Results 11  20
of
43
Filter Bank Fusion Frames
, 2010
"... In this paper we characterize and construct novel oversampled filter banks implementing fusion frames. A fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way. When properly designed, fusion frames can provide redundant encodings of signa ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In this paper we characterize and construct novel oversampled filter banks implementing fusion frames. A fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way. When properly designed, fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures. However, up to this point, few implementable constructions of such frames were known; we show how to construct them using oversampled filter banks. In this work, we first provide polyphase domain characterizations of filter bank fusion frames. We then use these characterizations to construct filter bank fusion frame versions of discrete wavelet and Gabor transforms, emphasizing those specific finite impulse response filters whose frequency responses are wellbehaved.
Compressed sensing for fusion frames
 in Proc. SPIE, Wavelets XIII
"... Compressed Sensing (CS) is a new signal acquisition technique that allows sampling of sparse signals using significantly fewer measurements than previously thought possible. On the other hand, a fusion frame is a new signal representation method that uses collections of subspaces instead of vectors ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Compressed Sensing (CS) is a new signal acquisition technique that allows sampling of sparse signals using significantly fewer measurements than previously thought possible. On the other hand, a fusion frame is a new signal representation method that uses collections of subspaces instead of vectors to present signals. This work combines these exciting new fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standards CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications. With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it needs not be sparse within each of the subspaces it occupies. We define a mixed l1/l2 norm for fusion frames. A signal sparse in the subspaces of the fusion frame can thus be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed l1/l2 norm. The sampling conditions we derive are very similar to the coherence and RIP conditions used in standard CS theory.
FUSION FRAMES AND THE RESTRICTED ISOMETRY PROPERTY
"... Abstract. We show that RIP frames, tight frames satisfying the restricted isometry property, give rise to nearly tight fusion frames which are nearly orthogonal and hence are nearly equiisoclinic. We also show how to replace parts of the RIP frame with orthonormal sets while maintaining the restric ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We show that RIP frames, tight frames satisfying the restricted isometry property, give rise to nearly tight fusion frames which are nearly orthogonal and hence are nearly equiisoclinic. We also show how to replace parts of the RIP frame with orthonormal sets while maintaining the restricted isometry property. 1.
Sparse recovery of fusion frame structured signals
 In preparation
"... Fusion frames are collection of subspaces which provide a redundant representation of signal spaces. They generalize classical frames by replacing frame vectors with frame subspaces. This paper considers the sparse recovery of a sparse signal in fusion frame. We use a block sparsity model for fusion ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Fusion frames are collection of subspaces which provide a redundant representation of signal spaces. They generalize classical frames by replacing frame vectors with frame subspaces. This paper considers the sparse recovery of a sparse signal in fusion frame. We use a block sparsity model for fusion frames and then show that sparse signals under this model can be compressively sampled and reconstructed in ways similar to standard Compressed Sensing (CS). In particular we invoke a mixed `1/`2 norm minimization in order to reconstruct sparse signals. In our work, we show that assuming a certain incoherence property of the subspaces and the apriori knowledge of it allows us to improve recovery when compared to usual block sparsity case. 1
Depth Sensing Using Active Coherent Illumination
, 2012
"... We examine the use of active coherent sensingan increasingly available technology for sensing the depth of scenes. A scene is a sparse signal but also exhibits significant structure which cannot be exploited using standard sparse recovery algorithms. Instead, inspired by the modelbased compressive ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We examine the use of active coherent sensingan increasingly available technology for sensing the depth of scenes. A scene is a sparse signal but also exhibits significant structure which cannot be exploited using standard sparse recovery algorithms. Instead, inspired by the modelbased compressive sensing literature we develop a scene model that incorporates occlusion constraints in recovering the depth map. Our model is computationally tractable; we develop a variation of the wellknown modelbased Compressive Sampling Matching Pursuit (CoSaMP) algorithm, and we demonstrate that our approach significantly improves reconstruction performance.
Constructing Fusion Frames with Desired Parameters
"... A fusion frame is a framelike collection of subspaces in a Hilbert space. It generalizes the concept of a frame system for signal representation. In this paper, we study the existence and construction of fusion frames. We first introduce two general methods, namely the spatial complement and the Na ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
A fusion frame is a framelike collection of subspaces in a Hilbert space. It generalizes the concept of a frame system for signal representation. In this paper, we study the existence and construction of fusion frames. We first introduce two general methods, namely 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 tight.
Finite Frames for Sparse Signal Processing
"... Over the last decade, considerable progress has been made towards developing new signal processing methods to manage the deluge of data caused by advances in sensing, imaging, storage, and computing technologies. Most of these methods are based on a simple but fundamental observation. That is, hig ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Over the last decade, considerable progress has been made towards developing new signal processing methods to manage the deluge of data caused by advances in sensing, imaging, storage, and computing technologies. Most of these methods are based on a simple but fundamental observation. That is, highdimensional data sets are typically highly redundant and live on lowdimensional manifolds or subspaces. This means that the collected data can often be represented in a sparse or parsimonious way in a suitably selected finite frame. This observation has also led to the development of a new sensing paradigm, called compressed sensing, which shows that highdimensional data sets can often be reconstructed, with high fidelity, from only a small number of measurements. Finite frames play a central role in the design and analysis of both sparse representations and compressed sensing methods. In this chapter, we highlight this role primarily in the context of compressed sensing for estimation, recovery, support detection, regression, and detection of sparse signals. The recurring theme is that frames with small spectral norm and/or small worstcase coherence, average coherence, or sum coherence are wellsuited for making measurements of sparse signals.
Compressed Subspace Matching on the Continuum
, 2014
"... We consider the general problem of matching a subspace to a signal in RN that has been observed indirectly (compressed) through a random projection. We are interested in the case where the collection of Kdimensional subspaces is continuously parameterized, i.e. naturally indexed by an interval from ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We consider the general problem of matching a subspace to a signal in RN that has been observed indirectly (compressed) through a random projection. We are interested in the case where the collection of Kdimensional subspaces is continuously parameterized, i.e. naturally indexed by an interval from the real line, or more generally a region of RD. Our main results show that if the dimension of the random projection is on the order of K times a geometrical constant that describes the complexity of the collection, then the match obtained from the compressed observation is nearly as good as one obtained from a full observation of the signal. We give multiple concrete examples of collections of subspaces for which this geometrical constant can be estimated, and discuss the relevance of the results to the general problems of template matching and source localization. 1
Uniform recovery of fusion frame structured sparse signals
, 2014
"... We consider the problem of recovering fusion frame sparse signals from incomplete measurements. These signals are composed of a small number of nonzero blocks taken from a family of subspaces. First, we show that, by using apriori knowledge of a coherence parameter associated with the angles betwe ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We consider the problem of recovering fusion frame sparse signals from incomplete measurements. These signals are composed of a small number of nonzero blocks taken from a family of subspaces. First, we show that, by using apriori knowledge of a coherence parameter associated with the angles between the subspaces, one can uniformly recover fusion frame sparse signals with a signicantly reduced number of vectorvalued (sub)Gaussian measurements via mixed ℓ1/ℓ2minimization. We prove this by establishing an appropriate version of the restricted isometry property. Our result complements previous nonuniform recovery results in this context, and provides stronger stability guarantees for noisy measurements and approximately sparse signals. Second, we determine the minimal number of scalarvalued measurements needed to uniformly recover all fusion frame sparse signals via mixed ℓ1/ℓ2minimization. This bound is achieved by scalarvalued subgaussian measurements. In particular, our result shows that the number of scalarvalued subgaussian measurements cannot be further reduced using knowledge of the coherence parameter. As a special case it implies that the best known uniform recovery result for block sparse signals using subgaussian measurements is optimal.