# Harmonic wavelet transform and image approximation,
*Journal of Mathematical Imaging and Vision*, vol.38, no.1, pp.14-34, 2010.

## Abstract

In 2006, Saito and Remy proposed a new transform called Laplace Local
Sine Transform (LLST) in image processing as follows.
Let *f* be a twice continuously differentiable function on a domain
*Ω*.
First we approximate *f* by a harmonic function *u* such that the
residual component *v=f-u* vanishes on the boundary of *Ω*.
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 a biorthonormal periodic wavelet basis with the symmetric
filter banks. We call this the *Harmonic Wavelet Transform* (HWT).
HWT has an advantage over both LLST and the conventional wavelet transforms.
On 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 *Ω*. 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.
**Keywords:** Harmonic wavelet transform, Laplace local sine
transform, biorthonormal wavelets, periodic wavelets, folded wavelets,
wavelets on interval, periodic wavelet coefficient, symmetry, odd extension.

Get the full paper (Revised on 04/20/10): PDF file.
Get the official version via doi:10.1007/s10851-010-0202-x.

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