The scalar shrinkage-thresholding operator (SSTO) is a key ingredient of many modern statistical signal processing algorithms including: sparse inverse problem solutions, wavelet denoising, and JPEG2000 image compression. In these applications, it is customary to select the threshold of the operator by solving a scalar sparsity penalized quadratic optimization. In this work, we present a natural multidimensional extension of the scalar shrinkage thresholding operator. Similarly to the scalar case, the threshold is determined by the minimization of a convex quadratic form plus an euclidean penalty, however, here the optimization is performed over a domain of dimension N ≥ 1. The solution to this convex optimization problem is called the multidimensional shrinkage threshold operator (MSTO). The MSTO reduces to the standard SSTO in the special case of N = 1. In the general case of N > 1 the optimal MSTO threshold can be found by a simple convex line search. We present three illustrative applications of the MSTO in the context of non-linear regression: l 2 -penalized linear regression, Group LASSO linear regression and Group LASSO logistic regression. In this paper, we introduce a multidimensional generalization of the scalar shrinkage thresholding operator. We define this operator as the minimization of a convex quadratic form plus an Euclidean norm penalty. We analyze this nondifferentiable optimization problem and discuss its properties. In particular, in analogy to the scalar shrinkage operator, we show that this generalization yields a multidimensional Shrinkage Thresholding Operator (MSTO) which takes a vector as an input and shrinks it or thresholds it depending on its Euclidean norm. For this purpose, we reformulate the problem as a constrained quadratic problem with a conic constraint. This principle leads to a theoretical result that transforms this multidimensional optimization problem into a simple line search which can be efficiently implemented. We show by simulations that evaluating the MSTO using line search is competitive with state-of-the-art convex solvers.In the second part of the paper, we discuss applications of the MSTO to statistical regression. First, we consider the Euclidean-norm penalized least squares and discuss its relation to ridge regression and robust regression [5]. Next, we address group LASSO linear regression [6]. In the special case of a block-orthogonal design matrix, we show that the problem can be reduced to evaluating the MSTO for each block. For other Group LASSO problems, we propose two iterative applications of the MSTO. In the first approach, we use Block Coordinate Descent to solve the linear regression problem with an arbitrary design matrix. The second approach tackles more complicated cost functions such as the logistic regression objective. Due to its similarity to the wellknown class of Iterative Thresholding Algorithms [1], we name the latter Iterative Group Shrinkage-Thresholding (IGST). In both cases, the MSTO enables one to solve large scale Group LASS...

Abstract.In this paper we propose an identification of morphological factors that may impact the Whole Body Specific Absorption Rate (WBSAR). The study is conducted for the case of an exposure to a front plane wave at the 2100MHz frequency carrier. This study is based on the development of different regression models for estimating the WBSAR as a function of morphological factors morphology. For this manner, a database of twelve anatomical human models (phantoms) has been considered. Also, eighteen supplementary phantoms obtained using morphing technique were generated to build the requested relation. The paper presents three models based on external morphological factors like the Body Surface Area (BSA), the Body Mass Index (BMI) or the body mass. These models show good results for families obtained by morphing technique on the estimation of the WBSAR (< 10%) but still less accurate (30%) when applied for different original phantoms. This study stresses the importance of the internal morphological factors such as muscle and fat proportions in the characterization of the WBSAR. The regression models are then improved using internal morphological factors with an estimation error around 10% on the WBSAR. Finally, this study is suited for establishing the statistical distribution of the WBSAR for a given population characterized by its morphology.

In this paper, we address three-dimensional tomographic reconstruction of rotational angiography acquisitions. In clinical routine, angular subsampling commonly occurs, due to the technical limitations of C-arm systems or possible improper injection. Standard methods such as filtered backprojection yield a reconstruction that is deteriorated by sampling artifacts, which potentially hampers medical interpretation. Recent developments of compressed sensing have demonstrated that it is possible to significantly improve reconstruction of subsampled datasets by generating sparse approximations through 1-penalized minimization. Based on these results, we present an extension of the iterative filtered backprojection that includes a sparsity constraint called soft background subtraction. This approach is shown to provide sampling artifact reduction when reconstructing sparse objects, and more interestingly, when reconstructing sparse objects over a non-sparse background. The relevance of our approach is evaluated in cone-beam geometry on real clinical data.

A flexible compressed-sensing-based algorithmic approach is proposed that is able to accommodate for a wide range of constraints. It is successfully applied to C-arm CBCT images that may not be so well approximated by piecewise constant functions.

The characterisation of population exposure to a 50-Hz magnetic field (MF) is important for assessing health effects of electromagnetic fields. With the aim of estimating and characterising the exposure of the French population to 50-Hz MFs, two representative samples of the population were made. A random selection method based on the distribution of households in different regions of France was used. The samples were carried out starting from a random polling of telephone numbers of households (listed, unlisted fixed phones and cell phones only). A total of 95,362 telephone numbers were dialed to have 2148 volunteers (1060 children and 1088 adults). They all agreed to carrying an EMDEX II meter, measuring and recording MFs, and to filling out a timetable for a 24-hour period. In this article, the methodology of the sample selection and the collection of all necessary information for the realisation of this study are presented.