This chapter discusses several types of transforms that are commonly used in image processing. The most common transform, the Fourier transform, will be presented first, followed by several closely related transforms such as the Hadamard, Walsh, and discrete cosine transforms, which are used in the area of image compression. The chapter will end with the Hough transform, which is used to find straight lines in a binary image, and the Hotelling transform, which is commonly used to find the orientation of the maximum dimension of an object.
A short review of one-dimensional Fourier transforms will be presented to give the reader the necessary background to understand the properties of the two-dimensional Fourier transform used in electronic image processing. The Fourier transform is the most commonly used transform in image processing because of its relationship to linear system theory. Given an input to a linear system, the output is predicted using the complex operation of convolution or by simple multiplication using the Fourier transform. The Fourier transform/series was originally developed by the French mathematician Baptiste Joseph Fourier (1768-1830) to solve Laplace's differential equation (known as the Dirichlet problem) that described the conduction of heat (distribution of temperature) along an infinite conducting sheet. Fourier used a series of sine and cosine functions as a solution to the differential equation by solving for a set of unknown coefficients.
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