Polyharmonic Wavelet Transform and Image Approximation

In 2006, Saito and Remy presented a new algorithm called polyharmonic local sine transform (PHLST) in image processing as follows. Let f be a twice continuously differentiable function on a domain Ω. First we approximate f by a polyharmonic function u such that the residual component v = f −u vanishes in the boundary of Ω. 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 a biorthonormal periodic wavelet basis with the symmetric filter bank. We call this algorithm the polyharmonic wavelet transform (PHWT) algorithm. The PHWT algorithm has an advantage over both the PHLST algorithm and the conventional wavelet transform. On one hand, it removes the boundary mismatches as PHLST. On the other hand, PHWT coefficients reflect the local smoothness of f in the interior of Ω. So the PHWT algorithm compresses data better than the PHLST algorithm, the periodic wavelet transform, and the folded wavelet transform.