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A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

Daniel Cohen-Or, Chen Greif, Tao Ju, Niloy J. Mitra, Ariel Shamir, Olga Sorkine-Hornung, Hao (Richard) Zhang
Chapman & Hall/CRC
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
David S. Mazel
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Here is a useful book (the title is correct!) that covers a wealth of processing tools in a single volume accessible to experts and novices alike. Topics include analytic geometry, linear algebra, least squares solutions, principal component analysis and singular value decomposition, spectral transforms, solution of linear systems, and graphs and images, to name a few. All topics are applicable to image processing and most are applicable to other areas as well.

Each chapter is stand-alone, so the reader can learn the material quickly without the need to study earlier chapters. If the reader doesn’t know what he needs, the table of contents shows the reader what’s in each chapter and whether the material might apply to his problem. The text is lucid, easy to read, and follows a logical progression. Equations are explained and the authors provide ample figures and pictures to illustrate the concepts and discussions.

For example, the authors discuss how to determine whether two lines intersect in 3-space with a step-by-step development of the mathematics and resulting algorithm. Next, they show how to fit a line to data points in the least-squares sense and give pictorial examples in 2- and 3-space. The chapter on Least-Squares Solutions shows how to fit, say, a curve to data points from an edge of an image and how to weight the data points to compensate for outliers. The chapter on spectral transforms provides an excellent discussion of image compression techniques such as the Discrete Cosine Transform and Laplacian smoothing.

Incidentally, the book intuitively discusses how one can use Laplacian equations for image reconstruction and image manipulation. There is an example where most of the image is missing, but with an image completeness algorithm based on smoothness (that is, find the best image that is smooth relative to the known data) one obtains a fitted image quite close to the original. (Incidentally, it has always amazed me how much information one can discard from an image and yet still recover the original image almost exactly.)

What is more, the book discusses Poisson panoramic image stitching and shows a beautiful example of a city landscape before and after the processing. This tells part of the story of current digital cameras that allow one to take multiple pictures of a scene with each picture angularly offset from the other. The processed image is a smooth panoramic view that would be unobtainable otherwise. The stitching and smooth transition between images is both natural and unnoticeable.

On a slightly negative note, the chapter on topology is a bit weak. Of course, topology is a complicated and involved topic and while the gist of it can be explained succinctly, applications require more space than the authors had. Still, as a reference that at least mentions this topic the authors do an admirable job summarizing the details.

In conclusion, this book is a good collection of algorithms and tools for computer graphics. The topics are useful, well-presented, and collected in a single book for easy access. If you work in this area, you’ll find this book beneficial.

David S. Mazel is an engineer and lives just outside of Washington, DC. He welcomes your thoughts and feedback. He can be reached at mazeld at gmail dot com.

Analytical Geometry Olga Sorkine-Hornung and Daniel Cohen-Or

Linear Algebra Daniel Cohen-Or, Olga Sorkine-Hornung, and Chen Greif

Least Squares Solutions Niloy J. Mitra

PCA and SVD Olga Sorkine-Hornung

Spectral Transform Hao (Richard) Zhang

Solution of Linear Systems Chen Greif

Laplace and Poisson Daniel Cohen-Or and Gil Hoffer

Curvatures: A Differential Geometry Tool Niloy J. Mitra and Daniel Cohen-Or

Dimensionality Reduction Hao (Richard) Zhang and Daniel Cohen-Or

Scattered Data Interpolation Tao Ju

Topology: How Are Objects Connected? Niloy J. Mitra

Graphs and Images Ariel Shamir

Skewing Scheme Daniel Cohen-Or