Skip to main content

Gitta Kutyniok

Followers

51

Following

12

Co-authors

12

Public Views

Interests

Uploads

Papers by Gitta Kutyniok

Erasure-Proof Coding with Fusion Frames

The main goal of this paper is the design of frames for transmitting vectors through a memoryless... more The main goal of this paper is the design of frames for transmitting vectors through a memoryless analog erasure channel. The channel transmits the frame coefficients perfectly or discards them, depending on the outcomes of Bernoulli trials with a failure probability q. For sufficiently small q, we construct frames which encode above a fixed non-zero rate and allow the receiver to recover part of the erased coefficients so that the remaining mean-square error vanishes as the frame size increases. We give examples for which the mean-square reconstruction error remaining after corrections are applied decays faster than any inverse power of the number of frame vectors.

Analysis of lscr<sub>1</sub> minimization in the Geometric Separation Problem

2008 42nd Annual Conference on Information Sciences and Systems, 2008

Modern data are often composed of two (or more) morphologically distinct constituents -for instan... more Modern data are often composed of two (or more) morphologically distinct constituents -for instance, pointlike and curvelike structures in astronomical imaging of galaxies. Although it seems impossible to extract those components -as there are two unknowns for every datum -suggestive empirical results have already been obtained especially by Jean-Luc Starck and collaborators. In this paper we develop a theoretical viewpoint, defining a Geometric Separation Problem and analyzing a model procedure. This procedure is inspired by work relating ℓ1 minimization and sparsity. The procedure uses two deliberately overcomplete systems which sparsify the different components and decomposes by ℓ1 minimization of the analysis (rather than synthesis) frame coefficients. We formalize two concepts -cluster coherence in place of the now-traditional singleton coherence and ℓ1 minimization in frame settings, including those where singleton coherence within one frame may be high -and develop all the needed machinery to make these into fruitful tools. Our general approach applies to the problem of geometric separation of pointlike and curvelike structures in images by employing frames of radial wavelets and curvelets or orthonormal wavelets and shearlets. Our theoretical results show that at all sufficiently fine scales, nearly-perfect separation is achieved. We use microlocal analysis to understand heuristically why separation might be possible and to organize a rigorous analysis.

<title>Coarse quantization with the fast digital shearlet transform</title>

Wavelets and Sparsity XIV, 2011

The fast digital shearlet transform (FDST) was recently introduced as a means to analyze natural ... more The fast digital shearlet transform (FDST) was recently introduced as a means to analyze natural images efficiently, owing to the fact that those are typically governed by cartoon-like structures. In this paper, we introduce and discuss a first-order hybrid sigma-delta quantization algorithm for coarsely quantizing the shearlet coefficients generated by the FDST. Radial oversampling in the frequency domain together with

Image Separation Using Wavelets and Shearlets

Lecture Notes in Computer Science, 2012

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 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. We further provide and discuss an efficient numerical scheme to solve the associated optimization problem. Finally, we present some experimental results showing the effectiveness of our algorithm.

A Survey of Compressed Sensing

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...

A fundamental identity for Parseval frames

by Gitta Kutyniok, Radu Balan, and Dan Edidin

Proceedings of the American Mathematical Society

In this paper we establish a suprising fundamental identity for Parseval frames in a Hilbert spac... more

Dualizable Shearlet Frames and Sparse Approximation

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 ...

Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis

SIAM Journal on Mathematical Analysis, 2009

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.

Sparse Proteomics Analysis - a Compressed Sensing-based Approach for Feature Selection and Classification of High-Dimensional Proteomics Mass Spectrometry Data

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.

Classification of Edges Using Compactly Supported Shearlets

We analyze the detection and classification of singularities of functions $f = \chi_B$, where $B ... more We analyze the detection and classification of singularities of functions $f = \chi_B$, where $B \subset \mathbb{R}^d$ and $d = 2,3$. It will be shown how the set $\partial B$ can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if $\partial S$ is a $d-1$ dimensional piecewise smooth manifold with $d=2$ or $3$, we will classify smooth and non-smooth components of $\partial S$. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in $3$ dimensions by compactly supported shearlets. Finally, geometric properties of $\partial S$ such as curvature are described in terms of the continuous shearlet transform of $f$.

From Wavelets to Shearlets and back again

In this paper we will study the Continuous Shearlet Transform from a wavelet point of view, and s... more

Optimally Sparse Fusion Frames: Existence and Construction

Digital Shearlet Transform

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

Shearlets: Theory and Applications

GAMM-Mitteilungen, 2014

Image interpolation using shearlet based iterative refinement

by Gitta Kutyniok and Detlev Marpe

Signal Processing: Image Communication, 2015

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.

<title>Modeling sensor networks with fusion frames</title>

Wavelets XII, 2007

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

Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements

SIAM Journal on Mathematical Analysis, 2015

In this paper we analyze two-dimensional wavelet reconstructions from Fourier samples within the ... more In this paper we analyze two-dimensional wavelet reconstructions from Fourier samples within the framework of generalized sampling. For this, we consider both separable compactly-supported wavelets and boundary wavelets. We prove that the number of samples that must be acquired to ensure a stable and accurate reconstruction scales linearly with the number of reconstructing wavelet functions. We also provide numerical experiments that corroborate our theoretical results.

Guest Editorial: Mathematics and Image Analysis

by Gitta Kutyniok and Gerlind Plonka-hoch

Journal of Mathematical Imaging and Vision, 2015

ABSTRACT This special issue highlights some recent developments in the field of mathematical imag... more ABSTRACT This special issue highlights some recent developments in the field of mathematical image analysis. The emphasis of the issue is on the interplay between advanced mathematical methods (such as non-smooth optimization, variational methods and regularization, inverse problems, and partial differential equations) and their application to various problems in image processing and computer vision, including restoration, segmentation, and classification. The special issue comprises nine papers which cover a wide range of topics in mathematical image analysis as outlined below.Variational imaging: The paper “Preconditioned Douglas-Rachford algorithms for TV and TGV regularized variational imaging problems” (DOI JMIV-D-14-00124), by Bredies and Sun, applies the preconditioned Douglas-Rachford scheme, introduced previously by the same authors, to solve structured saddle-point problems involving total variation (TV) and total generalized variation (TGV) regularizers. A sublinear ...

Shearlets on Bounded Domains

Springer Proceedings in Mathematics, 2011

Shearlet systems have so far been only considered as a means to analyze L 2 -functions defined on... more Shearlet systems have so far been only considered as a means to analyze L 2 -functions defined on R 2 , which exhibit curvilinear singularities. However, in applications such as image processing or numerical solvers of partial differential equations the function to be analyzed or efficiently encoded is typically defined on a non-rectangular shaped bounded domain. Motivated by these applications, in this paper, we first introduce a novel model for cartoon-like images defined on a bounded domain. We then prove that compactly supported shearlet frames satisfying some weak decay and smoothness conditions, when orthogonally projected onto the bounded domain, do provide (almost) optimally sparse approximations of elements belonging to this model class.

Measures of Scalability

by Gitta Kutyniok and Kasso Okoudjou

IEEE Transactions on Information Theory, 2015

Scalable frames are frames with the property that the frame vectors can be rescaled resulting in ... more Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three novel quantitative measures of the closeness to scalability for frames in finite dimensional real Euclidean spaces. Besides the natural measure of scalability given by the distance of a frame to the set of scalable frames, another measure is obtained by optimizing a quadratic functional, while the third is given by the volume of the ellipsoid of minimal volume containing the symmetrized frame. After proving that these measures are equivalent in a certain sense, we establish bounds on the probability of a randomly selected frame to be scalable. In the process, we also derive new necessary and sufficient conditions for a frame to be scalable.