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Compressed sensing
 IEEE Trans. Inform. Theory
"... Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measureme ..."
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Cited by 1730 (18 self)
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Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements can be dramatically smaller than the size. Thus, certain natural classes of images with pixels need only = ( 1 4 log 5 2 ()) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual pixel samples. More specifically, suppose has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)—so the coefficients belong to an ball for 0 1. The most important coefficients in that expansion allow reconstruction with 2 error ( 1 2 1
Highly Flexible Image Coding Using NonLinear Representations
, 2003
"... This paper presents a new image representation method based on a Matching Pursuit expansion over a dictionary built on anisotropically refined atoms. New breakthroughs in image coding certainly rely on truly multidimensional signal decompositions, and the coder proposed in this paper provides an ada ..."
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Cited by 5 (3 self)
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This paper presents a new image representation method based on a Matching Pursuit expansion over a dictionary built on anisotropically refined atoms. New breakthroughs in image coding certainly rely on truly multidimensional signal decompositions, and the coder proposed in this paper provides an adaptive way of representing images as a sum of twodimensional features. It is shown to provide very competitive results with the state of the art in image compression, as represented by JPEG2000. The visual quality also clearly favors the Matching Pursuit scheme, whose coding artifacts are less annoying than the ringing introduced by wavelets at very low bit rate. In addition to good compression performance at low bit rate, the new coder has the great advantage of producing a high flexibility scalable bitstream, which is becoming a very important feature in today's visual communication applications. The Matching Pursuit stream can easily be decoded at any spatial resolution, different from the original image, and the bitstream can very simply be truncated at any point to match diverse bandwidth requirements. These spatial and rate scalability features are shown to be way more flexible and less complex than transcoding operations generally applied to state of the art streams. Thanks to both its capacity for efficient representation of multidimensional signals, and its very flexible structure, the novel image coder proposed in this paper certainly opens interesting new perspectives in image coding and compression for visual communication services.
The Morphlet Transform: A Multiscale Representation for Diffeomorphisms
"... We describe a multiscale representation for diffeomorphisms. Our representation allows synthesis – e.g. generate random diffeomorphisms – and analysis – e.g. identify the scales and locations where the diffeomorphism has behavior that would be unpredictable based on its coarsescale behavior. Our re ..."
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Cited by 2 (0 self)
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We describe a multiscale representation for diffeomorphisms. Our representation allows synthesis – e.g. generate random diffeomorphisms – and analysis – e.g. identify the scales and locations where the diffeomorphism has behavior that would be unpredictable based on its coarsescale behavior. Our representation has a forward transform with coefficients that are organized dyadically, in a way that is familiar from wavelet analysis, and an inverse transform that is nonlinear, and generates true diffeomorphisms when the underlying object satisfies a certain sampling condition. Although both the forward and inverse transforms are nonlinear, it is possible to operate on the coefficients in the same way that one operates on wavelet coefficients; they can be shrunk towards zero, quantized, and can be randomized; such procedures are useful for denoising, compressing, and stochastic simulation. Observations include: (a) if a template image with edges is morphed by a complex but known transform, compressing the morphism is far more effective than compressing the morphed image. (b) One can create random morphisms with and desired selfsimilarity exponents by inverse transforming scaled Gaussian noise. (c) Denoising morpishms in a sense smooths the underlying level sets of the object. 1
THE SpEnt METHOD FOR LOSSY SOURCE CODING †
"... At present, the most successful methods for lossy source compression are samplefunction adaptive coders. Prominent examples of these techniques are the still image compression methods utilizing wavelet expansions and tree structures, such as the zerotree method or the SPIHT algorithm, and variable ..."
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At present, the most successful methods for lossy source compression are samplefunction adaptive coders. Prominent examples of these techniques are the still image compression methods utilizing wavelet expansions and tree structures, such as the zerotree method or the SPIHT algorithm, and variable rate speech coders that allocate bits to parameters within a frame based upon the classification of the current frame. All of these techniques can be classified as nonlinear approximation methods. In this work, we use Campbell’s coefficient rate, and the spectral entropy of the source random process, as a guide to formulate a new nonlinear approximation method to lossy source compression. We call this new approach, the spectral entropy (SpEnt) method, and we develop and report on the promise of SpEnt based coders for the lossy compression of still images and wideband speech (50 Hz to 7 kHz). 1.
Analysis of Fractals, Image Compression, Entropy Encoding, KarhunenLoève Transforms
, 2008
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