Results 1  10
of
226
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinato ..."
Abstract

Cited by 202 (31 self)
 Add to MetaCart
A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Random projections of smooth manifolds
 Foundations of Computational Mathematics
, 2006
"... We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N ..."
Abstract

Cited by 83 (23 self)
 Add to MetaCart
We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N → R M, M < N, on a smooth wellconditioned Kdimensional submanifold M ⊂ R N. As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on M are wellpreserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in R N. Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifoldmodeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data.
Ratedistortion optimized tree structured compression algorithms for piecewise smooth images
 IEEE Trans. Image Processing
, 2005
"... IEEE Transactions on Image Processing This paper presents novel coding algorithms based on tree structured segmentation, which achieve the correct asymptotic ratedistortion (RD) behavior for a simple class of signals, known as piecewise polynomials, by using an RD based prune and join scheme. Fo ..."
Abstract

Cited by 68 (16 self)
 Add to MetaCart
IEEE Transactions on Image Processing This paper presents novel coding algorithms based on tree structured segmentation, which achieve the correct asymptotic ratedistortion (RD) behavior for a simple class of signals, known as piecewise polynomials, by using an RD based prune and join scheme. For the one dimensional (1D) case, our scheme is based on binary tree segmentation of the signal. This scheme approximates the signal segments using polynomial models and utilizes an RD optimal bit allocation strategy among the different signal segments. The scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying RD behavior � D(R) ∼ c02 −c1R � , thus improving over classic wavelet schemes. We also prove that the computational complexity of the scheme is of O (N log N). We then show the extension of this scheme to the two dimensional (2D) case using a quadtree. This quadtree coding scheme also achieves an exponentially decaying RD behavior, for the polygonal image model composed of a white polygon shaped object against a uniform black background, with low computational cost of O (N log N). Again, the key is an RD optimized prune and join strategy. Finally, we conclude with numerical results, which show that the proposed quadtree coding scheme outperforms JPEG2000 by about 1 dB for real images, like cameraman, at low rates of around 0.15 bpp.
Liftingbased invertible motion adaptive transform (LIMAT) framework for highly scalable video compression
 IEEE Transactions on Image Processing
, 2003
"... We propose a new framework for highly scalable video compression, using a Liftingbased Invertible Motion Adaptive Transform (LIMAT). We use motioncompensated lifting steps to implement the temporal wavelet transform, which preserves invertibility, regardless of the motion model. By contrast, the i ..."
Abstract

Cited by 53 (2 self)
 Add to MetaCart
We propose a new framework for highly scalable video compression, using a Liftingbased Invertible Motion Adaptive Transform (LIMAT). We use motioncompensated lifting steps to implement the temporal wavelet transform, which preserves invertibility, regardless of the motion model. By contrast, the invertibility requirement has restricted previous approaches to either blockbased or global motion compensation. We show that the proposed framework effectively applies the temporal wavelet transform along a set of motion trajectories. An implementation demonstrates high coding gain from a finely embedded, scalable compressed bitstream. Results also demonstrate the effectiveness of temporal wavelet kernels other than the simple Haar, and the benefits of complex motion modeling, using a deformable triangular mesh. These advances are either incompatible or difficult to achieve with previously proposed strategies for scalable video compression. Video sequences reconstructed at reduced framerates, from subsets of the compressed bitstream, demonstrate the visually pleasing properties expected from lowpass filtering along the motion trajectories. The paper also describes a compact representation for the motion parameters, having motion overhead comparable to that of motioncompensated predictive coders. Our experimental results compare favourably with others reported in the literature, however, the principle objective of this paper is to motivate a new framework for highly scalable video compression.
ATOMS OF ALL CHANNELS, UNITE! AVERAGE CASE ANALYSIS OF MULTICHANNEL SPARSE RECOVERY USING GREEDY ALGORITHMS
, 2007
"... ..."
On the uniqueness of overcomplete dictionaries, and a practical way to retrieve them
, 2006
"... ..."
Image Compression by Linear Splines over Adaptive Triangulations
"... This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function, by a linear spline over an adapted triangulation, D(Y ), which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the d ..."
Abstract

Cited by 35 (7 self)
 Add to MetaCart
This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function, by a linear spline over an adapted triangulation, D(Y ), which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the distance to the image, measured by the mean square error, among all linear splines over D(Y ). The significant pixels in Y are selected by an adaptive thinning algorithm, which recursively removes less significant pixels in a greedy way, using a sophisticated criterion for measuring the significance of a pixel. The proposed compression method combines the approximation scheme with a customized scattered data coding scheme. We demonstrate that our compression method outperforms JPEG2000 on two geometric images and performs competitively with JPEG2000 on three popular test cases of real images.
Perceptual blur and ringing metrics: Application to JPEG2000,” Signal Process
 Image Commun
, 2004
"... We present a full and noreference blur metric as well as a fullreference ringing metric. These metrics are based on an analysis of the edges and adjacent regions in an image and have very low computational complexity. As blur and ringing are typical artifacts of wavelet compression, the metrics a ..."
Abstract

Cited by 35 (0 self)
 Add to MetaCart
We present a full and noreference blur metric as well as a fullreference ringing metric. These metrics are based on an analysis of the edges and adjacent regions in an image and have very low computational complexity. As blur and ringing are typical artifacts of wavelet compression, the metrics are then applied to JPEG2000 coded images. Their perceptual significance is corroborated through a number of subjective experiments. The results show that the proposed metrics perform well over a wide range of image content and distortion levels. Potential applications include source coding optimization and network resource management. r 2003 Elsevier B.V. All rights reserved.
Video Coding with MotionCompensated Lifted Wavelet Transforms
, 2004
"... This article explores the e#ciency of motioncompensated threedimensional transform coding, a compression scheme that employs a motioncompensated transform for a group of pictures. We investigate this coding scheme experimentally and theoretically. The practical coding scheme employs in temporal d ..."
Abstract

Cited by 34 (13 self)
 Add to MetaCart
This article explores the e#ciency of motioncompensated threedimensional transform coding, a compression scheme that employs a motioncompensated transform for a group of pictures. We investigate this coding scheme experimentally and theoretically. The practical coding scheme employs in temporal direction a wavelet decomposition with motioncompensated lifting steps. Further, we compare the experimental results to that of a predictive video codec with singlehypothesis motion compensation and comparable computational complexity. The experiments show that the 5/3 wavelet kernel outperforms both the Haar kernel and, in many cases, the reference scheme utilizing singlehypothesis motioncompensated predictive coding. The theoretical investigation models this motioncompensated subband coding scheme for a group of K pictures with a signal model for K motioncompensated pictures that are decorrelated by a linear transform. We utilize the KarhunenLoeve Transform to obtain theoretical performance bounds at high bitrates and compare to both optimum intraframe coding of individual motioncompensated pictures and singlehypothesis motioncompensated predictive coding. The investigation shows that motioncompensated threedimensional transform coding can outperform predictive coding with singlehypothesis motion compensation by up to 0.5 bits/sample. Preprint submitted to Elsevier Science 30 April 2004 Key words: Video Coding, Motion Compensation, Adaptive Wavelets, Lifting, ThreeDimensional Subband Coding of Video 1