TABLE OF CONTENTS A number of mathematical transformations can be applied to images to obtain information that is not readily available in the raw image. The Fourier transform is the most popular although other transforms, such as wavelets and the Gabor transform, are being increasingly used. The Fourier transform converts the spatial domain representation of an image into an alternative representation in the Fourier domain, in terms of spatial frequencies. Convolution of the input data with the point spread function of an imaging system results in the formation of an image. The convolution operation in the spatial domain is equivalent to multiplication in the Fourier domain, which is a more efficient method of performing smoothing or sharpening of an image.
After reading this chapter you will be able to:
describe how periodic waveforms consist of a linear superposition of sinusoids;
explain how the Fourier transform is derived from the Fourier series;
illustrate the concept of the discrete Fourier transform in two dimensions, with its dependence on sample and hold;
describe the phenomenon of aliasing and apply appropriate procedures to eliminate it;
outline the properties of the Fourier transform;
use cross-correlation to perform template matching;
obtain the spatial resolution of an imaging system both from its point spread function (PSF) and from its modulation transfer function (MTF), and show that they are equivalent;
use frequency domain filters to smooth or sharpen an image while avoiding ringing artifacts;
explain the...
