International Conference onInternational Conference onApplied Harmonic Analysis and Applied Harmonic Analysis and MultiscaleMultiscale ComputingComputing

J l 25J l 25 28 201128 2011July 25July 25--28, 201128, 2011

Summer SchoolSummer SchoolJuly 29July 29--31, 201131, 2011

U i i f AlbU i i f AlbUniversity of AlbertaUniversity of AlbertaEdmontonEdmontonCanada Canada

CITIZEN

International Conference on Applied Harmonic Analysis and Multiscale Computing 

 July 25—28, 2011 and  

Summer School, July 29—31, 2011 at the University of Alberta, Edmonton, Canada 

 Welcome to the University of Alberta and the City of Edmonton for this international conference on Applied Harmonic Analysis and Multiscale Computing (AHAMC 2011) and afterwards summer school. The purpose of the conference is to bring together mathematicians, computer scientists, and engineers in the area of applied and computational harmonic analysis to exchange ideas, collaborate among participants, and explore recent exciting developments.  The conference has over one hundred participants and features 67 presentations: 6 one‐hour plenary talks, 37 forty‐minute invited talks, 5 twenty‐minute invited talks, and 19 twenty‐minute contributed talks. This international conference concentrates on recent advances on applied and computational harmonic analysis, multiscale‐based methods, and their various applications in the broad sense. The topics of interest include, but not limited to, 

o Applied harmonic analysis o Approximation theory o Compressive sampling and sparsity o Computational methods o Image and signal processing o Learning theory and algorithms o Multiscale‐related numerical algorithms o Sampling theorems in signal processing  o Subdivision schemes o Wavelets and framelets 

 Scientific Committee: 

o Carl de Boor (University of Wisconsin ‐ Madison) o Ronald DeVore (Texas A&M University) o Thomas J. R. Hughes (The University of Texas at Austin) o Stanley Osher (University of California at Los Angeles) o Gilbert Strang (Massachusetts Institute of Technology) 

 

Organizing Committee: o Elena Braverman (University of Calgary) o Bin Han (University of Alberta) o Rong‐Qing Jia (University of Alberta) o Yau Shu Wong (University of Alberta) o Ozgur Yilmaz (University of British Columbia) 

 

Computer facilities: o CNS computer room at MEC 3‐3 West (third floor, near elevator, neighboring building MEC) o Wireless access to UWS (University Wireless Service): use provided username and password  

 Transportation between hotel and conference site at University of Alberta by conference bus: 

o July 25 (Monday):       7:30am & 8:00am (to conference site).  8:30pm & 9:00pm (back to hotel) o July 26 (Tuesday):       7:30am & 8:00am (to conference site).  6:00pm & 6:30pm (back to hotel) o July 27 (Wednesday): 7:30am & 8:00am (to conference site).  9:00pm & 9:30pm (back to hotel) o July 28 (Thursday):     7:30am & 8:00am (to conference site).  4:00pm  (back to hotel) 

Acknowledgement  This international conference and its afterwards summer school have been generously and mainly supported by Pacific Institute for the Mathematical Sciences (PIMS) through a two‐year PIMS Collaborative Research Group (CRG) in Applied and Computational Harmonic Analysis. The major purpose of this CRG is to bring together researchers in western Canada and USA to collaborate in the broad area of applied and computational harmonic analysis.  The AHAMC 2011 international conference and its afterwards summer school are two of the major activities of this CRG during the period of 2011—2013.  The members of this CRG on Applied and Computational Harmonic Analysis are:  

• Michael Adams (University of Victoria) • Len Bos (University of Calgary) • Elena Braverman (University of Calgary, CRG leader) • Tom Duchamp (University of Washington) • Bin Han (University of Alberta, CRG main leader) • Felix J. Herrmann (University of British Columbia) • Rong‐Qing Jia (University of Alberta, CRG leader) • Xiaobo Li (University of Alberta) • Peter D. Minev (University of Alberta) • Bernard Shizgal (University of British Columbia) • Yau Shu Wong (University of Alberta) • Ozgur Yilmaz (University of British Columbia, CRG leader) 

 The PIMS web page of this CRG and its activities can be found at http://www.pims.math.ca/scientific under Collaborative Research Groups: Applied and Computational Harmonic Analysis.  

Sponsors  

We gratefully acknowledge the generous funding by the following organizations and institutions for our conference and summer school:  

• The Pacific Institute for the Mathematical Sciences (PIMS) • The Mathematics of Information Technology and Complex Systems (MITACS) • Alberta Innovates ‐ Technology Futures • Grant MacEwan University • JackTeK System Ltd. • Citizen Travel Ltd.  • University of Alberta:  

o China Institute o Faculty of Sciences o Department of Mathematical and Statistical Sciences o Applied Mathematics Institute (AMI) 

  Conference Web Page: http://www.ualberta.ca/~ahamc        Conference Email Address: ahamc@ualberta.ca Conference Contact: Bin Han and Xiaosheng Zhuang 

Monday, July 25, 2011 

 o ETLE=ETL=ETLC=Engineering Teaching and Learning Complex o Each room has a computer, data projector, overhead projector, whiteboard. o No food or drink inside all lecture rooms ETLE 1‐003, 1‐013, 1‐017, and 2‐001. 

 

8:30am‐8:45am ETLE 1‐013 

Welcome Address 

8:45am‐9:45am ETLE 1‐013    Chair: Yau Shu Wong 

Steve Smale: Scales in Vision and Immunology 

9:45am‐10:30am Registration and 

Coffee/Tea Break (45 minutes) 

  ETLE 1‐013  Chair: Charles K. Chui  ETLE 1‐017  Chair: Ding‐Xuan Zhou 

10:30am‐11:10am 

Serge Dubuc Palindromic Matrices of Order Two and Three‐Point Subdivision Schemes 

Douglas P. Hardin Wavelets Centered on a Knot Sequence 

11:10am‐11:50am 

Kurt Jetter Non‐negative Subdivision and Non‐

homogeneous Markov Chains 

Palle E. T.  Jorgensen Ideas from Engineering of Signals Interacting with Approximation in Mathematics (Wavelets and More) 

11:50am‐1:45pm  Lunch Break (1 hour 55 minutes) 

1:45pm‐2:45pm ETLE 1‐013   Chair: Rong‐Qing Jia 

Gilbert Strang: Factoring Banded Matrices and Matrix Polynomials 

  ETLE 1‐013   Chair: Kurt Jetter  ETLE 1‐017   Chair: Jean‐Pierre Gabardo 

2:50pm‐3:30pm Jorg Peters 

Constructing Parametric Surfaces 

Gerlind Plonka Sparse Data Representation by the Easy 

Path Wavelet Transform 

3:30pm‐4:00pm  Coffee/Tea Break (30 minutes) 

4:00pm‐4:40pm 

Qing‐Tang Jiang Bi‐frames with 6‐fold Axial Symmetry for Hexagonal Data and Triangle Surface Multiresolution Processing 

Ivan Selesnick Sparse Signal Representation and the Tunable Q‐factor Wavelet Transform 

4:40pm‐5:20pm 

Michael Adams Triangle Meshes for Image 

Representation 

Bernie Shizgal Resolution of the Gibbs Phenomenon 

5:30pm‐9:30pm  Welcome Reception  at Main Quad between CAB and Athabasca Hall  

Tuesday, July 26, 2011 

8:30am‐9:30am 

ETLE 1‐013    Chair: Bin Han Charles K. Chui:   Approximation of Functions on Unknown Manifolds Defined by 

High‐Dimensional Unstructured Data 

  ETLE 1‐013   Chair: Gilbert Strang  ETLE 1‐017  Chair: Zuowei Shen 

9:35am‐10:15am 

Rong‐Qing Jia Explicit Iteration Schemes of the Uzawa Algorithm for Minimization Problems 

Arising from Image Processing 

Keith F. Taylor Wavelets with Crystal Symmetry Shifts 

10:15am‐10:40am  Coffee/Tea Break (25 minutes) 

10:40am‐11:20am 

Gregory Beylkin Approximations and Fast Algorithms for 

Green's Functions 

Gitta Kutyniok Image Inpainting and Sparse 

Approximation 

11:20am‐11:40am 

Wei Zhao Mathematical Modeling and Multilevel Computation of Dispersed Drug Release 

from Polymeric Matrix Systems 

Xiaosheng Zhuang The Fast Digital Shearlet Transform and Applications to Denoising and Coarse 

Quantization 

11:40am‐12:00pm 

Md. Kamrujjaman Numerical Approximations and Stability 

Analysis of 1D Heat Equation 

Julia Dobrosotskaya Shearlet Ginzurg‐Landau Energy, Anisotropic Analogues, Associated 

Operators and Applications 

12:00pm‐1:45pm  Lunch Break (1 hour 45 minutes) 

1:45pm‐2:45pm ETLE 1‐013     Chair:  Elena Braverman 

John J. Benedetto: Constructive Digital Harmonic Analysis 

  ETLE 1‐013   Chair: Douglas Hardin  ETLE 1‐017  Chair: Keith F. Taylor 

2:50pm‐3:30pm 

Amir Averbuch Multiscale Data Sampling and Function 

Extension 

Christopher Heil Modulation Spaces, BMO, BLT, and 

Rectangular Partial Sums 

3:30pm‐4:00pm  Coffee/Tea Break (30 minutes) 

4:00pm‐4:40pm 

Qiyu Sun Wiener's Lemma  and Two Nonlinear 

Sampling Problems in Signal Processing 

Jean‐Pierre Gabardo Convolution Inequalities Associated with Irregular Gabor and Wavelet Systems 

4:40pm‐5:20pm 

Say Song Goh Ambiguity Functions, Sampling and Uncertainty Principles in Parameter 

Estimation 

Russell Greiner An Alternative Approach to Designing Clinical Trials: Budgeted Learning of 

Effective Classifiers 

5:20pm‐5:40pm 

Matthew Hamilton High‐Performance Time‐Warped Multiscale Signal Reconstruction 

Xin Guo An Empirical Feature‐based Learning 

Algorithm Producing Sparse Approximation 

Wednesday, July 27, 2011 

 

8:30am‐9:30am ETLE 1‐013    Chair:  Ozgur Yilmaz 

Ding‐Xuan Zhou:   Error Analysis and Sparsity of Some Learning Algorithms 

  ETLE 1‐013  Chair:  Palle E. T. Jorgensen  ETLE 1‐017  Chair:  Michael Adams 

9:35am‐10:15am 

Zongmin Wu On the Different Kinds of Quasi‐

interpolation 

Hao‐Min Zhou Iterative Filtering, EMD and 

Instantaneous Frequency Analysis 

10:15am‐10:40am  Coffee/Tea Break (25 minutes) 

10:40am‐11:20am 

Seng Luan Lee Appell Sequences from Scaling 

Functions 

Peter Binev Sparse Tree Approximation in High 

Dimensions 

11:20am‐11:40am 

Rayan Saab Root‐Exponential Accuracy for Coarse 

Quantization of Finite Frame Expansions

Qaiser Jahan Wavelet packets and wavelet frame 

packets on local fields 

11:40am‐12:00pm 

Hassan Mansour Weighted  Minimization: Support 

Recovery Guarantees and an Iterative Algorithm 

Kavita Goyal Sparse Evolution of Functions Using 

Diffusion Wavelets  

12:00pm‐1:45pm  Lunch Break (1 hour 45 minutes) 

1:45pm‐2:45pm ETLE 1‐013     Chair: Sherman D. Riemenschneider 

Zuowei Shen:  MRA Based Wavelet Frame and Applications 

  ETLE 1‐013   Chair:  Amir Averbuch  ETLE 1‐017  Chair: Christopher Heil 

2:50pm‐3:30pm 

Ozgur Yilmaz Noise‐shaping Quantizers for 

Compressed Sensing 

Maria Skopina Approximation by Wavelet and Scaling Expansions 

3:30pm‐4:00pm  Coffee/Tea Break (30 minutes) 

4:00pm‐4:40pm 

Song Li Some Results on Low‐Rank Matrix 

Recovery 

Marcin Bownik Affine and Quasi‐affine Frames for 

Rational Dilations 

4:40pm‐5:20pm Bin Han 

Frequency‐based Directional Framelets 

Radu Balan Reconstruction from Magnitudes of 

Frame Coefficients 

6:00pm‐9:30pm  Conference Banquet at Faculty Club 

Thursday, July 28, 2011   ETLE 1‐013   Chair: John J. Benedetto  ETLE 1‐017  Chair: Seng Luan Lee 

8:30am‐9:10am 

Elena Braverman Efficient Computation of Oscillatory Integrals with Local Fourier Bases 

Zeev Ditzian Smoothness and Best Approximation on 

the Sphere 

9:10am‐9:50am 

Igor Novikov On Stationary and Nonstationary 

Biorthogonal Wavelets 

Feng Dai Moduli of Smoothness and 

Approximation on the Unit Sphere 

9:50am‐10:10am 

Laurent Simons A Note about Non Stationary Multiresolution Analysis 

Rasel Biswas Improve the Results of Gaussian 

Integration Points and Weights with High Precision Degree of Gauss Quadrature Rules

10:10am‐10:40am  Coffee/Tea Break (30 minutes)   ETLE 1‐013   Chair: Russell Greiner  ETLE 1‐017  Chair: Tom Duchamp 

10:40am‐11:20am 

Dale Schuurmans Convex Sparse Coding, Subspace 

Learning, & Semi‐Supervised Extensions 

Thomas Yu A Survey of Subdivision Algorithms of 

Manifold‐Valued Data 

11:20am‐11:40am 

Sharmin Nilufar Multiple Kernel Learning for Scale‐Space based Feature Selection 

Aleksandr Krivoshein On Construction of Symmetric MRA‐based Frame‐like Wavelet Systems 

11:40am‐12:00pm 

Lei Shi Concentration Estimates for Learning 

with  –Regularizer and Data Dependent Hypothesis Spaces 

David Jimenez Multidimensional A/D Conversion and 

Directional Bias 

12:00pm‐1:45pm  Lunch Break (1 hour 45 minutes) 

  ETLE 1‐013  Chair: Qing‐tang Jiang  ETLE 1‐017   Chair:  Bernie Shizgal 

1:45pm‐2:05pm 

Jostein Bratlie Mapping Tensor‐product Wavelet Bases 

and GPGPU‐computing of DWTs 

Michael Joya An Introduction to Synchrosqueezed 

Transforms 

2:05pm‐2:25pm 

Mpfareleni R. Gavhi Interpolatory Subdivision Schemes with 

Fractal Limit Curves 

Kedarnath Senapati Automatic Removal of Ocular Artifacts from EEG Signals Using S‐transform 

2:25pm‐2:45pm  

Brock Hargreaves Numerical Techniques for 

Approximating Fourier Integral Operators 

Md. Khademul Islam Molla Bivariate Empirical Mode Decomposition (BEMD) Based Data Adaptive Approach to 

EOG Suppression from EEG Signals 

2:45pm‐3:05pm Qun Mo 

Some New Bounds of the R.I.C. in Compressed Sensing 

Jin‐Xin Zhang Interpolation of Missing Values in Time 

Series Based on its Periodicity 

3:05pm‐3:25pm   

  Conference Ends 

Summer School  on Applied and Computational Harmonic Analysis 

July 29—31, 2011 at ETLE 2‐001, University of Alberta, Edmonton, Canada 

 The main goal of this summer school on applied and computational harmonic analysis is to enable interested graduate students and postdoctoral fellows to enlarge their scope of research areas, to foster collaboration among participants, and to be aware of most recent developments in various branches of applied and computational harmonic analysis. This summer school consists of tutorial lectures concentrating on the following topics:  

• Learning theory by Steve Smale and Ding‐Xuan Zhou on Friday, July 29, 2011 • MRA based wavelet frame and applications by Zuowei Shen on Saturday, July 30, 2011 • Algorithms of wavelets and framelets by Bin Han in the morning of Sunday, July 31, 2011 

 

Friday, July 29, 2011:    9:00am—12:00pm, 2:00pm—5:00pm Topics: Learning theory Tutorial lecturers: Steve Smale, Ding‐Xuan Zhou Link of tutorial materials and references online: http://www.ualberta.ca/~ahamc/toturial.html 

9:00am-10:00am: Tutorial lecture by Steve Smale Framework of the least squares regression, empirical risk minimization, hypothesis space, reproducing kernel Hilbert space Sample error, probability inequalities, covering number, approximation error. 10:00am-10:30am: break/discussion/questions

10:30am-11:30am: Tutorial lecture by Ding-Xuan Zhou Kernel methods in learning theory, regularization scheme, representor theorem, reduction of optimization problems, binary classification, support vector machine Misclassification error, Bayes rule, separable distributions, comparison theorem, regularization error, error bounds. 11:30am-12:00am: break/discussion/questions

2:00pm-3:00pm: Tutorial lecture by Ding-Xuan Zhou Dimensionality reduction, Laplacian eigenmap, spectral clustering, kernel PCA, Semi-supervised learning on manifolds, online learning. 3:00pm-3:30pm break/discussion/questions

3:30pm-4:30pm: Tutorial lecture by Ding-Xuan Zhou LASSO, elastic net, sparsity kernel projection machine, empirical feature, gradient learning and variable selection. 4:30pm-5:00pm: break/discussion/questions

Saturday, July 30, 2011: 9:00am—12:00pm, 2:00pm—5:00pm

Topics: MRA based wavelet frame and applications Tutorial lecturer: Zuowei Shen Link of tutorial materials online: http://www.math.nus.edu.sg/~matzuows/IASLectureNotes.pdf  One of the major driving forces in the area of applied and computational harmonic analysis during the last two decades is the development and the analysis of redundant systems that produce sparse approximations for classes of functions of interest.  Such redundant systems include wavelet frames, ridgelets, curvelets and shearlets, to name a few.  This series of talks focuses on tight wavelet frames that are derived from multiresolution analysis and their applications in imaging.  The pillar of this theory is the unitary extension principle and its various generalizations, hence we will first give in details on the theory that leads to the unitary extension principles.           The extension principles allow for systematic  constructions of wavelet frames that can be tailored to, and effectively used in, various problems in imaging science. We will discuss some of these applications of wavelet frames in details.  The discussion will include frame‐based image analysis and restorations, image inpainting, image denoising, image deblurring and blind deblurring, image decomposition, segmentation and CT image reconstruction. 

 

Sunday, July 31, 2011:      9:00am—12:00am Topics: Algorithms of wavelets and framelets Tutorial lecturer: Bin Han Link of tutorial materials online: http://www.ualberta.ca/~bhan/notes.pdf  9:00am‐10:00am: Some basic aspects on wavelets and framelets. More precisely, discrete wavelet/framelet transform, properties, multilevel wavelet transform, oblique extension principle, variants of discrete wavelet transform, wavelet transform for data on bounded interval.     10:00am‐10:30am: break/discussion/questions  10:30am‐11:30am: Design of wavelet and framelet filter banks. Interpolatory filters, orthogonal filter banks, symmetric complex orthogonal wavelet filter banks, biorthogonal wavelet filter banks, tight framelet filter banks, and dual framelet filter banks. 11:30am‐12:00pm: break/discussion/questions  

Abstracts of AHAMC 2011

Triangle Meshes for Image Representation

Michael Adamsmdadams@ece.uvic.ca

Department of Electrical and Computer EngineeringUniversity of Victoria, Victoria, BC, Canada V8W 3P6

In the last several years, image representations based on adaptive (i.e., irregular) samplinghave been receiving an increasing amount of attention from the research community. Thisis largely due to the fact that, in many applications, such representations have numerousadvantages over those based on traditional lattice-based sampling. Some of the many applica-tions that can benefit from adaptive sampling include: feature detection, pattern recognition,computer vision, restoration, tomographic reconstruction, filtering, and image/video coding.Amongst the many classes of image representations based on adaptive sampling, trianglemeshes (especially those based on Delaunay triangulations) have proven to be highly effec-tive. In order to employ a triangle-mesh representation of an image, a means is needed toconstruct such a representation given an arbitrary lattice-sampled image. Finding effectivesolutions to this mesh-generation problem is a challenging task, due to the conflicting goals ofobtaining the highest quality mesh (i.e., with low approximation error) at minimal computa-tional/memory cost. In this talk, the speaker will discuss the use of triangle meshes for imagerepresentation and consider primarily the problem of mesh generation.

Multiscale Data Sampling and Function Extension

Amir Averbuch1∗ Amit Bermanis2 Ronald R. Coifman3

amir@math.tau.ac.il1School of Computer Science, 2 School of Mathematical Sciences

Tel Aviv University, Tel Aviv 69978, Israel3Department of Mathematics, Program in Applied Mathematics

Yale University, New Haven, CT 06510, USA

Many kernel based methods, which are used for dimensionality reduction and data miningapplications, involve an application of a SVD to a kernel matrix, whose dimensions are pro-portional to the size of the data. When data is accumulated over time, a method for functionextension is required. We introduce a multiscale scheme for data sampling and function ex-tension, which can be applied in any metric space, not necessarily a vector space. The schemeis based on mutual distances between datapoints. It makes use of a coarse-to-fine hierarchyof the multiscale decomposition of a Gaussian kernel. It generates a sequence of subsamples,which we refer to as adaptive grids, and a sequence of approximations to a given empiricalfunction on the data, as well as their extensions to any newly-arrived datapoint. The sub-sampling is done by a special decomposition of the associated Gaussian kernel matrix in eachscale in the hierarchical decomposition. In each scale, the data is sampled by an interpolativedecomposition of a low-rank Gaussian kernel matrix that is defined on the data.

1

Reconstruction from Magnitudes of Frame Coefficients

Radu Balanrvbalan@math.umd.edu

Department of Mathematics and CSCAMMUniversity of Maryland, College Park, MD 20742, USA

In this paper we present and algorithm for signal reconstruction from absolute value offrame coefficients. Then we compare its performance to the Cramer-Rao Lower Bound (CRLB)at high signal-to-noise ratio. To fix notations, assume {fi; 1 ≤ i ≤ m} is a spanning set (henceframe) in Rn. Given noisy measurements di = |〈x, fi〉|2 + νi, 1 ≤ i ≤ m, the problem is torecover x ∈ Rn up to a global sign. In this paper the reconstruction algorithm solves aregularized least squares criterion of the form

I(x) =m∑

i=1

||〈x, fi〉|2 − di|2 + λ‖x‖2

This criterion is modified in the following way: 1) the vector x is replaced by a n× r matrixL; 2) the criterion is augmented to allow an iterative procedure. Once the matrix L has beenobtained, an estimate for x is obtained through an SDV factorization.

Constructive Digital Harmonic Analysis

John J. Benedettojjb@math.umd.edu

Norbert Wiener Center, Department of MathematicsUniversity of Maryland, College Park, USA

Construction: Constant Amplitude Zero Autocorrelation (CAZAC) sequences are a staplefor waveform design of the type used in radar and communications theory. We constructnumber theoretic CAZACs whose discrete narrow-band ambiguity functions have optimalbehavior to effect best possible localization.

Algorithm: Using our frame potential energy theory, a meaningful and effective classifica-tion algorithm is constructed for multi-spectral and hyper-spectral imagery data.

Theory: Linear and non-linear sampling formulas are fundamental in harmonic analysis.The linear theory, i.e., the Classical Sampling Formula, goes back to Cauchy and was appliedextensively in the first half of the 20th century by the likes of Hadamard, Wiener, Shannon, etal. We extend this in the context of Beurling’s balayage theorem, which we generalize in termsof parameterization of the space of Radon measures. The non-linear approach is formulateddeterministically for sparse sets of coefficients in signal representation by means of recursivequantization.

The Construction is a collaboration with Joseph Woodworth; the Algorithm is a collab-oration with Wojciech Czaja; the Linear Theory is a collaboration with Enrico Au-Yeung;the Non-linear Theory is a reformulation of results with Onur Oktay, which themselves areinspired by work with Powell and Yilmaz.

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Approximations and Fast Algorithms for Green’s Functions

Gregory Beylkinbeylkin@colorado.edu

Department of Applied Mathematics, University of Colorado at Boulder526 UCB, Boulder, CO 80309-0526, USA

Multiresolution methods in high dimensions use separated representation of Green’s func-tions to obtain efficient fast algorithms designed to yield any finite accuracy requested by theuser. This talk provides a brief overview of the approach with an emphasis on approximationsused in the construction of convolution kernels.

We first consider separated representations of non-oscillatory free-space Green’s functionsvia a near optimal linear combination of decaying Gaussians. We then extend the approachto Green’s functions satisfying Dirichlet, Neumann or mixed boundary conditions on simpledomains. We also briefly elucidate some delicate theoretical issues related to the constructionof periodic Green’s functions for Poisson’s equation.

For oscillatory Green’s functions, we split the application of the kernel between the spatialand the Fourier domains. In the spatial domain we use a near optimal linear combination ofdecaying Gaussians with positive coefficients and, in the Fourier domain, a multiplication bya band-limited kernel obtained by using new quadratures appropriate for the singularity inthe Fourier domain. Applying this approach to the free space and the quasi-periodic Green’sfunctions, as well as those with Dirichlet, Neumann or mixed boundary conditions on simpledomains, we obtain fast algorithms in dimensions two and three for computing volumetricintegrals.

We also briefly describe some extensions to non-convolution kernels.

Sparse Tree Approximation in High Dimensions

Peter Binevbinev@math.sc.edu

Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

In this talk, we will discuss methods of approximation in high dimensions based on adaptivepartitions. The information about the function f : X → Y to be approximated comes frompoint cloud data Z := {zi = (xi, yi)}N

i=1, where xi ∈ X and yi ∈ Y . We want to determine a

procedure f that for any query x ∈ X finds an approximation y = f(x) to f(x). The analysisof the point cloud Z uses a special organization of the data called sparse occupancy trees andits complexity is only mildly dependent on the dimensionality of X. We present results interms of Learning Theory considering both regression and classification setups.

Improve the Results of Gaussian Integration Points and Weightswith High Precision Degree of Gauss Quadrature Rules

Rasel Biswasrb3247@mun.ca

Department of Mathematics and Statistics,Memorial University, St. John’s, Newfoundland, Canada, A1C 5S7

This paper concentrates to improve the integration rules in which both the accuracy andefficiency is assured. In order to improve the accuracy and efficiency it is found that theabscissae and corresponding weights are needed to be evaluated correctly. In this paper

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Gaussian integration points are calculated from the algebraic equations generated by Legenderpolynomials. Main effort is involved in this part and in the next, calculation of correspondingweights are followed. Manual calculations of weights are impossible for higher order formulaeand hence a program is written and weights are calculated and tested with their propertyand found correct. Finally, software in FORTTRAN is developed and then the formulae aredemonstrated through several authors. Thus, we believe that the development of the numericalintegration rules in this paper will provide the way of exact evaluation of integrals encounteredin continuum mechanics problems for the analysis of real world problems accurately with lesscomputational effort.

Affine and Quasi-affine Frames for Rational Dilations

Marcin Bownikmbownik@uoregon.edu

Department of Mathematics, University of Oregon, Eugene, OR 97403–1222, U.S.A.

Quasi-affine systems were originally introduced by Ron and Shen [J. Funct. Anal. 148(1997), 408–447] for integer, expansive dilations. In this talk we extend the definition of quasi-affine systems to the class of rational, expansive dilations. Friday, June 17, 2011 at 12:56 pmWe show that an affine system is a frame if and only if the corresponding family of quasi-affinesystems are frames with uniform frame bounds. We also prove a similar equivalence resultbetween pairs of dual affine frames and dual quasi-affine frames. Finally, we uncover somefundamental differences between the integer and rational settings by exhibiting an exampleof a quasi-affine frame such that its affine counterpart is not a frame. This talk is based on ajoint work with Jakob Lemvig.

Mapping Tensor-product Wavelet Bases and GPGPU-computing of DWTs

Lubomir T. Dechevsky, Jostein Bratlie∗ and Joakim Gundersenltd@hin.no, jostein.bratlie@gmail.com, joag@hin.no

Faculty of Technology, Narvik University College, Narvik, Nordland, N-8505, Norway

An algorithm for computation of multivariate wavelet transforms (DWTs) on graphics pro-cessing units (GPUs) was proposed by some of the authors in Dechevsky, L.T., Gundersen, J.,Bang, B., Computing n-Variate Orthogonal Discrete Wavelet Transforms on Graphics Pro-cessing Units, in: I. Lirkov, S. Margenov, and J. Wasniewski (Eds.) LSSC’2009, LNCS 5910,Springer-Verlag, Berlin-Heidelberg, 2010, 730–737. This algorithm was based on mappingthe indices of orthonormal tensor-product wavelet bases of different number of variables anda tradeoff between the number of variables versus the resolution level, so that the resultingwavelet bases of different number of variables are with different resolution, but the overalldimension of the bases is the same. In the above-said paper, the algorithm was developedonly up to mapping of the indices of blocks of wavelet basis functions. This was sufficientto prove the consistency of the algorithm, but not enough for the mapping of the individualbasis functions in the bases needed for a programming implementation of the algorithm. Inthe present communication we upgrade this construction by passing from block-matrix indexmapping on to the detailed index mapping of the individual basis functions. We consider someexamples illustrating the results of the algorithm. The present work is part of the researchof the R&D Group for Mathematical Modelling, Numerical Simulation and Computer Visu-alization at Narvik University College within two consecutive Strategic Projects of the Nor-wegian Research Council –’GPGPU (General Purpose computation on Graphics ProcessingUnits (GPGPU): http://gpgpu.org/) – Graphics Hardware as a High-end ComputationalResource’ (2004-2007) and ’Heterogeneous Computing’ (2008-2010).

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Efficient Computation of Oscillatory Integrals with Local Fourier Bases

Elena Bravermanmaelena@ucalgary.ca

Department of Mathematics and StatisticsUniversity of Calgary, Calgary, AB, Canada T2N 1N4

Computation of oscillatory integrals is required in the numerical solution of wave equa-tions, as well as in many applications including reflection seismology, curvilinear tomography,seismic imaging. In this talk, I will discuss sparse representation and efficient computation ofoscillatory kernels and oscillatory integrals using adaptive multiscale local Fourier bases.

Approximation of Functions on Unknown Manifoldsdefined by High-Dimensional Unstructured Data

Charles Chuickchui@stanford.edu chui@arch.cs.umsl.edu

Stanford University and Department of Mathematics and Computer ScienceUniversity of Missouri-St. Louis, St. Louis, MO 63121, USA

With the recent rapid technological advancement and significantly lower manufacturingcost in such areas as image sensor and capture, satellite and medical imaging, memory de-vices, computing power, convenient internet access, low-cost wireless communication, andpowerful search engines, the tremendously huge amount of data information to be processedand understood is over-whelming. One of the most popular current approaches to this prob-lem is to represent each piece of information as a point in a high-dimensional Euclidean spaceIRs and consider the collection of such points as a point-cloud P that lies on some unknownmanifold X ⊂ IRs. For example, in application to photo library organization and image searchengine, each point in the point-cloud in IRs represents a digital image thumbnail, with thedimension s being the maximum resolution of the image collection. In general, when somepieces of information are only partially available or corrupted, or when the point-cloud is toolarge to handle, a subset C ⊂ P of reliable data, called training set, is used to process P .

Although the manifold X is unknown, whatever information available from the point-cloudP can be used to determine X through some symmetric positive semi-definite kernel K de-fined by the dataset. However, it is usually not economical or even not feasible to computethe spectral decomposition of K for a large point-cloud. To overcome this obstacle, we devel-oped a class of randomized algorithms for computing the ”anisotropic transformation” of thedataset to re-organize the data without the need of computing the eigenvalues directly. Thetransformed dataset then provides a hierarchal structure for manifold dimensionality reduc-tion, while preserving data topology and geometry. On the other hand, to apply this manifoldapproach to such application areas as pattern recognition, time series event prediction, andrecovery of corrupted or missing data values, certain appropriate functions of choice definedonly on some desired training sets C must be extended to the entire unknown manifold X.We will discuss how data geometry can be incorporated with spatial approximation to solvethis extension problem. In particular, we present a point-cloud interpolation formula thatprovides near-optimal degree of approximation to the (unknown) target functions.

The notion of anisotropic transformation and its corresponding randomized algorithmswere introduced and developed in a joint paper with Jianzhong Wang that appeared in therecently founded Springer ”International Journal of Geomathematics (GEM)” in 2010; andthe results on the extension of functions from training sets to the entire (unknown) manifoldrepresent a joint work with Hrushikesh Mhaskar.

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Moduli of Smoothness and Approximation on the Unit Sphere

Feng Daidfeng@math.ualberta.ca

Department of Mathematical and Statistical SciencesUniversity of Alberta, Edmonton, Alberta, Canada T6G 2G1

This is a joint work with Yuan Xu. A new modulus of smoothness based on the Euler anglesis introduced on the unit sphere and is shown to satisfy all the usual characteristic propertiesof moduli of smoothness, including direct and inverse theorem for the best approximationby polynomials and its equivalence to a K-functional, defined via partial derivatives in Eulerangles. Examples are given to show our new moduli of smoothness are computable.

Smoothness and Best Approximation on the Sphere

Zeev Ditzianzditzian@math.ualberta.ca

Department of Mathematical and Statistical SciencesUniversity of Alberta, Edmonton, Alberta, Canada T6G 2G1

The rate of best approximation of functions on the unit sphere in Rd by spherical harmonicpolynomials of degree n using different norms or quasinorms is related to various measures ofsmoothness. Recent results comparing classical and new moduli of smoothness on the sphereare described.

Shearlet Ginzurg-Landau Energy, Anisotropic Analogues,Associated Operators and Applications

Julia Dobrosotskaya∗, Wojciech Czaja

juliadob@math.umd.edu, wojtek@math.umd.edu

Department of Mathematics, Norbert Wiener Center,University of Maryland, College Park, College Park, MD 20742-4015, USA

The authors prove the Γ-convergence of a shearlet-adapted Ginzburg-Landau(-type) func-tional to a multiple of the TV seminorm. The design of the functional was inspired by thediffuse-interface wavelet Ginzburg-Landau(GL) energy introduced by J.D. in collaborationwith A.Bertozzi. The shearlet GL energy provides the isotropy that the wavelet GL lacks. Itopens a new perspective on the implementation and utilization of the TV-related methods.

The generalized notions of the direction-adaptive functionals (primarily related to varia-tional image processing) based on the energies of a similar class are considered. The problemof recovering a weighted-TV-like functional corresponding to any given Wulff shape is ad-dressed. The discussion is enriched with particular examples of modeling such energies usingshearlets and wavelets. Basic examples of the above applications are illustrated with numericalexperiment data; current/future research directions are outlined.

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Palindromic Matrices of Order Two and Three-Point Subdivision Schemes

Serge Dubucdubucs@dms.umontreal.ca

Departement de mathematiques et de statistique,Universite de Montreal, Montreal, Quebec, Canada H3C 3J7

We introduce a family of three-point subdivision schemes related to palindromic pairs ofmatrices of order 2. We apply the Moßner Theorem on palindromic matrices to the C0

convergence of these subdivision schemes. We study the Holder regularity of their limitfunctions. The Holder exponent which is found in the regular case is optimal for most limitfunctions. In the singular case, the modulus of continuity of the limit functions is of orderδ log δ. These results can be used for studying the C1 convergence of the Merrien family ofHermite subdivision schemes.

Convolution Inequalities Associated with Irregular Gabor and Wavelet Systems

Jean-Pierre Gabardogabardo@mcmaster.ca

Department of Mathematics & Statistics, McMaster University1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada

We consider certain convolution inequalities for translation-bounded, positive measures onRn and the a x+ b group. We show how such inequalities are related to the notion of Beurlingdensity in Euclidean spaces and how they can be used to define a corresponding notion ofdensity for the a x + b group. Applications of these results to irregular Gabor and waveletsystems will be given.

Interpolatory Subdivision Schemes with Fractal Limit Curves

Mpfareleni R. Gavhirejoya@sun.ac.za

Department of Mathematical Sciences (Mathematics Division),Stellenbosch University, Private Bag X1, 7602 Matieland, South Africa

We consider a one-parameter class of interpolatory subdivision schemes, and provide pa-rameter intervals on which subdivision convergence is guaranteed. It is then shown that, forparameter values inside certain sub-intervals of the convergence intervals, fractal subdivisioncurves are obtained. Graphical illustration are provided.

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Ambiguity Functions, Sampling and Uncertainty Principlesin Parameter Estimation

Say Song Gohmatgohss@nus.edu.sg

Department of Mathematics, National University of Singapore, Singapore 119260

The standard approach for joint estimation of time delay and Doppler shift of a signal is toestimate the point at which the cross ambiguity function of the original and modified signalsattains its maximum modulus. We shall present a fast and accurate method on band-limitedsignals for this parameter estimation problem. The method acts on approximated signalsobtained from discrete samples and given by truncated Shannon series. It uses Newton’smethod to estimate the time delay and Doppler shift by calculating a point at which the crossambiguity function of the approximated signals attains its maximum modulus. Numerical ex-periments demonstrated that the method generally outperformed other methods in estimatingboth time delay and Doppler shift. We shall also discuss extension of the ideas and setup toa Hilbert space context, and introduce the notion of a generalized cross ambiguity function.This extension unifies various problems of interest, including the joint estimation of time de-lay and time scale in wideband signal processing. Under the abstract generalization, errorbounds for estimating the parameters are derived, and we will reveal a connection betweenthese bounds and a new type of uncertainty principle. The estimation error is shown to liein an ellipsoidal region and the uncertainty principle gives an upper bound to the size of theregion. This is based on joint work with Tim N. T. Goodman and Fuchun Shang.

Sparse Evolution of Functions Using Diffusion Wavelets

Kavita Goyal∗ and Mani Mehrakavita.iit09@gmail.com, mmehra@maths.iitd.ac.in

Department of Mathematics, Indian Institute of Technology DelhiHauz Khas New Delhi-110 016, India

Its almost been 20 years since we are using wavelet theory for numerical solutions ofPDEs. It is well known fact that many nonlinear PDEs arising in real world have solutionscontaining local phenomena (e.g. formation of shocks) and interaction between several scales(e.g. Turbulence). Such solutions can often be well represented in wavelet spaces because ofthe nice properties of wavelets like compact support, vanishing moments etc. Traditionally,wavelet construction is in unbounded or periodic domains but its application for solving partialdifferential equations (PDEs) on general manifold is still in infancy stage. Diffusion wavelets(invented by Coifman et al. at Yale university) introduced the multiresolution geometricconstruction for the efficient computation of high powers of local diffusion operators, whichhave high powers with low numerical rank. Classes of operators satisfying these conditionsinclude discretization of differential operators, in any dimension, on manifolds, and in non-homogeneous media. The construction yields multiresolution scaling functions and waveletson domains, manifolds, graphs and other general classes of metric spaces. The aim of thispaper is to obtain sparse representations of functions which is essential for compression andis a first step towards solving PDEs using diffusion wavelet.

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An Alternative Approach to Designing Clinical Trials:Budgeted Learning of Effective Classifiers

R. Greinerrgreiner@ualberta.ca

Department of Computing ScienceUniversity of Alberta, Edmonton, Alberta, Canada T6G 2E8

Researchers often use clinical trials to collect the data needed to evaluate some hypothesis,or produce a classifier. During this process, they have to pay the cost of performing each test.Many studies will run a comprehensive battery of tests on each subject, for as many subjectsas their budget will allow – ie, ”round robin” (RR). We consider a more general model, wherethe researcher can sequentially decide which single test to perform on which specific individual;again subject to spending only the available funds. Our goal here is to use these funds mosteffectively, to collect the data that allows us to learn the most accurate classifier.

We first explore the simplified ”em coins version” of this task. After observing that this isNP-hard, we consider a range of heuristic algorithms, both standard and novel, and observethat our ”biased robin” approach is both efficient and much more effective than most otherapproaches, including the standard RR approach. We then apply these ideas to learning anaive-bayes classifier, and see similar behavior. Finally, we consider the most realistic model,where both the researcher gathering data to build the classifier, and the user (eg, physician)applying this classifier to an instance (patient) must pay for the features used — eg, theresearcher has $10,000 to acquire the feature values needed to produce an optimal $30/patientclassifier. Again, we see that our novel approaches are almost always much more effective thatthe standard RR model.

See http://webdocs.cs.ualberta.ca/ greiner/RESEARCH/BudgetedLearning/.This is jointwork with Aloak Kapoor, Dan Lizotte and Omid Madani.

An Empirical Feature-based Learning Algorithm Producing Sparse Approximations

Xin Guoxinguo2student.cityu.edu.hk

Department of Mathematics City University of Hong Kong83 Tat Chee Avenue Kowloon, Hong Kong, P. R. China

A learning algorithm for regression is studied. It is a modified kernel projection machinein the form of a least square regularization scheme with `1-regularizer in a data dependenthypothesis space based on empirical features (constructed by a reproducing kernel and thelearning data). The algorithm has three advantages. First, it does not involve any optimizationprocess. Second, it produces sparse representations with respect to empirical features undera mild condition, without assuming sparsity in terms of any basis or system. Third, theoutput function converges to the regression function in the reproducing kernel Hilbert spaceat a satisfactory rate. Our error analysis does not require any sparsity assumption about theunderlying regression function.

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High-Performance Time-Warped Multiscale Signal Reconstruction

Matthew Hamilton∗ and Pierre Boulangerhamilton@cs.ualberta.ca, pierreb@cs.ualberta.ca

Department of Computing Science, University of Alberta, Edmonton, Alberta, Canada

Multiscale methods are becoming increasingly interesting as computational scientists lookto model interesting physical phenomenon that span vast ranges of spatiotemporal scales.Often, these technique model inherently-connected phenomenon at separate scales in order toachieve a computationally more efficient solution than solving the full fine scale problem. Inother cases such as turbulent flows, self-similarity across scales is exploited to produce morecomputationally-efficient models.

At the core of these methods is some notion of scale which facilitates representation ofranges of spatiotemporal scales. Typical notions of scale used in simulations are based on theassumption of periodic functions or spatial homogeneity of solutions, represented in terms ofsignal-independent basis functions (Fourier, wavelets, etc.). In contrast, time-warped signalprocessing uses the signal itself in order to guide basis construction. This allows for a signal-dependent notion of scale that has nice properties with regards to nonstationary and aperiodicsignals.

In this talk, we explore applying the time-warping idea to the Fourier basis and experi-ment with performance and quality concerns with regards to multiscale reconstruction. Weshow that signals can be reconstructed using time-warped convolution techniques with higherquality than classical convolution reconstruction, with a only a relatively small computationalpenalty. Moreover, we explore how standard Fourier-based signal tools can be interpreted andcomputed in the time-warped context and analyze the high-performance computing proper-ties. Potential applications of this framework include multiscale visualization and simulationas well as various computational photography and image processing problems.

Frequency-based Directional Framelets

Bin Hanbhan@ualberta.ca

Department of Mathematical and Statistical SciencesUniversity of Alberta, Edmonton, Alberta, Canada T6G 2G1

In this talk, we discuss some recent theoretical developments on wavelets and framelets.First, we introduce the notion of frequency-based basic framelets. Next, we provide a com-plete characterization for frequency-based basic framelets and discuss their connections towavelet/framelet filter banks. We shall see that there is a one-one correspondence between awavelet/framelet filter bank and a frequency-based basic framelet. We provide examples ofbivariate directional framelets. Some connections to shearlets and curvelets will be discussed.Related papers are available at http://www.ualberta.ca/∼bhan

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Wavelets Centered on a Knot Sequence

B. Atkinson, D. Bruff, J. S. Geronimo, and D. P. Hardin∗

doug.hardin@vanderbilt.edu

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

We develop a general notion of orthogonal wavelets ‘centered’ on an irregular knot sequenceand present two families of orthogonal wavelets that are continuous and piecewise polynomial.As an application, we construct piecewise quadratic, orthogonal wavelet bases on the quasi-crystal lattice consisting of the τ -integers where τ is the golden-mean.

Numerical Techniques for Approximating Fourier Integral Operators

Brock Hargreaves∗ and Michael P. Lamoureuxbehargre@ucalgary.ca

Department of Mathematics and Statistics,University of Calgary, Calgary, Alberta, Canada T2N 1N4

Here we discuss recent and well known methods for approximating Fourier Integral Op-erators and a subset thereof called Pseudo-Differential Operators. Recent theoretical andcomputational work has resulted in methods which are accurate, robust, and fast. GaborMultipliers can exactly represent linear operators, and these computational methods can beused to approximate them. The butterfly algorithm has been adapted to be massively paralleland to handle general amplitude and phase functions. A brief discussion of these methodsis given, followed by applications to medical and seismic imaging, and solving linear partialdifferential equations.

Modulation Spaces, BMO, BLT, and Rectangular Partial Sums

Christopher Heilheil@math.gatech.edu

School of Mathematics, Georgia Tech, Atlanta, GA 30332-0160, USA

The modulation spaces quantify the time-frequency concentration of functions and distri-butions. We relate the modulation spaces, the space BMO of functions with bounded meanoscillation, and the Balian–Low Theorem. The embedding of modulation spaces into VMO(the space of functions of vanishing mean oscillation) is seen to lie behind the essential lim-itation of the time-frequency localization of Gabor systems that form Riesz bases. We alsoprove that a type of Balian–Low Theorem holds for Gabor Schauder bases, which raises inter-esting questions about the convergence of rectangular partial sums of Fourier series in higherdimensions.

This talk is based on joint work with Alex Powell (Vanderbilt University) and RamazanTinaztepe (Alabama A&M University).

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Wavelet packets and wavelet frame packets on local fields

Biswaranjan Behera and Qaiser Jahan∗

biswa@isical.ac.in, qaiser r@isical.ac.in

Theoretical Statistics and Mathematics Unit,Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India

Using a prime element of a local field K of positive characteristic p, the concepts ofmultiresolution analysis (MRA) and wavelet can be generalized to such a field. We prove aversion of the splitting lemma for this setup and using this lemma we have constructed thewavelet packets associated with such MRAs. We show that these wavelet packets generatean orthonormal basis by translations only. We also prove an analogue of splitting lemma forframes and construct the wavelet frame packets in this setting.

Non-negative Subdivision and Non-homogeneous Markov Chains

K. JetterKurt.Jetter@uni-hohenheim.de

Institut fur Angewandte Mathematik und StatistikUniversitat Hohenheim, D-70593 Stuttgart, Germany

Recent work by X. L. Zhou, see [4] and the references there, has settled a long-standingquestion of characterizing convergence of non-negative, univariate subdivision schemes. Werelate some of these results to methods used in the analysis of non-homogeneous Markovchains. In particular, the convergence result in [1] is a strong and so far less known basictheorem, from which convergence of non-negative subdivision can be characterized.

We will develop the main ideas and proofs following this approach through properties ofstochastic matrices, and of products of families of such matrices. In particular, the notion ofSIA matrices (coined by Wolfowitz [3]) is useful and applicable to multivariate non-negativesubdivision when the mask is finitely supported. In this case, an ’ergodic coefficient’ asintroduced by Hajnal [2] leads to a contraction argument for families of SIA matrices as theyappear in subdivision when looking at the corresponding submasks of the subdivision scheme.This is joint work with my PhD student Xianjun Li.

References:[1] J. M. Anthonisse and H. Tijms, Exponential convergence of products of stochastic matrices,J. Math. Anal.Appl. 59 (1977), 360–364.[2] J. Hajnal, Weak ergodicity in non-homogeneous Markov chains, Proc. Cambridge Phil. Soc.54 (1958), 233–246.[3] J. Wolfowitz, Products of indecomposable, aperiodic, stochastic matrices, Proc. Amer. Math. Soc.14 (1958), 733–737.[4] X.-L. Zhou, Positivity of refinable functions defined by nonnegative masks, Appl. Com-put. Harmonic Analysis 27 (2009), 133–156.

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Explicit Iteration Schemes of the Uzawa Algorithm forMinimization Problems Arising from Image Processing

Rong-Qing Jiarjia@ualberta.ca

Department of Mathematical and Statistical Sciences, University of Alberta, Canada

Recently the Uzawa algorithm has been widely used in image processing and compressivesampling. In this talk we indicate that many apparently different algorithms can be viewedas variations of the Uzawa algorithm. We discuss convergence of the Uzawa algorithm. Inparticular, we propose explicit iteration schemes based on matrix splitting. When the matrixsplitting is done by the symmetric Gauss-Seidel method, we establish convergence of thescheme with no restriction on the step size of the iteration.

Bi-frames with 6-fold Axial Symmetry for Hexagonal Dataand Triangle Surface Multiresolution Processing

Qingtang Jiangjiangq@umsl.edu

Department of Mathematics and Computer ScienceUniversity of Missouri–St. Louis, St. Louis, MO 63121, USA

In this talk we will discuss the construction of highly symmetric affine bi-frames for hexag-onal data/image and triangle surface multiresolution processing. Compared with the conven-tionally used square lattice, the hexagonal lattice has several advantages, including that ithas higher symmetry. It is desirable that the filter banks for hexagonal data also have highsymmetry which is pertinent to the symmetric structure of the hexagonal lattice. While inthe field of CAGD, when the filter banks are used for surface multiresolution processing, itis required that the corresponding decomposition and reconstruction algorithms for regularvertices have high symmetry so that these algorithms could be used to process surfaces withextraordinary vertices.

In this talk, we will show that the 6-fold axial symmetry is the desired symmetry which thefilter banks and bi-frames should possess when they are used for hexagonal data and trianglesurface multiresolution processing. We will also discuss the construction of bi-frames withsuch a symmetry. The multiresolution algorithms of the constructed b-frames are given bytemplates (stencils) so that it is easy to implement them for surface processing applicationssuch as surface progressive transmission, sparse representation and surface shrinkage.

Multidimensional A/D Conversion and Directional Bias

David Jimenez∗, Demetrio Labate and Manos Papadakisdjimenez@math.uh.edu

Department of Mathematics, University of Houston, Houston, Texas 77204-3008, USA

Digital Filters for d-dimensional signals are generated as the fourier transforms of square-integrable Zd-periodic functions. Depending on the variability of the decay rate of the filter’sFourier transform, which we call directional bias of the filter, the reconstruction errors of animput signal f may vary. This variation also depends on the rotations of the imput image.We study the effects on the truncation error EN(f) = infI{‖f − σI(f)‖ : I ⊂ Zd, |I| ≤ N} ofthe directional distribution of the decay of φ, where σI(f) =

∑n∈I〈f, τnφ〉τnφ, with τng(x) =

g(x− n).

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Ideas from Engineering of Signals Interacting withApproximation in Mathematics (Wavelets and More)

Palle E. T. Jorgensenpalle-jorgensen@uiowa.edu

Department of Mathematics, University of Iowa, USA.

We sketch how ideas are from engineering of signals (e.g., frequency bands) recently havingseen a renascence in computational harmonic analysis. They involve such mathematical toolsas operator theory, representation theory, infinite-dimensional groups, spectral analysis, andapproximation theory.

We sketch the use of recursive input-output models, and filters, adapted in mathematicswhere this has not been common fare until recently. One of our motivations is the desire toextend and refine these methods as they are used in wavelet algorithms, or more generally inmulti-scale resolutions.

In the simplest cases, this takes the form of operator valued functions (generating function)of a complex variable. And in many applications, it is possible to encode data as vectors ina Hilbert space H, and to do this in such a way that a finite selection of bands will thencorrespond to a cascading system of closed subspaces in H.

An Introduction to Synchrosqueezed Transforms

Michael Joyajoya@cs.ualberta.ca

Department of Computing ScienceUniversity of Alberta, Canada T6G 2E8

This talk reviews some recent developments in empirical harmonic analysis (Daubechies etal. 2011) which propose a “synchrosqueezed” approach for sharpening the resolution of a time-frequency representation R(t, f) by considering the behavior of its local gradient. We thenexamine variations on this approach and show results which demonstrate its effectiveness atresolving temporally co-located heart valve closure sounds collected by electronic stethoscope.

Numerical Approximations and Stability Analysis of 1D Heat Equation

Md. Kamrujjamanmkamrujj@ucalgary.ca

Department of Mathematics and Statistics,University of Calgary, Calgary, AB, Canada

Abstract: The present investigation deals with the solution of the 1D diffusion equations.We choose a thin, laterally insulated bar in which heat is constrained to flow along the barsaxis and no heat can enter or leave through the lateral surface. A solution of the initial-boundary value problem is obtained against along the x-axis on the entire surface of the bar[0, L]. Following methods are used to solve the equation such as (i) Analytically (ii) Crank-Nicholson Implicit Method (C-N) (iii) Explicit Forward-Difference Method (FTCS) and (iv)Implicit Backward-Difference Method (BTCS). The results are shown graphically as well astabulated form in different methods. Finally, Von-Neumann stability analysis has consideredwhether the methods are conditionally or unconditionally stable.

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On Construction of Symmetric MRA-based Frame-like Wavelet Systems

Aleksandr Krivosheinsan san@inbox.ru

Department of Applied Mathematics and Control Processes,St. Petersburg State University, St. Petersburg, Russia

Let M be a matrix dilation, m = | det M |. Mixed Extension Principle is a well knowngeneral scheme for the construction of dual wavelet frames. However, generally speaking,this scheme leads to MRA-based dual wavelet systems {ψ(ν)

ik }, {ψ(ν)ik } which are not necessary

frames in L2(Rd) and may even consist of tempered distributions. Some properties of these

systems such as frame-type expansion (with convergence in different senses) and their approx-imation order were investigated in [1]. The wavelet systems satisfying these properties werecalled frame-like.

For an arbitrary matrix dilation, any integer n and any integer/semi-integer s, we give anexplicit method for the construction of frame-like wavelet systems providing approximationorder n such that all wavelet functions {ψ(ν)}, {ψ(ν)} are symmetric/antisymmetric with re-spect to the point S = (M − I)−1s. For some class of 2 × 2 dilation matrices, given integern, we construct frame-like wavelet systems with approximation order n and axis symme-try/antisymmetry of all wavelet functions.This research was supported by Grant 09-01-00162 of RFBR.[1] Krivoshein A., Skopina M., Approximation by frame-like wavelet systems, accepted toACHA, available online.

Image Inpainting and Sparse Approximation

Emily King, Gitta Kutyniok∗, and Xiaosheng Zhuang

eking@math.umd.edu, kutyniok@uos.de, and xzhuang@uos.de

Institute for Mathematics, University of OsnabruckAlbrechtstrasse 28a, 49069 Osnabruck, Germany

One main problem in data processing is the reconstruction of missing data. In the situationof image data, this task is typically termed image inpainting. Recently, inspiring algorithmsusing sparse approximations and `1 minimization have been developed and have, for instance,been applied to seismic images. The main idea is to carefully select a representation sys-tem which sparsely approximates the governing features of the original image – curvilinearstructures in case of seismic data. The algorithm then computes an image, which coincideswith the known part of the corrupted image, by minimizing the `1 norm of the representationcoefficients.

In this talk, we will develop a mathematical framework to analyze why these algorithmssucceed and how accurate inpainting can be achieved. We will first present a general the-oretical approach. Then we will focus on the situation of images governed by curvilinearstructures, in which case shearlets will serve as the chosen representation system. Using thepreviously developed general theory and methods from microlocal analysis, under certain con-ditions on the size of the missing parts we will prove that such images can be arbitrarily wellreconstructed.

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Appell Sequences from Scaling Functions

S. L. Leematleesl@nus.edu.sg

Department of Mathematics, National University of SingaporeS17, 10 Kent Ridge Road, Singapore 119076

We consider Appell sequences of polynomials defined by generating functions

exz

φ(iz)=

∞∑m=0

Pm(x)

m!zm,

where φ is a scaling function. The Appel sequences exhibit some properties reminiscent ofHermite polynomials, which are generated by exz

G(iz), where G is the Gaussian function. For a

class of compactly supported sequences of scaling functions that approximate the Gaussianthe corresponding Appell sequences of polynomials converge to the Hermite polynomials. TheAppell sequence corresponding to a scaling function provides a representation for the recoveryof functions from their wavelet transforms with derivatives of the scaling function as themother wavelets.

Some Results on Low-Rank Matrix Recovery

Song Lisongli@zju.edu.cn

Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, P. R. China

In this talk, I shall investigate low-rank matrix recovery problem. Some characterizationconditions are obtained to guarantee that a matrix with rank at most r can be recoveredexactly by nuclear norm minimization.

Weighted `1 Minimization: Support Recovery Guaranteesand an Iterative Algorithm

Hassan Mansourhassanm@cs.ubc.ca

Mathematics and Computer Science Departments,University of British Columbia, Vancouver, BC, Canada V6T 1Z4

We study the support recovery conditions of weighted `1 minimization for signal recon-struction from compressed sensing measurements given multiple support estimate sets withdifferent accuracy. We also propose a sparse recovery algorithm based on iterative weighted`1 minimization that utilizes multiple weighting sets. We demonstrate experimentally thatthe proposed algorithm succeeds in recovering sparse signals when standard `1 minimizationfails. Moreover, we demonstrate through extensive experiments that the performance of theproposed algorithm is similar to that of the iterative re-weighted `1 algorithm proposed byCandes, Wakin, and Boyd. Finally, we identify signal classes for which, given a partial supportestimate, weighted `1 minimization is guaranteed to result in a new support estimate withimproved accuracy.

Joint work with Ozgur Yilmaz and Michael Friedlander.

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Some New Bounds of the R.I.C. in Compressed Sensing

Qun Mo∗, Song Li, Yi Shenmoqun@zju.edu.cn

Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China

The restricted isometry constant (R.I.C.) is important in compressed sensing. In this talk,I shall report our recent results on some new bounds of the R.I.C. First we shall give a newbound δ2s < 0.4931 as a sufficient condition for the `1-minimization problem to have an s-sparse solution. Next, we shall give a new bound δs+1 < 1/(

√s + 1) as a sufficient condition

for the orthogonal matching pursuit algorithm to recover each s-sparse signal in s iterations.Also, we shall use an example to show that bound is very tight. This example positivelyverifies the conjecture given by Dai and Milenkovic in 2009. These results are joint workswith Song Li and Yi Shen.

Bivariate Empirical Mode Decomposition (BEMD) Based Data AdaptiveApproach to EOG Suppression from EEG Signals

Md. Khademul Islam Mollamdkhadem@ualberta.ca

Department of Physics, University of Alberta, Edmonton, Alberta, Canada

A problem of eye-movement muscular interference removal from electroencephalogram(EEG) recordings is described. In many experiments in neuroscience it is crucial to separatedifferent sources of electrical activity within human body in a situation when a very limitedknowledge about nonlinear and nonstationary nature of the mixing process is available. Anew two step extension to bivariate empirical mode decomposition (BEMD) is proposed toremove ocular artifacts from EEG. The fractional Gaussian noise (fGn) is used as a referencefirst to preprocess electrooculogram (EOG) signal, which is next used in the second step as areference to clean EEG signals. Results with EEG experimental data validate the proposedapproach.

System description: This paper focuses on removal of eye movement related artifacts, whichcarry significant power in from of EOG contaminating much lower in power EEG. To tacklethese problems, the current study proposes to use BEMD, a new technique to decomposepairs of signals for which one is introduced as a reference. The aim of this paper is tointroduce a novel computational framework based on BEMD, convenient for simultaneous dataconditioning and information separation for neurophysiological signals with known interferencesources, in particular, to separate eye movements from EEG signals.

BEMD: The Empirical Mode Decomposition (EMD) is a signal processing decompositiontechnique that decomposes the signal into waveforms modulated in both amplitude and fre-quency by extracting all of the oscillatory modes embedded in the signal. The waveformsextracted by EMD are named Intrinsic Mode Functions (IMF). Each IMF is symmetric andit is assumed to yield a meaningful local frequency traces. Different IMFs do not exhibitthe same frequency at the same time. The traditional EMD is only suitable for univariatesignals. The Complex Empirical Mode Decomposition (Complex-EMD) is an extension ofthe basic EMD suitable for dealing with complex signal representations. The motivation toextend EMD was that a large number of signal processing applications have complex signalrepresentation. In addition, this extension could be applied on both the real and imaginaryparts simultaneously because complex signals have a mutual dependence between the real andimaginary parts. With separate decomposition, the mutual dependency is lost. The BEMD

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is a more generalized extension of the EMD to complex signals. It decomposes two variables(EEG and fGn) simultaneously based one their rotating properties.

Conclusion: The data-driven adaptive method proposed in this paper allowed us to sep-arate successfully EOG interference from multiple channel EEG recordings where a mixingmodel was not trivial. The two steps procedure allowed us first to filter out EOG from interfer-ing electrical and in general environmental noise which was modeled with fractional Gaussiannoise in form of a reference for first stage BEMD. Such a ”purified” EOG was later used as areference in a second stage BEMD pairwise with all EEG channels separately preventing anydata leakage or crosstalk among them, which is of high importance in biomedical applications.

Multiple Kernel Learning for Scale-Space based Feature Selection

Sharmin Nilufar∗, Nilanjan Ray and Hong Zhang

{sharmin, nray1, zhang}@cs.ualberta.caDepartment of Computing Science

University of Alberta, Edmonton, Alberta, Canada T6G 2E8

Scale-space representation of an image is a significant way to generate features for clas-sification. To select only the useful scales in the image scale-space, a framework of MultipleKernel Learning (MKL) is also proposed in the problem of large lump detection from oilsand. Presence of an undetected large lump can jam crushers and cause undesired productiondowntime in oil sands mining. Towards detecting large lumps by an automated image anal-ysis method, a novel data-dependent kernel function based on image scale-space features isdesigned. We utilize a 1-norm support vector machine (SVM) in the MKL optimization prob-lem for sparse selection of scales from the image scale-space. The optimized data-dependentkernel accommodates only a few scales that are most discriminatory according to the largemargin principle. With a 2-norm SVM this learned kernel is applied to detect large lumps inoil sand videos. We tested our method on three challenging oil sand data sets. Our methodyields encouraging results on these difficult-to-process images.

On Stationary and Nonstationary Biorthogonal Wavelets

Igor Novikovigor.nvkv@gmail.com

Department of Mathematics,Voronezh State University, Voronezh, Russia 394036

Basis and approximation properties of stationary and nonstationary biorthogonal com-pactly supported wavelets in different function spaces are considered. In particular station-ary biorthogonal compactly supported wavelets preserving localization with the growth ofsmoothness is examined. Two examples of infinitely differentiable nonstationary biorthogonalcompactly supported wavelets are investigated also.

Partially supported by RFBR Grant 11-01-00614-a.

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Constructing Parametric Surfaces

Jorg Petersjorg@cise.ufl.edu

Dept C.I.S.E., CSE BldgUniversity of Florida, Gainesville, FL 32611-6120, USA

Choice of atlas, in the form of patch layout, and change of variables, in the form ofreparameterization across patches, make the construction of parametric free-form surfaces fordesign an interesting and challenging problem even before determining the shape. This talkwill focus on recent insights into preferred patch layout, least reparameterizations, least degreeof spline representations and progress in design with rational splines.

Sparse Data Representation by the Easy Path Wavelet TransformGerlind Plonka

plonka@math.uni-goettingen.de

Institute for Numerical and Applied Mathematics,University of Gottingen, Germany

In the talk, recent methods for adaptive data representation will be summarized with afocus on the Easy Path Wavelet Transform (EPWT) and its generalizations. The EPWT hasrecently been proposed as a tool for sparse representations of bivariate functions from discretedata, in particular from image data. The EPWT is a locally adaptive wavelet transform. Itworks along pathways through the array of function values, and it exploits the local correlationsof the given data in a simple appropriate manner. In particular, we show that the EPWTleads, for a suitable choice of the pathways, to optimal N -term approximations for piecewiseHolder smooth functions with singularities along curves.

The results have been obtained jointly with Armin Iske (Hamburg), Daniela Rosca (Cluj-Napoca), and Stefanie Tenorth (Gottingen).

Root-Exponential Accuracy for Coarse Quantizationof Finite Frame Expansions

Rayan Saabrayans@math.duke.edu

Department of MathematicsDuke University, Durham, NC 27708, USA

In this talk, we show that by quantizing the N -dimensional frame coefficients of signalsin Rd using higher order Sigma-Delta quantization schemes, it is possible to achieve root-exponential accuracy in the oversampling rate λ := N/d. In particular, we construct a familyof finite frames tailored specifically for Sigma-Delta quantization that admit themselves asboth canonical duals and Sobolev duals. Our construction allows for error guarantees thatbehave as e−c

√λ, where under a mild restriction on the oversampling rate, the constants are

absolute. Moreover, we show that harmonic frames can be used to achieve the same guarantees,but with the constants now depending on d and that random frames achieve similar, albeitslightly weaker, guarantees (with high probability). Time permitting, we will discuss stabilityto non-quantization noise.

This is joint work with, in part with F. Krahmer and R. Ward, and in part with S. Gunturkand O. Yılmaz.

19

Convex Sparse Coding, Subspace Learning, and Semi-Supervised Extensions

Dale Schuurmansdale@cs.ualberta.ca

Department of Computing ScienceUniversity of Alberta, Canada

Automated feature discovery is a fundamental problem in data analysis. Although classicalfeature discovery methods do not guarantee optimal solutions in general, it has been recentlynoted that certain subspace learning and sparse coding problems can be solved efficiently,provided the number of features is not restricted beforehand. I will discuss an extendedcharacterization of this optimality result and describe the nature of the solutions under anexpanded set of practical conditions. In particular, I will demonstrate how the frameworkcan be applied to ”semi-supervised” prediction, and demonstrate that feature discovery canco-occur with input reconstruction and supervised training while admitting globally optimalsolutions. A comparison to existing semi- supervised feature discovery methods shows im-proved generalization and efficiency.

Joint work with Xinhua Zhang, Yaoliang Yu, Martha White and Ruitong Huang.

Sparse Signal Representation and the Tunable Q-factor Wavelet Transform

Ivan Selesnickselesi@poly.edu

Polytechnic Institute of New York UniversityElectrical and Computer Engineering

6 Metrotech Center, Brooklyn, NY 1121, USA

For the sparse wavelet representation of a signal, the Q-factor of the wavelet transformshould be chosen so as to match the signal’s oscillatory behavior. This talk describes a newwavelet transform, the ‘tunable Q-factor wavelet transform’ (TQWT), for which the Q-factoris continuously tunable. Therefore, the wavelet can be chosen according to the oscillatorybehavior of the signal, so as to enhance the sparsity of a sparse representation. The TQWT iswell suited for iterative algorithms for sparse representation as it is a fully-discrete tight framewhich can be efficiently implemented using radix-2 FFTs. Sparse TQWT representationsobtained by `1-norm minimization will be shown. [The Q-factor of a waveform is defined asthe ratio of its center frequency to its bandwidth.]

Automatic Removal of Ocular Artifacts from EEG Signals Using S-transform

Kedarnath Senapati∗, Sibsambhu Kar and Aurobinda Routray

kedar d2@yahoo.co.in, sibsambhukar@gmail.com, aroutray@ee.iitkgp.ernet.in∗Institute of Mathematics and Applications,

Bhubaneswar, 751 003, India

Department of Electrical Engineering,Indian Institute of Technology, Kharagpur, 721 302, India

Artifacts in electroencephalograph (EEG) signals are unwanted but unavoidable distur-bances in the recording process due to several reasons such as eye movements, body move-ments, power line interference etc. Among them the ocular artifacts are most significant.These are characterized by high amplitude but have overlapping frequency band with the use-ful signal. Hence, it is difficult to remove the ocular artifacts by traditional filtering methods.

20

This paper proposes a new approach of artifact removal using S -transform (ST). It providesan instantaneous time-frequency representation of a time-varying signal and generates highmagnitude S -coefficients at the instances of abrupt changes in the signal. A threshold func-tion has been defined in S -domain to detect the artifact zone in the signal. The artifact hasbeen attenuated by a suitable multiplying factor. The major advantage of ST-filtering is thatthe artifacts may be removed within a narrow time-window, while preserving the frequencyinformation at all other time points. It also preserves the absolutely referenced phase infor-mation of the signal after the removal of artifacts. Finally, a comparative study with wavelettransform (WT) demonstrates the effectiveness of the proposed approach.

MRA Based Wavelet Frame and Applications

Zuowei Shenmatzuows@nus.edu.sg

Department of MathematicsNational University of Singapore, Singapore 119076

One of the major driving forces in the area of applied and computational harmonic analysisduring the last two decades is the development and the analysis of redundant systems thatproduce sparse approximations for classes of functions of interest. Such redundant systemsinclude wavelet frames, ridgelets, curvelets and shearlets, to name a few. This talk focuses ontight wavelet frames that are derived from multiresolution analysis and their applications inimaging.

The pillar of this theory is the unitary extension principle and its various generalizations,hence we will first give a brief survey on the development of extension principles.

The extension principles allow for systematic constructions of wavelet frames that can betailored to, and effectively used in, various problems in imaging science. We will discuss someof these applications of wavelet frames. The discussion will include frame-based image analysisand restorations, image inpainting, image denosing, image deblurring and blind deblurring,image decomposition, segmentation and CT image reconstruction.

Concentration Estimates for Learning with `1-Regularizerand Data Dependent Hypothesis Spaces

Lei Shi∗ and Ding-Xuan Zhouleishi@cityu.edu.hk

Department of Mathematics, City University of Hong Kong, Hong Kong, P. R. China

We consider the regression problem by learning with a regularization scheme in a datadependent hypothesis space and `1-regularizer. The data dependence nature of the kernel-based hypothesis space provides flexibility for the learning algorithm. The regularizationscheme is essentially different from the standard one in a reproducing kernel Hilbert space:the kernel is not necessarily symmetric or positive semi-definite and the regularizer is the `1-norm of a function expansion involving samples. The differences lead to additional difficultyin the error analysis. We apply concentration techniques with `2-empirical covering numbersto improve the learning rates for the algorithm. Sparsity of the algorithm is studied based onour error analysis. We also show that a function space involved in the error analysis inducedby the `1-regularizer and non-symmetric kernel has nice behaviors in terms of the `2-empiricalcovering numbers of its unit ball.

21

Resolution of the Gibbs Phenomenon

Bernie Shizgalshizgal@chem.ubc.ca

Department of Chemistry, University of British Columbia2036 Main Mall, Vancouver, British Columbia V6T1Z1, Canada

The expansion of an analytic nonperiodic function on a finite interval in a Fourier seriesleads to spurious oscillations at the interval boundaries referred to as the Gibbs phenomenon.The present paper summarizes a new method for the resolution of the Gibbs phenomenon[1] which follows on the reconstruction method of Gottlieb and coworkers [2,3] based onGegenbauer polynomials orthogonal with respect to weight function (1 − x2)λ−1/2. We referto their approach as the direct method and to the new methodology as the inverse method.Both methods use the finite set of Fourier coefficients of some given function as input data inthe re-expansion of the function in Gegenbauer polynomials or in other orthogonal basis sets.The finite partial sum of the new expansion provides a spectrally accurate approximation tothe function [4]. In the direct method, this requires that certain conditions are met concerningthe parameter λ in the weight function, the number of Fourier coefficients, N and the numberof Gegenbauer polynomials, m. We show that the new inverse method can give exact resultsfor polynomials independent of λ and with m = N . The paper presents several numericalexamples applied to a single domain or to subdomains of the main domain so as to illustratethe inverse method in comparison with the direct method.

[1] B. D. Shizgal and J.-H. Jung, Towards the resolution of the Gibbs phenomena, J. Com-put. Appl. Math. 161, 41-65 (2003).

[2] D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon III: recovering exponential ac-curacy in a sub-interval from the spectral sum of piecewise analytic function, SIAM J.Numer. Anal. 33, 280-290 (1996).

[3] S. Gottlieb, J.-H. Jung and S. Kim, A Review of David Gottlieb’s Work on the Resolutionof the Gibbs Phenomenon, Commun. Comput. Phys. 9, 497-519 (2011).

[4] B. D. Shizgal and J.-H. Jung, On the numerical convergence with the inverse polynomialreconstruction method for the resolution of the Gibbs phenomenon, J. Comput. Phys.224, 477-488 (2007).

A Note about Non Stationary Multiresolution Analysis

Francoise Bastin and Laurent Simons∗

F.Bastin@ulg.ac.be, L.Simons@ulg.ac.be

University of Liege, Institute of Mathematics, B-4000 Liege, Belgium

An orthonormal basis of wavelets of L2(R) is an orthonormal basis of L2(R) of type

ψj,k = 2j/2ψ(2j · −k), j, k ∈ Z.

A classical method to obtain such bases consists in constructing a multiresolution analysis.When the mother wavelet ψ depends on the scale (i.e. the index j), a non stationary version ofmultiresolution analysis is then used. We generalize different characterizations of orthonormalbases of wavelets to the non stationary case (as main reference for the stationary case, weused results presented in “A First Course of Wavelets” of E. Hernandez and G. Weiss).

22

Approximation by Wavelet and Scaling Expansions

Maria Skopinaskopina@MS1167.spb.edu

Department of Applied Mathematics and Control Processes, St.PetersburgState University, Universitetskii pr.-35, St.Petersburg 198504, Russia

We study MRA-based compactly supported wavelets which are redundant representationsystems but not frames. Frame-type expansions with respect to such wavelet systems (withconvergence in different senses) and their approximation order are investigated. The advantageof these systems is in the simplicity of their construction. The wavelet functions shouldnot have vanishing moments, which is necessary for frames. Starting with any appropriatescaling function or scaling mask one can find a dual scaling mask and all wavelet masks byexplicit formulas for matrix extension. Also we investigate scaling expansions for band-limitedfunctions, in particular, the classical sampling theorem is extended to a wide class of functions.

Scales in Vision and Immunology

Steve Smalesmale@cityu.edu.hk, smale@math.berkeley.edu

Department of Mathematics, University of CaliforniaBerkeley, CA 94720-384, USA

and

Department of Mathematics, City University of Hong Kong83 Tat Chee Avenue, Kowloon Tong, Hong Kong

We will suggest an alternate perspective on multiscale computing with the goal of con-structing a “good kernel”.

Factoring Banded Matrices and Matrix Polynomials

Gilbert Stranggilstrang@gmail.com

Department of Mathematics, MIT77 Mass. Ave., Rm. 2-240, Cambridge, MA 02139-4307, USA

Banded matrices with banded inverses can be factored into tridiagonal matrices with tridi-agonal inverses. These matrices are rare but useful – wavelet matrices and ”CMV matrices”are leading examples. (The analysis and synthesis filters are all FIR, even for time-varyingfilters.) The number of tridiagonal factors depends on the bandwidth and not on the matrixsize.

When those matrices are block Toeplitz, with submatrices repeating down each (block)diagonal, all the information is in the matrix polynomial with those submatrices as coefficients.Suppose its determinant is 1. Then we look for linear factors with det = 1. This is a starton factoring more general doubly infinite matrices. I will look at the ordinary A = LU (orA = LPU) factorization for which the usual elimination process has no reasonable place tostart.

23

Wiener’s Lemma and Two Nonlinear Sampling Problemsin Signal Processing

Qiyu Sunqiyu.sun@ucf.edu

Department of MathematicsUniversity of Central Florida, Orlando, FL 32816, USA

Wiener’s lemma occurs in many fields of mathematics and engineering, such as functionalanalysis, harmonic analysis, signal processing, and numerical analysis. In this talk, I willdiscuss Wiener’s lemma for infinite matrices, and consider its nonlinear extension and appli-cations to nonlinear sampling problems in signal processing.

Wavelets with Crystal Symmetry Shifts

Josh MacArthur and Keith F. Taylor∗

kft@mathstat.dal.ca

Department of Mathematics & StatisticsDalhousie University, Halifax, NS, Canada B3H 3J5

We introduce the basics for a theory of wavelets on Rn where the usual lattice of shifts isreplaced by a more general noncommutative group and the dilation matrix must be compatiblewith the shift group. Our motivation is to provide a context which includes much of theinteresting work done recently on wavelets with composite dilations, but also admits thosecrystal symmetry groups of shifts which cannot be covered by the composite dilation theory.We include a number of examples on R2.

On the Different Kinds of Quasi-interpolation

Zongmin Wuzmwu@fudan.edu.cn

School of Mathematics, Fudan University, P. R. China

To analyze the data stemming from the application, one should find the underlying functionvia mathematical method such as interpolation or approximation. From the theory of theapproximation, we know that the choice of the basis should be data dependent. The quasi-interpolation is a typical method to find the simple basis as well as the approximant toapproximate the underlying function. The classical quasi-interpolation are almost all basedon the data of function values on grid. In this talk we will introduce the quasi-interpolationfor scattered date as well as for linear functional data. Then the quasi-interpolation can beused in the numerical solution of PDEs and different kinds of data analysis.

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Noise-shaping Quantizers for Compressed Sensing

Ozgur Yilmazoyilmaz@math.ubc.ca

Department of MathematicsUniversity of British Columbia, Vancouver, BC V6T 1Z2, Canada

Recent advances have established compressed sensing as an effective sampling theory foracquisition of high dimensional sparse vectors from few linear measurements. The vast major-ity of the results in the literature assume that the compressive samples are real numbers, anda comprehensive quantization theory for compressed sensing has been missing. In this talk, weshow that we can successfully employ noise-shaping Sigma-Delta quantizers for compressedsensing. We prove that, by adopting a two-stage approach involving the use of appropri-ate Sobolev dual frames in the reconstruction, Sigma-Delta quantizers utilize the inherentredundancy of compressed sensing more efficiently than ”any” round-off type quantization al-gorithm, at least in the case of Gaussian measurement matrices. This framework is especiallysuccessful if the underlying signals are exactly sparse. We will then consider the cases whenthere is measurement noise (in addition to the quantization error) and when the signal to beacquired is not strictly sparse, but compressible. We will introduce alternative reconstructionmethods that effectively handle this more general setup.

This is joint work with R. Saab, S. Gunturk, and in part with M. Lammers and A. Powell.

A Survey of Subdivision Algorithms of Manifold-Valued Data

Tom Duchamp and Gang Xie and Thomas Yu∗

yut@drexel.edu

Department of Mathematics,Drexel University, Philadelphia, USA

In this talk, I will first survey a number of different ways to adapt a given linear subdivisionrule to manifold-valued data. Of particular interest is the so-called single base-point scheme,which shows up in the construction of wavelet-like transform for manifold-valued data. I shallpresent a number of smoothness and approximation order results of these various subdivisionschemes.

Mathematical Modeling and Multilevel Computation ofDispersed Drug Release from Swellable and Erodible

Polymeric Matrix Systems

Dmitry Pelinovsky and Wei Zhao∗

dmpeli@math.mcmaster.ca, wzhao@math.mcmaster.ca

Department of Mathematics, McMaster University,1280 main street west, Hamilton, Ontario, Canada ON L8S 4K1, Canada

Many drugs possessing low to mediate water solubility would experience gradual dissolutionin modified release matrix systems. Thus drug release kinetics may be controlled by bothdissolution and diffusion. A mathematical model of drug release from swellable and erodiblematrix systems with initial drug loading higher than solubility was developed. The model wasverified by the existing exact solution with the assumption of dissolution much faster thandiffusion as a special case. Multilevel methods were introduced to solve the governing systemof diffusion equations in order to achieve better approximation with lower computational costs.

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Interpolation of Missing Values in Times Series Based on its Periodicity

Jin-Xin Zhangzhjinx@mail.sysu.edu.cn

Department of Medical Statistics and EpidemiologySchool of Public Health, Guangzhou, Guangdong,510080, P. R. China

In this talk, a method to deal with missing data in times series is introduced. The ap-plication of information from frequency domain to the estimation of missing values is theguideline of this method. The actual data and simulation data are used to evaluate the effi-ciency of filling, and to explore the necessary conditions of the process. The research involvestime-domain information extraction, frequency-domain information extraction, simulation,periodicity-weighted method, and evaluation to effectiveness of filling. We found that, in ran-domly missing pattern, periodicity method and spline method are both good, and the values ofRMSE and NRMSE are both small; while in continuously missing pattern, periodicity methodis more accurate than spline method.

Error Analysis and Sparsity of Some Learning Algorithms

Ding-Xuan Zhoumazhou@cityu.edu.hk

Department of Mathematics, City University of Hong Kong, Hong Kong

Learning theory studies learning function relations or data structures from samples. In thistalk we shall first introduce mathematical analysis of some learning algorithms generated byregularization schemes in reproducing kernel Hilbert spaces. Then we shall discuss two classesof kernel-based learning algorithms which produce sparse approximations for regression. Thefirst class is of kernel projection machine type and generated by least squares regularizationschemes with `q-regularizer (0 < q ≤ 1) in a data dependent hypothesis space based onempirical features (constructed by reproducing kernels and samples). The second class isspectral algorithms associated with high-pass filter functions. Learning rates and sparsityestimations will be provided based on properties of the kernel, the regression function, andthe probability measure.

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Iterative Filtering, EMD and Instantaneous Frequency Analysis

Hao-Min Zhouhmzhou@math.gatech.edu

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

The empirical mode decomposition (EMD) was a method pioneered by Huang et al as analternative technique to the traditional Fourier and wavelet techniques for studying signals.It decomposes a signal into several components called intrinsic mode functions (IMF), whichhave shown to admit better behaved instantaneous frequencies via Hilbert transforms. In thistalk we present our recent progress on an alternative algorithm for EMD based on iteratingcertain filters.

This approach yields similar results as the more traditional sifting algorithm for EMD.In some cases the convergence can be rigorously proved. The method is highly data depen-dent. It is designed to complement the classical Fourier transform or wavelets to better treatnon-stationary, nonlinear processes commonly used in practice. It returns adaptive sparserepresentations of original signals. Each component in the representations reflects certainintrinsic properties at any given moment. It has been demonstrated through many applica-tions that such type of algorithms compute effectively structured decompositions that provideoutstanding de-mixing ability to separate tangled signal components such as noise and back-ground information. (This presentation is based on collaborative work with Luan Lin (Brown),Jingfang Liu (Georgia Tech) and Yang Wang (Michigan State)).

The Fast Digital Shearlet Transform and Applicationsto Denoising and Coarse Quantization

Bernhard G. Bodmann, Gitta Kutyniok, Xiaosheng Zhuang∗

xzhuang@uos.de

Institut fur Mathematik, Universitat Osnabruck, Osnabruck, Germany

In signal processing, one of the primary goals is to obtain a [representation of the digital]signal of interest that is suitable for processing, storage, transmission, and recovery. Oneof the key features of a suitable representation system is redundancy, providing a represen-tation which is robust under noise, quantization, and data loss. Due to the fact that mosthigh-dimensional signals exhibit anisotropic features, directional representation systems likeshearlets and curvelets play a significant role, since they] have been proved to provide opti-mally sparse approximations of [such signals in contrast to] wavelet systems.

In this talk, we introduce and discuss a Fast Digital Shearlet Transform (FDST) which is afaithful digitization of the continuum domain shearlet transform. For this, we first introducepseudo-polar grids with oversampling along with the fast pseudo-polar Fourier transform.We show that there exist weightings of the pseudo-polar grid, which gives rise to an isometricpseudo-polar Fourier transform, allowing inversion by taking its adjoint. Based on this pseudo-polar Fourier transform, we shall provide a rationally designed FDST. We will prove that it isindeed the exact digitization of the shearlet transform in continuum domain, thereby showingthat shearlet theory provides a unified treatment of both the continuum and digital realms.In addition, we shall discuss the software package ShearLab that implements FDST alongsidewith various quantitative measures allowing one to tune parameters and objectively improvethe implementation as well as compare with other directional transform implementations.Finally, we will discuss several applications of the FDST, in particular, to denoising andcoarse quantization.

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List of Participants of AHAMC 2011 and Summer School 

 

Speakers of 1-hour Plenary Talks

John J. Benedetto University of Maryland, USA jjb@math.umd.edu

Charles K. Chui Stanford University and University of Missouri-St. Louis, USA

ckchui@stanford.edu

Zuowei Shen National University of Singapore,Singapore matzuows@nus.edu.sg

Steve Smale University of California, USA, and City University of Hong Kong, China

smale@cityu.edu.hk, smale@math.berkeley.edu

Gilbert Strang Massachusetts Inst. of Technology, USA gilstrang@gmail.com

Ding-Xuan Zhou City University of Hong Kong, China mazhou@cityu.edu.hk

Speakers of 40-minute Invited Talks Michael Adams University of Victoria, Canada mdadams@ece.uvic.ca

Amir Averbuch Tel Aviv University, Israel amir@math.tau.ac.il

Radu Balan University of Maryland, USA rvbalan@math.umd.edu

Gregory Beylkin University of Colorado at Boulder, USA Gregory.Beylkin@colorado.edu

Peter G. Binev University of South Carolina, USA binev@math.sc.edu

Marcin Bownik University of Oregon, USA mbownik@uoregon.edu

Elena Braverman University of Calgary, Canada maelena@math.ucalgary.ca

Feng Dai University of Alberta, Canada dfeng@math.ualberta.ca

Zeev Ditzian University of Alberta, Canada zditzian@math.ualberta.ca

Serge Dubuc University of Montreal, Canada dubucs@dms.umontreal.ca

Jean-Pierre Gabardo McMaster University, Canada gabardo@mcmaster.ca

Say Song Goh National University of Singapore,Singapore matgohss@nus.edu.sg

Russell Greiner University of Alberta, Canada rgreiner@ualberta.ca

Bin Han University of Alberta, Canada bhan@math.ualberta.ca

Douglas Hardin Vanderbilt University, USA doug.hardin@vanderbilt.edu

Christopher E. Heil Georgia Institute of Technology, USA heil@math.gatech.edu

Kurt Jetter University of Hohenheim, Germany Kurt.Jetter@uni-hohenheim.de

Rong-Qing Jia University of Alberta, Canada rjia@ualberta.ca

Qing-Tang Jiang University of Missouri-St. Louis, USA jiangq@umsl.edu

Palle Jorgensen University of Iowa, USA jorgen@math.uiowa.edu

Gitta Kutyniok University of Osnabrueck, Germany kutyniok@math.uni-osnabrueck.de

Seng Luan Lee National University of Singapore,Singapore matleesl@nus.edu.sg

Song Li Zhejiang University, China songli@zju.edu.cn

Qun Mo Zhejiang University, China moqun@zju.edu.cn

Igor Novikov Voronezh State University, Russia igor.nvkv@gmail.com

Jorg Peters University of Florida, USA jorg@cise.ufl.edu

Gerlind Plonka University of Goettingen, Germany plonka@math.uni-goettingen.de

Dale Schuurmans University of Alberta, Canada dale@cs.ualberta.ca

Ivan Selesnick Polytechnic University, USA selesi@poly.edu

Bernard Shizgal University of British Columbia, Canada shizgal@chem.ubc.ca

Maria A. Skopina St. Petersburg State University, Russia skopina@MS1167.spb.edu

Qiyu Sun University of Central Florida, USA qsun@mail.ucf.edu

Keith Taylor Dalhousie University, Canada kft@mathstat.dal.ca

Zongmin Wu Fudan University, China zmwu@fudan.edu.cn

Ozgur Yilmaz University of British Columbia, Canada oyilmaz@math.ubc.ca

Thomas P-Y Yu Drexel University, USA yut@drexel.edu

Hao-Min Zhou Georgina Institute of Technology, USA hmzhou@math.gatech.edu

Speakers of 20-minuted Invited Talks Md Kamrujjaman University of Calgary, Canada mkamrujj@ucalgary.ca

Hassan Mansour University of British Columbia, Canada hassanm@cs.ubc.ca

Rayan Saab Duke University, USA rayans@math.duke.edu

Wei Zhao McMaster University, Canada wzhao@math.mcmaster.ca

Xiaosheng Zhuang University of Osnabrueck, Germany xzhuang@uos.de

Speakers of 20-minute Contributed Talks Rasel Biswas Memorial University, Canada rb3247@mun.ca

Jostein Bratlie Narvik University College, Norway jostein.bratlie@gmail.com

Julia Dobrosotskaya University of Maryland, USA juliadob@math.umd.edu

Mpfareleni R. Gavhi University of Stellenbosch, South Africa rejoya@sun.ac.za

Kavita Goyal Indian Institute of Technology,India goyalkavita9@gmail.com

Xin Guo City University of Hong Kong,China xinguo2@student.cityu.edu.hk

Matthew Hamilton University of Alberta, Canada hamilton@cs.ualberta.ca

Brock Hargreaves University of Calgary, Canada behargre@ucalgary.ca

Qaiser Jahan Indian Statistical Institute, India qaiser_r@isical.ac.in

David Adrian Jimenez University of Houston, USA djimenez@math.uh.edu

Michael Joya University of Alberta, Canada joya@cs.ualberta.ca

Alexander Krivoshein St. Petersburg State University, Russia san_san@inbox.ru

Md. Khademul Islam Molla

University of Alberta, Canada mdkhadem@ualberta.ca

Sharmin Nilufar University of Alberta, Canada sharmin@ualberta.ca

Kedarnath Senapati Institute of Math and Applications, India kedar_d2@yahoo.co.in

Lei Shi City University of Hong Kong,China leishi@cityu.edu.hk

Laurent Simons University of Liege, Belgium L.Simons@ulg.ac.be

Jin Xin Zhang Sun Yat-Sen University, China zhjinx@mail.sysu.edu.cn

Participants Yasin Abbasi-Yadkori University of Alberta, Canada abbasiya@ualberta.ca

Gabor Bartok University of Alberta, Canada bartik@ualberta.ca

Meysam Bastani University of Alberta, Canada bastani@ualberta.ca

Catherine Beneteau University of South Florida, USA cbenetea@usf.edu

Ion Bica Grant MacEwan University, Canada bicai@macewan.ca

Pierre Boulanger University of Alberta, Canada pierreb@cs.ualberta.ca

Menglu Che University of Alberta, Canada mche1@ualberta.ca

Katherine Chen University of Alberta, Canada kchen4@ualberta.ca

Norman C. Corbett Okanagan College, Canada NCorbett@okanagan.bc.ca

Thomas Degris University of Alberta, Canada degris@ualberta.ca

Jian Deng University of Alberta, Canada jdeng@math.ualberta.ca

Thomas Duchamp University of Washington, USA duchamp@math.washington.edu

Ken Dwyer University of Alberta, Canada dwyer@cs.ualberta.ca

Koosha Golmohammadi University of Alberta, Canada golmoham@ualberta.ca

Kiana Hajebi University of Alberta, Canada hajebi@cs.ualberta.ca

Mohsen Hajiloo University of Alberta, Canada Hajiloo@ualberta.ca

Hua He University of Alberta, Canada hhe@cs.ualberta.ca

Shahin Jabbari University of Alberta, Canada sjabbari@cs.ualberta.ca

Amin Jorati University of Alberta, Canada jorati@cs.ualberta.ca

Mordi Raymond Ketuojo Lagos State University, Nigeria johnpatcompany@gmail.com

Shunjie Lau University of Alberta, Canada shunjie@ualberta.ca

Jian Jun Liu Chinese Academy of Comp. Math, China

Ashique R. Mahmood University of Alberta, Canada ashique@ualberta.ca

David McLaughlin Grant MacEwan University, Canada mclaughlind@macewan.ca

Peter D. Minev University of Alberta, Canada pminev@math.ualberta.ca

Farzaneh Mirzazadeh University of Alberta, Canada mirzazad@ualberta.ca

Chukwuemeka Ndubisi University of Aberdeen, United Kingdom c.kelly.ndubisi.10@aberdeen.ac.uk

Edmond C. Noumbisi Johannesburg City College, South Africa umbisiedmond@yahoo.com

Abiodun Oladimeji Lagos State University, Nigeria olonardabbey@yahoo.com

Bernardo Avila Pires University of Alberta, Canada bpires@ualberta.ca 

Sherman D. Riemenschneider

West Virginia University, USA sherm@math.wvu.edu

Lei Sha TerraNotes Ltd. GEOPHYSICS, Canada lei.terranotes@gmail.com

Yi Shi University of Alberta, Canada ys3@ualberta.ca

Csaba Szepesvari University of Alberta, Canada szepesva@ualberta.ca

Saman Vaisipour University of Alberta, Canada vaisipou@ualberta.ca

Heping Wang Capital Normal University, China wanghp@yahoo.cn

Martha White University of Alberta, Canada whitem@ualberta.ca

Yau Shu Wong University of Alberta, Canada yaushu.wong@ualberta.ca

Quan-Wu Xiao City University of Hong Kong, China qwxiao@cityu.edu.hk

Yaoliang Yu University of Alberta, Canada yaoliang@ualberta.ca

Zhenpeng Zhao University of Alberta, Canada zhenzhao@math.ualberta.ca

Navid Zolghadr University of Alberta, Canada zolghadr@ualberta.ca

   

UofA Map and Buildings: CAB (D5), ETL (D3), Faculty Club (A3), HUB (E7), SUB (E4), and MEC (C3). 

International Conference on

Applied H

armonic A

nalysis and Multiscale C

omputing

University of A

lberta, Edm

onton, Canada

July 25 – 28, 2011

Schedule of Summ

er School at ETLE 2-001:

o Friday 09:00-12:00 &

14:00-17:00, July 29, 2011: Learning theory by Steve Smale and D

ing-Xuan Zhou

o Saturday 09:00-12:00 &

14:00-17:00, July 30, 2011: MRA

based wavelet fram

e and applications by Zuowei Shen

o Sunday 09:00-11:30, July 31, 2011: A

lgorithms of w

avelets and framelets by Bin H

an

Computer facilities:

o CN

S computer room

at MEC 3-3 W

est (third floor, near elevator, neighboring building MEC)

o W

ireless access to UW

S (University W

ireless Service): use provided username and passw

ord

Places for lunch/breakfast: HU

B mall (near university bus station), SU

B (student union building), Lower M

ain floor of CAB.

Places for dinner: many restaurants in the dow

ntown area and a few

restaurants near the University of A

lberta.

Monday, July 25

Tuesday, July 26

Wedsday, July 27

Thursday, July 28

8:30-8:45 ETLE 1-013

Opening

8:45-9:45 ETLE 1-013

Steve Smale

9:45-10:30 Registration

Coffee/Tea Break

ETLE 1-013 ETLE 1-017

10:30-11:10 D

ubuc H

ardin 11:10-11:50

Jetter Jorgensen

11:50-1:45 Lunch Break

1:45-2:45 ETLE 1-013

Gilbert Strang

ETLE 1-013

ETLE 1-017 2:50-3:30

Peters Plonka

3:30-4:00 Coffee/Tea Break

4:00-4:40 Jiang

Selesnick 4:40-5:20

Adam

s Shizgal

5:30-9:30 W

elcome Reception

at Main Q

uad between CA

B and A

thabasca Hall

8:30-9:30 ETLE 1-013

Charles K. Chui

ETLE 1-013 ETLE 1-017

9:35-10:15 Jia

Taylor 10:15-10:40

Coffee/Tea Break

ETLE 1-013 ETLE 1-017

10:40-11:20 Beylkin

Kutyniok 11:20-11:40

Zhao Zhuang

11:40-12:00 Kam

rujja-m

an D

obrosotsk-aya

12:00-1:45 Lunch Break

1:45-2:45 ETLE 1-013

John J. Benedetto

ETLE 1-013 ETLE 1-017

2:50-3:30 A

verbuch H

eil 3:30-4:00

Coffee/Tea Break 4:00-4:40

Sun G

abardo 4:40-5:20

Goh

Greiner

5:20-5:40 H

amilton

Guo

8:30-9:30 ETLE 1-013

Ding-Xuan Zhou

ETLE 1-013

ETLE 1-017 9:35-10:15

Wu

Zhou 10:15-10:40

Coffee/Tea Break

ETLE 1-013 ETLE 1-017

10:40-11:20 Lee

Binev 11:20-11:40

Saab Jahan

11:40-12:00 M

ansour G

oyal

12:00-1:45 Lunch Break

1:45-2:45 ETLE 1-013

Zuowei Shen

ETLE 1-013

ETLE 1-017 2:50-3:30

Yilmaz

Skopina 3:30-4:00

Coffee/Tea Break 4:00-4:40

Li Bow

nik 4:40-5:20

Han

Balan

6:00-9:30 Conference Banquet

at Faculty Club

ETLE 1-013

ETLE 1-017 8:30-9:10

Braverman

Ditzian

9:10-9:50 N

ovikov D

ai 9:50-10:10

Simons

Biswas

10:10-10:40 Coffee/Tea Break

ETLE 1-013

ETLE 1-017 10:40-11:20

Schuurmans

Yu 11:20-11:40

Nilufar

Krivoshein 11:40-12:00

Shi Jim

enez

12:00-1:45 Lunch Break

ETLE 1-013

ETLE 1-017 1:45-2:05

Bratlie Joya

2:05-2:25 G

avhi Senapati

2:25-2:45 H

argreaves M

olla 2:45-3:05

Mo

Zhang 3:05-3:25

Conference Ends