## Basic Wavelet Theory and Multiresolution Analysis

Image transformation relies on using a set ofbasis functions on which the image is projected to form the transformed image. In the cosine transform, the basis functions are a series of cosine functions and the resulting domain is the frequency domain. In the case of the wavelet transform, the basis functions are derived from a mother wavelet function by dilation and translation. In the one-dimensional (1D) wavelet transform the basis functions are obtained using the mother wavelet function ^(x) such that

where a and b are the dilation and translation factors,

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respectively. The continuous wavelet transform of a function /(x) is expressed as

where * is the complex conjugate operator.

The basis functions given in Eq. (1) are redundant when a and b are continuous. It is possible, however, to discretize a and b so as to form an orthonormal basis. One way of discretizing a and b is to let a = 2P and b = 2pq, so that Eq. (1) becomes

where p and q are integers. The wavelet transform in Eq. (2) then becomes fw (p, q) = 2

Since p and q are integers, Eq. (4) is called a wavelet series. This representation indicates that the wavelet transform contains both spatial and frequency information.

The wavelet transform also relies on the concept of multiresolution analysis, which decomposes a signal into a series of smooth signals and their associated detailed signals at different resolution levels. The smooth signal at level "m" can be reconstructed from the "m + 1" level smooth signal and the associated "m + 1" detailed signals.