Harmonic Wavelet Transform and Image Approximation
In 2006, Saito and Remy proposed a new transform called the Laplace Local Sine Transform (LLST) in image processing as follows. Let f be a twice continuously differentiable function on a domain Omega. First we approximate f by a harmonic function u such that the residual component v=f-u vanishes on the boundary of Omega. Next, we do the odd extension for v, and then do the periodic extension, i.e. we obtain a periodic odd function v (*). Finally, we expand v (*) into Fourier sine series. In this paper, we propose to expand v (*) into a periodic wavelet series with respect to biorthonormal periodic wavelet bases with the symmetric filter banks. We call this the Harmonic Wavelet Transform (HWT). HWT has an advantage over both the LLST and the conventional wavelet transforms. On the one hand, it removes the boundary mismatches as LLST does. On the other hand, the HWT coefficients reflect the local smoothness of f in the interior of Omega. So the HWT algorithm approximates data more efficiently than LLST, periodic wavelet transform, folded wavelet transform, and wavelets on interval. We demonstrate the superiority of HWT over the other transforms using several standard images.
harmonic wavelet transform laplace local sine transform odd extension periodic wavelets folded wavelets wavelets on interval biorthonormal wavelets periodic wavelet coefficient symmetry