Haar Basis Wavelets

The Haar transform, which is one of the earliest transform functions proposed, was proposed in 1910 by a Hungarian mathematician Alfred Haar. It is found effective as it provides a simple approach for analysing the local aspects of a signal. Say we start with an image slice (one dimensional) of size , so we can write the image as Recursive Process of Decomposing an Image in terms of Sums and Differences Let us now group the image by consecutive pairs, i.e., We can apply the following linear transformation for each pair , i.e., So stores the sum of the two elements in a pair and stores the difference of the two elements in a pair. The upper index " 1 " refers to this transformation being the first step in a process that follows. Of course, this is invertible, i.e., So we can get back the original set of pairs, the original image. Thus, we can represent the image , of size , as pairs of the form We can then store the set of size and we decompose further the complementary set We interpret as a new image of half the size of the original one and we represent the set of pairs