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Digital Image Processing (2000)

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Image Transforms

8.1 Introduction

It is well known that a periodic signal may be written as a series expansion of sinusoids (or equivalently complex exponentials). More generally a signal f(t) may be expressed as the superposition
where k is an integer index of the finite or infinite summation, the ak are the real (or complex) valued expansion coefficients, and the k(t) are members of a set of real (or complex) functions called the expansion functions. If the expansion in Equation (8.1.1) is unique, the set is called a basis for the class of signals under investigation. The set members k(t) are known as the basis functions of the series expansion. It is desirable to have an orthogonal basis, meaning
since the coefficients can then be calculated by the inner product
where * denotes the complex conjugate operator. For a continuous-time Fourier series, the orthogonal basis functions are of the form k(t)= j 0 k t, with 0 as the fundamental frequency. For a Taylor series, expansion k(t) = tk. The series coefficients ak in Equation (8.1.3) give an alternative representation of the signal f(t) and are useful in filtering, data compression, feature extraction, and other applications. Equivalent formulations exist for the practical case of finite extent, 1D and 2D sampled signals. In the case of 1D sequences, a general orthogonal series expansion for a sequence f[n] of length N is a pair of transformations of the form
where k define a discrete set of so-called basis sequences. For example, in the case of the discrete Fourier transform the basis sequences take the form , for 0≤n≤N-1. All image transforms found in the package share this general formulation. They also share a common matrix product formulation:
where f, F are vectors of length N representing the signal and its transform, respectively, and , * are N×N matrices, the transform matrix, and its complex conjugate, respectively. For example, the Fourier transform matrix has the form:
In the context of image processing, a general orthogonal series expansion for an N × N image f [n,m] is a pair of transformations of the form
where k, l define a discrete set of basis arrays called basis images. The corresponding matrix formulations are as follows:
where , *, and  are the transpose, complex cojugate, and conjugate transpose of the matrix , respectively.
The next sections present four discrete transforms widely used in image processing and the important topics of computing the power spectrum, image restoration, also called deconvolution, and image compression. In Section 8.2, the discrete Fourier transform and in Section 8.3 the related power spectrum are presented. Subsequent sections describe other important image transforms such as the discrete cosine transform (Section 8.4), discrete Hadamard transform (Section 8.5), and discrete wavelet transform (Section 8.6). Image transforms are important in a variety of image processing applications, but are especially prominent in image compression. The last two sections of this chapter present in some detail two of the most important applications of discrete transforms in the area of image processing, namely image restoration and image compression. In Section 8.7 the important topic of image restoration is presented. Finally, in Section 8.8, we present a short introduction to block transform coding, a data compression technique that has found application in compression of still grayscale and color images.