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Selected Topics in Signal Processing, IEEE Journal of

Issue 4 • Date Aug. 2013

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Displaying Results 1 - 25 of 26
  • Table of contents

    Page(s): C1 - C4
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  • IEEE Journal of Selected Topics in Signal Processing publication information

    Page(s): C2
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  • Introduction to the issue on differential geometry in signal processing

    Page(s): 573 - 575
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  • Minkovskian Gradient for Sparse Optimization

    Page(s): 576 - 585
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (2446 KB) |  | HTML iconHTML  

    Information geometry is used to elucidate convex optimization problems under L1 constraint. A convex function induces a Riemannian metric and two dually coupled affine connections in the manifold of parameters of interest. A generalized Pythagorean theorem and projection theorem hold in such a manifold. An extended LARS algorithm, applicable to both under-determined and over-determined cases, is studied and properties of its solution path are given. The algorithm is shown to be a Minkovskian gradient-descent method, which moves in the steepest direction of a target function under the Minkovskian L1 norm. Two dually coupled affine coordinate systems are useful for analyzing the solution path. View full abstract»

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  • Learning Ancestral Atom via Sparse Coding

    Page(s): 586 - 594
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    Sparse signal models have been the focus of recent research. In sparse coding, signals are represented with a linear combination of a small number of elementary signals called atoms, and the collection of atoms is called a dictionary. Design of the dictionary has strong influence on the signal approximation performance. Recently, to put prior information into dictionary learning, several methods imposing a certain kind of structure on the dictionary are proposed. In this paper, like wavelet analysis, a dictionary for sparse signal representation is assumed to be generated from an ancestral atom, and a method for learning the ancestral atom is proposed. The proposed algorithm updates the ancestral atom by iterating dictionary update in unstructured dictionary space and projection of the updated dictionary onto the structured dictionary space. The algorithm allows a simple differential geometric interpretation. Numerical experiments are performed to show the characteristics and advantages of the proposed algorithm. View full abstract»

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  • Riemannian Medians and Means With Applications to Radar Signal Processing

    Page(s): 595 - 604
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    We develop a new geometric approach for high resolution Doppler processing based on the Riemannian geometry of Toeplitz covariance matrices and the notion of Riemannian p -means. This paper summarizes briefly our recent work in this direction. First of all, we introduce radar data and the problem of target detection. Then we show how to transform the original radar data into Toeplitz covariance matrices. After that, we give our results on the Riemannian geometry of Toeplitz covariance matrices. In order to compute p-means in practical cases, we propose deterministic and stochastic algorithms, of which the convergence results are given, as well as the rate of convergence and error estimates. Finally, we propose a new detector based on Riemannian median and show its advantage over the existing processing methods. View full abstract»

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  • Spinor Fourier Transform for Image Processing

    Page(s): 605 - 613
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    We propose in this paper to introduce a new spinor Fourier transform for both gray-level and color image processing. Our approach relies on the three following considerations: mathematically speaking, defining a Fourier transform requires to deal with group actions; vectors of the acquisition space can be considered as generalized numbers when embedded in a Clifford algebra; the tangent space of the image surface appears to be a natural parameter of the transform we define by means of so-called spin characters. The resulting spinor Fourier transform may be used to perform frequency filtering that takes into account the Riemannian geometry of the image. We give examples of low-pass filtering interpreted as diffusion process. When applied to color images, the entire color information is involved in a really non marginal process. View full abstract»

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  • Robust Independent Component Analysis via Minimum \gamma -Divergence Estimation

    Page(s): 614 - 624
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    Independent component analysis (ICA) has been shown to be useful in many applications. However, most ICA methods are sensitive to data contamination. In this article we introduce a general minimum U-divergence framework for ICA, which covers some standard ICA methods as special cases. Within the U-family we further focus on the γ-divergence due to its desirable property of super robustness for outliers, which gives the proposed method γ-ICA. Statistical properties and technical conditions for recovery consistency of γ-ICA are studied. In the limiting case, it improves the recovery condition of MLE-ICA known in the literature by giving necessary and sufficient condition. Since the parameter of interest in γ-ICA is an orthogonal matrix, a geometrical algorithm based on gradient flows on special orthogonal group is introduced. Furthermore, a data-driven selection for the γ value, which is critical to the achievement of γ-ICA, is developed. The performance, especially the robustness, of γ-ICA is demonstrated through experimental studies using simulated data and image data. View full abstract»

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  • Existence and Uniqueness of Hyperhelical Array Manifold Curves

    Page(s): 625 - 633
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1855 KB) |  | HTML iconHTML  

    A number of significant problems, arising frequently in array signal processing, have been successfully tackled using methods based on the concept of the array manifold. These approaches take advantage of the inherent information about the array system which is encapsulated in the geometry of the array manifold. Array ambiguities, array uncertainties, array design and performance characterization are just some of the areas that have benefited from this approach. However, the investigation of the geometry of the array manifold itself for most array geometries has been proven to be a complex problem, especially when higher order geometric properties need to be calculated. Nevertheless, special array geometries have been identified, for which the array manifold curve assumes a specific “hyperhelical” shape. This property of the array manifold greatly simplifies its geometric analysis and, consequently, the analysis of the associated array geometries. Hence, the goal of this paper is twofold; to provide the necessary and sufficient conditions for the existence of array manifold curves of hyperhelical shape; and to determine which array geometries can actually give rise to manifold curves of this shape. View full abstract»

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  • Newton Algorithms for Riemannian Distance Related Problems on Connected Locally Symmetric Manifolds

    Page(s): 634 - 645
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    The squared distance function is one of the standard functions on which an optimization algorithm is commonly run, whether it is used directly or chained with other functions. Illustrative examples include center of mass computation, implementation of k-means algorithm and robot positioning. This function can have a simple expression (as in the Euclidean case), or it might not even have a closed form expression. Nonetheless, when used in an optimization problem formulated on non-Euclidean manifolds, the appropriate (intrinsic) version must be used and depending on the algorithm, its gradient and/or Hessian must be computed. For many commonly used manifolds a way to compute the intrinsic distance is available as well as its gradient, the Hessian however is usually a much more involved process, rendering Newton methods unusable on many standard manifolds. This article presents a way of computing the Hessian on connected locally-symmetric spaces on which standard Riemannian operations are known (exponential map, logarithm map and curvature). Although not a requirement for the result, describing the manifold as naturally reductive homogeneous spaces, a special class of manifolds, provides a way of computing these functions. The main example focused in this article is centroid computation of a finite constellation of points on connected locally symmetric manifolds since it is directly formulated as an intrinsic squared distance optimization problem. Simulation results shown here confirm the quadratic convergence rate of a Newton algorithm on commonly used manifolds such as the sphere, special orthogonal group, special Euclidean group, symmetric positive definite matrices, Grassmann manifold and projective space. View full abstract»

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  • Frenet-Serret and the Estimation of Curvature and Torsion

    Page(s): 646 - 654
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    In this paper we approach the problem of analyzing space-time curves. In terms of classical geometry, the characterization of space-curves can be summarized in terms of a differential equation involving functional parameters curvature and torsion whose origins are from the Frenet-Serret framework. In particular, curvature measures the rate of change of the angle which nearby tangents make with the tangent at some point. In the situation of a straight line, curvature is zero. Torsion measures the twisting of a curve, and the vanishing of torsion describes a curve whose three dimensional range is restricted to a two-dimensional plane. By using splines, we provide consistent estimators of curves and in turn, this provides consistent estimators of curvature and torsion. We illustrate the usefulness of this approach on a biomechanics application. View full abstract»

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  • Riemannian Distances for Signal Classification by Power Spectral Density

    Page(s): 655 - 669
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    Signal classification is an important issue in many branches of science and engineering. In signal classification, a feature of the signals is often selected for similarity comparison. A distance metric must then be established to measure the dissimilarities between different signal features. Due to the natural characteristics of dynamic systems, the power spectral density (PSD) of a signal is often used as a feature to facilitate classification. We reason in this paper that PSD matrices have structural constraints and that they describe a manifold in the signal space. Thus, instead of the widely used Euclidean distance (ED), a more appropriate measure is the Riemannian distance (RD) on the manifold. Here, we develop closed-form expressions of the RD between two PSD matrices on the manifold and study some of the properties. We further show how an optimum weighting matrix can be developed for the application of RD to signal classification. These new distance measures are then applied to the classification of electroencephalogram (EEG) signals for the determination of sleep states and the results are highly encouraging. View full abstract»

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  • Array Manifold Curves in {Fraktur {C}}^{N} and Their Complex Cartan Matrix

    Page(s): 670 - 680
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    The differential geometry of array manifold curves has been investigated extensively in the literature, leading to numerous applications. However, the existing differential geometric framework restricts the Cartan matrix to be purely real and so the vectors of the moving frame BBU(s) are found to be orthogonal only in the wide sense (i.e. only the real part of their inner product is equal to zero). Imaginary components are then accounted for separately using the concept of the inclination angle of the manifold. The purpose of this paper is therefore to present an alternative theoretical framework which allows the manifold curve in FrakturCN to be characterized in a more convenient and direct manner. A continuously differentiable strictly orthonormal basis is established and forms a platform for deriving a generalized complex Cartan matrix with similar properties to those established under the previous framework. Concepts such as the radius of circular approximation, the manifold curve radii vector and the frame matrix are also revisited and rederived under this new framework. View full abstract»

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  • A Primer on Stochastic Differential Geometry for Signal Processing

    Page(s): 681 - 699
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    This primer explains how continuous-time stochastic processes (precisely, Brownian motion and other Itô diffusions) can be defined and studied on manifolds. No knowledge is assumed of either differential geometry or continuous-time processes. The arguably dry approach is avoided of first introducing differential geometry and only then introducing stochastic processes; both areas are motivated and developed jointly. View full abstract»

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  • Exploiting Information Geometry to Improve the Convergence Properties of Variational Active Contours

    Page(s): 700 - 707
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    In this paper we seek to exploit information geometry in order to define the Riemannian structure of the statistical manifold associated with the Chan-Vese active contour model. This Riemannian structure is obtained through a relationship between the contour's Mumford-Shah energy functional and the likelihood of the categorical latent variables of a Gaussian mixture model. Accordingly the natural metric of the statistical manifold formed by the contours is determined by their Fisher information matrix. Mathematical developments show that this matrix has a closed-form analytic expression and is diagonal. Based on this, we subsequently develop a natural gradient algorithm for the Chan-Vese active contour, with application to image segmentation. Because the proposed method performs optimization on the parameters natural manifold it attains dramatically faster convergence rates than the Euclidean gradient descent algorithm commonly used in the literature. Experiments performed on standard test images from the active contour literature are presented and confirm that the proposed natural gradient algorithm delivers accurate segmentation results in few iterations. Comparisons with methods from the state of the art show that the proposed method converges extremely fast, and could improve significantly the speed of many existing image segmentation methods based on the Chan-Vese active contour as well as enable its application to new problems. A MATLAB implementation of the proposed method is available at http://www.stats.bris.ac.uk/~p12320/code/SmoothNaturalGradient4ChanVese.rar. View full abstract»

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  • Signal Detection on Euclidean Groups: Applications to DNA Bends, Robot Localization, and Optical Communication

    Page(s): 708 - 719
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    Three problems from disparate application areas are presented and solved here using a unified framework: 1) estimating the bend angle induced in DNA by a bound ligand such as a transcription factor or anti-cancer drug; 2) determining the intent of a mobile robot by observing its trajectories corrupted by environmental noise; 3) estimating the bit-error probability function associated with phase noise in optical communication systems, and the associated problem of filter design. In all three problems, probability densities on the group of proper rigid-body motions of the plane contain a hidden signal that needs to be detected in order to advance the particular application area. Stochastic differential equations and corresponding Fokker-Planck equations describing random processes that evolve on this group (the Euclidean group) are used to model each of these problems, and methods from harmonic analysis and Lie theory are used to write approximate solutions. From these `forward' models the desired `signal' (i.e., an element, or a path, in the Euclidean group is extracted) to infer desired physical parameters from data. View full abstract»

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  • Stable Manifold Embeddings With Structured Random Matrices

    Page(s): 720 - 730
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    The fields of compressed sensing (CS) and matrix completion have shown that high-dimensional signals with sparse or low-rank structure can be effectively projected into a low-dimensional space (for efficient acquisition or processing) when the projection operator achieves a stable embedding of the data by satisfying the Restricted Isometry Property (RIP). It has also been shown that such stable embeddings can be achieved for general Riemannian submanifolds when random orthoprojectors are used for dimensionality reduction. Due to computational costs and system constraints, the CS community has recently explored the RIP for structured random matrices (e.g., random convolutions, localized measurements, deterministic constructions). The main contribution of this paper is to show that any matrix satisfying the RIP (i.e., providing a stable embedding for sparse signals) can be used to construct a stable embedding for manifold-modeled signals by randomizing the column signs and paying reasonable additional factors in the number of measurements, thereby generalizing previous stable manifold embedding results beyond unstructured random matrices. We demonstrate this result with several new constructions for stable manifold embeddings using structured matrices. This result allows advances in efficient projection schemes for sparse signals to be immediately applied to manifold signal models. View full abstract»

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  • IEEE Journal of Selected Topics in Signal Processing information for authors

    Page(s): 731 - 732
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  • J-STSP call for special issue proposals

    Page(s): 733
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  • Call for papers special issue on perception inspired video processing

    Page(s): 734
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  • Special issue on signal processing for large-scale mimo communications

    Page(s): 735
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  • Special issue on signal processing for social networks

    Page(s): 736
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  • Special issue on signal processing in smart electric power grid

    Page(s): 737
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  • Special issue on visual signal processing for wireless networks

    Page(s): 738
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  • 2013 IEEE membership application

    Page(s): 739 - 740
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Aims & Scope

The Journal of Selected Topics in Signal Processing (J-STSP) solicits special issues on topics that cover the entire scope of the IEEE Signal Processing Society including the theory and application of filtering, coding, transmitting, estimating, detecting, analyzing, recognizing, synthesizing, recording, and reproducing signals by digital or analog devices or techniques.

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Meet Our Editors

Editor-in-Chief
Fernando Pereira
Instituto Superior Técnico