are designed to enhance the sparsity. L1-SPFI significantly accelerates the ordinary CS based Fourier reconstruction method. This is performed by using SPF basis pair in CS estimation process which avoids the numerical Fourier transform in each iteration step. By considering high order basis in L1 optimization, L1-SPFI improves EAP reconstruction especially for the angular resolution. The proposed L1-SPFI was validated by synthetic, phantom and real data. The CS EAP and ODF estimations are discussed in detail and we show that recovering the angular information from CS EAP requires much less samples than exact CS EAP reconstruction. Various experiments on synthetic, phantom and real data validate the fact that SPF basis can sparsely represent DW-MRI signals and L1-SPFI largely improves the CS EAP reconstruction especially the angular resolution

Impact of radial and angular sampling on multiple shells acquisition in diffusion MRI

International audienceWe evaluate the impact of radial and angular sampling on multiple shells (MS) acquisition in diffusion MRI. The validation of our results is based on a new and efficient method to accurately reconstruct the Ensemble Average Propagator (EAP) in term of the Spherical Polar Fourier (SPF) basis from very few diffusion weighted magnetic resonance images (DW-MRI). This approach nicely exploits the duality between SPF and a closely related basis in which one can respectively represent the EAP and the diffusion signal using the same coefficients. We efficiently combine this relation to the recent acquisition and reconstruction technique called Compressed Sensing (CS). Based on results of multi-tensors models reconstruction, we show how to construct a robust acquisition scheme for both neural fibre orientation detection and attenuation signal/EAP reconstruction

Diffusion and multiple orientations from 1.5 MR systems with limited gradient tables

International audienceDiffusion MRI (dMRI) enables the quantification of water diffusion, influenced by the structure of biological tissues, from the acquisition of diffusion weighted magnetic resonance images (DW-MRI). While recent advances enable to recover complex fiber geometries using diffusion measurements along various sampling schemes of high order [4], some older MR systems work with limited gradient tables (ex: maximum of 6 or 12 directions). These systems are designed for Diffusion Tensor Imaging (DTI). Several hospitals and research institutes in the world are limited by these fixed DTI gradient sets. Therefore, groups that want to perform state-of-the-art tractography using high angular resolution diffusion imaging (HARDI) data are pernalized and can only perform DTI tractography on their old system. The Gaussian assumption of the tensor model, in DTI, is an over simplification of the diffusion phenomenon of water molecules in the brain and thus cannot resolve crossing fibers. In this work, we show that new diffusion signal modeling and processing techniques enable to capture complex angular structure of the diffusion process even from a reduced gradient direction set arising from an older MR system

Ensemble Average Propagator Reconstruction via Compressed Sensing: Discrete or Continuous Bases ?

International audienceIn this work, we propose to compare the sparsity of two classes of representations for the EAP : The discrete representations, via the Haar, Daubechie-Cohen-Fauveau (DCF) 5-3, DCF 9-7 wavelets bases , and the continuous representations, via Spherical Polar Fourier (SPF) and 3D-SHORE bases

Spherical Polar Fourier EAP and ODF Reconstruction via Compressed Sensing in Diffusion MRI

International audienceIn diffusion magnetic resonance imaging (dMRI), the Ensemble Average Propagator (EAP), also known as the propagator, describes completely the water molecule diffusion in the brain white matter without any prior knowledge about the tissue shape. In this paper, we describe a new and efficient method to accurately reconstruct the EAP in terms of the Spherical Polar Fourier (SPF) basis from very few diffusion weighted magnetic resonance images (DW-MRI). This approach nicely exploits the duality between SPF and a closely related basis in which one can respectively represent the EAP and the diffusion signal using the same coefficients, and efficiently combines it to the recent acquisition and reconstruction technique called Compressed Sensing (CS). Our work provides an efficient analytical solution to estimate, from few measurements, the diffusion propagator at any radius. We also provide a new analytical solution to extract an important feature characterising the tissue microstructure: the Orientation Distribution Function (ODF). We illustrate and prove the effectiveness of our method in reconstructing the propagator and the ODF on both noisy multiple q-shell synthetic and phantom data

Parametric Dictionary Learning in Diffusion MRI

International audienceIn this work, we propose an approach to exploit the ability of compressive sensing to recover diffusion MRI signal and its characteristics from a limited number of samples. Our approach is threefold. First, we learn and design a parametric dictionary from a set of training diffusion data. This provides a highly sparse representation of the diffusion signal. The use of a parametric method presents several advantages: we design a continuous representation of the signal, from which we can analytically recover some features such as the ODF; besides, the dictionary we train is acquisition-independant. Next, we use this sparse representation to reconstruct the signal of interest, using cross-validation to assess the optimal regularization parameter for each signal reconstruction. The use of cross-validation is critical in the L1 minimization problem, as the choice of the parameter is sensitive to the noise level, the number of samples, and the data sparsity. Third, we use a polynomial approach to accurately extract ODF maxima. In the last section, we motivate and describe the choice of experimental parameters for the HARDI contest

Tractography via the Ensemble Average Propagator in diffusion MRI

International audienceIt's well known that in diffusion MRI (dMRI), fibre crossing is an important problem for most existing diffusion tensor imaging (DTI) based tractography algorithms. To overcome these limitations, High Angular Resolution Diffusion Imaging (HARDI) based tractography has been proposed with a particular emphasis on the the Orientation Distribution Function (ODF). In this paper, we advocate the use of the Ensemble Average Propagator (EAP) instead of the ODF for tractography in dMRI and propose an original and efficient EAP-based tractography algorithm that outperforms the classical ODF-based tractography, in particular, in the regions that contain complex fibre crossing configurations. Various experimental results including synthetic, phantom and real data illustrate the potential of the approach and clearly show that our method is especially efficient to handle regions where fiber bundles are crossing, and still well handle other fiber bundle configurations such as U-shape and kissing fibers

Compressive Sensing Ensemble Average Propagator Estimation via L1 Spherical Polar Fourier Imaging

International audienceIn diffusion MRI (dMRI) domain, many High Angular Resolution Diffusion Imaging (HARDI) methods were proposed to estimate Ensemble Average Propagator (EAP) and Orientation Distribution Function (ODF). They normally need many samples, which limits their applications. Some Compressive Sensing (CS) based methods were proposed to estimate ODF in Q-Ball Imaging (QBI) from limited samples. However EAP estimation is much more difficult than ODF in QBI. Recently Spherical Polar Fourier Imaging (SPFI) was proposed to represent diffusion signal using Spherical Polar Fourier (SPF) basis without specific assumption on diffusion signals and analytically obtain EAP and ODF via the Fourier dual SPF (dSPF) basis from arbitrarily sampled signal. Normally the coefficients of SPF basis are estimated via Least Square with weighted L2 norm regularization (L2-SPFI). However, L2-SPFI needs a truncated basis to avoid overfitting, which brings some estimation errors. By considering the Fourier relationship between EAP and signal and the Fourier basis pair provided in SPFI, we propose a novel EAP estimation method, named L1-SPFI, to estimate EAP from limited samples using CS technique, and favorably compare it to the classical L2-SPFI method. L1-SPFI estimates the coefficients in SPFI using least square with weighted L1 norm regularization. The weights are designed to enhance the sparsity. L1-SPFI significantly accelerates the ordinary CS based Fourier reconstruction method. This is performed by using SPF basis pair in CS estimation process which avoids the numerical Fourier transform in each iteration step. By considering high order basis in L1 optimization, L1-SPFI improves EAP reconstruction especially for the angular resolution. The proposed L1-SPFI was validated by synthetic, phantom and real data. The CS EAP and ODF estimations are discussed in detail and we show that recovering the angular information from CS EAP requires much less samples than exact CS EAP reconstruction. Various experiments on synthetic, phantom and real data validate the fact that SPF basis can sparsely represent DW-MRI signals and L1-SPFI largely improves the CS EAP reconstruction especially the angular resolution

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