Harmonic Wavelet Transform and Image Approximation |
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Authors: | Zhihua Zhang Naoki Saito |
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Affiliation: | (1) Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-3139, USA;(2) School of Instrument Science and Engineering, Southeast University, Nanjing, Jiangsu, China, People’s Republic |
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Abstract: | In 2006, Saito and Remy proposed a new transform called the 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 biorthonormal periodic wavelet bases with the symmetric filter banks. We call
this the Harmonic Wavelet Transform (HWT). HWT has an advantage over both the LLST and the conventional wavelet transforms. On the 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. |
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Keywords: | |
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