Redundancy And Signal Denoising

Frame theory is a major tool for wavelet analysis and it was originally derived from the signal processing. It was invented by Duffin and Schaffer in the study of non-harmonic Fourier series in 1952, and it was also called Hilbert space frame.The framework theory in the development of wavelet analysis has been played a very important role. In the further development of wavelet theory, Daubechies, Grossmann, and Meyer defined the affine framework (also called wavelet frame) when they combined the continuous wavelet transform theory and frame theory in 1986. Today, the framework theory of wavelet analysis was not only been applied to the study, but also been used in signal processing, image processing, Sobolev space theory, numerical calculation of theoretical and applied other research in the field.In this article, the frame redundancy has been portrayed through signal reconstruction. The use of redundancy which has been possible to reduce larger frame of the noise coefficient has been first discussed, also the use conclusion of the frame for the design of projectors to eliminate noise has been further refined and proved, especially in the use of standardized frame to eliminate white noise vector etc.The transformation of the redundant including binary wavelet transform and curvelet transformation has been firstly discussed.The application of redundancy to optimize the transformation of the image de-noising has been secondly discussed:To preserve more detail during image denoising, a combined wavelet and curvelet approach 1 and 2 are presented based on optimization in this article.Method 1:When the noised image has been decomposd by wavelet transformation, each scale on the three high frequency sub-band of the detail component has been going to single inverse transform for obtainning the details of the scale image. Each detail image is firstly denoised with curvelet threshold method, then decomposed with wavelet.The obtained wavelet coefficients are used as substitution for ones of the noise image. Finally the denoised image is obtained by inverse transformation of the substitution wavelet coefficients. Expermental results showed better denoising promerfence for image with straight-line character compared with only wavelets or curvelets.Method 2:When the noised image has been decomposd by wavelet transformation, each scale on the three high frequency sub-band of the detail component has been going to single inverse transform for obtainning the detaiis of the scale image.Each detail image is first denoised with cycle spinning and fast curvelet threshold method,and then decomposed with wavelet.The obtained wavelet coefficients are used as substitution for ones of the noise image. Finally, the denoised image is obtained by inverse transformation of the substitution wavelet coefficients.This paper is composed of five parts.The chapter 1 is an introduction which has been summarized the emergence,development of wavelet analysis and frames theory.The second part is redundant frame which has been successive introduced a framework for redundant redundancy framework and its framework for characterization of such knowledge.The third part is redundant transform. In this section, the discrete transform ridge redundant wavelet transform, dyadic wavelet transform, curvelet transformation, curvelet transformation have been introduced.The fourth part is the signal de-noising, and been divided into two parts. The first part is the framework for de-noising including the use of redundancy which can reduce the noise factor of increasing the framework to further improve and prove the use of frame design to eliminate noise, especially the use of projection normative framework for the design of the vector projection eliminate the white noise, etc. the relevant conclusions. To preserve more detail during image denoising, a combined wavelet and curvelet approach 1 and 2 are presented based on optimization in this article.PartⅤis a summary.