Wavelet and Frame Theory: Frame Bound Gaps, Generalized Shearlets, Grassmannian Fusion Frames, and P -Adic Wavelets
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Sums of Hilbert Space Frames 1
SUMS OF HILBERT SPACE FRAMES SOFIAN OBEIDAT, SALTI SAMARAH, PETER G. CASAZZA AND JANET C. TREMAIN Abstract. We give simple necessary and sufficient conditions on Bessel sequences {fi} and {gi} and operators L1,L2 on a Hilbert space H so that {L1fi + L2gi} is a frame for H. This allows us to construct a large number of new Hilbert space frames from existing frames. 1. Introduction Frames for Hilbert spaces were introduced by Duffin and Schaeffer [8] as a part of their research in non-harmonic Fourier series. Their work on frames was somewhat forgotten until 1986 when Daubechies, Grossmann and Meyer [13] brought it all back to life during their fundamental work on wavelets. Today, frame theory plays an important role not just in signal processing, but also in dozens of applied areas (See [1, 2]). Holub [12] showed that if {xn} is any normalized basis for a Hilbert space H and {fn} is the associated dual basis of coefficient functionals, then the sequence {xn + fn} is again a basis for H. In this paper we study cases in which new frames can be obtained from old ones. Throughout H denotes a separable Hilbert space and HN is the Nth dimensional Hilbert space. A frame for H is a family of vectors fi ∈ H, for i ∈ I for which there exist constants A, B > 0 satisfying: 2 X 2 2 (1.1) Akfk ≤ |hf, fii| ≤ Bkfk i∈I for all f ∈ H. A and B are called the lower and upper frame bounds respectively. If A = B, this is called an A-tight frame. -
Designing Structured Tight Frames Via an Alternating Projection Method Joel A
1 Designing Structured Tight Frames via an Alternating Projection Method Joel A. Tropp, Student Member, IEEE, Inderjit S. Dhillon, Member, IEEE, Robert W. Heath Jr., Member, IEEE and Thomas Strohmer Abstract— Tight frames, also known as general Welch-Bound- tight frame to a given ensemble of structured vectors; then Equality sequences, generalize orthonormal systems. Numer- it finds the ensemble of structured vectors nearest to the ous applications—including communications, coding and sparse tight frame; and it repeats the process ad infinitum. This approximation—require finite-dimensional tight frames that pos- sess additional structural properties. This paper proposes a technique is analogous to the method of projection on convex versatile alternating projection method that is flexible enough to sets (POCS) [4], [5], except that the class of tight frames is solve a huge class of inverse eigenvalue problems, which includes non-convex, which complicates the analysis significantly. Nev- the frame design problem. To apply this method, one only needs ertheless, our alternating projection algorithm affords simple to solve a matrix nearness problem that arises naturally from implementations, and it provides a quick route to solve difficult the design specifications. Therefore, it is fast and easy to develop versions of the algorithm that target new design problems. frame design problems. This article offers extensive numerical Alternating projection is likely to succeed even when algebraic evidence that our method succeeds for several important cases. constructions are unavailable. We argue that similar techniques apply to a huge class of To demonstrate that alternating projection is an effective tool inverse eigenvalue problems. for frame design, the article studies some important structural There is a major conceptual difference between the use properties in detail. -
Dirichlet Character Difference Graphs 1
DIRICHLET CHARACTER DIFFERENCE GRAPHS M. BUDDEN, N. CALKINS, W. N. HACK, J. LAMBERT and K. THOMPSON Abstract. We define Dirichlet character difference graphs and describe their basic properties, includ- ing the enumeration of triangles. In the case where the modulus is an odd prime, we exploit the spectral properties of such graphs in order to provide meaningful upper bounds for their diameter. 1. Introduction Upon one's first encounter with abstract algebra, a theorem which resonates throughout the theory is Cayley's theorem, which states that every finite group is isomorphic to a subgroup of some symmetric group. The significance of such a result comes from giving all finite groups a common ground by allowing one to focus on groups of permutations. It comes as no surprise that algebraic graph theorists chose the name Cayley graphs to describe graphs which depict groups. More formally, if G is a group and S is a subset of G closed under taking inverses and not containing the identity, then the Cayley graph, Cay(G; S), is defined to be the graph with vertex set G and an edge occurring between the vertices g and h if hg−1 2 S [9]. Some of the first examples of JJ J I II Cayley graphs that are usually encountered include the class of circulant graphs. A circulant graph, denoted by Circ(m; S), uses Z=mZ as the group for a Cayley graph and the generating set S is Go back m chosen amongst the integers in the set f1; 2; · · · b 2 cg [2]. -
FRAME WAVELET SETS in R §1. Introduction a Collection of Elements
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 7, Pages 2045{2055 S 0002-9939(00)05873-1 Article electronically published on December 28, 2000 FRAME WAVELET SETS IN R X. DAI, Y. DIAO, AND Q. GU (Communicated by David R. Larson) Abstract. In this paper, we try to answer an open question raised by Han and Larson, which asks about the characterization of frame wavelet sets. We completely characterize tight frame wavelet sets. We also obtain some nec- essary conditions and some sufficient conditions for a set E to be a (general) frame wavelet set. Some results are extended to frame wavelet functions that are not defined by frame wavelet set. Several examples are presented and compared with some known results in the literature. x1. Introduction A collection of elements fxj : j 2Jgin a Hilbert space H is called a frame if there exist constants A and B,0<A≤ B<1, such that X 2 2 2 (1) Akfk ≤ jhf;xj ij ≤ Bkfk j2J for all f 2H. The supremum of all such numbers A and the infimum of all such numbers B are called the frame bounds of the frame and denoted by A0 and B0 respectively. The frame is called a tight frame when A0 = B0 andiscalleda normalized tight frame when A0 = B0 = 1. Any orthonormal basis in a Hilbert space is a normalized tight frame. The concept of frame first appeared in the late 40's and early 50's (see [6], [13] and [12]). The development and study of wavelet theory during the last decade also brought new ideas and attention to frames because of their close connections. -
The Contourlet Transform: an Efficient Directional Multiresolution Image Representation Minh N
IEEE TRANSACTIONS ON IMAGE PROCESSING 1 The Contourlet Transform: An Efficient Directional Multiresolution Image Representation Minh N. Do, Member, IEEE, and Martin Vetterli, Fellow, IEEE Abstract— The limitations of commonly used separable ex- representation has to be obtained by structured transforms and tensions of one-dimensional transforms, such as the Fourier fast algorithms. and wavelet transforms, in capturing the geometry of image For one-dimensional piecewise smooth signals, like scan- edges are well known. In this paper, we pursue a “true” two- dimensional transform that can capture the intrinsic geometrical lines of an image, wavelets have been established as the right structure that is key in visual information. The main challenge tool, because they provide an optimal representation for these in exploring geometry in images comes from the discrete nature signals in a certain sense [1], [2]. In addition, the wavelet of the data. Thus, unlike other approaches, such as curvelets, representation is amenable to efficient algorithms; in particular that first develop a transform in the continuous domain and it leads to fast transforms and convenient tree data structures. then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence These are the key reasons for the success of wavelets in to an expansion in the continuous domain. Specifically, we many signal processing and communication applications; for construct a discrete-domain multiresolution and multidirection example, the wavelet transform was adopted as the transform expansion using non-separable filter banks, in much the same way for the new image-compression standard, JPEG-2000 [3]. -
Introducing the Mini-DML Project Thierry Bouche
Introducing the mini-DML project Thierry Bouche To cite this version: Thierry Bouche. Introducing the mini-DML project. ECM4 Satellite Conference EMANI/DML, Jun 2004, Stockholm, Sweden. 11 p.; ISBN 3-88127-107-4. hal-00347692 HAL Id: hal-00347692 https://hal.archives-ouvertes.fr/hal-00347692 Submitted on 16 Dec 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Introducing the mini-DML project Thierry Bouche Université Joseph Fourier (Grenoble) WDML workshop Stockholm June 27th 2004 Introduction At the Göttingen meeting of the Digital mathematical library project (DML), in May 2004, the issue was raised that discovery and seamless access to the available digitised litterature was still a task to be acomplished. The ambitious project of a comprehen- sive registry of all ongoing digitisation activities in the field of mathematical research litterature was agreed upon, as well as the further investigation of many linking op- tions to ease user’s life. However, given the scope of those projects, their benefits can’t be expected too soon. Between the hope of a comprehensive DML with many eYcient entry points and the actual dissemination of heterogeneous partial lists of available material, there is a path towards multiple distributed databases allowing integrated search, metadata exchange and powerful interlinking. -
The Canonical Dual Frame of a Wavelet Frame
Applied and Computational Harmonic Analysis 12, 269–285 (2002) doi:10.1006/acha.2002.0381 The Canonical Dual Frame of a Wavelet Frame Ingrid Daubechies 1 PACM, Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544 E-mail: [email protected] and Bin Han 2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada E-mail: [email protected] Communicated by Charles K. Chui Received July 11, 2000; revised December 20, 2001 In this paper we show that there exist wavelet frames that have nice dual wavelet frames, but for which the canonical dual frame does not consist of wavelets, i.e., cannot be generated by the translates and dilates of a single function. 2002 Elsevier Science (USA) Key Words: wavelet frame; the canonical dual frame; dual wavelet frame. 1. INTRODUCTION We start by recalling some notations and definitions. Let H be a Hilbert space. A set of elements {hk}k∈Z in H is said to be a frame (see [5]) in H if there exist two positive constants A and B such that 2 2 2 Af ≤ |f,hk| ≤ Bf , ∀f ∈ H, (1.1) k∈Z where A and B are called the lower frame bound and upper frame bound, respectively. In particular, when A = B = 1, we say that {hk}k∈Z is a (normalized) tight frame in H.The 1 Research was partially supported by NSF Grants DMS-9872890, DMS-9706753 and AFOSR Grant F49620- 98-1-0044. Web: http://www.princeton.edu/~icd. 2 Research was supported by NSERC Canada under Grant G121210654 and by Alberta Innovation and Science REE under Grant G227120136. -
Quantization in Signal Processing with Frame Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Vanderbilt Electronic Thesis and Dissertation Archive QUANTIZATION IN SIGNAL PROCESSING WITH FRAME THEORY By JIAYI JIANG Dissertation Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics May, 2016 Nashville, Tennessee Approved: Alexander Powell Akram Aldroubi Doug Hardin Brett Byram ©Copyright by Jiayi Jiang, 2016. All Rights Reserved ii ACKNOWLEDGEMENTS Firstly, I would like to express my sincere gratitude to my advisor, Alexander.M.Powell. He taught me too many thing in these five years which are not only on mathematics research but also on how to be a polite and organised person. Thanks a lot for his encouragement, patience and professional advices. I really appreciate for his supporting and helping when I decide to work in industry instead of academic. All of these mean a lot to me! My time in Vanderbilt has been a lot of fun. Thanks for the faculty of the math depart- ment. Thanks for all my teachers, Akram Aldroubi, Doug Hardin, Mark Ellingham, Mike Mihalik, Edward Saff. Each Professors in these list taught me at least one course which I will be benefited from them in my future carer. I really enjoy the different way of teaching from these professors. Thanks for my Vandy friends, Corey Jones, Yunxiang Ren, Sui tang and Zhengwei Liu. You guys give me too many helps and joys. I have to say thanks to my wife Suyang Zhao. -
Tight Wavelet Frame Construction and Its Application for Image Processing
Tight Wavelet Frame Construction and its application for Image Processing by Kyunglim Nam (Under the direction of Ming-Jun Lai) Abstract As a generalized wavelet function, a wavelet frame gives more rooms for different construc- tion methods. In this dissertation, first we study two constructive methods for the locally supported tight wavelet frame for any given refinable function whose Laurent polynomial satisfies the QMF or the sub-QMF conditions in Rd. Those methods were introduced by Lai and St¨ockler. However, to apply the constructive method under the sub-QMF condition we need to factorize a nonnegative Laurent polynomial in the multivariate setting into an expression of a finite square sums of Laurent polynomials. We find an explicit finite square sum of Laurent polynomials that expresses the nonnegative Laurent polynomial associated with a 3-direction or 4-direction box spline for various degrees and smoothness. To facilitate the description of the construction of box spline tight wavelet frames, we start with B-spline tight wavelet frame construction. For B-splines we find the sum of squares form by using Fej´er-Rieszfactorization theorem and construct tight wavelet frames. We also use the tensor product of B-splines to construct locally supported bivariate tight wavelet frames. Then we explain how to construct box spline tight wavelet frames using Lai and St¨ockler’s method. In the second part of dissertation, we apply some of our box spline tight wavelet frames for edge detection and image de-noising. We present a lot of images to compare favorably with other edge detection methods including orthonormal wavelet methods and six engineering methods from MATLAB Image Processing Toolbox. -
The Complex Shearlet Transform and Applications to Image Quality Assessment
Technische Universität Berlin Fachgruppe Angewandte Funktionalanalysis The Complex Shearlet Transform and Applications to Image Quality Assessment Master Thesis by Rafael Reisenhofer 18. Juli 2014 – 17. Oktober 2014 Primary reviewer: Prof. Dr. Gitta Kutyniok Secondary reviewer: Dr. Wang-Q Lim Supervisor: Prof. Dr. Gitta Kutyniok Rafael Reisenhofer Beusselstraße 64 10553 Berlin Hiermit erkläre ich, dass ich die vorliegende Arbeit selbstständig und eigenhändig sowie ohne unerlaubte fremde Hilfe und ausschließlich unter Verwendung der aufgeführten Quellen und Hilfsmittel angefertigt habe. Berlin, den 17. Oktober 2014 Rafael Reisenhofer Contents List of Figures 5 1 Introduction 7 2 Complex Shearlet Transforms 9 2.1 From Fourier to Shearlets via an Analysis of Optimality . ........ 10 2.1.1 FourierAnalysis........................... 14 2.1.2 Wavelets............................... 20 2.1.2.1 Wavelets in Two Dimensions . 26 2.1.3 Shearlets............................... 27 2.2 Wavelet and Shearlet Transforms . 33 2.3 Complex Wavelet and Shearlet Transforms . ... 34 3 Complex Shearlet Transforms and the Visual Cortex 41 4 Applications 45 4.1 EdgeDetection ............................... 45 4.1.1 PhaseCongruency ......................... 47 4.1.2 Detection of Singularities With Pairs of Symmetric and Odd- SymmetricWavelets ........................ 51 4.1.3 Edge Detection With Pairs of Symmetric and Odd-Symmetric Shearlets............................... 56 4.2 ImageQualityAssessment . 62 4.2.1 Structural Similarity Index (SSIM) . 63 4.2.2 Complex Shearlet-Based Image Similarity Measure . .... 67 4.2.3 NumericalResults ......................... 71 5 Discussion 75 A Some Definitions and Formulas 79 A.1.1 PolynomialDepthSearch . 81 References 83 3 4 List of Figures 2.1 A piecewise smooth function and a cartoon like image . ...... 11 2.2 ExampleofFouriermodes. -
201.1.8001 Fall 2013-2014 Lecture Notes Updated: December 24, 2014
INTRODUCTION TO PROBABILITY ARIEL YADIN Course: 201.1.8001 Fall 2013-2014 Lecture notes updated: December 24, 2014 Contents Lecture 1. 3 Lecture 2. 9 Lecture 3. 18 Lecture 4. 24 Lecture 5. 33 Lecture 6. 38 Lecture 7. 44 Lecture 8. 51 Lecture 9. 57 Lecture 10. 66 Lecture 11. 73 Lecture 12. 76 Lecture 13. 81 Lecture 14. 87 Lecture 15. 92 Lecture 16. 101 1 2 Lecture 17. 105 Lecture 18. 113 Lecture 19. 118 Lecture 20. 131 Lecture 21. 135 Lecture 22. 140 Lecture 23. 156 Lecture 24. 159 3 Introduction to Probability 201.1.8001 Ariel Yadin Lecture 1 1.1. Example: Bertrand's Paradox We begin with an example [this is known as Bertrand's paradox]. Joseph Louis Fran¸cois Question 1.1. Consider a circle of radius 1, and an equilateral triangle bounded in the Bertrand (1822{1900) circle, say ABC. (The side length of such a triangle is p3.) Let M be a randomly chosen chord in the circle. What is the probability that the length of M is greater than the length of a side of the triangle (i.e. p3)? Solution 1. How to chose a random chord M? One way is to choose a random angle, let r be the radius at that angle, and let M be the unique chord perpendicular to r. Let x be the intersection point of M with r. (See Figure 1, left.) Because of symmetry, we can rotate the triangle so that the chord AB is perpendicular to r. Since the sides of the triangle intersect the perpendicular radii at distance 1=2 from 0, M is longer than AB if and only if x is at distance at most 1=2 from 0. -
A48 Integers 11 (2011) Enumeration of Triangles in Quartic Residue
#A48 INTEGERS 11 (2011) ENUMERATION OF TRIANGLES IN QUARTIC RESIDUE GRAPHS Mark Budden Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC [email protected] Nicole Calkins Department of Mathematics, Armstrong Atlantic State University, Savannah, GA [email protected] William Nathan Hack Department of Mathematics, Armstrong Atlantic State University, Savannah, GA [email protected] Joshua Lambert Department of Mathematics, Armstrong Atlantic State University, Savannah, GA [email protected] Kimberly Thompson Department of Mathematics, Armstrong Atlantic State University, Savannah, GA [email protected] Received: 10/10/10, Accepted: 7/19/11, Published: 9/13/11 Abstract For a fixed prime p 1 (mod 4), we define the corresponding quartic residue graph and determine the n≡umber of triangles contained in such a graph. Our computation requires us to compute the number of pairs of consecutive quartic residues modulo p via the evaluation of certain quartic Jacobi sums. 1. Introduction Although Raymond Paley’s life passed by quickly, his impact resonated across mul- tiple mathematical disciplines such as complex analysis, combinatorics, number theory, and harmonic analysis [2, 8, 9, 10]. Perhaps one of Paley’s most notable contributions occurred when he brought the fields of combinatorics and number theory even closer together. One such mathematical achievement came about in INTEGERS: 11 (2011) 2 1933, when Paley [8] used the quadratic residues of the field with prime order p 3 ≡ (mod 4) to construct Hadamard matrices of order p + 1. While Paley passed away that same year, his results spurred a great deal of interest. One person that took notice of Paley’s achievements was Horst Sachs [12].