Sensitivity computation of the ℓ ₁ minimization problem and its application to dictionary design of ill-posed problems

Abstract

The 1 minimization problem has been studied extensively in the past few years. Recently, there has been a growing interest in its application for inverse problems. Most studies have concentrated in devising ways for sparse representation of a solution using a given prototype dictionary. Very few studies have addressed the more challenging problem of optimal dictionary construction, and even these were primarily devoted to the simplistic sparse coding application. In this paper, sensitivity analysis of the inverse solution with respect to the dictionary is presented. This analysis reveals some of the salient features and intrinsic difficulties which are associated with the dictionary design problem. Equipped with these insights, we propose an optimization strategy that alleviates these hurdles while utilizing the derived sensitivity relations for the design of a locally optimal dictionary. Our optimality criterion is based on local minimization of the Bayesian risk, given a set of training models. We present a mathematical formulation and an algorithmic framework to achieve this goal. The proposed framework offers the design of dictionaries for inverse problems that incorporate non-trivial, non-injective observation operators, where the data and the recovered parameters may reside in different spaces. We test our algorithm and show that it yields improved dictionaries for a diverse set of inverse problems in geophysics and medical imaging.

Designing Optimal Spectral Filters for Inverse Problems

“…The Bayes risk minimization formulation allows us to incorporate probabilistic information in the form of probability distributions in order to design optimal filters. It is important to note that problem (2.4) can also be interpreted as a stochastic programming problem, and we refer the interested reader to books [31,29,27,32], survey papers [8,30], other contributions [21,23,18,14], and references therein. In the next section, we discuss some approaches to approximate problem (2.4), and we describe numerical methods for optimization in this framework.…”

Learning regularization parameters for general-form Tikhonov

“…In this paper, we propose a learning approach for computing optimal regularization parameters for both Tikhonov problems (1.2) and (1.3). Previous work on learning approaches in the context of regularization methods for solving inverse problems can be found in [6,5,9,13,18,19,21,25]. However, none of these works specifically address the general-form Tikhonov and multi-parameter Tikhonov problems.…”

“…Training data is commonly used in scientific applications such as biomedical and geophysical imaging [10,12,33], and previous work on learning approaches in the context of regularization for solving inverse problems can be found in [6,7,11,15,16,21,22,25,32]. However, none of these works specifically address the general-form Tikhonov and multiparameter Tikhonov problems.…”

Optimization-Based Speckle Tracking Algorithm for Left Ventricle Strain Estimation: A Feasibility Study

“…The function is a regularization function which represents the difference between the displacement obtained by block matching and some estimate of the "true" displacement. The parameter allows to weight the regularization term versus the data term [47]. Under the assumption of relatively small speckle decorrelation noise, the resultant displacement field is approximately continuous.…”