The Research Of Complex Wavelets Theory And Their Applications In Image Denoising And Enhancement

Efficient sparse representation and processing of unstable signal are the main contents in mathematics and information science. Recently, the Discrete Wavelet Transform (DWT) has become efficient in the sparse representation of unstable signal and also is a powerful tool for signal and image processing. It, however, has some disadvanges, including, (1) It is shift sensitive because the input signal shift generates unpredictable changes in DWT coefficients; (2) It suffers from poor directionality because DWT coefficients reveal merely three spatial orientations; (3) It lacks of the phase information that accurately dscribes non-stationary signal behavior; that undermine its usage in many applications. Therefore, there is a strong motivation to study new types of wavelet transforms with better shift invariance and directionality. Due to the imperfection of image acquisition systems and transmission channels, the observed images are often in low-quality or degraded by noise. The goal of image denoising is to remove the noise while retaining as much as possible the important features (edges) and obtain acceptable image for vision. The image enhancement algorithms are to process a given image so the results are better than original image for their applications or objectives. Noise elimination and image enhancement are still the most fundamental, widely studied, and largely unsolved problems in computer vision and image processing.To overcome the disadvantages of the traditional DWT, this thesis mainly focus on two new types of complex wavelet transform: the dual-tree complex wavelet transform (DT-CWT) and the higher density dual tree DWT. The properties of the DT-CWT such as approximate shift-invariance, directionality and implementation issue are carefully investigigated. Furthermore, a new algorithm to construct wavelet filterbank of the DT-CWT is presented. At the same time, a new complex wavelet transform - the higher denstiy dual tree DWT is introduced and the corresponding characteristics are studies and a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed is developed. To better process the non-stational signal, the total variation and optimization based schemes for signal and image adaptive decomposition are preliminarily studied. Some classical applications of the proposed complex wavelet transforms are also further studies such as image denoising and enhancement. Results of experiments show that the proposed new algorithms perform better than the now existing methods.The main achievements in this paper are as follow:First, an approach for designing biorthogonal DT-CWT filters is proposed; where the two related wavelets pairs form approximate Hilbert transform pairs. Different from the existing design techniques, the two wavelet filterbanks obtained here are both of linear phases. By adjusting the parameters, wavelet fitlers with rational coefficients may be achieved, which can speed up the DT-CWT effectively.Then, to overcome the disadavanges of the DWT, we introduce the higher-density dual-tree DWT, which is a DWT that combines the higher-density DWT and the DT-CWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. We develop a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractinal-delay allpass filters, spectral factorization and the solutions have vanishing moments, compact support, a high degree of smootheness, intermediate scales, approximate Hilbert transform properties, and are nearly shift-invariant.In addition, we investigate the problem of adaptive deocompositon of the signal and image, the optimization method and total variation model are employed in the process. Experimental results show that the proposed methods are effective to a wide range of signals and images; when compared to the fixed wavelet bases method, the produced reconstruct images with our adaptive method are with better PSNR and visual quality.Based on the theory above, to address the problems of the image denoising and enhancement, we investigate the image denosing and enhancement in the new types of complex wavelet transform domain in detail. Three new image denoising algorithms based on the DT-CWT are proposed: (i) a new locally adaptive image denoising method, which exploits the intra-scale and inter-scale depencencies in the DT-CWT domain; (ii) a new non-tranining compelx wavelet Hidden Markov Tree (CHMT) model, which is based on the DT-CWT and a fast parameters estimation technique; (iii) a new denoising algorithm based on the Gaussian scale mixture (GSM) of the coefficients of the DT-CWT. These methods exploit the properties of the DT-CWT and the statistics of the coefficients and the obtained better denoising performances while reducing the computational complexity. At the same time, we introduce an effective integration of the intrascale correlations within the interscale SURE based orthonormal wavelet thresholding, which can solve the problem of the interscale method that is not very effective for those images that have substantial high-frequency contents.In addition, we also investigate the image enhancement based on the new types of wavelet transform and the statistical characters of visual representation and propose two new method of image enhancement: (i) a novel method for image enhancement, which exploits the properties of the double-density dual-tree DWT and the statistical characters of visual representation; (ii) a new method for noisy image enhancement, which is based on the GSM model of the DT-CWT coefficients and the combination of the DT-CWT and the statistical characters of visual representation, and can optimize the contrast of image features of while minimizing image denoise.