Ill-posed linear inverse problems (ILIP), such as restoration and reconstruction, are a core topic of signal/image processing. A standard approach to deal with ILIP uses a constrained optimization problem, where a regularization function is minimized under the constraint that the solution explains the observations sufficiently well. The regularizer and constraint are usually convex; however, several particular features of these problems (huge dimensionality, non-smoothness) preclude the use of off-the-shelf optimization tools and have stimulated much research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the category of augmented Lagrangian methods, can be used to deal with a variety of imaging ILIP, including deconvolution and reconstruction from compressive observations (such as MRI). Experiments testify for the effectiveness of the proposed method.
One-bit compressive sensing gains its popularity in signal processing and communications due to its low storage costs and low hardware complexity. However, it has been a challenging task to recover the signal only by exploiting the one-bit (the sign) information. In this paper, we appropriately formulate the one-bit compressive sensing into a double-sparsity constrained optimization problem. The first-order optimality conditions for this nonconvex and discontinuous problem are established via the newly introduced $τ$-stationarity, based on which, a gradient projection subspace pursuit (\texttt…
Advances in compressive sensing provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able to achieve optimal sample complexity. This has created an open problem in compressive phase retrieval: under generic, phaseless linear measurements, are there tractable reconstruction algorithms that suc…
In this work we study a constrained monotone inclusion involving the normal cone to a closed vector subspace and a priori information on primal solutions. We model this information by imposing that solutions belongs to the fixed point set of an averaged nonexpansive mapping. We characterize the solutions using an auxiliary inclusion that involves the partial inverse operator. Then, we propose the primal-dual partial inverse splitting and we prove its weak convergence to a solution of the inclusion, generalizing several methods in the literature. The efficiency of the proposed method is illustr…
Hard-thresholding-based algorithms have seen various advantages for sparse optimization in controlling the sparsity and allowing for fast computation. Recent research shows that when techniques of the Newton-type methods are integrated, their numerical performance can be improved surprisingly. This paper develops a gradient projection Newton pursuit algorithm that mainly adopts the hard-thresholding operator and employs the Newton pursuit only when certain conditions are satisfied. The proposed algorithm is capable of converging globally and quadratically under the standard assumptions. When i…
High fidelity models used in many science and engineering applications couple multiple physical states and parameters. Inverse problems arise when a model parameter cannot be determined directly, but rather is estimated using (typically sparse and noisy) measurements of the states. The data is usually not sufficient to simultaneously inform all of the parameters. Consequently, the governing model typically contains parameters which are uncertain but must be specified for a complete model characterization necessary to invert for the parameters of interest. We refer to the combination of the add…
In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging techology known as impedance acoustic tomography, for which we provide numerical experiments.
We apply the superiorization methodology to the intensity-modulated radiation therapy (IMRT) treatment planning problem. In superiorization, linear voxel dose inequality constraints are the fundamental modeling tool within which a feasibility-seeking projection algorithm will seek a feasible point. This algorithm is then perturbed with gradient descent steps to reduce a nonlinear objective function. Within the open-source inverse planning toolkit matRad, we implement a prototypical algorithmic framework for superiorization using the well-established Agmon, Motzkin, and Schoenberg (AMS) feasibi…
X-ray Computed Tomography (CT) reconstruction from a sparse number of views is a useful way to reduce either the radiation dose or the acquisition time, for example in fixed-gantry CT systems, however this results in an ill-posed inverse problem whose solution is typically computationally demanding. Approximate Message Passing (AMP) techniques represent the state of the art for solving under-sampling Compressed Sensing problems with random linear measurements but there are still not clear solutions on how AMP should be modified and how it performs with real world problems. This paper investiga…
