Maxwell parallel imaging.

PURPOSE: To develop a general framework for parallel imaging (PI) with the use of Maxwell regularization for the estimation of the sensitivity maps (SMs) and constrained optimization for the parameter-free image reconstruction. THEORY AND METHODS: Certain characteristics of both the SMs and the images are routinely used to regularize the otherwise ill-posed optimization-based joint reconstruction from highly accelerated PI data. In this paper, we rely on a fundamental property of SMs-they are solutions of Maxwell equations-we construct the subspace of all possible SM distributions supported in a given field-of-view, and we promote solutions of SMs that belong in this subspace. In addition, we propose a constrained optimization scheme for the image reconstruction, as a second step, once an accurate estimation of the SMs is available. The resulting method, dubbed Maxwell parallel imaging (MPI), works for both 2D and 3D, with Cartesian and radial trajectories, and minimal calibration signals. RESULTS: The effectiveness of MPI is illustrated for various undersampling schemes, including radial, variable-density Poisson-disc, and Cartesian, and is compared against the state-of-the-art PI methods. Finally, we include some numerical experiments that demonstrate the memory footprint reduction of the constructed Maxwell basis with the help of tensor decomposition, thus allowing the use of MPI for full 3D image reconstructions. CONCLUSION: The MPI framework provides a physics-inspired optimization method for the accurate and efficient image reconstruction from arbitrary accelerated scans.

Iterative Joint Image Demosaicking and Denoising using a Residual Denoising Network.

Modern digital cameras rely on the sequential execution of separate image processing steps to produce realistic images. The first two steps are usually related to denoising and demosaicking where the former aims to reduce noise from the sensor and the latter converts a series of light intensity readings to color images. Modern approaches try to jointly solve these problems, i.e. joint denoising-demosaicking which is an inherently ill-posed problem given that two-thirds of the intensity information is missing and the rest are perturbed by noise. While there are several machine learning systems that have been recently introduced to solve this problem, the majority of them relies on generic network architectures which do not explicitly take into account the physical image model. In this work we propose a novel algorithm which is inspired by powerful classical image regularization methods, large-scale optimization, and deep learning techniques. Consequently, our derived iterative optimization algorithm, which involves a trainable denoising network, has a transparent and clear interpretation compared to other black-box data driven approaches. Our extensive experimentation line demonstrates that our proposed method outperforms any previous approaches for both noisy and noise-free data across many different datasets. This improvement in reconstruction quality is attributed to the rigorous derivation of an iterative solution and the principled way we design our denoising network architecture, which as a result requires fewer trainable parameters than the current state-of-the-art solution and furthermore can be efficiently trained by using a significantly smaller number of training data than existing deep demosaicking networks.

Improved Computational Efficiency of Locally Low Rank MRI Reconstruction Using Iterative Random Patch Adjustments.

This paper presents and analyzes an alternative formulation of the locally low-rank (LLR) regularization framework for magnetic resonance image (MRI) reconstruction. Generally, LLR-based MRI reconstruction techniques operate by dividing the underlying image into a collection of matrices formed from image patches. Each of these matrices is assumed to have low rank due to the inherent correlations among the data, whether along the coil, temporal, or multi-contrast dimensions. The LLR regularization has been successful for various MRI applications, such as parallel imaging and accelerated quantitative parameter mapping. However, a major limitation of most conventional implementations of the LLR regularization is the use of multiple sets of overlapping patches. Although the use of overlapping patches leads to effective shift-invariance, it also results in high-computational load, which limits the practical utility of the LLR regularization for MRI. To circumvent this problem, alternative LLR-based algorithms instead shift a single set of non-overlapping patches at each iteration, thereby achieving shift-invariance and avoiding block artifacts. A novel contribution of this paper is to provide a mathematical framework and justification of LLR regularization with iterative random patch adjustments (LLR-IRPA). This method is compared with a state-of-the-art LLR regularization algorithm based on overlapping patches, and it is shown experimentally that results are similar but with the advantage of much reduced computational load. We also present theoretical results demonstrating the effective shift invariance of the LLR-IRPA approach, and we show reconstruction examples and comparisons in both retrospectively and prospectively undersampled MRI acquisitions, and in T1 parameter mapping.

Poisson image reconstruction with Hessian Schatten-norm regularization.

Poisson inverse problems arise in many modern imaging applications, including biomedical and astronomical ones. The main challenge is to obtain an estimate of the underlying image from a set of measurements degraded by a linear operator and further corrupted by Poisson noise. In this paper, we propose an efficient framework for Poisson image reconstruction, under a regularization approach, which depends on matrix-valued regularization operators. In particular, the employed regularizers involve the Hessian as the regularization operator and Schatten matrix norms as the potential functions. For the solution of the problem, we propose two optimization algorithms that are specifically tailored to the Poisson nature of the noise. These algorithms are based on an augmented-Lagrangian formulation of the problem and correspond to two variants of the alternating direction method of multipliers. Further, we derive a link that relates the proximal map of an l(p) norm with the proximal map of a Schatten matrix norm of order p. This link plays a key role in the development of one of the proposed algorithms. Finally, we provide experimental results on natural and biological images for the task of Poisson image deblurring and demonstrate the practical relevance and effectiveness of the proposed framework.

Hessian Schatten-norm regularization for linear inverse problems.

We introduce a novel family of invariant, convex, and non-quadratic functionals that we employ to derive regularized solutions of ill-posed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, which are computed at every pixel of the image. They can be viewed as second-order extensions of the popular total-variation (TV) semi-norm since they satisfy the same invariance properties. Meanwhile, by taking advantage of second-order derivatives, they avoid the staircase effect, a common artifact of TV-based reconstructions, and perform well for a wide range of applications. To solve the corresponding optimization problems, we propose an algorithm that is based on a primal-dual formulation. A fundamental ingredient of this algorithm is the projection of matrices onto Schatten norm balls of arbitrary radius. This operation is performed efficiently based on a direct link we provide between vector projections onto lq norm balls and matrix projections onto Schatten norm balls. Finally, we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data.

Constrained regularized reconstruction of X-ray-DPCI tomograms with weighted-norm.

In this paper we introduce a new reconstruction algorithm for X-ray differential phase-contrast Imaging (DPCI). Our approach is based on 1) a variational formulation with a weighted data term and 2) a variable-splitting scheme that allows for fast convergence while reducing reconstruction artifacts. In order to improve the quality of the reconstruction we take advantage of higher-order total-variation regularization. In addition, the prior information on the support and positivity of the refractive index is considered, which yields significant improvement. We test our method in two reconstruction experiments involving real data; our results demonstrate its potential for in-vivo and medical imaging.

Hessian-based norm regularization for image restoration with biomedical applications.

We present nonquadratic Hessian-based regularization methods that can be effectively used for image restoration problems in a variational framework. Motivated by the great success of the total-variation (TV) functional, we extend it to also include second-order differential operators. Specifically, we derive second-order regularizers that involve matrix norms of the Hessian operator. The definition of these functionals is based on an alternative interpretation of TV that relies on mixed norms of directional derivatives. We show that the resulting regularizers retain some of the most favorable properties of TV, i.e., convexity, homogeneity, rotation, and translation invariance, while dealing effectively with the staircase effect. We further develop an efficient minimization scheme for the corresponding objective functions. The proposed algorithm is of the iteratively reweighted least-square type and results from a majorization-minimization approach. It relies on a problem-specific preconditioned conjugate gradient method, which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time. We validate the overall proposed regularization framework through deblurring experiments under additive Gaussian noise on standard and biomedical images.

Bayesian inference on multiscale models for poisson intensity estimation: applications to photon-limited image denoising.

We present an improved statistical model for analyzing Poisson processes, with applications to photon-limited imaging. We build on previous work, adopting a multiscale representation of the Poisson process in which the ratios of the underlying Poisson intensities (rates) in adjacent scales are modeled as mixtures of conjugate parametric distributions. Our main contributions include: 1) a rigorous and robust regularized expectation-maximization (EM) algorithm for maximum-likelihood estimation of the rate-ratio density parameters directly from the noisy observed Poisson data (counts); 2) extension of the method to work under a multiscale hidden Markov tree model (HMT) which couples the mixture label assignments in consecutive scales, thus modeling interscale coefficient dependencies in the vicinity of image edges; 3) exploration of a 2-D recursive quad-tree image representation, involving Dirichlet-mixture rate-ratio densities, instead of the conventional separable binary-tree image representation involving beta-mixture rate-ratio densities; and 4) a novel multiscale image representation, which we term Poisson-Haar decomposition, that better models the image edge structure, thus yielding improved performance. Experimental results on standard images with artificially simulated Poisson noise and on real photon-limited images demonstrate the effectiveness of the proposed techniques.