Abstract. Several variations are given on the construction of orthonormal bases of wavelets with compact support. They have, respectively, more symmetry, more regularity, or more vanishing moments for the scaling function than the examples constructed in Daubechies [Comm. Pure Appl. Math., 41 (1988), pp. 909-996]. Key words, wavelets, orthonormal bases, regularity, symmetry AMS(MOS) subject classifications. 26A16, 26A18, 26A27, 39B12

We present a method for searching in an image database using a query image that is similar to the intended target. The query image may be a hand-drawn sketch or a (potentially low-quality) scan of the image to be retrieved. Our searching algorithm makes use of multiresolution wavelet decompositions of the query and database images. The coefficients of these decompositions are distilled into small "signatures" for each image. We introduce an "image querying metric" that operates on these signatures. This metric essentially compares how many significant wavelet coefficients the query has in common with potential targets. The metric includes parameters that can be tuned, using a statistical analysis, to accommodate the kinds of image distortions found in different types of image queries. The resulting algorithm is simple, requires very little storage overhead for the database of signatures, and is fast enough to be performed on a database of 20,000 images at interactive rates (on standard...

this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.

Radiosity methods have been shown to be an effective means to solve the global illumination problem in Lambertian diffuse environments. These methods approximate the radiosity integral equation by projecting the unknown radiosity function into a set of basis functions with limited support resulting in a set of n linear equations where n is the number of discrete elements in the scene. Classical radiosity methods required the evaluation of n 2 interaction coefficients. Efforts to reduce the number of required coefficients without compromising error bounds have focused on raising the order of the basis functions, meshing, accounting for discontinuities, and on developing hierarchical approaches, which have been shown to reduce the required interactions to O(n). In this paper we show that the hierarchical radiosity formulation is an instance of a more general set of methods based on wavelet theory. This general framework offers a unified view of both higher order element approaches to...

We describe a multiresolution curve representation, based on wavelets, that conveniently supports a variety of operations: smoothing a curve; editing the overall form of a curve while preserving its details; and approximating a curve within any given error tolerance for scan conversion. We present methods to support continuous levels of smoothing as well as direct manipulation of an arbitrary portion of the curve; the control points, as well as the discrete nature of the underlying hierarchical representation, can be hidden from the user. The multiresolution representation requires no extra storage beyond that of the original control points, and the algorithms using the representation are both simple and fast.

We present a generative model for isotropic bidirectional reflectance distribution functions (BRDFs) based on acquired reflectance data. Instead of using analytical reflectance models, we represent each BRDF as a dense set of measurements. This allows us to interpolate and extrapolate in the space of acquired BRDFs to create new BRDFs. We treat each acquired BRDF as a single high-dimensional vector taken from a space of all possible BRDFs. We apply both linear (subspace) and non-linear (manifold) dimensionality reduction tools in an effort to discover a lowerdimensional representation that characterizes our measurements. We let users define perceptually meaningful parametrization directions to navigate in the reduced-dimension BRDF space. On the low-dimensional manifold, movement along these directions produces novel but valid BRDFs.

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Abstract. This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so called N-term approximation), which would be obtained by keeping the N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N −s)in the energy norm, whenever such a rate is possible by N-term approximation. The range of s>0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to N. The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization. 1.

This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform [12, 10] in two and three dimensions. The first digital transformation is based on unequally-spaced fast Fourier transforms (USFFT) while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n 2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at

This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden