On some entropy functionals derived from R\'enyi information divergence

We consider the maximum entropy problems associated with R\'enyi $Q$-entropy, subject to two kinds of constraints on expected values. The constraints considered are a constraint on the standard expectation, and a constraint on the generalized expectation as encountered in nonextensive statistics. The optimum maximum entropy probability distributions, which can exhibit a power-law behaviour, are derived and characterized. The R\'enyi entropy of the optimum distributions can be viewed as a function of the constraint. This defines two families of entropy functionals in the space of possible expected values. General properties of these functionals, including nonnegativity, minimum, convexity, are documented. Their relationships as well as numerical aspects are also discussed. Finally, we work out some specific cases for the reference measure $Q(x)$ and recover in a limit case some well-known entropies

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle and its cases of equality. This Shannon entropic version of the well-known Heisenberg uncertainty principle can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Renyi entropies. We recall that in both Shannon and Renyi cases, and for a given dimension n, the only case of equality occurs for Gaussian random vectors. We show that as n grows, however, the bound is also asymptotically attained in the cases of n-dimensional Student-t and Student-r distributions. A complete analytical study is performed in a special case of a Student-t distribution. We also show numerically that this effect exists for the particular case of a n-dimensional Cauchy variable, whatever the Renyi entropy considered, extending the results of Abe and illustrating the analytical asymptotic study of the student-t case. In the Student-r case, we show numerically that the same behavior occurs for uniformly distributed vectors. These particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the vector when the dimension increases

3D nonrigid medical image registration using a new information theoretic measure.

International audienceThis work presents a novel method for the nonrigid registration of medical images based on the Arimoto entropy, a generalization of the Shannon entropy. The proposed method employed the Jensen-Arimoto divergence measure as a similarity metric to measure the statistical dependence between medical images. Free-form deformations were adopted as the transformation model and the Parzen window estimation was applied to compute the probability distributions. A penalty term is incorporated into the objective function to smooth the nonrigid transformation. The goal of registration is to optimize an objective function consisting of a dissimilarity term and a penalty term, which would be minimal when two deformed images are perfectly aligned using the limited memory BFGS optimization method, and thus to get the optimal geometric transformation. To validate the performance of the proposed method, experiments on both simulated 3D brain MR images and real 3D thoracic CT data sets were designed and performed on the open source elastix package. For the simulated experiments, the registration errors of 3D brain MR images with various magnitudes of known deformations and different levels of noise were measured. For the real data tests, four data sets of 4D thoracic CT from four patients were selected to assess the registration performance of the method, including ten 3D CT images for each 4D CT data covering an entire respiration cycle. These results were compared with the normalized cross correlation and the mutual information methods and show a slight but true improvement in registration accuracy

Tsallis Entropy In Bi-level And Multi-level Image Thresholding

International audienceThe maximum entropy principle has a relevant role in image processing, in particular for thresholding and image segmentation. Different entropic formulations are available to this purpose; one of them is based on the Tsallis non-extensive entropy. Here, we propose a discussion of its use for bi-and multi-level thresholding

Tsallis Entropy In Bi-level And Multi-level Image Thresholding

The maximum entropy principle has a relevant role in image processing, in particular for thresholding and image segmentation. Different entropic formulations are available to this purpose; one of them is based on the Tsallis non-extensive entropy. Here, we propose a discussion of its use for bi-and multi-level thresholding

Medical Image Registration and 3D Object Matching

The great challenge in image registration and 3D object matching is to devise computationally efficient algorithms for aligning images so that their details overlap accurately and retrieving similar shapes from large databases of 3D models. The first problem addressed is this thesis is medical image registration, which we formulate as an optimization problem in the information-theoretic framework. We introduce a viable and practical image registration method by maximizing an entropic divergence measure using a modified simultaneous perturbation stochastic approximation algorithm. The feasibility of the proposed image registration approach is demonstrated through extensive experiments. The rest of the thesis is devoted to a joint exploitation of geometry and topology of 3D objects for as parsimonious as possible representation of models and its subsequent application in 3D object representation, matching, and retrieval problems. More precisely, we introduce a skeletal graph for topological 3D shape representation using Morse theory. The proposed skeletonization algorithm encodes a 3D shape into a topological Reeb graph using a normalized mixture distance function. We also propose a novel graph matching algorithm by comparing the relative shortest paths between the skeleton endpoints. Moreover, we describe a skeletal graph for 3D object matching and retrieval. This skeleton is constructed from the second eigenfunction of the Laplace-Beltrami operator defined on the surface of the 3D object. Using the generalized eigenvalue decomposition, a matrix computational framework based on the finite element method is presented to compute the spectrum of the Laplace-Beltrami operator. Illustrating experiments on two standard 3D shape benchmarks are provided to demonstrate the feasibility and the much improved performance of the proposed skeletal graphs as shape descriptors for 3D object matching and retrieval

Edges Detection Based On Renyi Entropy with Split/Merge

Most of the classical methods for edge detection are based on the first and second order derivatives of gray levels of the pixels of the original image. These processes give rise to the exponential increment of computational time, especially with large size of images, and therefore requires more time for processing. This paper shows the new algorithm based on both the Rényi entropy and the Shannon entropy together for edge detection using split and merge technique. The objective is to find the best edge representation and decrease the computation time. A set of experiments in the domain of edge detection are presented. The system yields edge detection performance comparable to the classic methods, such as Canny, LOG, and Sobel.  The experimental results show that the effect of this method is better to LOG, and Sobel methods. In addition, it is better to other three methods in CPU time. Another benefit comes from easy implementation of this method. Keywords: Rényi Entropy, Information content, Edge detection, Thresholdin

Graph similarity through entropic manifold alignment

In this paper we decouple the problem of measuring graph similarity into two sequential steps. The first step is the linearization of the quadratic assignment problem (QAP) in a low-dimensional space, given by the embedding trick. The second step is the evaluation of an information-theoretic distributional measure, which relies on deformable manifold alignment. The proposed measure is a normalized conditional entropy, which induces a positive definite kernel when symmetrized. We use bypass entropy estimation methods to compute an approximation of the normalized conditional entropy. Our approach, which is purely topological (i.e., it does not rely on node or edge attributes although it can potentially accommodate them as additional sources of information) is competitive with state-of-the-art graph matching algorithms as sources of correspondence-based graph similarity, but its complexity is linear instead of cubic (although the complexity of the similarity measure is quadratic). We also determine that the best embedding strategy for graph similarity is provided by commute time embedding, and we conjecture that this is related to its inversibility property, since the inverse of the embeddings obtained using our method can be used as a generative sampler of graph structure.The work of the first and third authors was supported by the projects TIN2012-32839 and TIN2015-69077-P of the Spanish Government. The work of the second author was supported by a Royal Society Wolfson Research Merit Award

Multimodality and Nonrigid Image Registration with Application to Diffusion Tensor Imaging

The great challenge in image registration is to devise computationally efficient algorithms for aligning images so that their details overlap accurately. The first problem addressed in this thesis is multimodality medical image registration, which we formulate as an optimization problem in the information-theoretic setting. We introduce a viable and practical image registration method by maximizing a generalized entropic dissimilarity measure using a modified simultaneous perturbation stochastic approximation algorithm. The feasibility of the proposed image registration approach is demonstrated through extensive experiments. The rest of the thesis is devoted to nonrigid medical image registration. We propose an informationtheoretic framework by optimizing a non-extensive entropic similarity measure using the quasi-Newton method as an optimization scheme and cubic B-splines for modeling the nonrigid deformation field between the fixed and moving 3D image pairs. To achieve a compromise between the nonrigid registration accuracy and the associated computational cost, we implement a three-level hierarchical multi-resolution approach in such a way that the image resolution is increased in a coarse to fine fashion. The feasibility and registration accuracy of the proposed method are demonstrated through experimental results on a 3D magnetic resonance data volume and also on clinically acquired 4D computed tomography image data sets. In the same vein, we extend our nonrigid registration approach to align diffusion tensor images for multiple components by enabling explicit optimization of tensor reorientation. Incorporating tensor reorientation in the registration algorithm is pivotal in wrapping diffusion tensor images. Experimental results on diffusion-tensor image registration indicate the feasibility of the proposed approach and a much better performance compared to the affine registration method based on mutual information, not only in terms of registration accuracy in the presence of geometric distortions but also in terms of robustness in the presence of Rician noise