Volume 1:
This book is comprehensive in scope and one of the most well-written technical books in existence. In the preface the author states 'I love to write', and considering the exceptional quality of this book, this indeed shows through.
The first part of the book covers the human visual system, the understanding of which is fundamental to designing effective computer graphics. Several interesting topics are discussed, including Mach bands, color opponency, perceptual color matching, MacAdam ellipses, RGB color space, and gamut mapping.
The second part covers more technical matters, namely that of signal processing. The mathematical background assumed of the reader increases dramatically in this part; some exposure to elementary calculus and differential equations would suffice. The author does a good job of explaining such concepts as linear operators and the Dirac bracket notation. The pictorial representation he gives of the convolution operation is very helpful. In addition, Fourier analysis is presented at a level that makes it very clear exactly what is happening to signals, both discrete and continuous, when taking the Fourier transform. The Fast Fourier transform is not discussed however, dissapointingly. Suprisingly, a whole chapter is devoted to wavelet transforms, a topic usually not included at this level. Wavelets are used as a tool to deal with nonstationary signals. Usually discussed at a very abstract level, the presentation here is crystal clear and vey intutive, and the reader will take away a deeper appreciation of these objects than what could have been obtained from the usual presentations.
Chapter 7 is one of the most important in the book for it covers Monte Carlo techniques for evaluating the integrals that arise in image processing. The speed of convergance of Monte Carlo is addressed, along with how to estimate confidence levels when the parent distribution is normal. The author presents five different ways of doing 'blind' Monte Carlo, including rejection, blind stratified, weighted, and quasi Monte Carlo. Quasi Monte Carlo has taken on particular importance in recent years wherever Monte Carlo techniques are used. The author also presents four different ways of doing 'informed' Monte Carlo, i.e. when some information about the signal is known.
Uniform sampling of continuous signals is done in the next chapter. After discussing an example of sampling and reconstruction, the author outlines in detail the mathematical theory behind the uniform sampling and reconstruction of one-and two-dimensional signals. The chapter ends with a discussion of a technique to reduce aliasing artifacts called supersampling.
The next chapter covers nonuniform sampling and reconstruction. Naturally this is more complicated from a mathematical standpoint, due to the role of stochastic processes, but the author does a good job of discussing the relevant concepts. Most interesting is his treatment of the duality between aliasing and noise.
Chapter 10 surveys some of the more modern and practical techniques used for sampling and reconstruction of two-dimensional signals. Uniform sampling is discussed in terms of rectangular and hexagonal lattices; nonuniform sampling in terms of Poisson sampling and N-books sampling. Pseudocode is given for the decreasing radius algorithm. The concept of a refinement test is introduced and broken down into five categories, each of which is discussed in detail. The refinement test allows one to decide when more samples are needed in a neighborhood, and refinement geometry indicates where the samples are to be placed. Refinement geometry is discussed in this chapter also, with linear and area bisection techniques outlined, along with multiple-level and tree-based sampling. Techniques for interpolation and reconstruction, such as warping are also treated, and the author gives brief overviews of one-dimensional and two-dimensional sampling theorems. Numerous other methods, going by several different names are also discussed.
A very large set of references is given at the end of the book, covering a wide variety of topics in computer graphics and mathematical formalism. I have not read the second volume, but I am sure it respects the high quality of the first.
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