Books in this area tend to be dominated by theory, from Fourier analysis to wavelets. In its coverage of applications and different possibilities of multiresolution analysis in computer graphics, this book emphasizes the practical aspects of wavelets over theory.
The book includes an introduction and five parts. The first four parts, “Images,” “Curves,” “Surfaces,” and “Physical Simulation,” are divided into 13 chapters. The last part includes three appendices, “Linear Algebra Review,” “B-spline Wavelet Matrices,” and “Matlab Code for B-spline Wavelets.”
Chapter 1 is a useful introduction to wavelets. First, the difficulties of Fourier analysis are briefly mentioned. Next, the authors intuitively develop the idea of using wavelets for analysis. A succinct historical presentation, from the initial work of Weierstrass in 1873 to the multiresolution analysis of Meyer and Mallat, is given, but some important authors in this area, such as Daubechies and Chui, are not mentioned explicitly here. The authors explain that, unlike classical multiresolution analysis, which uses a shift-invariant theory, they present a shift-variant approach that is closely related to recursive subdivision. The chapter ends with an overview of the book.
Chapter 2 concerns the Haar wavelet basis. The authors give an example of a 1D image composed of four pixels. The image is decomposed in the Haar basis by recursively averaging and differencing coefficients called a filter bank. Some more abstract notions, such as vector space functions, are developed. Next, the orthogonality property of the Haar basis and normalization are explained. A pseudocode procedure accomplishing the decomposition of an array in Haar wavelets and one for the reconstruction of the original data are given. The last part of this chapter is concerned with wavelet compression. Unfortunately, a different example is used in this last part. Although this chapter is easy to read and contains appropriate figures, use of the same example for the entire chapter would have been preferable.
Chapter 3, “Image Compression,” begins with two different approaches for wavelet decomposition of a 2D image, each one generalizing the 1D wavelet transform described in the previous part. The authors then describe the standard and nonstandard constructions of 2D Haar basis functions, using an appropriate figure in each case. Next, they define wavelet image compression in three steps: computation of normalized 2D Haar basis coefficients, sorting them, and applying a threshold. The authors suggest using a binary search algorithm in order to improve the last two steps. Next, they briefly explain the compression of color images. In general, this chapter is insufficiently detailed and cannot be considered an introduction to this subject. The authors only mention the number of coefficients. Therefore, the effective recognition of the compression rate becomes difficult. The addition of an appendix recalling matrices and tensor products would have been helpful for readers unfamiliar with the subject.
Chapter 4 concerns an image editing and painting system developed in a previous paper. It begins with a presentation of the data structure used (a quadtree). Next, the three main goals (display, paint, and update) are described. A pseudocode for display is suggested. The parts covering paint and update will not be very clear to anyone unfamiliar with this system. Next, the boundary conditions (partially displayed image), display and editing at fractional resolution (using two types of workstations: with or without antialiased polygon drawing) and image editing examples are studied. Comprehending these examples using only fixed images is rather difficult.
Chapter 5, “Image Querying,” describes the use of a low-resolution image or a rough sketch painted by the user (a query) to search for an image (a target) in a large database. The principle consists of using wavelets to obtain a signature (the most significant information contained in the image). Next, the authors develop a metric for image querying using a wavelet approach. The important components of the metric (such as color space and wavelet type) are explained. After some simplifications, the metric is finally defined just as a weighted sum of the difference in the average color between the query and the target, and the number of matched coefficients. This approach is well illustrated by a simple and clear example. Unfortunately, the authors do not include enough information and detail from the previous paper on which this chapter is based, but the chapter is generally clear and well illustrated.
Chapter 6 develops the basic notions of subdivision for parametric curves. The general principle and Chikian’ subdivision algorithm are given. The relation with uniform quadratic B-splines is mentioned. Different masks are presented. Next, the principle of nonuniform subdivision is studied and the matrix form is given. In the later sections, the problems of evaluation masks, relations with vector spaces, and, finally, refinable scaling functions are described comprehensibly, with enough detail. In addition to introducing subdivision curves, this chapter can be considered a basis for the next.
Chapter 7, “The Theory of Multiresolution Analysis,” begins with theoretical details of shift-variant multiresolution analysis. Then orthogonal wavelets, in particular Daubechies wavelets and quadrature mirror filters, are studied. Semiorthogonal and biorthogonal wavelet constructions are also developed. The chapter ends with a summary using a table of basic function constructions and matrix constraints for these three kinds of wavelets. Readers familiar with this area will find the chapter rather simple but not really new. Readers unfamiliar with the subject will find the chapter difficult, and for them, a preliminary study of some references, some of which are mentioned by the authors, is highly recommended.
Chapter 8, on multiresolution curves, begins with a very brief survey of some work on the subject. Next, the problem of smoothing curves is studied, and the authors show that a linear interpolation allows a continuous variation with a representation in wavelets. Editing a curve, in particular in different levels of resolution; scan conversion; and curve compression are other problems studied in the next sections. This chapter is also based on a previous paper but is sufficiently detailed and is illustrated by several appropriate figures that are helpful in comprehending the subject.
Multiresolution tiling is the subject of chapter 9. First, the tiling problem (construction of the “best” surface between two contours) and previous works concerning this problem, as well as their limitations (essentially the computation time), are studied. Then the authors develop their multiresolution tiling approach, which can be summarized as using a wavelet decomposition to find low-resolution approximations of the contours and improving the tiling by local optimizations. This chapter is clear and presents an interesting application of wavelets.
Surface wavelets are studied in chapter 10. First, multiresolution analysis for surfaces is overviewed. The main difficulty is the design of four analysis and synthesis filters matching some special conditions. Then, subdivision surfaces (analogous with curves), selection of a particular inner product, and construction of biorthogonal surface wavelets (using the notion of lifting) are developed. Finally, multiresolution representation of surfaces using k-disc wavelets is studied. This chapter is clear and contains some useful figures and examples, especially for the construction of biorthogonal surface wavelets.
Chapter 11, “Surface Applications,” concerns the use of the k-disc surface wavelets developed in the previous chapter. The conversion of a polyhedral surface to the multiresolution form is studied. Next, surface compression, including texture map compression, is presented. Continuous level of detail control, wavelet presentation for progressive transmission, multiresolution editing, and some future directions for surface wavelets are the other parts of this short chapter that contains many color images as helpful examples.
Chapter 12 covers variational modeling. This approach consists of specifying an objective function and a few constraints. The computer then finds the “best” geometric primitive that matches the constraints. The authors explain the use of finite elements in variational modeling. They then show that variational modeling using wavelets can converge more rapidly on a solution. The chapter ends by presenting an adaptive approach that consists of beginning with a few rough functions and adding finer basis functions where they are needed.
Chapter 13 is about global illumination, an important and widely studied area in computer graphics. The authors recall the well-known radiosity problem. They suggest using Haar wavelets instead of box basis functions. A wavelet-based radiosity algorithm is described, and possible improvements are explained. Readers familiar with this subject will easily comprehend the chapter, but it is not sufficiently precise for readers unfamiliar with it.
In chapter 14, “Further Reading,” the authors very briefly survey some additional material associated with each part of the book.
Appendix A is a review of linear algebra basics. It is insufficient for novice readers and too elementary for those familiar with the material. The next two appendices, “B-spline Wavelet Matrices” and “Matlab Code for B-spline Wavelets,” are more useful.
The originality of this book lies in its emphasis on the practical aspects of wavelets over the theoretical. Unfortunately, it is an assembly of previous papers by the authors, without additional information and details. Some interesting topics, such as wavelets for fractals, are not mentioned. Still, even if some parts of the book seem more or less superficial, it can generally be considered a valuable reference for wavelet applications in computer graphics.
