Mathematical Image Processing

The book aims to explain the mathematics of image processing. The natural objects in this approach are continuous images, in contrast with discrete digital images. Even if the tools are ultimately applied to digital images, the continuous point of view unifies concepts and allows the application of powerful, interesting, well-understood mathematics. A book on image processing with explicit emphasis on continuous images is very welcome to complement the classic books on digital image processing such as Digital Image Processing by Gonzalez and Woods. 

 

The book starts with the main tools of interpolation, filtering and convolutions, and morphology. It goes on to present multiscale methods, such as Fourier and wavelet transforms, and the application of partial differential equations and variational methods in image processing. The continuous point of view is natural and essential here. The mathematical tools are developed and discussed for performing the basic image processing tasks, such as denoising, deblurring, enhancing, detecting edges, and inpainting.

 

The preface says that the book targets "third-year students" and assumes knowledge of linear algebra, calculus, and numerical analysis, and perhaps the basics of real analysis and functional analysis.  My impression is that such students will find this book difficult to follow unless they already have a strong background in analysis or in image processing. Learning both subjects at the same time will hard.

 

Despite this, the book presents the mathematics very well, developing the mathematical tools are they are needed. Undergraduate students will need mathematical maturity to follow and appreciate all this. Chapter 2 is a brief summary of functional analysis and measure and integration; it disrupts the flow and would be better placed as an appendix. More advanced readers will find the book a solid reference.

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Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.