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Elad M 2003 Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization
 Proc. Natl Acad. Sci. USA 100 2197–202
"... Given a ‘dictionary ’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considere ..."
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Cited by 362 (32 self)
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Given a ‘dictionary ’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is an overcomplete system consisting of exactly two orthobases, and has shown that, under a condition of mutual incoherence of the two bases, and assuming that S has a sufficiently sparse representation, this representation is unique and can be found by solving a convex optimization problem: specifically, minimizing the ℓ1 norm of the coefficients γ. In this paper, we obtain parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems. We introduce the Spark, ameasure of linear dependence in such a system; it is the size of the smallest linearly dependent subset (dk). We show that, when the signal S has a representation using less than Spark(D)/2 nonzeros, this representation is necessarily unique.
Uncertainty principles and ideal atomic decomposition
 IEEE Transactions on Information Theory
, 2001
"... Suppose a discretetime signal S(t), 0 t
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Cited by 354 (19 self)
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Suppose a discretetime signal S(t), 0 t<N, is a superposition of atoms taken from a combined time/frequency dictionary made of spike sequences 1ft = g and sinusoids expf2 iwt=N) = p N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discretetime signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time/frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the `1 norm of the coe cients among all decompositions. Here \highly sparse " means that Nt + Nw < p N=2 where Nt is the number of time atoms, Nw is the number of frequency atoms, and N is the length of the discretetime signal.
Stable recovery of sparse overcomplete representations in the presence of noise
 IEEE TRANS. INFORM. THEORY
, 2006
"... Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes t ..."
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Cited by 287 (20 self)
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Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimalsparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
Performance Evaluation of Texture Segmentation Algorithms based on Wavelets
, 1996
"... In this paper we consider a large number of experiments (some 800 in total) using a variety of different methods for texture segmentation based upon Wavelets. The experiments also consider ten different wavelet filters and use as a testing bed a variety of composite images taken from the Brodatz dat ..."
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Cited by 12 (1 self)
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In this paper we consider a large number of experiments (some 800 in total) using a variety of different methods for texture segmentation based upon Wavelets. The experiments also consider ten different wavelet filters and use as a testing bed a variety of composite images taken from the Brodatz database. Ground truth for the texture segmentation is known for these images. We present a method for evaluating how well the different textures are separated in the selected feature spaces of these different algorithms, as well as measuring the final performance of the segmentation boundary compared to ground truth. We introduce the twopoint correlation function as a performance measure, as well as a tool for selecting features, and show that it can quantify performance in a way that correlates well with ground truth measures. We show that among the methods we have tested, one stands out clearly as superior to all the others, and that the choice of filters plays little role. 1 Introduction ...
Performance measures for Waveletbased Segmentation Algorithms
, 1997
"... This thesis is concerned with the performance measures for the waveletbased texture segmentation algorithms. After, a brief introduction to wavelets and various texture segmentation algorithms, we present four wavelet detection transformation techniques. We then introduce the distance histogram and ..."
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Cited by 5 (0 self)
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This thesis is concerned with the performance measures for the waveletbased texture segmentation algorithms. After, a brief introduction to wavelets and various texture segmentation algorithms, we present four wavelet detection transformation techniques. We then introduce the distance histogram and the twopoint correlation function as quality measures in feature space. We use the distance histogram as a performance measure, as well as a tool for selecting features, and show that it can quantify performance in a way that correlates with ground truth measures. We show the results of 4 different possible waveletbased feature detection methods combined by 10 wavelet filters. Brodatz images are used as test images. We show that among the methods we have tested, one stands out clearly as superior to all the others, and that the choice of filters plays little role. There are, however, cases where the distance histogram does not indicate the presence of any distinct clusters in the feature ...
Wavelet based acoustic detection of moving vehicles
 Multidimensional Systems and Signal Processing
"... We propose a robust algorithm to detect the arrival of a vehicle of arbitrary type when other noises are present. It is done via analysis of its acoustic signature against an existing database of recorded and processed acoustic signals. To achieve it with minimum number of false alarms, we combine a ..."
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Cited by 2 (2 self)
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We propose a robust algorithm to detect the arrival of a vehicle of arbitrary type when other noises are present. It is done via analysis of its acoustic signature against an existing database of recorded and processed acoustic signals. To achieve it with minimum number of false alarms, we combine a construction of a training database of acoustic signatures signals emitted by vehicles using the distribution of the energies among blocks of wavelet packet coefficients with a procedure of random search for a nearoptimal footprint (RSNOFP). The number of false alarms in the detection is minimized even under severe conditions such as: the signals emitted by vehicles of different types differ from each other, whereas the set of nonvehicle recordings (the training database) contains signals emitted by planes, helicopters, wind, speech, steps, etc. The proposed algorithm is robust even when the tested conditions are completely different from the conditions where the training signals were recorded. The proposed technique has many algorithmic variations. For example, it can be used to distinguish among different types of vehicles. The proposed algorithm is a generic solution for process control that is based on a learning phase (training) followed by an automatic real time detection. 1
Harvard Medical School,
"... Abstract: We present a Bayesian approach for nonparametric curve estimation based on a continuous wavelet dictionary, where the unknown function is modeled by a random sum of wavelet functions at arbitrary locations and scales. By avoiding the dyadic constraints for orthonormal wavelet bases, the co ..."
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Abstract: We present a Bayesian approach for nonparametric curve estimation based on a continuous wavelet dictionary, where the unknown function is modeled by a random sum of wavelet functions at arbitrary locations and scales. By avoiding the dyadic constraints for orthonormal wavelet bases, the continuous overcomplete wavelet dictionary has greater flexibility to adapt to the structure of the data, and leads to sparse representations. The price for this flexibility is the computational challenge of searching over an infinite number of potential dictionary elements. We develop a reversible jump Markov Chain Monte Carlo algorithm which utilizes local features in the proposal distributions and leads to better mixing of the Markov chain. Performance comparison in terms of sparsity and mean square error is carried out on standard wavelet test functions. Results on a nonequally spaced example show that our method compares favorably to methods using interpolation or imputation. Key words and phrases: overcomplete dictionaries; Bayesian inference; wavelets; nonparametric regression; reversible jump Markov chain Monte Carlo; stochastic expansions; 1.
J Math Imaging Vis (2010) 38: 197–225 DOI 10.1007/s1085101002244 Block Based Deconvolution Algorithm Using Spline Wavelet Packets
, 2010
"... Abstract This paper presents robust algorithms to deconvolve discrete noised signals and images. The idea behind the algorithms is to solve the convolution equation separately in different frequency bands. This is achieved by using spline wavelet packets. The solutions are derived as linear combinat ..."
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Abstract This paper presents robust algorithms to deconvolve discrete noised signals and images. The idea behind the algorithms is to solve the convolution equation separately in different frequency bands. This is achieved by using spline wavelet packets. The solutions are derived as linear combinations of the wavelet packets that minimize some parameterized quadratic functionals. Parameters choice, which is performed automatically, determines the tradeoff between the solution regularity and the initial data approximation. This technique, which id called Spline Harmonic Analysis, provides a unified computational scheme for the design of orthonormal spline wavelet packets, fast implementation of the algorithm and an explicit representation of the solutions. The presented algorithms provide stable solutions that accurately approximate the original objects.