PHLST5: A Practical and Improved Version of Polyharmonic Local Sine Transform

Abstract

We introduce a practical and improved version of the Polyharmonic Local Sine Transform (PHLST) called PHLST5. After partitioning an input image into a set of rectangular blocks, the original PHLST decomposes each block into a polyharmonic component and a residual. Each polyharmonic component solves a polyharmonic equation with the boundary conditions that match the values and normal derivatives of even orders along the boundary of the corresponding block with those of the original image block. Thanks to these boundary conditions, the residual component can be expanded into a Fourier sine series without facing the Gibbs phenomenon, and its Fourier sine coefficients decay faster than those of the original block. Due to the difficulty of estimating normal derivatives of higher orders, however, only the harmonic case (i.e., Laplace’s equation) has been implemented to date, which was called Local Laplace Sine Transform (LLST). In that case, the Fourier sine coefficients of the residual decay in the order O(‖k‖⁻³), where k is the frequency index vector. Unlike the original PHLST, PHLST5 only imposes the boundary values and the first order normal derivatives as the boundary conditions, which can be estimated using the information of neighbouring image blocks. In this paper, we derive a fast algorithm to compute a 5th degree polyharmonic function that satisfies such boundary conditions. Although the Fourier sine coefficients of the residual of PHLST5 possess the same decaying rate as in LLST, by using additional information of first order normal derivative from the boundary, the blocking artifacts are largely suppressed in PHLST5 and the residual component becomes much smaller than that of LLST. Therefore PHLST5 provides a better approximation result. We shall also show numerical experiments that demonstrate the superiority of PHLST5 over the original LLST in terms of the efficiency of approximation.