J. F. James. Cambridge University Press, UK, 2002.
ISBN 0-521-80826-X
Reviewed by: Pradeep Luther and Carlo Knupp, Imperial College, London, UK
Published in Microscopy & Analysis, November 200
For readers of this magazine, Fourier transforms are important because they are at the heart of the theory behind image formation, microscope design and function. They can also be extensively used to treat and analyse the collected images. Therefore, a good understanding of Fourier theory should be part of a microscopist’s forte. J. F. James introduces this essential topic in this short book. James’s book is addressed to undergraduate (physical sciences and engineering) students who need to learn about everyday applications of Fourier transforms without getting submerged in complex mathematics. It certainly fulfils this aim, but it will also be useful to researchers for reference and as a repository of useful equations.
The book starts with an introduction to one-dimensional Fourier series and transforms, with special attention to their application in the most widely used functions such as Top-hat, Gaussian, Delta functions and Dirac combs. The author’s enthusiasm for the subject is evident as he declares that the convolution theorem is the most astonishing result in Fourier theory. In Chapter 2, the mathematical basis is further developed through the discussion of convolution theorem, autocorrelation and other fundamental theorems, such as sampling theorem, aliasing and interpolation.
Following the first two theoretical chapters, the remainder of the book is devoted to the applications of Fourier transforms. Chapter 3 explains the application of Fourier theory to Fraunhofer diffraction. Chapter 4 is dedicated to signal analysis and communication theory where the concept of noise and filters are treated. Chapter 5 is devoted to spectroscopy and spectral line shapes, while Chapter 6 deals with the theory of 2D FTs with example applications on Fraunhofer diffraction in 2 dimensions.
Chapter 7 introduces multi-dimensional FTs and its applications in computerised axial tomography. 3D reconstruction from projections is an important tool in different fields of physics, like astronomy, medical imaging and electron tomography. The chapter has an excellent description of Radon transforms and the use of Fourier methods to recover the original image.
In Chapter 8 complex Fourier transforms are discussed. Finally, in Chapter 9, discrete and digital FTs are discussed. The theory of the Fast Fourier Transform algorithm is presented. For a more graphical illustration of this algorithm, the book by Brigham (The Fast Fourier Transform) is recommended. A BASIC program for calculating a 1D Fast Fourier transform is presented. However, this could be more useful presented as an algorithm, which a programmer can translate into a modern language.
The description of the Fourier synthesizer is one of the little gems tucked away in this book. For example, a top-hat function is generated in an oscilloscope as a finite sum of a series. As it is band limited, there are fine ripples at the edges. In 1898, Michelson and Stratton constructed a mechanical Fourier synthesizer comprising an assembly of 80 coupled gear-wheels. When it was used to generate a top-hat function, they observed ripples adjacent to the top-hat, and ascribed them to imperfections in the mechanical construction. It was Gibbs who showed that limited bandwidth caused this effect, now termed the Gibbs phenomenon.
On the whole, this is an excellent book to initiate students who possess a reasonable mathematical background to the use of Fourier transforms in certain physical and engineering applications. For further detail on this subject, the books listed in the bibliography should be consulted.