Image restoration and reconstruction using projections onto epigraph set of convex cost fuchtions

Cataloged from PDF version of article.This thesis focuses on image restoration and reconstruction problems. These inverse problems are solved using a convex optimization algorithm based on orthogonal Projections onto the Epigraph Set of a Convex Cost functions (PESC). In order to solve the convex minimization problem, the dimension of the problem is lifted by one and then using the epigraph concept the feasibility sets corresponding to the cost function are defined. Since the cost function is a convex function in R N , the corresponding epigraph set is also a convex set in R N+1. The convex optimization algorithm starts with an arbitrary initial estimate in R N+1 and at each step of the iterative algorithm, an orthogonal projection is performed onto one of the constraint sets associated with the cost function in a sequential manner. The PESC algorithm provides globally optimal solutions for different functions such as total variation, `1-norm, `2-norm, and entropic cost functions. Denoising, deconvolution and compressive sensing are among the applications of PESC algorithm. The Projection onto Epigraph Set of Total Variation function (PES-TV) is used in 2-D applications and for 1-D applications Projection onto Epigraph Set of `1-norm cost function (PES-`1) is utilized. In PES-`1 algorithm, first the observation signal is decomposed using wavelet or pyramidal decomposition. Both wavelet denoising and denoising methods using the concept of sparsity are based on soft-thresholding. In sparsity-based denoising methods, it is assumed that the original signal is sparse in some transform domain such as Fourier, DCT, and/or wavelet domain and transform domain coefficients of the noisy signal are soft-thresholded to reduce noise. Here, the relationship between the standard soft-thresholding based denoising methods and sparsity-based wavelet denoising methods is described. A deterministic soft-threshold estimation method using the epigraph set of `1-norm cost function is presented. It is demonstrated that the size of the `1-ball can be determined using linear algebra. The size of the `1-ball in turn determines the soft-threshold. The PESC, PES-TV and PES-`1 algorithms, are described in detail in this thesis. Extensive simulation results are presented. PESC based inverse restoration and reconstruction algorithm is compared to the state of the art methods in the literature.Tofighi, MohammadM.S

Denoising using projections onto the epigraph set of convex cost functions

A new denoising algorithm based on orthogonal projections onto the epigraph set of a convex cost function is presented. In this algorithm, the dimension of the minimization problem is lifted by one and feasibility sets corresponding to the cost function using the epigraph concept are defined. As the utilized cost function is a convex function in RN, the corresponding epigraph set is also a convex set in RN+1. The denoising algorithm starts with an arbitrary initial estimate in RN+1. At each step of the iterative denoising, an orthogonal projection is performed onto one of the constraint sets associated with the cost function in a sequential manner. The method provides globally optimal solutions for total-variation, ℓ1, ℓ2, and entropic cost functions.1 © 2014 IEEE

A Non-Local Structure Tensor Based Approach for Multicomponent Image Recovery Problems

Non-Local Total Variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the Structure Tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the non-local variations, jointly for the different components, through various $\ell_{1,p}$ matrix norms with $p \ge 1$. To facilitate the choice of the hyper-parameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the ST-NLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented thanks to the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for multispectral and hyperspectral images. The results demonstrate the interest of introducing a non-local structure tensor regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods

Range-doppler radar target detection using denoising within the compressive sensing framework

Compressive sensing (CS) idea enables the reconstruction of a sparse signal from a small set of measurements. CS approach has applications in many practical areas. One of the areas is radar systems. In this article, the radar ambiguity function is denoised within the CS framework. A new denoising method on the projection onto the epigraph set of the convex function is also developed for this purpose. This approach is compared to the other CS reconstruction algorithms. Experimental results are presented1. © 2014 EURASIP

Proximity Operators of Discrete Information Divergences

Information divergences allow one to assess how close two distributions are from each other. Among the large panel of available measures, a special attention has been paid to convex $\varphi$-divergences, such as Kullback-Leibler, Jeffreys-Kullback, Hellinger, Chi-Square, Renyi, and I$_{\alpha}$ divergences. While $\varphi$-divergences have been extensively studied in convex analysis, their use in optimization problems often remains challenging. In this regard, one of the main shortcomings of existing methods is that the minimization of $\varphi$-divergences is usually performed with respect to one of their arguments, possibly within alternating optimization techniques. In this paper, we overcome this limitation by deriving new closed-form expressions for the proximity operator of such two-variable functions. This makes it possible to employ standard proximal methods for efficiently solving a wide range of convex optimization problems involving $\varphi$-divergences. In addition, we show that these proximity operators are useful to compute the epigraphical projection of several functions of practical interest. The proposed proximal tools are numerically validated in the context of optimal query execution within database management systems, where the problem of selectivity estimation plays a central role. Experiments are carried out on small to large scale scenarios

Epigraphical splitting for solving constrained convex optimization problems with proximal tools

International audienceWe propose a proximal approach to deal with a class of convex variational problems involving nonlinear constraints. A large family of constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different blocks of the linearly-transformed signal. For such constraints, the associated projection operator generally does not have a simple form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. In particular, we focus on constraints involving q-norms with q ≥ 1, distance functions to a convex set, and L1,p-norms with p ∈ {2, +∞}. The proposed approach is validated in the context of image restoration by making use of constraints based on Non-Local Total Variation. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems. A second application to a pulse shape design problem is provided in order to illustrate the flexibility of the proposed approach

Range resolution improvement in passive bistatic radars using nested FM channels and least squares approach

One of the main disadvantages of using commercial broadcasts in a Passive Bistatic Radar (PBR) system is the range resolution. Using multiple broadcast channels to improve the radar performance is offered as a solution to this problem. However, it suffers from detection performance due to the side-lobes that matched filter creates for using multiple channels. In this article, we introduce a deconvolution algorithm to suppress the side-lobes. The two-dimensional matched filter output of a PBR is further analyzed as a deconvolution problem. The deconvolution algorithm is based on making successive projections onto the hyperplanes representing the time delay of a target. Resulting iterative deconvolution algorithm is globally convergent because all constraint sets are closed and convex. Simulation results in an FM based PBR system are presented. © 2015 SPIE