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Convex Analysisの二,三の進展について

Image reconstruction is essential for imaging applications across the physical and life sciences, including optical and radar systems, magnetic resonance imaging, X-ray computed tomography, positron emission tomography, ultrasound imaging and radio astronomy. During image acquisition, the sensor encodes an intermediate representation of an object in the sensor domain, which is subsequently reconstructed into an image by an inversion of the encoding function. Image reconstruction is challenging because analytic knowledge of the exact inverse transform may not exist a priori, especially in the presence of sensor non-idealities and noise. Thus, the standard reconstruction approach involves approximating the inverse function with multiple ad hoc stages in a signal processing chain, the composition of which depends on the details of each acquisition strategy, and often requires expert parameter tuning to optimize reconstruction performance. Here we present a unified framework for image reconstruction-automated transform by manifold approximation (AUTOMAP)-which recasts image reconstruction as a data-driven supervised learning task that allows a mapping between the sensor and the image domain to emerge from an appropriate corpus of training data. We implement AUTOMAP with a deep neural network and exhibit its flexibility in learning reconstruction transforms for various magnetic resonance imaging acquisition strategies, using the same network architecture and hyperparameters. We further demonstrate that manifold learning during training results in sparse representations of domain transforms along low-dimensional data manifolds, and observe superior immunity to noise and a reduction in reconstruction artefacts compared with conventional handcrafted reconstruction methods. In addition to improving the reconstruction performance of existing acquisition methodologies, we anticipate that AUTOMAP and other learned reconstruction approaches will accelerate the development of new acquisition strategies across imaging modalities.

Image reconstruction by domain-transform manifold learning

The continuous development and extensive use of computed tomography (CT) in medical practice has raised a public concern over the associated radiation dose to the patient. Reducing the radiation dose may lead to increased noise and artifacts, which can adversely affect the radiologists’ judgment and confidence. Hence, advanced image reconstruction from low-dose CT data is needed to improve the diagnostic performance, which is a challenging problem due to its ill-posed nature. Over the past years, various low-dose CT methods have produced impressive results. However, most of the algorithms developed for this application, including the recently popularized deep learning techniques, aim for minimizing the mean-squared error (MSE) between a denoised CT image and the ground truth under generic penalties. Although the peak signal-to-noise ratio is improved, MSE- or weighted-MSE-based methods can compromise the visibility of important structural details after aggressive denoising. This paper introduces a new CT image denoising method based on the generative adversarial network (GAN) with Wasserstein distance and perceptual similarity. The Wasserstein distance is a key concept of the optimal transport theory and promises to improve the performance of GAN. The perceptual loss suppresses noise by comparing the perceptual features of a denoised output against those of the ground truth in an established feature space, while the GAN focuses more on migrating the data noise distribution from strong to weak statistically. Therefore, our proposed method transfers our knowledge of visual perception to the image denoising task and is capable of not only reducing the image noise level but also trying to keep the critical information at the same time. Promising results have been obtained in our experiments with clinical CT images.

Low-Dose CT Image Denoising Using a Generative Adversarial Network With Wasserstein Distance and Perceptual Loss

Model-based reconstruction is a powerful framework for solving a variety of inverse problems in imaging. In recent years, enormous progress has been made in the problem of denoising, a special case of an inverse problem where the forward model is an identity operator. Similarly, great progress has been made in improving model-based inversion when the forward model corresponds to complex physical measurements in applications such as X-ray CT, electron-microscopy, MRI, and ultrasound, to name just a few. However, combining state-of-the-art denoising algorithms (i.e., prior models) with state-of-the-art inversion methods (i.e., forward models) has been a challenge for many reasons. In this paper, we propose a flexible framework that allows state-of-the-art forward models of imaging systems to be matched with state-of-the-art priors or denoising models. This framework, which we term as Plug-and-Play priors, has the advantage that it dramatically simplifies software integration, and moreover, it allows state-of-the-art denoising methods that have no known formulation as an optimization problem to be used. We demonstrate with some simple examples how Plug-and-Play priors can be used to mix and match a wide variety of existing denoising models with a tomographic forward model, thus greatly expanding the range of possible problem solutions.

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Plug-and-Play priors for model based reconstruction

In this paper, we introduce a new CT image denoising method based on the generative adversarial network (GAN) with Wasserstein distance and perceptual similarity. The Wasserstein distance is a key concept of the optimal transform theory, and promises to improve the performance of the GAN. The perceptual loss compares the perceptual features of a denoised output against those of the ground truth in an established feature space, while the GAN helps migrate the data noise distribution from strong to weak. Therefore, our proposed method transfers our knowledge of visual perception to the image denoising task, is capable of not only reducing the image noise level but also keeping the critical information at the same time. Promising results have been obtained in our experiments with clinical CT images.

/pdf/low-dose-ct-image-denoising-using-a-generative-adversarial-4zbqq2u1is.pdf

Low Dose CT Image Denoising Using a Generative Adversarial Network with Wasserstein Distance and Perceptual Loss

Statistical image reconstruction using penalized weighted least-squares (PWLS) criteria can improve image-quality in X-ray computed tomography (CT). However, the huge dynamic range of the statistical weights leads to a highly shift-variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly. We propose to alleviate the problem by using a variable-splitting scheme that separates the shift-variant and ("nearly") invariant components of the statistical data model and also decouples the regularization term. This leads to an equivalent constrained problem that we tackle using the classical method-of-multipliers framework with alternating minimization. The specific form of our splitting yields an alternating direction method of multipliers (ADMM) algorithm with an inner-step involving a "nearly" shift-invariant linear system that is suitable for FFT-based preconditioning using cone-type filters. The proposed method can efficiently handle a variety of convex regularization criteria including smooth edge-preserving regularizers and non- smooth sparsity-promoting ones based on the l1-norm and total variation. Numerical experiments with synthetic and real in vivo human data illustrate that cone-filter preconditioners accelerate the proposed ADMM resulting in fast convergence of ADMM compared to conventional (nonlinear conjugate gradient, ordered subsets) and state-of-the-art (MFISTA, split-Bregman) algorithms that are applicable for CT.

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A Splitting-Based Iterative Algorithm for Accelerated Statistical X-Ray CT Reconstruction

Magnetic resonance image (MRI) reconstruction using SENSitivity Encoding (SENSE) requires regularization to suppress noise and aliasing effects. Edge-preserving and sparsity-based regularization criteria can improve image quality, but they demand computation-intensive nonlinear optimization. In this paper, we present novel methods for regularized MRI reconstruction from undersampled sensitivity encoded data-SENSE-reconstruction-using the augmented Lagrangian (AL) framework for solving large-scale constrained optimization problems. We first formulate regularized SENSE-reconstruction as an unconstrained optimization task and then convert it to a set of (equivalent) constrained problems using variable splitting. We then attack these constrained versions in an AL framework using an alternating minimization method, leading to algorithms that can be implemented easily. The proposed methods are applicable to a general class of regularizers that includes popular edge-preserving (e.g., total-variation) and sparsity-promoting (e.g., -norm of wavelet coefficients) criteria and combinations thereof. Numerical experiments with synthetic and in vivo human data illustrate that the proposed AL algorithms converge faster than both general-purpose optimization algorithms such as nonlinear conjugate gradient (NCG) and state-of-the-art MFISTA.

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Parallel MR Image Reconstruction Using Augmented Lagrangian Methods

Regularized iterative reconstruction algorithms for imaging inverse problems require selection of appropriate regularization parameter values. We focus on the challenging problem of tuning regularization parameters for nonlinear algorithms for the case of additive (possibly complex) Gaussian noise. Generalized cross-validation (GCV) and (weighted) mean-squared error (MSE) approaches [based on Stein's unbiased risk estimate (SURE)] need the Jacobian matrix of the nonlinear reconstruction operator (representative of the iterative algorithm) with respect to the data. We derive the desired Jacobian matrix for two types of nonlinear iterative algorithms: a fast variant of the standard iterative reweighted least-squares method and the contemporary split-Bregman algorithm, both of which can accommodate a wide variety of analysis- and synthesis-type regularizers. The proposed approach iteratively computes two weighted SURE-type measures: predicted-SURE and projected-SURE (which require knowledge of noise variance σ2), and GCV (which does not need σ2) for these algorithms. We apply the methods to image restoration and to magnetic resonance image (MRI) reconstruction using total variation and an analysis-type l1-regularization. We demonstrate through simulations and experiments with real data that minimizing predicted-SURE and projected-SURE consistently lead to near-MSE-optimal reconstructions. We also observe that minimizing GCV yields reconstruction results that are near-MSE-optimal for image restoration and slightly suboptimal for MRI. Theoretical derivations in this paper related to Jacobian matrix evaluations can be extended, in principle, to other types of regularizers and reconstruction algorithms.

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Regularization Parameter Selection for Nonlinear Iterative Image Restoration and MRI Reconstruction Using GCV and SURE-Based Methods

Statistical X-ray computed tomography (CT) reconstruction can improve image quality from reduced dose scans, but requires very long computation time. Ordered subsets (OS) methods have been widely used for research in X-ray CT statistical image reconstruction (and are used in clinical PET and SPECT reconstruction). In particular, OS methods based on separable quadratic surrogates (OS-SQS) are massively parallelizable and are well suited to modern computing architectures, but the number of iterations required for convergence should be reduced for better practical use. This paper introduces OS-SQS-momentum algorithms that combine Nesterov's momentum techniques with OS-SQS methods, greatly improving convergence speed in early iterations. If the number of subsets is too large, the OS-SQS-momentum methods can be unstable, so we propose diminishing step sizes that stabilize the method while preserving the very fast convergence behavior. Experiments with simulated and real 3D CT scan data illustrate the performance of the proposed algorithms.

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Combining Ordered Subsets and Momentum for Accelerated X-Ray CT Image Reconstruction

To reduce blur in noisy images, regularized image restoration methods have been proposed that use nonquadratic regularizers (like l1 regularization or total-variation) that suppress noise while preserving edges in the image. Most of these methods assume a circulant blur (periodic convolution with a blurring kernel) that can lead to wraparound artifacts along the boundaries of the image due to the implied periodicity of the circulant model. Using a noncirculant model could prevent these artifacts at the cost of increased computational complexity. In this paper, we propose to use a circulant blur model combined with a masking operator that prevents wraparound artifacts. The resulting model is noncirculant, so we propose an efficient algorithm using variable splitting and augmented Lagrangian (AL) strategies. Our variable splitting scheme, when combined with the AL framework and alternating minimization, leads to simple linear systems that can be solved noniteratively using fast Fourier transforms (FFTs), eliminating the need for more expensive conjugate gradient-type solvers. The proposed method can also efficiently tackle a variety of convex regularizers, including edge-preserving (e.g., total-variation) and sparsity promoting (e.g., l1-norm) regularizers. Simulation results show fast convergence of the proposed method, along with improved image quality at the boundaries where the circulant model is inaccurate.

Sathish Ramani

Papers

A Splitting-Based Iterative Algorithm for Accelerated Statistical X-Ray CT Reconstruction

Parallel MR Image Reconstruction Using Augmented Lagrangian Methods

Regularization Parameter Selection for Nonlinear Iterative Image Restoration and MRI Reconstruction Using GCV and SURE-Based Methods

Combining Ordered Subsets and Momentum for Accelerated X-Ray CT Image Reconstruction

Accelerated Edge-Preserving Image Restoration Without Boundary Artifacts