Murali Rao

University of Florida, Gainesville, Florida, United States

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Publications (63)62.06 Total impact

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    ABSTRACT: Structural heterogeneity of particles can be investigated by their three-dimensional principal components. This paper addresses the question of whether, and with what algorithm, the three-dimensional principal components can be directly recovered from cryo-EM images. The first part of the paper extends the Fourier slice theorem to covariance functions showing that the three-dimensional covariance, and hence the principal components, of a heterogeneous particle can indeed be recovered from two-dimensional cryo-EM images. The second part of the paper proposes a practical algorithm for reconstructing the principal components directly from cryo-EM images without the intermediate step of calculating covariances. This algorithm is based on maximizing the (posterior) likelihood using the Expectation-Maximization algorithm. The last part of the paper applies this algorithm to simulated data and to two real cryo-EM data sets: a data set of the 70S ribosome with and without Elongation Factor-G (EF-G), and a data set of the influenza virus RNA dependent RNA Polymerase (RdRP). The first principal component of the 70S ribosome data set reveals the expected conformational changes of the ribosome as the EF-G binds and unbinds. The first principal component of the RdRP data set reveals a conformational change in the two dimers of the RdRP. Copyright © 2015. Published by Elsevier Inc.
    No preview · Article · Jun 2015 · Journal of Structural Biology
  • Hemant D. Tagare · Murali Rao
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    ABSTRACT: This paper proposes a deterministic explanation for mutual-information-based image registration (MI registration). The explanation is that MI registration works because it aligns certain image partitions. This notion of aligning partitions is new, and is shown to be related to Schur- and quasi-convexity. The partition-alignment theory of this paper goes beyond explaining mutual- information. It suggests other objective functions for registering images. Some of these newer objective functions are not entropy-based. Simulations with noisy images show that the newer objective functions work well for registration, lending support to the theory. The theory proposed in this paper opens a number of directions for further research in image registration. These directions are also discussed.
    No preview · Article · Jun 2015 · IEEE Transactions on Pattern Analysis and Machine Intelligence
  • Yunmei Chen · Jiangli Shi · Murali Rao · Jin-Seop Lee
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    ABSTRACT: A novel variational model for deformable multi-modal 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.
    No preview · Article · Feb 2015 · Inverse Problems and Imaging
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    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 non-parametrically 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.
    Full-text · Article · Nov 2012 · IEEE Transactions on Information Theory
  • Sangun Park · Murali Rao · Dong Wan Shin
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    ABSTRACT: Some extensions of entropy and KL information to the survival function have been recently proposed. We first compare some extensions of KL information and provide a criterion in choosing one among those extensions. Then we study moment constraints for maximum cumulative residual entropy distribution (Rao et al., 2004) in view of the relation between the cumulative residual entropy difference and cumulative residual Kullback–Leibler (KL) information. We further discuss the estimation methods and suggest a weighted cumulative residual entropy. Finally, we discuss the application of the estimated cumulative residual KL information as a goodness of fit test statistic with numerical examples.
    No preview · Article · Nov 2012 · Statistics [?] Probability Letters
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    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 positive-definite 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 positive-definite 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.
    No preview · Article · Apr 2012 · Neural Computation
  • Sohan Seth · Murali Rao · Il Park · José C. Principe
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    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 positive-definite 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 positive-definite 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.
    No preview · Article · Sep 2011 · IEEE Transactions on Signal Processing
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    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.
    Full-text · Conference Paper · May 2011
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    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 state-of-the-art techniques.
    No preview · Article · Jan 2011 · Signal Processing
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    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-
    Preview · Article · Jan 2011 · SIAM Journal on Applied Mathematics
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    ABSTRACT: In multi-hop packet radio networks with low mobility, forwarding strategies based on the location of the radios are considered to be quite robust and scalable since they consume less network resources for the exchange and storage of network state information than conventional routing. When such geographical forwarding strategies are used over fading channels, the presence of multiple potential forwarding agents provides multi-user diversity. However, the number of radios that should keep their receivers active during a transmission may need to be limited to conserve the energy of mobile devices. Since receivers at different distances from the transmitter have different probabilities of recovering a message, we have previously proposed node activation based on link distance (NA-BOLD) schemes to increase the transmission distance and/or transport capacity, with a constraint on the energy used in reception (which depends on the number of nodes that activate to receive a transmission). In this work, we consider a total energy constraint, i.e., a constraint on the sum of the energies used in transmission and reception. We optimize the allocation of energy between transmission and reception when nodes activate using NA-BOLD approaches.
    No preview · Conference Paper · Nov 2010
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    ABSTRACT: In wireless multi-hop 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 node-activation based on link-distance (NA-BOLD). We investigate optimal and sub-optimal NA-BOLD 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 NA-BOLD schemes achieve significantly larger transmission distances than conventional schemes.
    Preview · Article · Jun 2010 · IEEE Transactions on Communications
  • Jiangli Shi · Yunmei Chen · Murali Rao · Jinseop Lee
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    ABSTRACT: We present a novel variational framework for deformable multi-modal 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.
    No preview · Article · Mar 2010 · Proceedings of SPIE - The International Society for Optical Engineering
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    ABSTRACT: In mobile ad hoc networks (MANETs), conventional packet forwarding schemes that pre-select the next-hop receivers for a packet may fail if the channel coherence time is on the order of the typical packet duration because the pre-selected 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 next-hop 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 multi-user 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 next-hop 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 node-activation strategies and transmission rates to maximize the expected value of transport capacity over a Nakagami-m 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.
    Preview · Conference Paper · Jan 2009
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    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 non-penetration 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 semi-derivative in domains with cracks are presented. Of particular interest is the so-called 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 non-linear boundary conditions. The results obtained use non-linear potential theory and are interesting on their own. In Chapter 5, several techniques for the study of cracked domains with non-penetration 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, non-penetration conditions on the crack faces are considered, which leads to a non-linear 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 non-linear condition on the crack faces. We give applications of this method to some classical problems. Finally, in Chapter 7 we study integro-differential equations arising from bridged crack models. This is a classical technique, but we introduce a few modern approaches to it for completeness sake.
    No preview · Article · Jan 2009 · Dissertationes Mathematicae

  • No preview · Conference Paper · Jan 2009
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    Pengwen Chen · Yunmei Chen · Murali Rao
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    ABSTRACT: Bregman divergences are generalizations of the well known Kullback Leibler diver-gence. 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 noise-corrupted 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 binary-valued or integer-valued 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 log-likelihood 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 Kullback-Leibler divergence(defined in Definition 1.4) [8]. It is not a metric. However, as proved in [11] the square root of the Jensen-Shannon 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, Jensen-Shannon 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 so-called 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 NS052831-01 A1.
    Full-text · Article · Dec 2008 · Communications in mathematical sciences
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    Pengwen Chen · Yunmei Chen · Murali Rao
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    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.
    Full-text · Article · Jun 2008 · Journal of Mathematical Analysis and Applications
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    ABSTRACT: Information theoretical learning (ITL) is a signal processing technique that goes far beyond the traditional techniques based on second order statistics which highly relies on the linearity and Gaussinarity assumptions. Information potential (IP) and symmetric information potential (SIP) are very important concepts in ITL used for system adaptation and data inference. In this paper, a mathematical analysis of the bias and the variance of their estimators is presented. Our results show that the variances decrease as the sample size N increases at the speed of O(N<sup>-1</sup>) and a bound exists for the biases. A simple numerical simulation is demonstrated to support our analysis.
    No preview · Conference Paper · Sep 2007
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    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.
    No preview · Article · Mar 2007 · International Journal of Computer Vision