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Fusion frames: existence and construction
, 2009
"... Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a framelike collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we st ..."
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Cited by 22 (12 self)
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a framelike collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods – the spatial complement and the Naimark complement – for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given M, N, m ∈ N and {λj} M j=1, does there exist a fusion frame in RM with N subspaces of dimension m for which {λj} M j=1 are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m ∈ N and {λj} M j=1. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity
, 2011
"... The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smoot ..."
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Cited by 21 (8 self)
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The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multiorientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to nonEuclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping “pictures”.
Sparse representation for target detection in hyperspectral imagery
 IEEE J. Sel. Topics Signal Process
"... Abstract—In this paper, we propose a new sparsitybased algorithm for automatic target detection in hyperspectral imagery (HSI). This algorithm is based on the concept that a pixel in HSI lies in a lowdimensional subspace and thus can be represented as a sparse linear combination of the training sa ..."
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Cited by 11 (5 self)
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Abstract—In this paper, we propose a new sparsitybased algorithm for automatic target detection in hyperspectral imagery (HSI). This algorithm is based on the concept that a pixel in HSI lies in a lowdimensional subspace and thus can be represented as a sparse linear combination of the training samples. The sparse representation (a sparse vector corresponding to the linear combination of a few selected training samples) of a test sample can be recovered by solving an 0norm minimization problem. With the recent development of the compressed sensing theory, such minimization problem can be recast as a standard linear programming problem or efficiently approximated by greedy pursuit algorithms. Once the sparse vector is obtained, the class of the test sample can be determined by the characteristics of the sparse vector on reconstruction. In addition to the constraints on sparsity and reconstruction accuracy, we also exploit the fact that in HSI the neighboring pixels have a similar spectral characteristic (smoothness). In our proposed algorithm, a smoothness constraint is also imposed by forcing the vector Laplacian at each reconstructed pixel to be minimum all the time within the minimization process. The proposed sparsitybased algorithm is applied to several hyperspectral imagery to detect targets of interest. Simulation results show that our algorithm outperforms the classical hyperspectral target detection algorithms, such as the popular spectral matched filters, matched subspace detectors, adaptive subspace detectors, as well as binary classifiers such as support vector machines. Index Terms—Hyperspectral imagery, sparse recovery, sparse representation, spatial correlation, target detection. I.
EQUIANGULAR TIGHT FRAMES FROM COMPLEX SEIDEL MATRICES CONTAINING CUBE ROOTS OF UNITY
, 805
"... Abstract. We derive easily verifiable conditions which characterize when complex Seidel matrices containing cube roots of unity have exactly two eigenvalues. The existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame ve ..."
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Cited by 10 (0 self)
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Abstract. We derive easily verifiable conditions which characterize when complex Seidel matrices containing cube roots of unity have exactly two eigenvalues. The existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame vectors is always a common multiple of the cube roots of unity. We also exhibit a relationship between these equiangular tight frames, complex Seidel matrices, and highly regular, directed graphs. We construct examples of such frames with arbitrarily many vectors. 1.
Frequencydomain design of overcomplete rationaldilation wavelet transforms
 IEEE Trans. on Signal Processing
, 2009
"... The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Qfactor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible fami ..."
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Cited by 10 (5 self)
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The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Qfactor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied. The new wavelet transform can attain higher Qfactors (desirable for processing oscillatory signals) or the same low Qfactor of the dyadic wavelet transform. The new wavelet transform is modestly overcomplete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible ‘constantQ’ discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for L2(R). The wavelet can be made to resemble a Gabor function and can hence have good concentration in the timefrequency plane. The construction of the new wavelet transform depends on the judicious use of both the transform’s redundancy and the flexibility allowed by frequencydomain filter design. I.
Compressive sampling of swallowing accelerometry signals using timefrequency dictionaries based on modulated discrete prolate spheroidal sequences
 EURASIP Journal on Advances in Signal Processing
, 2012
"... Monitoring physiological functions such as swallowing often generates large volumes of samples to be stored and processed, which can introduce computational constraints especially if remote monitoring is desired. In this paper, we propose a compressive sensing (CS) algorithm to alleviate some of th ..."
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Cited by 9 (2 self)
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Monitoring physiological functions such as swallowing often generates large volumes of samples to be stored and processed, which can introduce computational constraints especially if remote monitoring is desired. In this paper, we propose a compressive sensing (CS) algorithm to alleviate some of these issues while acquiring dualaxis swallowing accelerometry signals. The proposed CS approach uses a timefrequency dictionary where the members are modulated discrete prolate spheroidal sequences (MDPSS). These waveforms are obtained by modulation and variation of discrete prolate spheroidal sequences (DPSS) in order to reflect the timevarying nature of swallowing acclerometry signals. While the modulated bases permit one to represent the signal behavior accurately, the matching pursuit algorithm is adopted to iteratively decompose the signals into an expansion of the dictionary bases. To test the accuracy of the proposed scheme, we carried out several numerical experiments with synthetic test signals and dualaxis swallowing accelerometry signals. In both cases, the proposed CS approach based on the MDPSS yields more accurate representations than the CS approach based on DPSS. Specifically, we show that dualaxis swallowing accelerometry signals can be accurately reconstructed
MultipleBases BeliefPropagation Decoding of HighDensity Cyclic Codes
, 2009
"... We introduce a new method for decoding short and moderate length linear block codes with dense paritycheck matrix representations of cyclic form, termed multiplebases beliefpropagation (MBBP). The proposed iterative scheme makes use of the fact that a code has many structurally diverse parityche ..."
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Cited by 8 (3 self)
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We introduce a new method for decoding short and moderate length linear block codes with dense paritycheck matrix representations of cyclic form, termed multiplebases beliefpropagation (MBBP). The proposed iterative scheme makes use of the fact that a code has many structurally diverse paritycheck matrices, capable of detecting different error patterns. We show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard BP decoding. Furthermore, we describe how to choose sets of paritycheck matrices of cyclic form amenable for multiplebases decoding, based on analytical studies performed for the binary erasure channel. For several cyclic and extended cyclic codes, the MBBP decoding performance can be shown to closely follow that of maximumlikelihood decoders.
SemiSupervised Multiresolution Classification Using Adaptive Graph Filtering with Application to Indirect Bridge Structural Health Monitoring
"... We present a multiresolution classification framework with semisupervised learning on graphs with application to the indirect bridge structural health monitoring. Classification in realworld applications faces two main challenges: reliable features can be hard to extract and few labeled signals a ..."
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Cited by 7 (6 self)
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We present a multiresolution classification framework with semisupervised learning on graphs with application to the indirect bridge structural health monitoring. Classification in realworld applications faces two main challenges: reliable features can be hard to extract and few labeled signals are available for training. We propose a novel classification framework to address these problems: we use a multiresolution framework to deal with nonstationarities in the signals and extract features in each localized timefrequency region and semisupervised learning to train on both labeled and unlabeled signals. We further propose an adaptive graph filter for semisupervised classification that allows for classifying unlabeled as well as unseen signals and for correcting mislabeled signals. We validate the proposed framework on indirect bridge structural health monitoring and show that it performs significantly better than previous approaches.
Optimally Sparse Frames
 IEEE Trans. Inform. Theory
, 2011
"... Dedicated to the memory of Nigel J. Kalton, who was a great person, friend, and mathematician. ..."
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Cited by 7 (5 self)
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Dedicated to the memory of Nigel J. Kalton, who was a great person, friend, and mathematician.
SpectrumAdapted Tight Graph Wavelet and VertexFrequency Frames
, 2013
"... We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spect ..."
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Cited by 7 (5 self)
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We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues. We use this approach to construct computationally efficient, spectrumadapted, tight vertexfrequency and graph wavelet frames. We give numerous examples of the resulting spectrumadapted graph filters, and also present an illustrative example of vertexfrequency analysis using the proposed construction.