Don Hong

Wavelet Image Compressor—MinImage

Nowadays, still images are used everywhere in the digital world. The shortages of storage capacity and transmission bandwidth make efficient compression solutions essential. A revolutionary mathematics tool, wavelet transform, has already shown its power in image processing. MinImage, the major topic of this paper, is an application that compresses still images by wavelets. MinImage is used to compress grayscale images and true color images. It implements the wavelet transform to code standard BMP image files to LET wavelet image files, which is defined in MinImage. The code is written in C++ on the Microsoft Windows NT platform. This paper illustrates the design and implementation details in Min-Image according to the image compression stages. First, the preprocessor generates the wavelet transform blocks. Second, the basic wavelet decomposition is applied to transform the image data to the wavelet coefficients. The discrete wavelet transforms are the kernel component of MinImage and are discussed in detail. The different wavelet transforms can be plugged in to extend the functionality of MinImage. The third step is the quantization. The standard scalar quantization algorithm and the optimized quantization algorithm, as well as the dequantization, are described. The last part of MinImage is the entropy-coding schema. The reordering of the coefficients based on the Peano Curve and the different entropy coding methods are discussed. This paper also gives the specification of the wavelet compression parameters adjusted by the end user. The interface, parameter specification, and analysis of MinImage are shown in the final appendix.

Surface compression using a space of C 1 cubic splines with a hierarchical basis

A method for compressing surfaces associated with C1 cubic splines defined on triangulated quadrangulations is described. The method makes use of hierarchical bases, and does not require the construction of wavelets.

Chapter 9 – Wavelets in Signal Processing

This chapter discusses the application of wavelets to signal processing and their use in transforming signals. The objectives of signal processing are to analyze accurately, code efficiently, transmit rapidly, and then to reconstruct carefully at the receiver the delicate oscillations or fluctuations of this function of time. This is important because all of the information contained in the signal is effectively present and hidden in the complicated arabesques appearing in its graphical representation. To accomplish these tasks, signals need to be transformed (coded/decoded) into a particular form for a certain task. Fast wavelet transformation provides effective algorithms for signal coding and decoding. Mathematically, a one-dimensional signal appears as a function of time. If the time variable is changed continuously, then the signal is called an analog signal or a continuous signal. If the time variable runs through a discrete set, then it is called a discrete signal, or a digital signal, which is the numerical representation of an analog signal. Changing an analog signal to a discrete signal is called discretization. A popular way to discretize an analog signal is by sampling.

Construction of wavelets and prewavelets over triangulations

Constructions of wavelets and prewavelets over triangulations with an emphasis of the continuous piecewise polynomial setting are discussed. Some recent results on piecewise linear prewavelets and orthogonal wavelets are presented.

Orthogonal multiwavelets of multiplicity four

Abstract We consider solutions of a system of refinement equations with a 4 × 1 function vector and three nonzero 4 × 4 coefficient matrices. We give explicit expressions of coefficient matrices such that the refinement function vector and the corresponding wavelet vector have properties of short support [0, 2], symmetry or antisymmetry, and orthogonality. The properties of convergence of the subdivision scheme, approximation order, and smoothness of the refinement functions are also discussed.

Algorithm for Optimal Triangulations in Scattered Data Representation and Implementation

Scattered data collected at sample points may be used to determine simple functions to best fit the data. An ideal choice for these simple functions is bivariate splines. Triangulation of the sample points creates partitions over which the bivariate splines may be defined. But the optimality of the approximation is dependent on the choice of triangulation. An algorithm, referred to as an Edge Swapping Algorithm, has been developed to transform an arbitrary triangulation of the sample points into an optimal triangulation for representation of the scattered data. A Matlab package has been completed that implements this algorithm for any triangulation on a given set of sample points.

An explicit local basis for C 1 cubic spline spaces over a triangulated quadrangulation

Let S31 (♦) be the bivariate C1-cubic spline space over a triangulated quadrangulation ♦. In this paper, an explicit representation of a locally supported basis of S31 (♦) is given using the interpolation conditions at vertices.

Optimal-order approximation by mixed three-directional spline elements

This paper is concerned with a study of approximation order and construction of locally supported elements for the space S14(Δ) of C1 quartic pp (piecewise polynomial) functions on a triangulation Δ of a connected polygonal domain Ω in R2. It is well known that, when Δ is a three-directional mesh Δ(1), the order of approximation of S14(Δ(1)) is only 4, not 5. Though a local Clough-Tocher refinement procedure of an arbitrary triangulation Δ yields the optimal (fifth) order of approximation from the space S14(Δ) (see [1]), it needs more data points in addition to the vertex set of the triangulation Δ. In this paper, we will introduce a particular mixed three-directional mesh Δ(3) and construct so-called mixed three-directional elements. We prove that the space S14(Δ(3)) achieves its optimal-order of approximation by constructing an interpolation scheme using mixed three-directional elements.

Some new formulations of smoothness conditions and conformality conditions for bivariate splines

Abstract This paper is concerned with a study of some new formulations of smoothness conditions and conformality conditions for multivariate splines in terms of B -net representation. In bivariate setting, a group of new parameters of bivariate cubic polynomials over a planar simplex is introduced, and smoothness conditions and conformality conditions of bivariate cubic C 1 splines are simplified.

Chapter 5 - Vector Spaces, Hilbert Spaces, and the L 2 Space

This chapter reviews the definitions and properties of finite dimensional vector spaces based on an algebraic approach. Isomorphism is a one-to-one and onto mapping between two mathematical entities, which preserves the structure of those entities (whether the structure is connectivity in a graph or the binary operation in a group). The structure in a vector space consists of scalar multiplication and vector addition, and that is why isomorphism is defined from this perspective. The Fundamental Theorem of Finite-Dimensional Vector Spaces postulates that an n -dimensional vector space over scalar field R is isomorphic to R n . Similarly, an n -dimensional vector space over scalar field C is isomorphic to C n . As opposed to writing any element of the vector space as a finite linear combination of basis vectors, it is desirable to write any element as a series of basis vectors. Hilbert spaces have much of the associated geometry of familiar vector spaces because they are endowed with an inner product. A vector space with a norm is a normed linear space. A normed linear space, which is complete with respect to the norm is a Banach space. That is, a Banach space is a normed linear space in which Cauchy sequences converge.