Algorithm Improvement Of Discrete Ridgelet Transform And Its Application In Image Processing

Wavelet transform has attracted attention in various application areas as a useful time-frequency analysis theory. Its success is mainly due to the perfection of monovariant polynomial factorization theory in mathematics and the simplicity of Mallat's one-dimensional (1D) signal decomposition structure in signal analysis engineering. However, two-dimensional discrete wavelet transform (2D-DWT) generated by a tensor-product of two perpendicular 1D wavelets is sensitive to horizontal, vertical and diagonal directions only, thus encounters difficulty in effective representation of geometrical structures in images. To overcome the weakness of the wavelet in higher dimensions, the multiscale geometric analysis theory which is developed relatively independently from the subjects of harmonic analysis, computer vision, and statistical analysis, etc. has provided several novel transform methods, such as ridgelet transform, curvelet transform, and has been intensively discussed, recently.Aimed at an efficient representation of linear singularities in images, the main idea of ridgelet transform is to map linear singularities in the image domain to point singularities via Radon transform, and then the 1D-DWT is applied to handle the point singularities in the Radon domain along radial directions. This dissertation focuses on algorithm improvement of the discrete ridgelet transform (DRIT) and its applications in image processing, where the discrete periodic Radon transform (DPRT) is adopted for implementation of the discrete Radon transform in DRIT.The main work of this dissertation includes:(1) The main properties of a transform, such as invertibility, coefficient sparsity and orthogonality are investigated, and the basic theory and research background of several non-adaptive multiscale geometric transform methods nowadays are reviewed. Based on the analysis of the wavelet theory and the weakness of 2D separable wavelet in high dimensional signal processing, continuous ridgelet transform is introduced. Implementation methods of discrete Radon transform (DRT) and transform schemes in DRT domains are studied. (2) The discrete periodic Radon transform is adopted to unify the implementation of DRT, and the "wrap around" effect which heavily restricts the application of DRIT is investigated. Based on analysis of the relationship between the "wrap around" effect and the distribution of DPRT coefficients, an angle-based orthogonal finite ridgelet transform (AFRIT) algorithm and an energy-based adaptive orthogonal DRIT (EDRIT) algorithm are proposed. Experiments using nonlinear approximation show the superiority of AFRIT and EDRIT over traditional transform methods in energy concentration property, while the "wrap around" effect is greatly reduced in reconstruction images by them.(3) The denoising problem of additive white Gaussian noise (AWGN) using DRIT is modeled. Based on analysis of statistical characteristics of DRIT coefficients, a novel colomnwise threshold (CT) method is proposed. Denoising experiments are carried out on abundant images containing strong linear singularities and texture components with varying levels of AWGN, and the results show that adopting the CT threshold, both AFRIT and EDRIT achieve prominent improvement in terms of signal to noise ratio (SNR) and visual quality. Comparison of denoising ability of different transform schemes and thresholding methods is discussed in detail.(4) FRIT is applied to texture classification as it provides effective representation of directional and multiscale characteristics of images and its coefficient matrix is compact. Based on the analysis of energy contribution of FRIT coefficients, a novel subband division method is proposed for feature extraction in texture classification. A 'one-against-one' multi-class SVM with RBF kernel is adopted as the classifier. Experiments carried out on abundant databases with varying sizes demonstrated its validity.