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PHLST5: A practical and improved version of
polyharmonic local sine transform (with J. Zhao and Y. Wang), *Journal of Mathematical Imaging and Vision*, vol.30, no.1, pp.23-41, 2008.

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Abstract

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(1/|***k**|^{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.

Get the full paper (revised as of Aug. 20, 2007): PDF file.
Get the official version via doi:10.1007/s10851-007-0044-3.

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