Filters of wavelets on invariant sets for image denoising

Abstract

Aiming at overcoming the shortcomings of existing wavelet denoising methods, we propose an image denoising algorithm based on wavelets on invariant sets. These wavelets, in comparison with classical wavelets, have the following features: they have vanishing moments of a high order and at the same time a short filter length. Moreover, boundary extension normally required for classical wavelets in wavelet transformations is not needed for wavelets on invariant sets. We identify a class of discrete orthogonal transforms, such as the discrete cosine transform of the second type, the Hadamard transform, the Slant transform and the Hartley transform with the filters of wavelets on invariant sets. This viewpoint gives us an insightful understanding of these transforms in the framework of the multiscale analysis. In turn, it leads to more efficient algorithms to image denoising. We demonstrate the performance of our algorithm on images with varying noise levels. The numerical results show that our proposed algorithm offers effective noise removal in noisy images.

Keywords:

• filter
• wavelet
• invariant set
• image denoising

AMS Subject Classifications::

• 42C40
• 65T60
• 68U10
• 94A08

Acknowledgements

The authors would like to thank the referees for their valuable suggestions, which greatly improve the readability of the article. Q. Lian was supported by National Science Foundation of China under grant 10671008 and 10901013, and by Beijing Natural Science Foundation under grant 1092001, L. Shen was supported by the US National Science Foundation under grant DMS-0712827. Y. Xu was supported in part by the US National Science Foundation under grant DMS-0712827, by the Natural Science Foundation of China under grants 10371122 and 10631080 and by Ministry of Education, People's Republic of China under the Changjiang Scholarship program through Zhongshan University, China.