We consider estimating a random vector from its noisy projections onto low dimensional subspaces ... more We consider estimating a random vector from its noisy projections onto low dimensional subspaces constituting a fusion frame. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first determine the minimum mean-squared error (MSE) in linearly estimating
Fusion frames were recently introduced to model applications un- der distributed processing requi... more Fusion frames were recently introduced to model applications un- der distributed processing requirements. In this paper we study the behavior of fusion frames under erasures of subspaces and of local frame vectors. We derive results on su-cient conditions for a fusion frame to be robust to such erasures as well as results on the design of fusion frames which are
If ψ ∈ L2(R), Λ is a discrete subset of the affine groupA =R + ×R, and w: Λ →R + is a weight func... more If ψ ∈ L2(R), Λ is a discrete subset of the affine groupA =R + ×R, and w: Λ →R + is a weight function, then the weighted wavelet system generated by ψ, Λ, and w is $$\mathcal{W}(\psi ,\Lambda ,\omega ) = \{ \omega (a,b)^{1/2} a^{ - 1/2} \psi (\frac{x}{a} - b):(a,b) \in \Lambda \} $$ . In this article we define lower and upper weighted densities D w − (Λ) and D w + (Λ) of Λ with respect to the geometry of the affine group, and prove that there exist necessary conditions on a weighted wavelet system in order that it possesses frame bounds. Specifically, we prove that if W(ψ, Λ, w) possesses an upper frame bound, then the upper weighted density is finite. Furthermore, for the unweighted case w = 1, we prove that if W(ψ, Λ, 1) possesses a lower frame bound and D w + (Λ−1) < ∞, then the lower density is strictly positive. We apply these results to oversampled affine systems (which include the classical affine and the quasi-affine systems as special cases), to co-affine wavelet systems, and to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems.
Recent Advances in Computational Sciences - Selected Papers from the International Workshop on Computational Sciences and Its Education, 2008
Wiener amalgam spaces are a class of spaces of functions or distributions defined by a norm which... more Wiener amalgam spaces are a class of spaces of functions or distributions defined by a norm which amalgamates a local criterion for membership in the space with a global criterion. This article presents a proof of a useful convolution relation for amalgam spaces on the affine group.
In this paper, we present an image separation method for separating images into point-and curveli... more In this paper, we present an image separation method for separating images into point-and curvelike parts by employing a combined dictionary consisting of wavelets and compactly supported shearlets utilizing the fact that they sparsely represent point and curvilinear singularities, respectively. Our methodology is based on the very recently introduced mathematical theory of geometric separation, which shows that highly precise separation of the morphologically distinct features of points and curves can be achieved by ℓ 1 minimization. Finally, we present some experimental results showing the effectiveness of our algorithm, in particular, the ability to accurately separate points from curves even if the curvature is relatively large due to the excellent localization property of compactly supported shearlets.
Regularization techniques for the numerical solution of nonlinear inverse scattering problems in ... more Regularization techniques for the numerical solution of nonlinear inverse scattering problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions. Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for the regularization in a Tikhonov method. We examine both directly the nonlinear problem and a linearized problem obtained by the Born approximation technique. As problem classes we study the acoustic inverse scattering problem and the electromagnetic inverse scattering problem. We show that this approach introduces a sparse regularization for the nonlinear setting and we present a result describing the behavior of the local regularity of a scatterer under linearization, which shows that the linearization does not affect the sparsity of the problem. The analytical results are illustrated by...
A basic task in signal analysis is to characterize data in a meaningful way for analysis and clas... more A basic task in signal analysis is to characterize data in a meaningful way for analysis and classification purposes. Time-Frequency transforms are powerful strategies for signal decomposition, and important recent generalizations have been achieved in the setting of frame theory. In parallel recent developments, tools from algebraic topology, traditionally developed in purely abstract settings, have provided new insights in applications to data analysis. In this report, we investigate some interactions of these tools, both theoretically and with numerical experiments in order to characterize signals and their corresponding adaptive frames. We explain basic concepts in persistent homology as an important new subfield of computational topology, as well as formulations of time-frequency analysis in frame theory. Our objective is to use persistent homology for constructing topological signatures of signals in the context of frame theory for classification and analysis purposes. The mai...
Compressed sensing was introduced some ten years ago as an effective way of acquiring signals, wh... more Compressed sensing was introduced some ten years ago as an effective way of acquiring signals, which possess a sparse or nearly sparse representation in a suitable basis or dictionary. Dueto its solid mathematical backgrounds, it quickly attracted the attention of mathematicians from several different areas, so that the most important aspects of the theory are nowadays very well understood. In recent years, its applications started to spread out through applied mathematics, signal processing, and electrical engineering. The aim of this chapter is to provide an introduction into the basic concepts of compressed sensing. In the first part of this chapter, we present the basic mathematical concepts of compressed sensing, including the Null Space Property, Restricted Isometry Property, their connection to basis pursuit and sparse recovery, and construction of matrices with small restricted isometry constants. This presentation is easily accessible, largely self-contained, and includes p...
The new notion of fusion frames will be presented in this article. Fusion frames provide an exten... more The new notion of fusion frames will be presented in this article. Fusion frames provide an extensive framework not only to model sensor networks, but also to serve as a means to improve robustness or develop efficient and feasible reconstruction algorithms. Fusion frames can be regarded as sets of redundant subspaces each of which contains a spanning set of local frame vectors, where the subspaces have to satisfy special overlapping properties. Main aspects of the theory of fusion frames will be presented with a particular focus on the design of sensor networks. New results on the construction of Parseval fusion frames will also be discussed.
In this paper we establish a suprising fundamental identity for Parseval frames in a Hilbert spac... more In this paper we establish a suprising fundamental identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.
This paper proves that every frame of windowed exponentials satises a Strong Homogeneous Approxim... more This paper proves that every frame of windowed exponentials satises a Strong Homogeneous Approximation Property with respect to its canonical dual frame, and a Weak Homogeneous Approximation Property with respect to an arbitrary dual frame. As a consequence, a simple proof of the Nyquist density phenomenon satised by frames of windowed expo- nentials with one or nitely many generators is obtained. The more delicate cases of Schauder bases and exact systems of windowed exponentials are also studied. New results on the relationship between density and frame bounds for frames of windowed exponentials are obtained. In particular, it is shown that a tight frame of windowed exponentials must have uniform Beurling density.
Shearlet systems have been introduced as directional representation systems, which provide optima... more Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations by a unified treatment of the continuum and digital realm. They are redundant systems, and their frame properties have been extensively studied. In contrast to certain band-limited shearlets, compactly supported shearlets provide high spatial localization, but do not constitute Parseval frames. Thus reconstruction of a signal from shearlet coefficients requires knowledge of a dual frame. However, no closed and easily computable form of any dual frame is known. In this paper, we introduce the class of dualizable shearlet systems, which consist of compactly supported elements and can be proven to form frames for $L^2(\R^2)$. For each such dualizable shearlet system, we then provide an explicit construction of an associated dual frame, which can be ...
In this paper, we propose a solution for a fundamental problem in computational harmonic analysis... more In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of non-stationary bivariate subdivision schemes, which allow to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.
High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce v... more High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce very high-dimensional data-sets. In a clinical setting one is often interested how MS spectra differ between patients of different classes, for example spectra from healthy patients vs. spectra from patients having a particular disease. Machine learning algorithms are needed to (a) identify these discriminating features and (b) classify unknown spectra based on this feature set. Since the acquired data is usually noisy, the algorithms should be robust to noise and outliers, and the identified feature set should be as small as possible.
Sparse representations have emerged as a powerful tool in signal and information processing, culm... more Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. This work combines these exciting fields to introduce a new sparsity model
In this paper we will study the Continuous Shearlet Transform from a wavelet point of view, and s... more In this paper we will study the Continuous Shearlet Transform from a wavelet point of view, and show how this per-spective can be used to derive a new geometric interpretation of this transform providing the possibility for FFT-based fast methods to compute the Continuous Shearlet Transform.
Optimally Sparse Fusion Frames: Existence and Construction
Fusion frame theory is an emerging mathematical theory that provides a natural framework for perf... more Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. In this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a ‘uniform basis’ over all subspaces, thereby enabling low-complexity fusion frame decompositions. We then provide an algorithmic construction to compute fusion frames with desired fusion frame operators, including tight fusion frames. Surprisingly, we can even prove that our algorithm constructs optimally sparse fusion frames. Keywords— Computational complexity, frame decompositions, frame operator, frames, redundancy, sparse approximations, sparse matrices, tight frames.
Over the past years, various representation systems which sparsely approximate functions governed... more Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were provided. However, one of the most common shortcomings of these frameworks is the
Many important problem classes are governed by anisotropic features such as singularities concent... more Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic structures is obviously the more important the higher the number of spatial variables is, the principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known wavelets are not capable of efficiently encoding such anisotropic features, various directional representation systems were suggested during the last years. Of those, shearlets are the most widely used today due to their optimal sparse approximation properties in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations. This article shall serve as an introduction to and a survey about shearlets.
This paper proposes an image interpolation algorithm exploiting sparse representation for natural... more This paper proposes an image interpolation algorithm exploiting sparse representation for natural images. It involves three main steps: (a) obtaining an initial estimate of the high resolution image using linear methods like FIR filtering, (b) promoting sparsity in a selected dictionary through iterative thresholding, and (c) extracting high frequency information from the approximation to refine the initial estimate. For the sparse modeling, a shearlet dictionary is chosen to yield a multiscale directional representation. The proposed algorithm is compared to several state-of-the-art methods to assess its objective as well as subjective performance. Compared to the cubic spline interpolation method, an average PSNR gain of around 0.8 dB is observed over a dataset of 200 images.
We consider estimating a random vector from its noisy projections onto low dimensional subspaces ... more We consider estimating a random vector from its noisy projections onto low dimensional subspaces constituting a fusion frame. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first determine the minimum mean-squared error (MSE) in linearly estimating
Fusion frames were recently introduced to model applications un- der distributed processing requi... more Fusion frames were recently introduced to model applications un- der distributed processing requirements. In this paper we study the behavior of fusion frames under erasures of subspaces and of local frame vectors. We derive results on su-cient conditions for a fusion frame to be robust to such erasures as well as results on the design of fusion frames which are
If ψ ∈ L2(R), Λ is a discrete subset of the affine groupA =R + ×R, and w: Λ →R + is a weight func... more If ψ ∈ L2(R), Λ is a discrete subset of the affine groupA =R + ×R, and w: Λ →R + is a weight function, then the weighted wavelet system generated by ψ, Λ, and w is $$\mathcal{W}(\psi ,\Lambda ,\omega ) = \{ \omega (a,b)^{1/2} a^{ - 1/2} \psi (\frac{x}{a} - b):(a,b) \in \Lambda \} $$ . In this article we define lower and upper weighted densities D w − (Λ) and D w + (Λ) of Λ with respect to the geometry of the affine group, and prove that there exist necessary conditions on a weighted wavelet system in order that it possesses frame bounds. Specifically, we prove that if W(ψ, Λ, w) possesses an upper frame bound, then the upper weighted density is finite. Furthermore, for the unweighted case w = 1, we prove that if W(ψ, Λ, 1) possesses a lower frame bound and D w + (Λ−1) < ∞, then the lower density is strictly positive. We apply these results to oversampled affine systems (which include the classical affine and the quasi-affine systems as special cases), to co-affine wavelet systems, and to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems.
Recent Advances in Computational Sciences - Selected Papers from the International Workshop on Computational Sciences and Its Education, 2008
Wiener amalgam spaces are a class of spaces of functions or distributions defined by a norm which... more Wiener amalgam spaces are a class of spaces of functions or distributions defined by a norm which amalgamates a local criterion for membership in the space with a global criterion. This article presents a proof of a useful convolution relation for amalgam spaces on the affine group.
In this paper, we present an image separation method for separating images into point-and curveli... more In this paper, we present an image separation method for separating images into point-and curvelike parts by employing a combined dictionary consisting of wavelets and compactly supported shearlets utilizing the fact that they sparsely represent point and curvilinear singularities, respectively. Our methodology is based on the very recently introduced mathematical theory of geometric separation, which shows that highly precise separation of the morphologically distinct features of points and curves can be achieved by ℓ 1 minimization. Finally, we present some experimental results showing the effectiveness of our algorithm, in particular, the ability to accurately separate points from curves even if the curvature is relatively large due to the excellent localization property of compactly supported shearlets.
Regularization techniques for the numerical solution of nonlinear inverse scattering problems in ... more Regularization techniques for the numerical solution of nonlinear inverse scattering problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions. Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for the regularization in a Tikhonov method. We examine both directly the nonlinear problem and a linearized problem obtained by the Born approximation technique. As problem classes we study the acoustic inverse scattering problem and the electromagnetic inverse scattering problem. We show that this approach introduces a sparse regularization for the nonlinear setting and we present a result describing the behavior of the local regularity of a scatterer under linearization, which shows that the linearization does not affect the sparsity of the problem. The analytical results are illustrated by...
A basic task in signal analysis is to characterize data in a meaningful way for analysis and clas... more A basic task in signal analysis is to characterize data in a meaningful way for analysis and classification purposes. Time-Frequency transforms are powerful strategies for signal decomposition, and important recent generalizations have been achieved in the setting of frame theory. In parallel recent developments, tools from algebraic topology, traditionally developed in purely abstract settings, have provided new insights in applications to data analysis. In this report, we investigate some interactions of these tools, both theoretically and with numerical experiments in order to characterize signals and their corresponding adaptive frames. We explain basic concepts in persistent homology as an important new subfield of computational topology, as well as formulations of time-frequency analysis in frame theory. Our objective is to use persistent homology for constructing topological signatures of signals in the context of frame theory for classification and analysis purposes. The mai...
Compressed sensing was introduced some ten years ago as an effective way of acquiring signals, wh... more Compressed sensing was introduced some ten years ago as an effective way of acquiring signals, which possess a sparse or nearly sparse representation in a suitable basis or dictionary. Dueto its solid mathematical backgrounds, it quickly attracted the attention of mathematicians from several different areas, so that the most important aspects of the theory are nowadays very well understood. In recent years, its applications started to spread out through applied mathematics, signal processing, and electrical engineering. The aim of this chapter is to provide an introduction into the basic concepts of compressed sensing. In the first part of this chapter, we present the basic mathematical concepts of compressed sensing, including the Null Space Property, Restricted Isometry Property, their connection to basis pursuit and sparse recovery, and construction of matrices with small restricted isometry constants. This presentation is easily accessible, largely self-contained, and includes p...
The new notion of fusion frames will be presented in this article. Fusion frames provide an exten... more The new notion of fusion frames will be presented in this article. Fusion frames provide an extensive framework not only to model sensor networks, but also to serve as a means to improve robustness or develop efficient and feasible reconstruction algorithms. Fusion frames can be regarded as sets of redundant subspaces each of which contains a spanning set of local frame vectors, where the subspaces have to satisfy special overlapping properties. Main aspects of the theory of fusion frames will be presented with a particular focus on the design of sensor networks. New results on the construction of Parseval fusion frames will also be discussed.
In this paper we establish a suprising fundamental identity for Parseval frames in a Hilbert spac... more In this paper we establish a suprising fundamental identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.
This paper proves that every frame of windowed exponentials satises a Strong Homogeneous Approxim... more This paper proves that every frame of windowed exponentials satises a Strong Homogeneous Approximation Property with respect to its canonical dual frame, and a Weak Homogeneous Approximation Property with respect to an arbitrary dual frame. As a consequence, a simple proof of the Nyquist density phenomenon satised by frames of windowed expo- nentials with one or nitely many generators is obtained. The more delicate cases of Schauder bases and exact systems of windowed exponentials are also studied. New results on the relationship between density and frame bounds for frames of windowed exponentials are obtained. In particular, it is shown that a tight frame of windowed exponentials must have uniform Beurling density.
Shearlet systems have been introduced as directional representation systems, which provide optima... more Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations by a unified treatment of the continuum and digital realm. They are redundant systems, and their frame properties have been extensively studied. In contrast to certain band-limited shearlets, compactly supported shearlets provide high spatial localization, but do not constitute Parseval frames. Thus reconstruction of a signal from shearlet coefficients requires knowledge of a dual frame. However, no closed and easily computable form of any dual frame is known. In this paper, we introduce the class of dualizable shearlet systems, which consist of compactly supported elements and can be proven to form frames for $L^2(\R^2)$. For each such dualizable shearlet system, we then provide an explicit construction of an associated dual frame, which can be ...
In this paper, we propose a solution for a fundamental problem in computational harmonic analysis... more In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of non-stationary bivariate subdivision schemes, which allow to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.
High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce v... more High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce very high-dimensional data-sets. In a clinical setting one is often interested how MS spectra differ between patients of different classes, for example spectra from healthy patients vs. spectra from patients having a particular disease. Machine learning algorithms are needed to (a) identify these discriminating features and (b) classify unknown spectra based on this feature set. Since the acquired data is usually noisy, the algorithms should be robust to noise and outliers, and the identified feature set should be as small as possible.
Sparse representations have emerged as a powerful tool in signal and information processing, culm... more Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. This work combines these exciting fields to introduce a new sparsity model
In this paper we will study the Continuous Shearlet Transform from a wavelet point of view, and s... more In this paper we will study the Continuous Shearlet Transform from a wavelet point of view, and show how this per-spective can be used to derive a new geometric interpretation of this transform providing the possibility for FFT-based fast methods to compute the Continuous Shearlet Transform.
Optimally Sparse Fusion Frames: Existence and Construction
Fusion frame theory is an emerging mathematical theory that provides a natural framework for perf... more Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. In this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a ‘uniform basis’ over all subspaces, thereby enabling low-complexity fusion frame decompositions. We then provide an algorithmic construction to compute fusion frames with desired fusion frame operators, including tight fusion frames. Surprisingly, we can even prove that our algorithm constructs optimally sparse fusion frames. Keywords— Computational complexity, frame decompositions, frame operator, frames, redundancy, sparse approximations, sparse matrices, tight frames.
Over the past years, various representation systems which sparsely approximate functions governed... more Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were provided. However, one of the most common shortcomings of these frameworks is the
Many important problem classes are governed by anisotropic features such as singularities concent... more Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic structures is obviously the more important the higher the number of spatial variables is, the principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known wavelets are not capable of efficiently encoding such anisotropic features, various directional representation systems were suggested during the last years. Of those, shearlets are the most widely used today due to their optimal sparse approximation properties in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations. This article shall serve as an introduction to and a survey about shearlets.
This paper proposes an image interpolation algorithm exploiting sparse representation for natural... more This paper proposes an image interpolation algorithm exploiting sparse representation for natural images. It involves three main steps: (a) obtaining an initial estimate of the high resolution image using linear methods like FIR filtering, (b) promoting sparsity in a selected dictionary through iterative thresholding, and (c) extracting high frequency information from the approximation to refine the initial estimate. For the sparse modeling, a shearlet dictionary is chosen to yield a multiscale directional representation. The proposed algorithm is compared to several state-of-the-art methods to assess its objective as well as subjective performance. Compared to the cubic spline interpolation method, an average PSNR gain of around 0.8 dB is observed over a dataset of 200 images.
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