ABSTRACT
Image transforms [52] are used for decomposing images in different structures or components so that a linear combination of these components provide the image itself. We may consider an image as a 2-D real function f(x, y) ∈ L2(R2), (x, y) ∈ R2, where R is the set of real numbers and L2(R2) is the space of all square integrable functions. In a transform, the function is represented as a linear combination of a family (or a set) of functions, known as basis functions. The number of basis functions in the set may be finite or infinite. Let B = {bi(x, y)| − ∞ < i < ∞} be a set of basis functions, where bi(x, y) may be in real (R) or complex space (C). In that case, a transform of the function f(x, y) with respect to the basis set B is defined by the following expression:
f(x, y) = ∞∑
i=−∞ aibi(x, y), (2.1)
where ai’s are in R or C depending upon the space of the basis functions. They are known as transform coefficients. This set of coefficients provides an alternative description of the function f(x, y).
