An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and recovering than they do in any other orthogonal basis. In fact, simple thresholding in an unconditional basis works essentially better for recovery and estimation than other methods, period. (Performance is measured in an asymptotic minimax sense.) As an application, we formalize and prove Mallat's Heuristic, which says that wavelet bases are optimal for representing functions containing singularities, when there may be an arbitrary number of singularities, arbitrarily distributed.

Introduction Scientific data is often sampled or computed over a dense mesh in order to capture high frequency components or achieve a desired error bound. Interactive display and navigation of such large meshes is impeded by the sheer number of triangles required to sufficiently model highly complex data. A number of simplification techniques have been developed which reduce the number of triangles to a particular desired triangle count or until a particular error Preprint submitted to Elsevier Preprint 3 December 1997 threshold is met. Given an initial triangulation M of a domain D and a function F(x) defined over the triangulation, the simplified mesh can be called M 0 and the resulting function F 0 (x). The measure of error in a simplified mesh M i is usually represented as: ffl(M 0<F1

We describe segmented multiresolution analyses of [0; 1]. Such multiresolution analyses lead to segmented wavelet bases which are adapted to discontinuities, cusps, etc., at a given location 2 [0; 1]. Our approach emphasizes the idea of average-interpolation { synthesizing a smooth function on the line having prescribed boxcar averages. This particular approach leads to methods with subpixel resolution and to wavelet transforms with the advantage that, for a signal of length n, all n pixel-level segmented wavelet transforms can be computed simultaneously in a total time and space which are both O(n log(n)). We consider the search for a segmented wavelet basis which, among all such segmented bases, minimizes the \entropy " of the resulting coe cients. Fast access to all segmentations enables fast search for a best segmentation. When the \entropy " is Stein's Unbiased Risk Estimate, one obtains a new method of edge-preserving de-noising. When the \entropy " is the ` 2-energy, one obtains a new multi-resolution edge detector, which works not only for step discontinuities but also for cusp and higherorder discontinuities, and in a near-optimal fashion in the presence of noise. We describe an iterative approach, Segmentation Pursuit, for identifying edges by the fast segmentation algorithm and removing them from the data. 1 Key Words and Phrases. Segmented Multi-Resolution analysis. Edge-Preserving Image processing methods. Edge detection. Sub-pixel resolution.

Wavelets and multiresolution analysis are instrumental for developing efficient methods for representing, storing and manipulating functions at various levels of detail. Although alternative methods such as hierarchical quadtrees or pyramidal models have been used to that effect as well, wavelets have picked up increasing popularity in recent years due to their energy compactness, efficiency, and speed. Wavelet representations have achieved a great success in a wide variety of applications, including graphics, data compression, signal processing, physical simulation, hierarchical optimization, and numerical analysis, among others. This paper gives an overview of wavelets and their construction, as well as some applications to graphics and 3-D mesh processing at multiple levels of detail. The emphasis is on meshes of arbitrary topology and their multiresolution analysis by means of subdivision wavelets and their generalizations. In particular, both the traditional (first generation) wav...

this paper we will construct wavelets on a bounded interval. Multiple scaling functions are used, instead of a single one. In this paper, there are taken as as a finite sequence of polynomials which are orthogonal on a bounded interval. Following a similar way to that in the book [3], wavelets can be established formally. In practice, we can construct the wavelets by setting a fixed number of conditions. By setting vanishing moments equally for the scaling functions and wavelets, we can get fast wavelet approximation. Image compression can attain high compression rate by using 2-dimensional wavelets([4], [7], [10], [12], [13]). The last section fully explain how to obtain the 2D wavelet and then use them in image compression. Complexity of our method are given and compared with that of the well-known Haar wavelet. Lastly, examples are present to show how the image compression effect is. Notations that will be often used later are Inner product is defined as usual (f; g) =

We consider classes of smooth functions on [0; 1] with mean square norm. We present a wavelet-based method for obtaining approximate pointwise reconstruction of every function with nearly minimal cost without substantially increasing the amount of data stored. In more detail: each function f of a class is supplied with a binary code of minimal (up to a constant factor) length, where the minimal length equals the "-entropy of the class, " ? 0.