On the determination of optimal tuning parameters for a space-variant LASSO problem using geometric and convex analysis techniques

Compressed Sensing (CS) comprises a wide range of theoretical and applied techniques to recover signals given a partial knowledge of their coefficients. It finds its applications in several fields, such as mathematics, physics, engineering, and many medical sciences, to name a few. Driven by our interest in the mathematics behind Magnetic Resonance Imaging (MRI) and Compressed Sensing (CS), we use convex analysis techniques to determine analytically the optimal tuning parameters of the space-variant LASSO with voxel-wise weighting, under assumptions on the fidelity term, either on the sign of its gradient or orthogonality-like conditions on its matrix. Finally, we conclude conjecturing what the explicit form of optimal parameters should be in the most general setting (hypotheses-free) of the space-variant LASSO

Structural Variability from Noisy Tomographic Projections

In cryo-electron microscopy, the 3D electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy 2D images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low-rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For $n$ images of size $N$-by-$N$ pixels, we propose an algorithm for calculating this covariance estimator with computational complexity $\mathcal{O}(nN^4+\sqrt{\kappa}N^6 \log N)$, where the condition number $\kappa$ is empirically in the range $10$--$200$. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in 6D. This is then solved by the conjugate gradient method with an appropriate circulant preconditioner. The result is the first computationally efficient algorithm for consistent estimation of 3D covariance from noisy projections. It also compares favorably in runtime with respect to previously proposed non-consistent estimators. Motivated by the recent success of eigenvalue shrinkage procedures for high-dimensional covariance matrices, we introduce a shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We evaluate our methods on simulated datasets and achieve classification results comparable to state-of-the-art methods in shorter running time. We also present results on clustering volumes in an experimental dataset, illustrating the power of the proposed algorithm for practical determination of structural variability.Comment: 52 pages, 11 figure

Acceleration Methods for MRI

Acceleration methods are a critical area of research for MRI. Two of the most important acceleration techniques involve parallel imaging and compressed sensing. These advanced signal processing techniques have the potential to drastically reduce scan times and provide radiologists with new information for diagnosing disease. However, many of these new techniques require solving difficult optimization problems, which motivates the development of more advanced algorithms to solve them. In addition, acceleration methods have not reached maturity in some applications, which motivates the development of new models tailored to these applications. This dissertation makes advances in three different areas of accelerations. The first is the development of a new algorithm (called B1-Based, Adaptive Restart, Iterative Soft Thresholding Algorithm or BARISTA), that solves a parallel MRI optimization problem with compressed sensing assumptions. BARISTA is shown to be 2-3 times faster and more robust to parameter selection than current state-of-the-art variable splitting methods. The second contribution is the extension of BARISTA ideas to non-Cartesian trajectories that also leads to a 2-3 times acceleration over previous methods. The third contribution is the development of a new model for functional MRI that enables a 3-4 factor of acceleration of effective temporal resolution in functional MRI scans. Several variations of the new model are proposed, with an ROC curve analysis showing that a combination low-rank/sparsity model giving the best performance in identifying the resting-state motor network.PhDBiomedical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120841/1/mmuckley_1.pd

Structure-preserving machine learning for inverse problems

Inverse problems naturally arise in many scientific settings, and the study of these problems has been crucial in the development of important technologies such as medical imaging. In inverse problems, the goal is to estimate an underlying ground truth u∗, typically an image, from corresponding measurements y, where u∗ and y are related by y = N(A(u∗)) (1) for some forward operator A and noise-generating process N (both of which are generally assumed to be known). Variational regularisation is a well-established approach that can be used to approximately solve inverse problems such as Problem (1). In this approach an image is reconstructed from measurements y by solving a minimisation problem such as uˆ = argmin d(A(u),y) +αJ(u). (2) While this approach has proven very successful, it generally requires the parts that make up the optimisation problem to be carefully chosen, and the optimisation problem may require considerable computational effort to solve. There is an active line of research into overcoming these issues using data-driven approaches, which aim to use multiple instances of data to inform a method that can be used on similar data. In this dissertation we investigate ways in which favourable properties of the variational regularisation approach can be combined with a data-driven approach to solving inverse problems. In the first chapter of the dissertation, we propose a bilevel optimisation framework that can be used to optimise sampling patterns and regularisation parameters for variational image reconstruction in accelerated magnetic resonance imaging. We use this framework to learn sampling patterns that result in better image reconstructions than standard random variable density sampling patterns that sample with the same rate. In the second chapter of the dissertation, we study the use of group symmetries in learned reconstruction methods for inverse problems. We show that group invariance of a functional implies that the corresponding proximal operator satisfies a group equivariance property. Applying this idea to model proximal operators as roto-translationally equivariant in an unrolled iterative reconstruction method, we show that reconstruction performance is more robust when tested on images in orientations not seen during training (compared to similar methods that model proximal operators to just be translationally equivariant) and that good methods can be learned with less training data. In the final chapter of the dissertation, we propose a ResNet-styled neural network architecture that is provably nonexpansive. This architecture can be thought of as composing discretisations of gradient flows along learnable convex potentials. Appealing to a classical result on the numerical integration of ODEs, we show that constraining the operator norms of the weight operators is sufficient to give nonexpansiveness, and additional analysis in the case that the numerical integrator is the forward Euler method shows that the neural network is an averaged operator. This guarantees that its fixed point iterations are convergent, and makes it a natural candidate for a learned denoiser in a Plug-and-Play approach to solving inverse problemsCantab Capital Institute for the Mathematics of Informatio

Model-based reconstruction of accelerated quantitative magnetic resonance imaging (MRI)

Quantitative MRI refers to the determination of quantitative parameters (T1,T2,diffusion, perfusion etc.) in magnetic resonance imaging (MRI). The ’parameter maps’ are estimated from a set of acquired MR images using a parameter model, i.e. a set of mathematical equations that describes the MR images as a function of the parameter(s). A precise and accurate highresolution estimation of the parameters is needed in order to detect small changes and/or to visualize small structures. Particularly in clinical diagnostics, the method provides important information about tissue structures and respective pathologic alterations. Unfortunately, it also requires comparatively long measurement times which preclude widespread practical applications. To overcome such limitations, approaches like Parallel Imaging (PI) and Compressed Sensing (CS) along with the model-based reconstruction concept has been proposed. These methods allow for the estimation of quantitative maps from only a fraction of the usually required data. The present work deals with the model-based reconstruction methods that are applicable for the most widely available Cartesian (rectilinear) acquisition scheme. The initial implementation was based on accelerating the T*2 mapping using Maximum Likelihood estimation and Parallel Imaging (PI). The method was tested on a Multiecho Gradient Echo (MEGE) T*2 mapping experiment in a phantom and a human brain with retrospective undersampling. Since T*2 is very sensitive to phase perturbations as a result of magnetic field inhomogeneity further work was done to address this. The importance of coherent phase information in improving the accuracy of the accelerated T*2 mapping fitting was investigated. Using alternating minimization, the method extends the MLE approach based on complex exponential model fitting which avoids loss of phase information in recovering T*2 relaxation times. The implementation of this method was tested on prospective(real time) undersampling in addition to retrospective. Compared with fully sampled reference scans, the use of phase information reduced the error of the accelerated T*2 maps by up to 20% as compared to baseline magnitude-only method. The total scan time for the four times accelerated 3D T*2 mapping was 7 minutes which is clinically acceptable. The second main part of this thesis focuses on the development of a model-based super-resolution framework for the T2 mapping. 2D multi-echo spin-echo (MESE) acquisitions suffer from low spatial resolution in the slice dimension. To overcome this limitation while keeping acceptable scan times, we combined a classical super-resolution method with an iterative model-based reconstruction to reconstruct T2 maps from highly undersampled MESE data. Based on an optimal protocol determined from simulations, we were able to reconstruct 1mm3 isotropic T2 maps of both phantom and healthy volunteer data. Comparison of T2 values obtained with the proposed method with fully sampled reference MESE results showed good agreement. In summary, this thesis has introduced new approaches to employ signal models in different applications, with the aim of either accelerating an acquisition, or improving the accuracy of an existing method. These approaches may help to take the next step away from qualitative towards a fully quantitative MR imaging modality, facilitating precision medicine and personalized treatment

Wavelet-Regularized Reconstruction For Rapid MRI

We propose a reconstruction scheme adapted to MRI that takes advantage of a sparsity constraint in the wavelet domain. We show that, artifacts are significantly reduced compared to conventional reconstruction methods. Our approach is also competitive with Total Variation regularization both in terms of MSE and computation time. We show that l(1) regularization allows partial recovery of the missing k-space regions. We also present a multi-level version that significantly reduces the computational cost