# Digital Image Reconstruction: Deblurring and Denoising

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Amos Yahil, Jan 13, 2014 Available from:- [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the problem of reconstructing signals and images from a subsampled convolution of masked snapshots and a known filter. The problem is studied in the context of coded imaging systems, where the diversity provided by the random masks makes the deconvolution problem significantly better conditioned than it would be from a set of direct measurements. We start by studying the conditioning of the forward linear measurement operator that describes the system in terms of the number of masks $K$, the dimension of image $L$, the number of sensors $N$, and certain characteristics of the blur kernel. We show that stable deconvolution is possible when $KN \geq L\log L$, meaning that the total number of sensor measurements is within a logarithmic factor of the image size. Next, we consider the scenario where the target image is known to be sparse. We show that under mild conditions on the blurring kernel, the linear system is a restricted isometry when the number of masks is within a logarithmic factor of the number of active components, making the image recoverable using any one of a number of sparse recovery techniques. - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we propose a new statistical stopping rule for constrained maximum likelihood iterative algorithms applied to ill-posed inverse problems. To this aim we extend the definition of Tikhonov regularization in a statistical framework and prove that the application of the proposed stopping rule to the Iterative Space Reconstruction Algorithm (ISRA) in the Gaussian case and Expectation Maximization (EM) in the Poisson case leads to well defined regularization methods according to the given definition. We also prove that, if an inverse problem is genuinely ill-posed in the sense of Tikhonov, the same definition is not satisfied when ISRA and EM are optimized by classical stopping rule like Morozov's discrepancy principle, Pearson's test and Poisson discrepancy principle. The stopping rule is illustrated in the case of image reconstruction from data recorded by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). First, by using a simulated image consisting of structures analogous to those of a real solar flare we validate the fidelity and accuracy with which the proposed stopping rule recovers the input image. Second, the robustness of the method is compared with the other classical stopping rules and its advantages are shown in the case of real data recorded by RHESSI during two different flaring events. -
##### Article: Level Set Estimation from Projection Measurements: Performance Guarantees and Fast Computation

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**ABSTRACT:**Estimation of the level set of a function (i.e., regions where the function exceeds some value) is an important problem with applications in digital elevation mapping, medical imaging, astronomy, etc. In many applications, the function of interest is not observed directly. Rather, it is acquired through (linear) projection measurements, such as tomographic projections, interferometric measurements, coded-aperture measurements, and random projections associated with compressed sensing. This paper describes a new methodology for rapid and accurate estimation of the level set from such projection measurements. The key defining characteristic of the proposed method, called the projective level set estimator, is its ability to estimate the level set from projection measurements without an intermediate reconstruction step. This leads to significantly faster computation relative to heuristic "plug-in" methods that first estimate the function, typically with an iterative algorithm, and then threshold the result. The paper also includes a rigorous theoretical analysis of the proposed method, which utilizes the recent results from the non-asymptotic theory of random matrices results from the literature on concentration of measure and characterizes the estimator's performance in terms of geometry of the measurement operator and 1-norm of the discretized function.SIAM Journal on Imaging Sciences 09/2012; 6(4). DOI:10.1137/120891927 · 2.87 Impact Factor