
[Show abstract] [Hide abstract]
ABSTRACT: A novel variational model for deformable multimodal image registration is presented in this work. As an alternative to the models based on maximizing mutual information, the Renyi's statistical dependence measure of two random variables is proposed as a measure of the goodness of matching in our objective functional. The proposed model does not require an estimation of the continuous joint probability density function. Instead, it only needs observed independent instances. Moreover, the theory of reproducing kernel Hilbert space is used to simplify the computation. Experimental results and comparisons with several existing methods are provided to show the effectiveness of the model. Inverse Problems and Imaging 02/2015; 9(1):79103. DOI:10.3934/ipi.2015.9.79 · 1.13 Impact Factor

Source Available from: Luis Sanchez Giraldo
[Show abstract] [Hide abstract]
ABSTRACT: Information theory provides principled models to analyze different inference
and learning problems such as hypothesis testing, clustering, dimensionality
reduction, classification, among others. However, the use of information
theoretic quantities as test statistics, that is, as quantities obtained from
empirical data, posses a challenging estimation problem that often leads to
strong simplifications such as Gaussian models, or the use of plug in density
estimators that are restricted to certain representation of the data. In this
paper, a framework to nonparametrically obtain measures of entropy directly
from data using infinitely divisible kernels is presented. In resemblance to
quantum information theory, functionals on positive definite matrices that
satisfy similar properties to the ones given in Renyi's axiomatic definition of
entropy are defined. Therefore, the estimation of the probability law
underlying the data is avoided, capitalizing on the representation power that
positive definite kernels bring. In the proposed framework, analogues to
quantities such as conditional entropy and mutual information are obtained.
Numerical validation using the proposed quantities to test independence is
provided. In the considered examples, the proposed framework can achieve state
of the art performances. IEEE Transactions on Information Theory 11/2012; 61(1). DOI:10.1109/TIT.2014.2370058 · 2.33 Impact Factor

[Show abstract] [Hide abstract]
ABSTRACT: Exploratory tools that are sensitive to arbitrary statistical variations in spike train observations open up the possibility of novel neuroscientific discoveries. Developing such tools, however, is difficult due to the lack of Euclidean structure of the spike train space, and an experimenter usually prefers simpler tools that capture only limited statistical features of the spike train, such as mean spike count or mean firing rate. We explore strictly positivedefinite kernels on the space of spike trains to offer both a structural representation of this space and a platform for developing statistical measures that explore features beyond count or rate. We apply these kernels to construct measures of divergence between two point processes and use them for hypothesis testing, that is, to observe if two sets of spike trains originate from the same underlying probability law. Although there exist positivedefinite spike train kernels in the literature, we establish that these kernels are not strictly definite and thus do not induce measures of divergence. We discuss the properties of both of these existing nonstrict kernels and the novel strict kernels in terms of their computational complexity, choice of free parameters, and performance on both synthetic and real data through kernel principal component analysis and hypothesis testing. Neural Computation 04/2012; 24(8):222350. DOI:10.1162/NECO_a_00309 · 2.21 Impact Factor

[Show abstract] [Hide abstract]
ABSTRACT: This paper proposes a unified framework for several available measures of independence by generalizing the concept of information theoretic learning (ITL). The key component of ITL is the use of inner product between two density functions as a measure of similarity between two random variables. We show that by generalizing the inner product using a symmetric strictly positivedefinite kernel and by choosing appropriate kernels, it is possible to reproduce a number of popular measures of independence. This unified framework also allows the design of new strictly positivedefinite kernels and corresponding measures of independence. Following this framework we explore a new measure of independence and apply it in the context of linear independent component analysis (ICA). An attractive property of the proposed method is that it does not involve any free parameter and we demonstrate that it performs equally well compared to the existing methods for ICA. IEEE Transactions on Signal Processing 09/2011; 59(859):3624  3635. DOI:10.1109/TSP.2011.2153197 · 2.79 Impact Factor

Source Available from: psu.edu
[Show abstract] [Hide abstract]
ABSTRACT: In wireless networks, the channels are often subject to random variations that limit the reliability of communications between any two radios. Geographic transmission strategies can improve the performance in such networks, by allowing any of a transmitter's neighbors that success fully receive a transmission and are in the direction of the packet's destination to forward the packet on to the destination. However, requiring all of the radios in a network to keep their receivers on to receive geographic transmissions will significantly shorten the network lifetime by depleting the energy of the radios in the network. In this work, we investigate an optimal strategy for deciding which neighbors of a transmitter should activate to try to recover a transmission. We find a solution to this problem by solving for a related measure in a constrained optimization problem. We present results that compare the performance of the optimal approach, our previous suboptimal eorts, and a conventional approach. time to deploy such resources. Such scenarios typically arise in tactical military communications and during disaster recovery operations. The communication signals in wireless MANETs usually experi ence significant losses from both electromagnetic attenuation over distance and mul tipath fading. Multipath fading, often just called fading, is a random phenomenon caused when the transmitted signal reflects o objects in the environment. The sig SIAM Journal on Applied Mathematics 01/2011; 71(2):586604. DOI:10.1137/090756879 · 1.43 Impact Factor

Source Available from: Jose C Principe
[Show abstract] [Hide abstract]
ABSTRACT: This paper addresses the estimation of symmetric χ<sup>2</sup>divergence between two point processes. We propose a novel approach by, first, mapping the space of spike trains in an appropriate functional space, and then, estimating the divergence in this functional space using a least square regression approach. We compare the proposed approach with other available methods on simulated data, and discuss its pros and cons. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011, May 2227, 2011, Prague Congress Center, Prague, Czech Republic; 01/2011

[Show abstract] [Hide abstract]
ABSTRACT: In this paper, we propose a novel test of independence based on the concept of correntropy. We explore correntropy from a statistical perspective and discuss its properties in the context of testing independence. We introduce the novel concept of parametric correntropy and design a test of independence based on it. We further discuss how the proposed test relaxes the assumption of Gaussianity. Finally, we discuss some computational issues related to the proposed method and compare it with stateoftheart techniques. Signal Processing 01/2011; 91(1):1527. DOI:10.1016/j.sigpro.2010.06.002 · 2.21 Impact Factor

Source Available from: psu.edu
[Show abstract] [Hide abstract]
ABSTRACT: In wireless multihop packet radio networks (MPRNs) that employ geographic transmissions, sleep schedules or node activation techniques may be used to power off some nodes to conserve energy. We consider the problem of selecting which nodes should power on to listen to a scheduled transmission when the channel suffers from random fading. We choose the objective of maximizing the expected value of the distance covered in a single transmission between a transmitter and the farthest receiver that successfully receives the packet, under a constraint on the expected number of receivers that turn on. Since there is a tradeoff between the distance of a node from the transmitter and the probability that the node receives the message correctly, we propose to use nodeactivation based on linkdistance (NABOLD). We investigate optimal and suboptimal NABOLD schemes and compare their performance with that of schemes that use a constant sleep schedule for every node within some radius of the transmitter. Our results show that the proposed NABOLD schemes achieve significantly larger transmission distances than conventional schemes. IEEE Transactions on Communications 06/2010; DOI:10.1109/TCOMM.2010.05.080695 · 1.99 Impact Factor

[Show abstract] [Hide abstract]
ABSTRACT: We present a novel variational framework for deformable multimodal image registration. Our approach is based on Renyi's statistical dependence measure of two random variables with the use of reproducing kernel Hilbert spaces associated with Gaussian kernels to simplify the computation. The popularly used method of maximizing mutual information based optimization algorithms are complex and sensitive to the quantization of the intensities, because it requires the estimation of continuous joint probability density function (pdf). The proposed model does not deal with joint pdf but instead observed independent samples. Experimental results are provided to show the effectiveness of the model. Proceedings of SPIE  The International Society for Optical Engineering 03/2010; DOI:10.1117/12.844553 · 0.20 Impact Factor

Source Available from: wireless.ece.ufl.edu
[Show abstract] [Hide abstract]
ABSTRACT: In mobile ad hoc networks (MANETs), conventional packet forwarding schemes that preselect the nexthop receivers for a packet may fail if the channel coherence time is on the order of the typical packet duration because the preselected node may often suffer a deep fade for the duration of the packet. An alternative approach is geographic transmission, in which the packet is transmitted in the direction of the destination, but the nexthop forwarding node is selected among those nodes that are in the direction of the destination and that correctly recover the message. This approach takes advantage of multiuser diversity to significantly improve the probability of the packet being correctly received by a forwarding agent. However, this approach places additional burden on the energies of the mobile nodes if the forwarding scheme requires all of the nexthop neighbors of the transmitter (that are in the direction of the destination) attempt to receive a transmitted message. In this paper, we consider the joint design of nodeactivation strategies and transmission rates to maximize the expected value of transport capacity over a Nakagamim channel under a constraint on the expected number of nodes that attempt to receive a packet. We show that our approach offers better performance than other approaches. Global Telecommunications Conference, 2008. IEEE GLOBECOM 2008. IEEE; 01/2009

[Show abstract] [Hide abstract]
ABSTRACT: Problems involving cracks are of particular importance in structural mechanics, and gave rise to many interesting mathematical techniques to treat them. The difficulties stem from the singularities of domains, which yield lower regularity of solutions. Of particular interest are techniques which allow us to identify cracks and defects from the mechanical properties. Long before advent of mathematical modeling in structural mechanics, defects were identified by the fact that they changed the sound of a piece of material when struck. These techniques have been refined over the years. This volume gives a compilation of recent mathematical methods used in the solution of problems involving cracks, in particular problems of shape optimization. It is based on a collection of recent papers in this area and reflects the work of many authors, namely Gilles Frémiot (Nancy), Werner Horn (Northridge), Jiří Jarušek (Prague), Alexander Khludnev (Novosibirsk), Antoine Laurain (Graz), Murali Rao (Gainesville), Jan Sokołowski (Nancy) and Carol Ann Shubin (Northridge). We review the techniques which can be used for numerical analysis and shape optimization of problems with cracks and of the associated variational inequalities. The mathematical results include sensitivity analysis of variational inequalities, based on the concept of conical differential introduced by Mignot. We complete results on conical differentiability obtained for obstacle problems, by results derived for cracks with nonpenetration condition and parabolic variational inequalities. Numerical methods for some problems are given as an illustration. From the point of view of applied mathematics numerical analysis is a necessary ingredient of applicability of the models proposed. We also extend the result on conical differentiability to the case of some evolution variational inequalities. The same mathematical model can be represented in different ways, like primal, dual or mixed formulations for an elliptic problem. We use such possibilities for models with cracks. For the shape sensitivity analysis, in Chapters 1 to 3 we give a thorough introduction to the use of first and second order shape derivatives and their application to problems involving cracks. In Chapter 1, for the convenience of the reader, we provide classical results on shape sensitivity analysis in smooth domains. In Chapter 2, the results on the first order Eulerian semiderivative in domains with cracks are presented. Of particular interest is the socalled structure theorem for the shape derivative. In Chapter 3, the results on the Fréchet derivative in domains with cracks are presented as well, for first and second order derivatives, using a technique different from that in Chapter 2. In Chapter 4, we extend those ideas to Banach spaces, and give some applications of this extended theory. The polyhedricity of convex sets is considered in the spirit of {ignot}, {rs01}, in the most general setting. These abstract results can be applied to sensitivity analysis of crack problems with nonlinear boundary conditions. The results obtained use nonlinear potential theory and are interesting on their own. In Chapter 5, several techniques for the study of cracked domains with nonpenetration conditions on the crack faces in elastic bodies are presented. The classical crack theory in elasticity is characterized by linear boundary conditions which do not correspond to the physical reality since the crack faces can penetrate each other in this model. In this chapter, nonpenetration conditions on the crack faces are considered, which leads to a nonlinear problem. The model is presented and the shape sensitivity analysis is performed. Chapter 6 is devoted to the newly developed smooth domain method for cracks. In that chapter the problem on a domain with a crack is transformed into a new problem on a smooth domain. This approach is useful for numerical methods. In {belh} this formulation is used combined with mixed finite elements, and some error estimates are derived for the finite element approximation of variational inequalities with nonlinear condition on the crack faces. We give applications of this method to some classical problems. Finally, in Chapter 7 we study integrodifferential equations arising from bridged crack models. This is a classical technique, but we introduce a few modern approaches to it for completeness sake. Dissertationes Mathematicae 01/2009; 462. DOI:10.4064/dm46201 · 0.44 Impact Factor

Frontiers in Systems Neuroscience. Conference Abstract: Computational and systems neuroscience (COSYNE); 01/2009

Source Available from: P. Chen
[Show abstract] [Hide abstract]
ABSTRACT: We present a novel classifier for a collection of nonnegative L1L1 functions. Given two sets of data, one set coming from “similar” distributions labeled as normal, and the other unspecified labeled as abnormal. To understand the structure of normality, and further to classify new data with minimal errors, we propose to find the smallest CKL spheres (based on Csiszar divergences) including as many normal data as possible and excluding as many abnormal data as possible. We prove the existence and uniqueness of such a classifier. Journal of Mathematical Analysis and Applications 06/2008; 342(2):915–930. DOI:10.1016/j.jmaa.2007.12.071 · 1.12 Impact Factor

Source Available from: P. Chen
[Show abstract] [Hide abstract]
ABSTRACT: Bregman divergences are generalizations of the well known Kullback Leibler divergence. They are based on convex functions and have recently received great attention. We present a class of "squared root metrics" based on Bregman divergences. They can be regarded as natural generalization of Euclidean distance. We provide necessary and sufficient conditions for a convex function so that the square root of its associated average Bregman divergence is a metric. Analysis of noisecorrupted data, is difficult without interpreting the data to have been randomly drawn from some unknown distribution with unknown parameters. The most common assumption on noise is Gaussian distribution. However, it may be inappropriate if data is binaryvalued or integervalued or nonnegative. Gaussian is a member of the exponential family. Other members of this family for example the Poisson and the Bernoulli are better suited for integer and binary data. Exponential families and Bregman divergences (Definition 1.1) have a very intimate relationship. There exists a unique Bregman divergence corresponding to every regular exponential family [13][3]. More precisely, the loglikelihood of an exponential family distribution can be represented by a sum of a Bregman divergence and a parameter unrelated term. Hence, Bregman divergence provides a likelihood distance for exponential family in some sense. This property has been used in generalizing principal component analysis to the Exponential family [7]. The Bregman divergence however is not a metric, because it is not symmetric, and does not satisfy the triangle inequality. Consider the case of KullbackLeibler divergence(defined in Definition 1.4) [8]. It is not a metric. However, as proved in [11] the square root of the JensenShannon divergence 1 2 (KL(f, 1 2 (f + g)) + KL(g, 1 2 (f + g))). is a metric. Moreover, it is always finite for any two densities. In fact, JensenShannon divergence is nothing but an averaged Bregman divergence associated with the convex function xlogx. It is very natural to ask whether square roots of other averaged Bregman divergences also are metric? This is the main motivation of this work. We will provide a sufficient and necessary condition on the associated convex function, such that the square root of the corresponding averaged Bregman divergence is a metric. Clearly the justification of the triangle inequality is the only nontrivial part. One of the most critical properties of a metric is the triangle inequality, which ensures that if both a, b and b, c are "close", so are a, c. This property has many applications. For instance, an important task in pattern recognition is the searching of the nearest neighbor in a multidimensional vector space. One of the efficient methods of finding nearest neighbors is through the construction of a socalled metric tree. Given a metric space with N objects we can arrange them into a metric tree with height ≈ log 2 N . The triangle inequality, then saves a lot of effort in finding the nearest * The second author is supported by NIH R01 NS05283101 A1. Communications in mathematical sciences 01/2008; 6(2008). DOI:10.4310/CMS.2008.v6.n4.a7 · 1.12 Impact Factor

[Show abstract] [Hide abstract]
ABSTRACT: We present a coupled minimization problem for image segmentation using prior shape and intensity profile. One part of the
model minimizes a shape related energy and the energy of geometric active contour with a parameter that balances the influence
from these two. The minimizer corresponding to a fixed parameter in this minimization gives a segmentation and an alignment
between the segmentation and prior shape. The second part of this model optimizes the selection of the parameter by maximizing
the mutual information of image geometry between the prior and the aligned novel image over all the alignments corresponding
to different parameters in the first part. By this coupling the segmentation arrives at higher image gradient, forms a shape
similar to the prior, and captures the prior intensity profile. We also propose using mutual information of image geometry
to generate intensity model from a set of training images. Experimental results on cardiac ultrasound images are presented.
These results indicate that the proposed model provides close agreement with expert traced borders, and the parameter determined
in this model for one image can be used for images with similar properties. International Journal of Computer Vision 03/2007; 71(3):259272. DOI:10.1007/s1126300685242 · 3.81 Impact Factor

[Show abstract] [Hide abstract]
ABSTRACT: In this paper, we propose a variational model for curve matching based on KullbackLeibler(KL) divergence. This framework
accomplishes the difficult task of finding correspondences for a group of curves simultaneously in a symmetric and transitive
fashion. Moreover the distance in the energy functional has the metric property. We also introduce a location weighted model
to handle noise, distortion and occlusion. Numerical results indicate the effective of this framework. The existence of this
model is also provided. Scale Space and Variational Methods in Computer Vision, First International Conference, SSVM 2007, Ischia, Italy, May 30  June 2, 2007, Proceedings; 01/2007

International Journal of Pure and Applied Mathematics 01/2007; 39(3).

Source Available from: Renming Song
[Show abstract] [Hide abstract]
ABSTRACT: Let X be a Lévy process in
\mathbbRd \mathbb{R}^{d} ,
d \geqslant 3d \geqslant 3, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X. Potential Analysis 07/2006; 25(1):127. DOI:10.1007/s111180059003z · 0.99 Impact Factor

04/2006: pages 135144;

01/2006: pages 236245;