Filters of wavelets on invariant sets for image denoising

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.