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56
Constrained Overcomplete Analysis Operator Learning for Cosparse Signal Modelling
"... We consider the problem of learning a lowdimensional signal model from a collection of training samples. The mainstream approach would be to learn an overcomplete dictionary to provide good approximations of thetraining samples using sparsesynthesis coefficients. This famous sparse model has a less ..."
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We consider the problem of learning a lowdimensional signal model from a collection of training samples. The mainstream approach would be to learn an overcomplete dictionary to provide good approximations of thetraining samples using sparsesynthesis coefficients. This famous sparse model has a less well known counterpart, in analysis form, called the cosparse analysis model. In this new model, signals are characterised by their parsimony in a transformed domain using an overcomplete (linear) analysis operator. We propose to learn an analysis operator from a training corpus using a constrained optimisation framework based on L1 optimisation. The reason for introducing a constraint in the optimisation framework is to exclude trivial solutions. Although there is no final answer here for which constraint is the most relevant constraint, we investigate some conventional constraints in the model adaptation field and use the uniformly normalised tight frame (UNTF) for this purpose. We then derive a practical learning algorithm, based on projected subgradients and DouglasRachford splitting technique, and demonstrate its ability to robustly recover a ground truth analysis operator, when provided with a clean training set, of sufficient size. We also find an analysis operator for images, using some noisy cosparse signals, which is indeed a more realistic experiment. As the derived optimisation problem is not a convex program, we often find a local minimum using such variational methods. For two different settings, we provide preliminary theoretical support for the wellposedness of the learning problem, which can be practically used to test the local identifiability conditions of learnt operators.
An Investigation of 3D DualTree Wavelet Transform for Video Coding
 in Proceedings of International Conference on Image Processing (ICIP
, 2004
"... This paper examines the properties of a recently introduced 3D dualtree discrete wavelet transform (DDWT) for video coding. The 3D DDWT is an attractive video representation because it isolates motion along different directions in separate sub bands. However, it is an over complete transform w ..."
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Cited by 10 (4 self)
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This paper examines the properties of a recently introduced 3D dualtree discrete wavelet transform (DDWT) for video coding. The 3D DDWT is an attractive video representation because it isolates motion along different directions in separate sub bands. However, it is an over complete transform with 8:1 redundancy. We examine the effectiveness of the iterative projectivebased noiseshaping scheme proposed by Kingsbury [3] on reducing the number of coefficients. We also investigate the correlation between sub bands at the same spatial/temporal location, both in the significance map and in actual coefficient value.
Independent Multiresolution Component Analysis and Matching Pursuit
 Comput. Stat. Data Anal
, 2001
"... We show that decomposing a class of signals with overcomplete dictionaries of functions and combining multiresolution and independent component analysis allow for feature detection in complex nonstationary high frequency time series. Computational learning techniques are then designed through the M ..."
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Cited by 7 (3 self)
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We show that decomposing a class of signals with overcomplete dictionaries of functions and combining multiresolution and independent component analysis allow for feature detection in complex nonstationary high frequency time series. Computational learning techniques are then designed through the Matching Pursuit algorithm, whose performance is monitored so to extract relevant information about the structure of the volatility function. We refer to wavelet and cosine packet dictionaries due to the fact that with intradaily time series some features of the underlying stochastic processes may remain undetected when standard volatility models are applied to the observed data. Independent component analysis results are particularly encouraging and suggest a better compromise between time and frequency resolutions, and thus a more efficient and accurate Matching Pursuit performance.
DeNoising via Wavelet Transforms Using Steerable Filters
, 1995
"... Feature extraction remains an important part of lowlevel vision. Traditional oriented filters have been effective tools to identify features, such as lines and edges. Steerable filters, which can be adjusted at arbitrary orientation, have made decisions of feature orientations more precise. Combine ..."
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Cited by 5 (1 self)
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Feature extraction remains an important part of lowlevel vision. Traditional oriented filters have been effective tools to identify features, such as lines and edges. Steerable filters, which can be adjusted at arbitrary orientation, have made decisions of feature orientations more precise. Combined with a pyramid structure of a multiscale representation, these filters can provide a reliable and efficient tool for image analysis. This paper takes advantage of multiscale steerable filters in the context of denoising. First a set of novel filters are designed, that decompose the frequency plane into distinct directional bands. Next, we identify the dominant direction and strength at each point of an image from quadrature pairs of steerable filters. A nonlinear threshold function is then applied to the filtered coefficients to suppress noise. The denoised image is restored from coefficients modified at each level of transform space. We demonstrate the benefits of multiscale steerable fi...
Applications of sparse approximation in communications
 in IEEE Int. Symp. Inf. Theory, 2005
"... Abstract—Sparse approximation problems abound in many ..."
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Abstract—Sparse approximation problems abound in many
SPARSE REPRESENTATION FOR IMAGE PREDICTION
"... This paper addresses the problem of closedloop spatial image prediction based on sparse signal representation techniques. The basis functions which best approximate a causal neighborhood are used to extrapolate the signal in the region to predict. Two iterative algorithms for sparse signal represen ..."
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This paper addresses the problem of closedloop spatial image prediction based on sparse signal representation techniques. The basis functions which best approximate a causal neighborhood are used to extrapolate the signal in the region to predict. Two iterative algorithms for sparse signal representation are considered: the Matching Pursuit algorithm and the Global Matched Filter. The predicted signal PSNR achieved with these two methods are compared against those obtained with the directional predictive modes of H.264/AVC. 1.
ATOMIC DECOMPOSITION DEDICATED TO AVC AND SPATIAL SVC PREDICTION
"... In this work, we propose the use of sparse signal representation techniques to solve the problem of closedloop spatial image prediction. The reconstruction of signal in the block to predict is based on basis functions selected with the Matching Pursuit (MP) iterative algorithm, to best match a caus ..."
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In this work, we propose the use of sparse signal representation techniques to solve the problem of closedloop spatial image prediction. The reconstruction of signal in the block to predict is based on basis functions selected with the Matching Pursuit (MP) iterative algorithm, to best match a causal neighborhood. We evaluate this new method in terms of PSNR and bitrate in a H264/AVC encoder. Experimental results indicate an improvement of ratedistortion performance. In this paper, we also present results concerning the use of this technique for intrainter layer prediction refinement, in a scalable video coding (SVC) like scheme. Index Terms — intraprediction, atomic decomposition, extrapolation
On growth and formlets: Sparse multiscale coding of planar shape
 Image and Vision Computing
, 2012
"... This paper presents a sparse representation of 2D planar shape through the composition of warping functions, termed formlets, localized in scale and space. Each formlet subjects the 2D space in which the shape is embedded to a localized isotropic radial deformation. By constraining these localized w ..."
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This paper presents a sparse representation of 2D planar shape through the composition of warping functions, termed formlets, localized in scale and space. Each formlet subjects the 2D space in which the shape is embedded to a localized isotropic radial deformation. By constraining these localized warping transformations to be diffeomorphisms, the topology of shape is preserved, and the set of simple closed curves is closed under any sequence of these warpings. A generative model based on a composition of formlets applied to an embryonic shape, e.g., an ellipse, has the advantage of synthesizing only those shapes that could correspond to the boundaries of physical objects. To compute the set of formlets that represent a given boundary, we demonstrate a greedy coarsetofine formlet pursuit algorithm that serves as a noncommutative generalization of matching pursuit for sparse approximations. We evaluate our method by pursuing partially occluded shapes, comparing performance against a contourbased sparse shape coding framework. 1.
Sparse Approximations for HighFidelity Compression of Network Traffic Data
 In Proceedings of ACM/USENIX Internet Measurement Conference (IMC
, 2005
"... An important component of traffic analysis and network monitoring is the ability to correlate events across multiple data streams, from different sources and from different time periods. Storing such a large amount of data for visualizing traffic trends and for building prediction models of “normal ..."
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Cited by 2 (1 self)
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An important component of traffic analysis and network monitoring is the ability to correlate events across multiple data streams, from different sources and from different time periods. Storing such a large amount of data for visualizing traffic trends and for building prediction models of “normal ” network traffic represents a great challenge because the data sets are enormous. In this paper we present the application and analysis of signal processing techniques for effective practical compression of network traffic data. We propose to use a sparse approximation of the network traffic data over a rich collection of natural building blocks, with several natural dictionaries drawn from the networking community’s experience with traffic data. We observe that with such natural dictionaries, high fidelity compression of the original traffic data can be achieved such that even with a compression ratio of around 1:6, the compression error, in terms of the energy of the original signal lost, is less than 1%. We also observe that the sparse representations are stable over time, and that the stable components correspond to welldefined periodicities in network traffic. 1
S+WAVELETS: An ObjectOriented Toolkit for Wavelet Analysis
 IEEE SPECTRUM
, 1995
"... S+WAVELETS is an objectoriented toolkit for wavelet analysis of signals, time series, images, and other data. It is a module of the SPlus language for data analysis, statistics, and scientific computing. The module is oriented towards engineers, mathemeticians, statisticians, and scientists in a b ..."
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Cited by 2 (1 self)
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S+WAVELETS is an objectoriented toolkit for wavelet analysis of signals, time series, images, and other data. It is a module of the SPlus language for data analysis, statistics, and scientific computing. The module is oriented towards engineers, mathemeticians, statisticians, and scientists in a broad range of disciplines. S+WAVELETS is being used for applications as diverse as data visualization and analysis, nonparametric statistical estimation, signal and image compression, signal processing, and prototyping of new fast algorithms. The toolkit offers a rich collection of transforms, ranging from the classical discrete wavelet transform to timefrequency decompositions such as wavelet packets and cosine packets. A variety of algorithms and tools for selecting transforms are available, including the Coifman and Wickerhauser "best basis" algorithm and the Mallat and Zhang "matching pursuit" algorithm. With over 500 functions, S+WAVELETS provides a complete computing environment for wavelet analysis, allowing the user to manipulate, visualize, synthesize, and analyze wavelet objects. The objectoriented design of S+WAVELETS makes these functions easy to use and provides an organizing framework for wavelet analysis.