Mathematics Colloquia and Seminars

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Description

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 polyharmonic equation (e.g., Laplace equation, biharmonic equation, etc.) given the boundary values which are the pixel values along the borders created by the characteristic functions. In 1D, the polyharmonic component is simply an odd order polynomial passing through the boundary points while in the higher dimension it is not a polynomial of n variables in general. Once this component is computed, this is subtracted from the original local piece to obtain the residual, whose Fourier sine expansion has quickly decaying coefficients since the boundary values (possibly with their normal derivatives) of the residual is zero. In fact, we can show that the Fourier sine coefficients of the residual after subtracting the polyharmonic component of order m, are of O(|k|^{-2m-1}). Using this transform, we can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. This ability allows us to approximately segment a given image into local pieces according to their smoothness and locations of singularities, which leads to an efficient image compression scheme. We will demonstrate the superior performance of this new transform to some of the conventional methods such as JPEG/DCT and the lapped orthogonal transform (also known as local cosine transform) using actual examples.