A New Local Sine Transform without Overlaps: A Combination of Computational Harmonic Analysis and PDE

摘要

We introduce a new local sine transform that can completely localize image information in both the space and spatial frequency domains. Instead of constructing a basis, we first segment an image into local pieces using the characteristic functions, then decompose each piece into two components: the polyharmonic component and the residual. The polyharmonic component is obtained by solving the elliptic boundary value problem associated with the so-called polyharmonic equation (e.g., Laplace equation, biharmonic equation, etc.) given the boundary values (the pixel values along the borders created by the characteristic functions) possibly with the estimates of normal derivatives at the boundaries. Once this component is obtained, this is subtracted from the original local piece to obtain the residual, whose Fourier sine series expansion has quickly decaying coefficients since the boundary values of the residual (possibly with their normal derivatives) vanish. Using this transform, we can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. We will demonstrate the superior performance of this new transform in terms of image compression to some of the conventional methods such as JPEG/DCT and the lapped orthogonal transform using actual examples.