An introduction to wavelets
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 Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Isotropic polyharmonic B-splines: scaling functions and wavelets
IEEE Transactions on Image Processing
Wavelet filter evaluation for image compression
IEEE Transactions on Image Processing
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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 驴. First we approximate f by a harmonic function u such that the residual component v=f驴u vanishes on the boundary of 驴. 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 驴. 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.