Model-based reconstruction is a powerful framework for solving a variety of inverse problems in imaging. The method works by combining a forward model of the imaging system with a prior model of the image itself, and the reconstruction is then computed by minimizing a functional consisting of the sum of two terms corresponding to the forward and prior models. 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. A wide range of methods including nonlocal means, dictionary-based methods, 3D block matching, TV minimization and kernel-based filtering have proven that it is possible to recover high fidelity images even after a great deal of noise has been added. 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, electronmicroscopy, 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 report, we propose a flexible framework that allows state-of-the-art forward models of imaging systems to be matched with state-of-the-art prior 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.

Abstract-Many material and biological samples in scientific imaging are characterized by non-local repeating structures. These are studied using scanning electron microscopy and electron tomography. Sparse sampling of individual pixels in a 2D image acquisition geometry, or sparse sampling of projection images with large tilt increments in a tomography experiment, can enable high speed data acquisition and minimize sample damage caused by the electron beam.In this paper, we present an algorithm for electron tomographic reconstruction and sparse image interpolation that exploits the non-local redundancy in images. We adapt a framework, termed plug-and-play (P&P) priors, to solve these imaging problems in a regularized inversion setting. The power of the P&P approach is that it allows a wide array of modern denoising algorithms to be used as a "prior model" for tomography and image interpolation. We also present sufficient mathematical conditions that ensure convergence of the P&P approach, and we use these insights to design a new non-local means denoising algorithm. Finally, we demonstrate that the algorithm produces higher quality reconstructions on both simulated and real electron microscope data, along with improved convergence properties compared to other methods.Index Terms-Plug-and-play, prior modeling, bright field electron tomography, sparse interpolation, non-local means, doublystochastic gradient non-local means, BM3D.

When applying sparse representation techniques to images, the standard approach is to independently compute the representations for a set of overlapping image patches. This method performs very well in a variety of applications, but results in a representation that is multi-valued and not optimized with respect to the entire image. An alternative representation structure is provided by a convolutional sparse representation, in which a sparse representation of an entire image is computed by replacing the linear combination of a set of dictionary vectors by the sum of a set of convolutions with dictionary filters. The resulting representation is both single-valued and jointly optimized over the entire image. While this form of a sparse representation has been applied to a variety of problems in signal and image processing and computer vision, the computational expense of the corresponding optimization problems has restricted application to relatively small signals and images. This paper presents new, efficient algorithms that substantially improve on the performance of other recent methods, contributing to the development of this type of representation as a practical tool for a wider range of problems.

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Replacing the l(2) data fidelity term of the standard Total Variation (TV) functional with an l(1) data fidelity term has been found to offer a number of theoretical and practical benefits. Efficient algorithms for minimizing this l(1)-TV functional have only recently begun to be developed, the fastest of which exploit graph representations, and are restricted to the denoising problem. We describe an alternative approach that minimizes a generalized TV functional, including both l(2)-TV and l(1)-TV as special cases, and is capable of solving more general inverse problems than denoising (e.g., deconvolution). This algorithm is competitive with the graph-based methods in the denoising case, and is the fastest algorithm of which we are aware for general inverse problems involving a nontrivial forward linear operator.

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Convolutional sparse representations are a form of sparse representation with a dictionary that has a structure that is equivalent to convolution with a set of linear filters. While effective algorithms have recently been developed for the convolutional sparse coding problem, the corresponding dictionary learning problem is substantially more challenging. Furthermore, although a number of different approaches have been proposed, the absence of thorough comparisons between them makes it difficult to determine which of them represents the current state of the art. The present work both addresses this deficiency and proposes some new approaches that outperform existing ones in certain contexts. A thorough set of performance comparisons indicates a very wide range of performance differences among the existing and proposed methods, and clearly identifies those that are the most effective. We do not consider the analysis form [2] of sparse representations in this work, focusing instead on the more common synthesis form. 2 We do not consider the very recent online CDL algorithms [18], [19], [20], [21] in this work.