A first course in the numerical analysis of differential equations
A first course in the numerical analysis of differential equations
A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions
SIAM Journal on Scientific Computing
A fast 3D Poisson solver of arbitrary order accuracy
Journal of Computational Physics
A Legendre Spectral Galerkin Method for the Biharmonic Dirichlet Problem
SIAM Journal on Scientific Computing
Image quality assessment: from error visibility to structural similarity
IEEE Transactions on Image Processing
Isotropic polyharmonic B-splines: scaling functions and wavelets
IEEE Transactions on Image Processing
Improvement of DCT-Based Compression Algorithms Using Poisson's Equation
IEEE Transactions on Image Processing
The polynomial Fourier transform with minimum mean square error for noisy data
Journal of Computational and Applied Mathematics
Periodic Plus Smooth Image Decomposition
Journal of Mathematical Imaging and Vision
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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(驴k驴驴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.