Regularized Spherical Polar Fourier Diffusion MRI with Optimal Dictionary Learning

International audienceOne important problem in diffusion MRI (dMRI) is to recover the diffusion weighted signal from only a limited number of samples in q-space. An ideal framework for solving this problem is Compressed Sensing (CS), which takes advantage of the signal's sparseness or compressibility, allowing the entire signal to be reconstructed from relatively few measurements. CS theory requires a suitable dictionary that sparsely represents the signal. To date in dMRI there are two kinds of Dictionary Learning (DL) methods: 1) discrete representation based DL (DR-DL), and 2) continuous representation based DL (CR-DL). Due to the discretization in q-space, DR-DL suffers from the numerical errors in interpolation and regridding. By considering a continuous representation using Spherical Polar Fourier (SPF) basis, this paper proposes a novel CR-DL based Spherical Polar Fourier Imaging, called DL-SPFI, to recover the diffusion signal as well as the Ensemble Average Propagator (EAP) in continuous 3D space with closed form. DL-SPFI learns an optimal dictionary from the space of Gaussian diffusion signals. Then the learned dictionary is adaptively applied for different voxels in a weighted LASSO framework to robustly recover the di ffusion signal and the EAP. Compared with the start-of-the-art CR-DL method by Merlet et al. and DRDL by Bilgic et al., DL-SPFI has several advantages. First, the learned dictionary, which is proved to be optimal in the space of Gaussian diffusion signal, can be applied adaptively for different voxels. To our knowledge, this is the first work to learn a voxel-adaptive dictionary. The importance of this will be shown theoretically and empirically in the context of EAP estimation. Second, based on the theoretical analysis of SPF basis, we devise an efficient learning process in a small subspace of SPF coefficients, not directly in q-space as done by Merlet et al.. Third, DL-SPFI also devises different regularization for different atoms in the learned dictionary for robust estimation, by considering the structural prior in the space of signal exemplars. We evaluate DL-SPFI in comparison to L1-norm regularized SPFI (L1-SPFI) with fixed SPF basis, and the DR-DL by Bilgic et al. The experiments on synthetic data and real data demonstrate that the learned dictionary is sparser than SPF basis and yields lower reconstruction error than Bilgic's method, even though only simple synthetic Gaussian signals were used for training in DL-SPFI in contrast to real data used by Bilgic et al

Single- and Multiple-Shell Uniform Sampling Schemes for Diffusion MRI Using Spherical Codes

In diffusion MRI (dMRI), a good sampling scheme is important for efficient acquisition and robust reconstruction. Diffusion weighted signal is normally acquired on single or multiple shells in q-space. Signal samples are typically distributed uniformly on different shells to make them invariant to the orientation of structures within tissue, or the laboratory coordinate frame. The Electrostatic Energy Minimization (EEM) method, originally proposed for single shell sampling scheme in dMRI, was recently generalized to multi-shell schemes, called Generalized EEM (GEEM). GEEM has been successfully used in the Human Connectome Project (HCP). However, EEM does not directly address the goal of optimal sampling, i.e., achieving large angular separation between sampling points. In this paper, we propose a more natural formulation, called Spherical Code (SC), to directly maximize the minimal angle between different samples in single or multiple shells. We consider not only continuous problems to design single or multiple shell sampling schemes, but also discrete problems to uniformly extract sub-sampled schemes from an existing single or multiple shell scheme, and to order samples in an existing scheme. We propose five algorithms to solve the above problems, including an incremental SC (ISC), a sophisticated greedy algorithm called Iterative Maximum Overlap Construction (IMOC), an 1-Opt greedy method, a Mixed Integer Linear Programming (MILP) method, and a Constrained Non-Linear Optimization (CNLO) method. To our knowledge, this is the first work to use the SC formulation for single or multiple shell sampling schemes in dMRI. Experimental results indicate that SC methods obtain larger angular separation and better rotational invariance than the state-of-the-art EEM and GEEM. The related codes and a tutorial have been released in DMRITool.Comment: Accepted by IEEE transactions on Medical Imaging. Codes have been released in dmritool https://diffusionmritool.github.io/tutorial_qspacesampling.htm

Denoising and fast diffusion imaging with physically constrained sparse dictionary learning

International audienceDiffusion-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

Spatially Regularizing High Angular Resolution Diffusion Imaging

Many recent high angular resolution diffusion imaging (HARDI) reconstruction techniques have been introduced to infer ensemble average propagator (EAP),describing the three-dimensional (3D) average diffusion process of water molecules or the angular structure information contained in EAP, orientation distribution function (ODF). Most of these methods perform reconstruction independently at each voxel, which essentially ignoring the functional nature of the HARDI data at different voxels in space. The aim of my thesis is to develop methods which can spatially and adaptively infer the EAP, or ODF of water diffusion in regions with complex fiber configurations. In Chapter 3, we propose a penalized multi-scale adaptive regression model (PMARM) framework to spatially and adaptively infer the ODF of water diffusion in regions with complex fiber configurations. We first represent DW-MRI signals using Spherical Harmonic (SH) basis, then apply PMARM on advanced statistical methods to calculate the coefficients of SH representation, from which ODF representation is calculated using Funk-Radon transformation. PMARM reconstructs the ODF at each voxel by adaptively borrowing the spatial information from the neighboring voxels. We show in the real and simulated data sets that PMARM can substantially reduce the noise level, while improving the ODF reconstruction. In Chapter 4, we propose a robust multi-scale adaptive and sequential smoothing (MASS) method framework to robustly, spatially and adaptively infer the EAP of water diffusion in regions with complex fiber configurations. We first calculate spherical polar Fourier basis representation of the DW-MRI signals, and then apply MASS adaptively and sequentially updating SPF representation by borrowing the spatial information from the neighboring voxels. We show in the real and simulated data sets that MASS can reduce the angle detection errors on fiber crossing area and provides more accurate reconstructions than standard voxel-wise methods and robust MASS performs very well with the presence of outliers. In Chapter 5, we extend multi-scale adaptive method framework to dictionary learning methods, and show that by adding smoothing technique, we can significantly improve the accuracy of EAP reconstruction and reduce the angle detection errors on fiber crossing, even in very low signal-to-noise ratio situation.Doctor of Philosoph

Sparse and Redundant Representations for Inverse Problems and Recognition

Sparse and redundant representation of data enables the description of signals as linear combinations of a few atoms from a dictionary. In this dissertation, we study applications of sparse and redundant representations in inverse problems and object recognition. Furthermore, we propose two novel imaging modalities based on the recently introduced theory of Compressed Sensing (CS). This dissertation consists of four major parts. In the first part of the dissertation, we study a new type of deconvolution algorithm that is based on estimating the image from a shearlet decomposition. Shearlets provide a multi-directional and multi-scale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. We develop a deconvolution algorithm that allows for the approximation inversion operator to be controlled on a multi-scale and multi-directional basis. Furthermore, we develop a method for the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation method. In the second part of the dissertation, we study a reconstruction method that recovers highly undersampled images assumed to have a sparse representation in a gradient domain by using partial measurement samples that are collected in the Fourier domain. Our method makes use of a robust generalized Poisson solver that greatly aids in achieving a significantly improved performance over similar proposed methods. We will demonstrate by experiments that this new technique is more flexible to work with either random or restricted sampling scenarios better than its competitors. In the third part of the dissertation, we introduce a novel Synthetic Aperture Radar (SAR) imaging modality which can provide a high resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. We demonstrate that this new imaging scheme, requires no new hardware components and allows the aperture to be compressed. Also, it presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced on-board storage requirements. The last part of the dissertation deals with object recognition based on learning dictionaries for simultaneous sparse signal approximations and feature extraction. A dictionary is learned for each object class based on given training examples which minimize the representation error with a sparseness constraint. A novel test image is then projected onto the span of the atoms in each learned dictionary. The residual vectors along with the coefficients are then used for recognition. Applications to illumination robust face recognition and automatic target recognition are presented

Statistical Learning Methods for Diffusion Magnetic Resonance Imaging

Diffusion Magnetic Resonance Imaging (dMRI) is a commonly used imaging technique to reveal white matter (WM) microstructure by probing the diffusion of water molecules. The diffusion of water molecules is constrained by the biological boundaries including nerves and tissues. Thus, quantifying the diffusion process is important to understand the WM microstructure. However, the development of efficient analytical methods for the reconstruction, lifespan structural connectome analysis, and surrogate variable analysis have fallenseriously behind the technological advances. This challenge motivates us to develop new statistical learning methods for dMRI. In the first project, we propose a two-stage sparse and adaptive smoothing model (TSASM) for two major image denoising tasks in neuroimaging data analysis, including image reconstruction from a series of noisy images within each subject and group analysis of images obtained from different subjects. Our TSASM consists of an initial smoothing stage of applying a penalized M-estimator and a refined smoothing stage of applying kernel-based smoothing methods. The key novelties of our TSASM are that it accounts for the sparse structure of imaging signals while preserving piecewise smooth regions with unknown edges. In the second project, we develop a scalable analytical method for mapping the lifespan human structural connectome. Specifically, we develop a novel lifespan population-based structural connectome (LPSC) framework that integrates fiber bundle and functional network information for hierarchically guiding the registration. Our LPSC is applicable to several neuroimaging studies of neuropsychiatric disorders as well as normal brain development. An improved understanding of human structural connectome has the potential to inspire new approaches to prevention, diagnosis, and treatment of many illnesses. In the third project, we propose an eigen-shrinkage projection (ESP) method to perform the surrogate variable analysis and solve the hidden confounder and harmonization problems in the neuroimaging studies. Our ESP can eliminate the signals from primary variable while preserving the eigenvalue-gap between hidden confounder and noises, which enables hidden confounders estimation from the projected data. We then investigate the statistical properties of the estimated hidden confounders and uncover the natural connection with ridge regression. Numerical experiments are used to illustrate the finite-sample performance.Doctor of Philosoph

Deep Convolutional Neural Network for Inverse Problems in Imaging

In this paper, we propose a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyperparameter selection. The starting point of this paper is the observation that unrolled iterative methods have the form of a CNN (filtering followed by pointwise nonlinearity) when the normal operator $(H ^{ * }$ H, where $H ^{ * }$ is the adjoint of the forward imaging operator, H) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms. The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a 512 × 512 image on the GPU