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Item Analysis of the DCS one-stage Greedy Algorothm for Common Sparse Supports(2005-11-01) Baron, Dror; Duarte, Marco F.; Wakin, Michael; Sarvotham, Shriram; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Analysis of the DCS one-stage Greedy Algorothm for Common Sparse SupportsShow more Item Approximation and Compression of Piecewise Smooth Images Using a Wavelet/Wedgelet Geometric Model(2003-09-01) Romberg, Justin; Wakin, Michael; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Inherent to photograph-like images are two types of structures: large smooth regions and geometrically smooth edge contours separating those regions. Over the past years, efficient representations and algorithms have been developed that take advantage of each of these types of structure independently: quadtree models for 2D wavelets are well-suited for uniformly smooth images (CÂ² everywhere), while quadtree-organized*wedgelet*approximations are appropriate for purely geometrical images (containing nothing but CÂ² contours). This paper shows how to*combine*the wavelet and wedgelet representations in order to take advantage of both types of structure simultaneously. We show that the asymptotic approximation and rate-distortion performance of a wavelet-wedgelet representation on piecewise smooth images mirrors the performance of both wavelets (for uniformly smooth images) and wedgelets (for purely geometrical images). We also discuss an efficient algorithm for fitting the wavelet-wedgelet representation to an image; the convenient quadtree structure of the combined representation enables new algorithms such as the recent WSFQ geometric image coder.Show more Item Compressing Piecewise Smooth Multidimensional Functions Using Surflets: Rate-Distortion Analysis(2004-03-01) Chandrasekaran, Venkat; Wakin, Michael; Baron, Dror; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Discontinuities in data often represent the key information of interest. Efficient representations for such discontinuities are important for many signal processing applications, including compression, but standard Fourier and wavelet representations fail to efficiently capture the structure of the discontinuities. These issues have been most notable in image processing, where progress has been made on modeling and representing one-dimensional edge discontinuities along*C²*curves. Little work, however, has been done on efficient representations for higher dimensional functions or on handling higher orders of smoothness in discontinuities. In this paper, we consider the class of*N*-dimensional Horizon functions containing a*C*smooth singularity in N-1 dimensions, which serves as a manifold boundary between two constant regions; we first derive the optimal rate-distortion function for this class. We then introduce the^{K}*surflet*representation for approximation and compression of Horizon-class functions. Surflets enable a multiscale, piecewise polynomial approximation of the discontinuity. We propose a compression algorithm using surflets that achieves the optimal asymptotic rate-distortion performance for this function class. Equally important, the algorithm can be implemented using knowledge of only the*N*-dimensional function, without explicitly estimating the (*N*-1)-dimensional discontinuity.*This technical report is a supplement to a CISS 2004 paper "Compression of Higher Dimensional Functions Containing Smooth Discontinuities". The body of the paper is the same, while the appendices contain additional details and proofs for all theorems.*Show more Item Compression of Higher Dimensional Functions Containing Smooth Discontinuities(2004-03-01) Chandrasekaran, Venkat; Wakin, Michael; Baron, Dror; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Discontinuities in data often represent the key information of interest. Efficient representations for such discontinuities are important for many signal processing applications, including compression, but standard Fourier and wavelet representations fail to efficiently capture the structure of the discontinuities. These issues have been most notable in image processing, where progress has been made on modeling and representing one-dimensional edge discontinuities along*C²*curves. Little work, however, has been done on efficient representations for higher dimensional functions or on handling higher orders of smoothness in discontinuities. In this paper, we consider the class of*N*-dimensional Horizon functions containing a*C*smooth singularity in N-1 dimensions, which serves as a manifold boundary between two constant regions; we first derive the optimal rate-distortion function for this class. We then introduce the^{K}*surflet*representation for approximation and compression of Horizon-class functions. Surflets enable a multiscale, piecewise polynomial approximation of the discontinuity. We propose a compression algorithm using surflets that achieves the optimal asymptotic rate-distortion performance for this function class. Equally important, the algorithm can be implemented using knowledge of only the*N*-dimensional function, without explicitly estimating the (*N*-1)-dimensional discontinuity.Show more Item Distributed Compressed Sensing of Jointly Sparse Signals(2005-11-01) Sarvotham, Shriram; Baron, Dror; Wakin, Michael; Duarte, Marco F.; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we expand our theory for distributed compressed sensing (DCS) that enables new distributed coding algorithms for multi-signal ensembles that exploit both intra- and inter-signal correlation structures. The DCS theory rests on a new concept that we term the joint sparsity of a signal ensemble. We present a second new model for jointly sparse signals that allows for joint recovery of multiple signals from incoherent projections through simultaneous greedy pursuit algorithms. We also characterize theoretically and empirically the number of measurements per sensor required for accurate reconstruction.Show more Item Edge Characteristics in Wavelet-Based Image Coding(2001-04-20) Wakin, Michael; Digital Signal Processing (http://dsp.rice.edu/)Show more Accurate prediction of wavelet coefficients relies on an understanding of the phase effects of edge alignment. This research examines techniques for uncovering edge information based on the available coefficients. These techniques are evaluated in the context of reconstructing an image from quantized wavelet coefficients. A predictor is described which can be trained on the coefficients to capture relationships among the pixels. Another method is presented where the quantized coefficients are interpolated to recreate an underlying continuous function. Based on this research, the interpolation process offers more promise in solving the dequantization problem.Show more Item A Geometric Hidden Markov Tree Wavelet Model(2003-08-01) Romberg, Justin; Wakin, Michael; Choi, Hyeokho; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more In the last few years, it has become apparent that traditional wavelet-based image processing algorithms and models have significant shortcomings in their treatment of edge contours. The standard modeling paradigm exploits the fact that wavelet coefficients representing smooth regions in images tend to have small magnitude, and that the multiscale nature of the wavelet transform implies that these small coefficients will persist across scale (the canonical example is the venerable zero-tree coder). The edge contours in the image, however, cause more and more large magnitude wavelet coefficients as we move down through scale to finer resolutions. But if the contours are smooth, they become simple as we zoom in on them, and are well approximated by straight lines at fine scales. Standard wavelet models exploit the grayscale regularity of the smooth regions of the image, but not the geometric regularity of the contours. In this paper, we build a model that accounts for this geometric regularity by capturing the dependencies between complex wavelet coefficients along a contour. The Geometric Hidden Markov Tree (GHMT) assigns each wavelet coefficient (or spatial cluster of wavelet coefficients) a hidden state corresponding to a linear approximation of the local contour structure. The shift and rotational-invariance properties of the complex wavelet transform allow the GHMT to model the behavior of each coefficient given the presence of a linear edge at a specified orientation --- the behavior of the wavelet coefficient given the state. By connecting the states together in a quadtree, the GHMT ties together wavelet coefficients along a contour, and also models how the contour itself behaves across scale. We demonstrate the effectiveness of the model by applying it to feature extraction.Show more Item Geometric Methods for Wavelet-Based Image Compression(SPIE, 2003-08-01) Wakin, Michael; Romberg, Justin; Choi, Hyeokho; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Natural images can be viewed as combinations of smooth regions, textures, and geometry. Wavelet-based image coders, such as the space-frequency quantization (SFQ) algorithm, provide reasonably efficient representations for smooth regions (using zerotrees, for example) and textures (using scalar quantization) but do not properly exploit the geometric regularity imposed on wavelet coefficients by features such as edges. In this paper, we develop a representation for wavelet coefficients in geometric regions based on the wedgelet dictionary, a collection of geometric atoms that construct piecewise-linear approximations to contours. Our*wedgeprint representation*implicitly models the coherency among geometric wavelet coefficients. We demonstrate that a simple compression algorithm combining wedgeprints with zerotrees and scalar quantization can achieve near-optimal rate-distortion performance*D*(*R*) ~ (log*R*)Â²/*R*Â² for the class of piecewise-smooth images containing smooth*C*Â² regions separated by smooth*C*Â² discontinuities. Finally, we extend this simple algorithm and propose a complete compression framework for natural images using a rate-distortion criterion to balance the three representations. Our Wedgelet-SFQ (WSFQ) coder outperforms SFQ in terms of visual quality and mean-square error.Show more Item Geometric Tools for Image Compression(2002-11-01) Wakin, Michael; Romberg, Justin; Choi, Hyeokho; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Images typically contain strong geometric features, such as edges, that impose a structure on pixel values and wavelet coefficients. Modeling the joint coherent behavior of wavelet coefficients is difficult, and standard image coders fail to fully exploit this geometric regularity. We introduce wedgelets as a geometric tool for image compression. Wedgelets offer piecewise-linear approximations of edge contours and can be efficiently encoded. We describe the fundamental challenges that arise when applying such a tool to image compression. To meet these challenges, we also propose an efficient rate-distortion framework for natural image compression using wedgelets.Show more Item High-Resolution Navigation on Non-Differentiable Image Manifolds(2005-03-01) Wakin, Michael; Donoho, David; Choi, Hyeokho; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more The images generated by varying the underlying articulation parameters of an object (pose, attitude, light source position, and so on) can be viewed as points on a low-dimensional*image parameter articulation manifold*(IPAM) in a high-dimensional ambient space. In this paper, we develop theory and methods for the inverse problem of estimating, from a given image on or near an IPAM, the underlying parameters that produced it. Our approach is centered on the observation that, while typical image manifolds are not differentiable, they have an intrinsic multiscale geometric structure. In fact, each IPAM has a family of approximate tangent spaces, each one good at a certain resolution. Putting this structural aspect to work, we develop a new algorithm for high-accuracy parameter estimation based on a coarse-to-fine Newton iteration through the family of approximate tangent spaces. We test the algorithm in several idealized registration and pose estimation problems.Show more Item Image Compression using an Efficient Edge Cartoon + Texture Model(IEEE Computer Society, 2002-04-01) Wakin, Michael; Romberg, Justin; Choi, Hyeokho; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Wavelet-based image coders optimally represent smooth regions and isolated point singularities. However, wavelet coders are less adept at representing perceptually important edge singularities, and coding performance suffers significantly as a result. In this paper, we propose a novel two-stage image coder framework based on modeling images as edge cartoons + textures. In stage 1, we infer and efficiently code the edge information from the image using a multiscale wedgelet decomposition. In stage 2, we code the residual, "edgeless" texture image using a standard wavelet coder. Our preliminary coder improves significantly over standard wavelet coding techniques in terms of visual quality.Show more Item Image Compression using Multiscale Geometric Edge Models(2002-05-20) Wakin, Michael; Digital Signal Processing (http://dsp.rice.edu/)Show more Edges are of particular interest for image compression, as they communicate important information, contribute large amounts of high-frequency energy, and can generally be described with few parameters. Many of today's most competitive coders rely on wavelets to transform and compress the image, but modeling the joint behavior of wavelet coefficients along an edge presents a distinct challenge. In this thesis, we examine techniques for exploiting the simple geometric structure which captures edge information. Using a multiscale wedgelet decomposition, we present methods for extracting and compressing a cartoon sketch containing the significant edge information, and we discuss practical issues associated with coding the residual textures. Extending these techniques, we propose a rate-distortion optimal framework (based on the Space-Frequency Quantization algorithm) using wedgelets to capture geometric information and wavelets to describe the rest. At low bitrates, this method yields compressed images with sharper edges and lower mean-square error.Show more Item A Markov Chain Analysis of Blackjack Strategy(2004-07-01) Wakin, Michael; Rozell, ChrisShow more Blackjack receives considerable attention from mathematicians and entrepreneurs alike, due to its simple rules, its inherent random nature, and the abundance of "prior" information available to an observant player. Many attempts have been made to propose card-counting systems that exploit such information to the player's advantage. Because blackjack is a complicated game, attempts to actually calculate the expected gain from a particular system often rely on simulation techniques. While such techniques may yield correct results, they may also fail to explore the interesting mathematical properties of the game. Despite the apparent complexity, there is a great deal of structure inherent in both the blackjack rules and the card-counting systems. Exploiting this structure and elementary results from the theory of Markov chains, we present a novel framework for analyzing the expected advantage of a card-counting system entirely without simulation. The method presented here requires only a few, mild simplifying assumptions, can account for many rule variations, and is applicable to a large class of counting systems. As a specific example, we verify the reported advantage provided by one of the earliest systems, the Complete Point-Count System, discussed in Edward Thorp's famous book,*Beat the Dealer*. While verifying this analysis is satisfying, in our opinion the primary value of this work lies in the exposition of an interesting mathematical framework for analyzing a complicated "real-world" problem.Show more Item Multiscale Geometric Image Processing(2003-07-01) Romberg, Justin; Wakin, Michael; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more Since their introduction a little more than 10 years ago, wavelets have revolutionized image processing. Wavelet based algorithms define the state-of-the-art for applications including image coding (JPEG2000), restoration, and segmentation. Despite their success, wavelets have significant shortcomings in their treatment of edges. Wavelets do not parsimoniously capture even the simplest geometrical structure in images, and wavelet based processing algorithms often produce images with ringing around the edges. As a first step towards accounting for this structure, we will show how to explicitly capture the geometric regularity of contours in cartoon images using the wedgelet representation and a multiscale geometry model. The wedgelet representation builds up an image out of simple piecewise constant functions with linear discontinuities. We will show how the geometry model, by putting a joint distribution on the orientations of the linear discontinuities, allows us to weigh several factors when choosing the wedgelet representation: the error between the representation and the original image, the parsimony of the representation, and whether the wedgelets in the representation form "natural" geometrical structures. Finally, we will analyze a simple wedgelet coder based on these principles, and show that it has optimal asymptotic performance for simple cartoon images.Show more Item The Multiscale Structure of Non-Differentiable Image Manifolds(SPIE, 2005-08-01) Wakin, Michael; Donoho, David; Choi, Hyeokho; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more In this paper, we study families of images generated by varying a parameter that controls the appearance of the object/scene in each image. Each image is viewed as a point in high-dimensional space; the family of images forms a low-dimensional submanifold that we call an image appearance manifold (IAM). We conduct a detailed study of some representative IAMs generated by translations/rotations of simple objects in the plane and by rotations of objects in 3-D space. Our central, somewhat surprising, finding is that IAMs generated by images with sharp edges are nowhere differentiable. Moreover, IAMs have an inherent multiscale structure in that approximate tangent planes fitted to*ps*-neighborhoods continually twist off into new dimensions as the scale parameter $\eps$ varies. We explore and explain this phenomenon. An additional, more exotic kind of local non-differentiability happens at some exceptional parameter points where occlusions cause image edges to disappear. These non-differentiabilities help to understand some key phenomena in image processing. They imply that Newton's method will not work in general for image registration, but that a multiscale Newton's method will work. Such a multiscale Newton's method is similar to existing coarse-to-fine differential estimation algorithms for image registration; the manifold perspective offers a well-founded theoretical motivation for the multiscale approach and allows quantitative study of convergence and approximation. The manifold viewpoint is also generalizable to other image understanding problems.Show more Item Multiscale Wedgelet Image Analysis: Fast Decompositions and Modeling(2002-06-01) Romberg, Justin; Wakin, Michael; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more The most perceptually important features in images are geometrical, the most prevalent being the smooth contours ("edges") that separate different homogeneous regions and delineate distinct objects. Although wavelet based algorithms have enjoyed success in many areas of image processing, they have significant short-comings in their treatment of edges. Wavelets do not parsimoniously capture even the simplest geometrical structure in images, and as a result wavelet based processing algorithms often produce images with ringing around the edges. The multiscale wedgelet framework is a first step towards explicitly capturing geometrical structure in images. The framework has two components: decomposition and representation. The multiscale wavelet decomposition divides the image into dyadic blocks at different scales and projects these image blocks onto wedgelets - simple piecewise constant functions with linear discontinuities. The multiscale wedgelet representation is an approximation of the image built out of wedgelets from the decomposition. In choosing the wedgelets to form the representation, we can weigh several factors: the error between the representation and the original image, the parsimony of the representation, and whether the wedgelets in the representation form "natural" geometrical structure. In this paper, we show that an efficient multiscale wedgelet decomposition is possible if we carefully choose the set of possible wedgelet orientations. We also present a modeling framework that makes it possible to incorporate simple geometrical constraints into the choice of wedgelet representation, resulting in parsimonious image approximations with smooth contours.Show more Item Non-Redundant, Linear-Phase, Semi-Orthogonal, Directional Complex Wavelets(2004-05-01) Fernandes, Felix; Wakin, Michael; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more The directionality and phase information provided by non-redundant complex wavelet transforms (NCWTs) provide significant potential benefits for image/video processing and compression applications. However, because existing NCWTs are created by downsampling filtered wavelet coefficients, the finest scale of these transforms has resolution 4x lower than the real input signal. In this paper, we propose a linear-phase, semi-orthogonal, directional NCWT design using a novel*triband*filter bank. At the finest scale, the resulting transform has resolution 3x lower than the real input signal. We provide a design example to demonstrate three important properties for image/video processing applications: directionality, magnitude coherency, and phase coherency.Show more Item On The Problem of Simultaneous Encoding of Magnitude and Location Information(2002-11-20) Castro, Rui; Wakin, Michael; Orchard, Michael; Digital Signal Processing (http://dsp.rice.edu/)Show more Modern image coders balance bitrate used for encoding the location of signicant transform coefficients, and bitrate used for coding their values. The importance of balancing location and value information in practical coders raises fundamental open questions about how to code even simple processes with joint uncertainty in coefficient location and magnitude. This paper studies the most basic example of such a process: a 2-D process studied earlier by Weidmann and Vetterli that combines Gaussian magnitude information with Bernoulli location uncertainty. The paper offers insight into the coding of this process by investigating several new coding strategies based on more general approaches to lossy compression of location. Extending these ideas to practical coding, we develop a trellis-coded quantization algorithm with performance matching the published theoretical bounds. Finally, we evaluate the quality of our strategies by deriving a rate-distortion bound using Blahut's algorithm for discrete sources.Show more Item Phase and Magnitude Perceptual Sensitivities in Nonredundant Complex Wavelet Representations(2003-11-01) Wakin, Michael; Orchard, Michael; Baraniuk, Richard G.; Chandrasekaran, Venkat; Digital Signal Processing (http://dsp.rice.edu/)Show more The recent development of a nonredundant complex wavelet transform allows a novel framework for image analysis. Work on this representation has recognized that the phase and magnitude of complex coefficients can be related to important geometric properties in images. Existing work on human visual system (HVS) sensitivity offers little guidance in understanding the relative importance of noise (e.g., introduced by lossy coding) in phase components and magnitude components. The distinct geometric significance of the two components would suggest that their respective errors relate to different types of image structure, and thus each would have its own unique HVS sensitivity. In this paper, we extend the study of just-noticeable-differences (JND) to magnitude/phase sensitivities in complex wavelet representations and outline and report on preliminary experiments characterizing them.Show more Item Random Filters for Compressive Sampling and Reconstruction(2006-05-01) Baraniuk, Richard G.; Wakin, Michael; Duarte, Marco F.; Tropp, Joel A.; Baron, Dror; Digital Signal Processing (http://dsp.rice.edu/)Show more We propose and study a new technique for efficiently acquiring and reconstructing signals based on convolution with a fixed FIR filter having random taps. The method is designed for sparse and compressible signals, i.e., ones that are well approximated by a short linear combination of vectors from an orthonormal basis. Signal reconstruction involves a non-linear Orthogonal Matching Pursuit algorithm that we implement efficiently by exploiting the nonadaptive, time-invariant structure of the measurement process. While simpler and more efficient than other random acquisition techniques like Compressed Sensing, random filtering is sufficiently generic to summarize many types of compressible signals and generalizes to streaming and continuous-time signals. Extensive numerical experiments demonstrate its efficacy for acquiring and reconstructing signals sparse in the time, frequency, and wavelet domains, as well as piecewise smooth signals and Poisson processes.Show more