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A new local sine transform without overlaps: A combination of computational harmonic analysis and PDE, (with J.-F. Remy), in *Wavelets X*
(M. A. Unser, A. Aldroubi, and A. F. Laine, eds.), Proc. SPIE vol.5207, pp.495-506, 2003.

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Abstract

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.

Get the full paper: PDF file.
Get the official version via doi:10.1117/12.503962.

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