In Chapter 2, we note that a digital image is represented by the function of discrete variables, f (m, n). This discrete function can be represented by a matrix, which can be transformed to a vector using stacked notation. This transformation leads to the representation of images by vectors; and optical blurring functions, discrete Fourier transforms and various other image operations as matrices. By representing many image processing operations as matrix-vector operations, we can use the powerful methods of linear algebra to address our problems and formulate concise solutions. Here we review the properties of matrix theory that we need for this text. This is a brief summary and does not attempt to derive results. For a more complete presentation, a text on matrix algebra is suggested, such as [174, 181, 236].

Basic matrix definitions and properties

To begin, let us summarize the important properties of matrix-vectors and their operations in Table B.1.We will then give more details of the less familiar definitions and operations and introduce the pseudoinverse and elementary matrix calculus.

Kronecker product

The Kronecker product is useful for representing 2-D transformations, such as the Fourier transform and other transforms, on images using stacked notation.