(October 3) Wojtek Czaja: The Balian--Low theorem for the symplectic form on R^2d - In this more general context we show that the choice of standard coordinates is not essential, and that one can work in any representation. In particular, we first prove the Balian--Low theorem for arbitrary quadratic forms. Then we generalize further and prove the Balian--Low theorem for differential operators associated with a symplectic basis for the symplectic form on R^2d.
(October 31) Shijun Zheng: Schrodinger operator Besov spaces, and Wavelet computations for thin film image processing - The sech^2 x potential is a celebrated object in soliton theory. In this talk We consider the Schroedinger operator on the real line, H=-d^2/dx^2+V, where V(x) is a negative hyperbolic sechant potential. Using biorthogonal dyadic system, we introduce Besov spaces and Triebel-Lizorkin spaces associated with H. Our approach is based on eigenfunction expansion method, where the eigenfunctions of H are chosen to be the solutions of the Lippman-Schwinger equation in scattering theory. We also prove a Mikhlin-Hormander type spectral operator theorem on these spaces, including the L^p boundedness result. As a brief overview of current developments in this area we shall compare the Hermite and Laguerre wavepacket case, as well as recent general spectral multiplier results based on heat kernel approach.
In the second part of my talk, I am going to give a short introduction of the conductivity model for our TAIP program (Thin Film Analog Imaging Processor). The TAIP device is a hybrid analog/digital device concept which promises to perform approximate image filtering operations very rapidly and with very low power consumption compared to standard digital processors. Our proposal is to provide efficient wavelet computational method of the solution of the anisotropic heat equation, which arises in the context of a static electromagnetic field for our model.
(November 7) Loukas Grafakos: The role of Gabor wavelets in the Carleson-Hunt theorem on almost everywhere convergence of the Fourier series. - A maximal dyadic sum operator which models the Carleson-Hunt operator using Gabor wavelets will be discussed. The proof of boundedness of this operator will be outlined and the role of the Gabor wavelets will be explained. An improved energy estimate for Bessel-type expressions associated with Gabor wavelets will be crucial in obtaining the boundedness of these operators.
(November 21) Darrin Speegle: Meyer type wavelet bases in R^n. - Meyer constructed a dyadic wavelet basis of functions on the line whose Fourier transforms are $C^\infty$ and supported in $[-8\pi/3, -2\pi/3] \cup [2\pi/3, 8\pi/3]$. This construction was generalized to rational dilations by Auscher; in particular to arbitrary integer dilations. In this talk, I will describe a sufficient geometric condition on expansive, integer dilations for there to exist these Meyer type wavelets in $R^n$. Using this condition, I will describe how one can show that for all expansive $2\times 2$ integer matrices, there exists Meyer type wavelet bases. Open problems relating to both the geometric condition on the matrices and the general existence question of Meyer type wavelet bases in $R^n$ will also be given.
(December 12) Joe Lakey: Uncertainty principle inequalities: progress and some problems - The uncertainty principle says that a function and its Fourier transform cannot both be arbitrarily well-localized. Different forms of this principle require different methods of proof. For example, the Heisenberg inequality just requires Plancherel's theorem and integration by parts -- real variable methods, while Hardy's theorem ultimately boils down to the maximum principle from complex analysis. Other versions still require `phase space' methods. This talk will be largely expository, pointing out relationships between various forms of the uncertainty principle and their proofs. Some open problems, including phase space versions and finite uncertainty principles will also be mentioned.
(February 20) Eitan Tadmor: On a new scale of regularity spaces for incompressible Euler equations. - We present a sharp local condition for the lack of concentration (and hence -- the $L^2$ convergence of) sequences of approximate solutions to the incompressible Euler equations. Our approach, based on an $H^{-1}$ stability condition, relies on using a generalized Div-Curl Lemma to replace the role that elliptic regularity theory has played previously in this problem. Our results identify the 'critical' regularity which prevent concentration, regularity which is quantified in terms of Lebesgue, Lorentz, Orlicz and Morrey spaces. In particular, the strong convergence criterion cast in terms of circulation logarithmic decay rates due to DiPerna \& Majda is simplified (-- removing the weak control of the vorticity at infinity) and extended (-- to any number of space dimensions).
Finally, we introduce a new scale of intermediate function spaces covering the gap between the weak $L^{p\infty}$ spaces and the larger Morrey spaces, and allowing us to make wavelet-based subtle distinctions in the $N$-dimensional borderline cases which separate between $H^{-1}$-compactness and the phenomena of concentration-cancelation. Expressed in terms of the new scale of spaces, these borderline cases are shown to be intimately related to uniform bounds of the coulomb energy and the configuration of the vorticity.
(April 10) Sinan Gunturk: How Accurate Can One-Bit Quantization Get? - One-bit quantization is a method of representing bounded signals by {+1,-1} sequences that are computed from regularly spaced samples of these signals; as the sampling density is increased, convolving these one-bit sequences with appropriately chosen averaging kernels must produce increasingly close approximations of the original signals. This method is widely used for analog-to-digital conversion of audio signals because of the many advantages its implementation presents over the classical and more familiar method of fine-resolution quantization. Despite its popular use, one-bit quantization is not well-understood in the approximation theoretical context. A fundamental open problem is the determination of the best possible behavior of the approximation error as a function of the sampling density for various function classes, and most importantly for the class of bandlimited functions, which is a model space for audio signals. Some of the other open problems ask for precise error bounds for particular popular one-bit quantization algorithms.
In this talk, we present the recent progress towards the solution of these problems, and the interplay of various types of mathematics in achieving these results. In particular, we give the first one-bit quantization algorithm that provides exponential accuracy for the class of bandlimited functions.
(April 18) Chan Woo Yang: Double Hilbert transforms along surfaces in $R^{2+n}$ - In this talk we will consider double Hilbert transforms $H$ along analytic surfaces defined by $$Hf(x,y,z) =\int_{|s|\le 1} \int_{|t|\le 1} f(x-t,y-t,z-P(s,t))\frac{ds dt}{st}$$ where $z=(z_1,...,z_n)$ and $P(s,t)=(P_1(s,t),...,P_n(s,t))$. $L^p$ boundedness of $H$ when $n=1$ and $P_1$ is a polynomial have been considered by Carbery, Wainger, and Wright in [1]. It turns out that $L^p$ boundedness of $H$ is related to the Newton Polygon of $P_1$. In this talk we will discuss about the extention of their result to the general dimension $2+n$. Necessary and sufficient conditions for $L^p$ boundedness of $H$ will be presented.
[1] A. Carbery, S. Wainger, and J. Wright, Double Hilbert transforms along polynomial surfaces in $R^3$, Duke Math. J. 101(2000), 499--513.
(April 24) Mauro Maggioni: Wavelet and Clustering Techniques for Analysis of Hyperspectral Data - We present some techniques for analysis of high-dimensional data based on wavelet packets, spectral graph theory and other statistical techniques for analyzing and clustering data in high-dimension. While the mathematical framework we will present is completely general, we will also consider an application of these techniques to hyperspectral imaging for a specific medical application.
(May 8) Dan Scholnik: Spatio-Temporal Delta-Sigma Modulation - Classical delta-sigma modulation gains resolution through temporal oversampling by using a high-speed, low-resolution quantizer and shaping the resulting quantization errors out of the signal band. Likewise, spatial delta-sigma modulation is often used in image halftoning to create high-visual-quality binary images. In this talk I will present an approach to joint spatial and temporal delta-sigma modulation uisng a general time-invariant, space-varying architecture. Although the primary application of interest is to reduce temporal oversampling requirements in high-SNR antenna transmit arrays, the technique also has potential application to halftoned movies on binary displays.
(September 18) Hamid Behmard: Sampling of Bandlimited Functions on Unions of Shifted Lattices - The classical sampling theorem permits reconstruction of a bandlimited function from its values on a set of equidistant points on the real line R. This theory has been extended in many directions. In one of these extensions, one considers sampling sets, which are unions of cosets of one subgroup, hence periodic sampling. We consider the case where the sampling set is a union of cosets of possibly different subgroups, hence noperiodic sampling. In our theory, a function f can be reconstructed from its samples provided the sampling set and the support of the Fourier transform of f satisfy certain compatibility conditions. New results show that these conditions can be relaxed. An explicit reconstruction formula is given for sampling sets which are unions of two shifted lattices. While explicit formulas for unions of more than two lattices are possible, it is more convenient to use a recursive algorithm. The analysis is presented in the general framework of locally compact abelian groups, but several specific examples are given on the real line. Examples in finite cyclic groups Z_N and Z_N X Z_N are presented and implemented in MATLAB.
(October 16) Glenn Easley: A New Class of Discrete Radon Transforms: Theory and Image Processing Applications - We introduce and study a new class of Radon transforms in a discrete setting for the purpose of applying them to the ridgelet and curvelet transforms. We give a detailed analysis of the p-adic case and provide a closed-form formula for an inverse of the p-adic Radon transform. We give conditions for a scaled version of the generalized discrete Radon transform to yield a tight frame, and discuss a direct Radon matrix method for the implementation of a local ridgelet transform. We then study the effectiveness of some types of the generalized Radon transforms in reducing a type of noise known as speckle that is present in synthetic aperture radar (SAR) imagery.
At the heart of the talk is the following. The Ridgelet transform is based on combining the Radon transform and the wavelet transform. There is currently some controversy on how to apply this transform in a computational setting for applications. The results we present give a generalized discrete Ridgelet transform that has as special cases examples given by others in the scientific community. Comparative results will be given to show how performances vary for different settings or different applications.
(February 5) Alfred S. Carasso: Singular integrals, image smoothness, and the recovery of texture in image deblurring - Total variation (TV) image deblurring is a PDE-based technique that preserves edges, but often eliminates vital small-scale information, or texture. This phenomenon re ects the fact that most natural images are not of bounded variation. The present paper reconsiders the image deblurring problem in Lipschitz spaces \Lambda(\alpha ; p; q), wherein a wide class of non-smooth images can be accomodated. A new and fast FFT-based deblurring method is developed that can recover texture in cases where TV deblurring fails completely. Singular integrals, such as the Poisson kernel, are used to create an e ective new image analysis tool that can calibrate the lack of smoothness in an image. It is found that a rich class of images in \Lambda(\alpha; 1; \infty) intersect \Lambda(\beta; 2; \infty) with 0.2<\alpha, \beta<0.7. The Poisson kernel is then used to regularize the deblurring problem by appropriately constraining its solutions in \Lambda(\alpha ; 2;\infty) spaces, leading to new L2 error bounds that substantially improve on the Tikhonov- Miller method. This so-called Poisson Singular Integral or PSI method is only one of an in nite variety of singular integral deblurring methods that can be constructed. The method is found to be well-behaved in both the L1 and L2 norms, producing results closely matching those obtained in the theoretically optimal, but practically unrealizable, case of true Wiener ltering. Deblurring experiments on synthetically defo- cused images illustrate the PSI method's very signi cant improvements over both the total variation and Tikhonov-Miller methods. In addi- tion, successful reconstructions with inexact prior Lipschitz space in- formation, highlight the robustness and practicality of the PSI method.
(February 26) Ramani Duraiswami: An Introduction to the Fast Multipole Method - The matrix vector product, or summation operation, lies at the core of most scientific computation. The product of a general dense $N\times N$ matrix with a $N$ vector requires $O(N^2)$ operations and $O(N^2)$ memory. One of the major achievements of computational harmonic analysis, the Fast Fourier Transform, reduced the complexity of this operation for uniformly sampled Fourier matrices to $O(N\log N)$ operations, and had significant scientific and industrial impacts. The fast multipole method, invented by Vladimir Rokhlin and Leslie Greengard in the late 1980s, allows the computation of the matrix vector product to a specified precision for a large class of dense matrices. Since its introduction it has had significant successes in the solution of integral equations of potential theory, of sums of special functions such as Gaussians, in nonuniform fast-Fourier transforms, and others.
In this talk we introduce the key ideas of the method, and describe some open questions and research areas in the method.
(March 4) David Brady: Spaces, bases and segmentation in radiation sensor systems - A radiation sensor system consists of:
an object-field mapping, a field propagation model, a field-data mapping, and a data analysis algorithm.
This talk overviews mappings, models and algorithms in conventional and emerging optical sensor systems and sensor networks and describes how non-local bases and multiplex codes may be used to improve data efficiency and feature specificity in sensor systems.
(March 18) Eitan Tadmor:
A multiscale image representation using
hierarchical $(BV,L^2)$
decompositions -
We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation of general images. The starting point is a variational decomposition of an image, f = u0+v0, where [u0,v0] is the minimizer of a J-functional, J(f,c0; X,Y)=inf{u+v=f} {||u||X + c0 ||v||Yp}. Such minimizers are standard tools for denoising, deblurring, compression, ... of images, e.g., [Mumford-Shah] and [Rudin-Osher-Fatemi]. Here, u0 should capture `essential features' of f, to be separated from the spurious components in v0, and c0 is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinement step [uj+1,vj+1] = arginf J(vj,c02j), leading to the hierarchical decomposition, f = ∑j=0k uj + vk. We focus our attention on the particular case of (X,Y)=(BV,L2) decomposition. The resulting hierarchical decomposition, f ~ ∑j uj, is essentially nonlinear. The questions of convergence, energy decomposition, localization and adaptivity are discussed. The decomposition is constructed by numerical solution of successive Euler-Lagrange equations. Numerical results illustrate applications to synthetic and real images (both grayscale and colored images).