Algorithm to Generate Wavelet Transform from an Orthogonal Transform

摘要

This paper proposes algorithm to generate discrete wavelet transform from any orthogonal transform. The wavelet analysis procedure is to adopt a wavelet prototype function, called an analyzing wave or mother wave. Other wavelets are produced by translation and contraction of the mother wave. By contraction and translation infinite set of functions can be generated. This set of functions must be orthogonal and this condition qualifies a transform to be a wavelet transform. Thus there are only few functions which satisfy this condition of orthogonality. To simplify this situation, this paper proposes a generalized algorithm to generate discrete wavelet transform from any orthogonal transform. For an NxN orthogonal transform matrix T, element of each row of T is repeated N times to generate N Mother waves. Thus rows of original transform matrix become wavelets. As an example we have illustrated the procedure of generating Walsh wavelet called ‘Walshlet’ from Walsh transform. Since data compression is one of the best applications of wavelets, we have implemented image compression using Walsh as well as Walshlet. Our experimental results show that performance of image compression technique using Walshlet is much better than that of standard Walsh transform. More over image reconstructed from Walsh transform has some blocking artifact, which is not present in the image reconstructed from Walshlet. Similarly image compression using DCT and DCT Wavelet has been implemented. Again the results of DCT Wavelet have been proved to perform better than normal DCT