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90
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”.
Detecting activations over graphs using spanning tree wavelet bases
 In Artificial Intelligence and Statistics (AISTATS
, 2013
"... ar ..."
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 8 (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.
Localized iterative methods for interpolation in graph structured data
 in Global Conference on Signal and Information Processing (GlobalSIP), 2013 IEEE
, 2013
"... In this paper, we present two localized graph filtering based methods for interpolating graph signals defined on the vertices of arbitrary graphs from only a partial set of samples. The first method is an extension of previous work on reconstructing bandlimited graph signals from partially observe ..."
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Cited by 8 (2 self)
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In this paper, we present two localized graph filtering based methods for interpolating graph signals defined on the vertices of arbitrary graphs from only a partial set of samples. The first method is an extension of previous work on reconstructing bandlimited graph signals from partially observed samples. The iterative graph filtering approach very closely approximates the solution proposed in the that work, while being computationally more efficient. As an alternative, we propose a regularization based framework in which we define the cost of reconstruction to be a combination of smoothness of the graph signal and the reconstruction error with respect to the known samples, and find solutions that minimize this cost. We provide both a closed form solution and a computationally efficient iterative solution of the optimization problem. The experimental results on the recommendation system datasets demonstrate effectiveness of the proposed methods. 1.
LEARNING OF STRUCTURED GRAPH DICTIONARIES
"... We propose a method for learning dictionaries towards sparse approximation of signals defined on vertices of arbitrary graphs. Dictionaries are expected to describe effectively the main spatial and spectral components of the signals of interest, so that their structure is dependent on the graph info ..."
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Cited by 7 (5 self)
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We propose a method for learning dictionaries towards sparse approximation of signals defined on vertices of arbitrary graphs. Dictionaries are expected to describe effectively the main spatial and spectral components of the signals of interest, so that their structure is dependent on the graph information and its spectral representation. We first show how operators can be defined for capturing different spectral components of signals on graphs. We then propose a dictionary learning algorithm built on a sparse approximation step and a dictionary update function, which iteratively leads to adapting the structured dictionary to the class of target signals. Experimental results on synthetic and natural signals on graphs demonstrate the efficiency of the proposed algorithm both in terms of sparse approximation and support recovery performance. Index Terms — dictionary learning, signal processing on graphs, sparse approximations 1.
Parametric Dictionary Learning for Graph Signals
"... Abstract—We propose a parametric dictionary learning algorithm to design structured dictionaries that sparsely represent graph signals. We incorporate the graph structure by forcing the learned dictionaries to be concatenations of subdictionaries that are polynomials of the graph Laplacian matrix. T ..."
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Cited by 7 (2 self)
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Abstract—We propose a parametric dictionary learning algorithm to design structured dictionaries that sparsely represent graph signals. We incorporate the graph structure by forcing the learned dictionaries to be concatenations of subdictionaries that are polynomials of the graph Laplacian matrix. The resulting atoms capture the main spatial and spectral components of the graph signals of interest, leading to adaptive representations with efficient implementations. Experimental results demonstrate the effectiveness of the proposed algorithm for the sparse approximation of graph signals. I.
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.
Multiresolution graph signal processing via circulant structures
 IN IEEE DSP/SPE WORSKHOP 2013. IEEE
, 2013
"... We offer a new paradigm for multiresolution analysis and processing of graph signals using circulant structures. Among the essential features of circulant graphs is that they accommodate fundamental signal processing operations, such as linear shiftinvariant filtering, downsampling, upsampling, an ..."
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We offer a new paradigm for multiresolution analysis and processing of graph signals using circulant structures. Among the essential features of circulant graphs is that they accommodate fundamental signal processing operations, such as linear shiftinvariant filtering, downsampling, upsampling, and reconstruction—features that we take to substantial advantage. We design twochannel, criticallysampled, perfectreconstruction, orthogonal latticefilter structures to process signals on circulant graphs. To extend our reach to more general graphs, we present a method to decompose a connected, undirected graph into a linear combination of circulant graphs. Our circulant decomposition is analogous to designing linear timevarying lattice filters by suitably adapting the coefficients of a linear timeinvariant filter. To evaluate the systems and methods that we have propounded, we offer examples of synthetic and realworld graph datasets and their multiscale decompositions.