Fractional Convolution Theory And Applications

As a powerful communication medium, image is one of the major ways for human beings to acquire and exchange information in our daily life and scientific research. But, due to the equipment imperfect and some external interferes or other factors, images may be affected by the noise from different sources during acquisition and transmission processing. This will lead to a certain degree of degradation of images. But high quality image is of great significance for subsequent processing and research. In order to remove the noise of image, a lot of image denoising methods have been developed. Each has his own strong and weak points. In this thesis, two fractional algorithm on the basis of the Non-Local Means filtering and wavelet analysis is proposed. Theatrical research and numerical experiments, show that our algorithm is of better performance.First, a new fractional wavelet transformation is discussed. As a very important method in image denoising, wavelet analysis uses a multi-scale decomposition of image. In fact it decomposes an image into low-frequency and high-frequency parts. And through the threshold procedure of high frequency coefficients and reconstruction, a denoised image is obtained. In this thesis, we propose a new definition of fractional convolution. Based on this fractional convolution, a new fractional wavelet transformation is given. We discuss its properties and apply it to image processing. The fractional wavelet transform method denoises image in fractional domain, which is a state between spatial and wavelet domain and it has a lot of special properties. The numerical experiments and their discussion show that this fractional wavelet transform method is of better performance compared with traditional wavelet transform methods.The other work in this thesis is Fractional Non-Local Mean filtering. Using the self-similarity of an image, Non-Local Means method find similar patches of an image in spatial domain. It estimates the clear image patches through average of these similar patches. The weights depend on the similarity between patches. Although this algorithm has very good denoising effects, it cannot protect well detail and texture information. Because this kind of information can be clearly reflected in high frequency domain, we propose to find similar patches and use Non-Local Means method both in spatial domain and in high-frequency domain. Then two results are used to obtain an estimated image in the fractional Fourier domain. This algorithm is called Fractional Non-Local Means algorithm because it is based on the fractional Fourier transformation. Numerical simulation results show the effectiveness of the proposed algorithm.