Complex Directional Wavelet Transforms: Representation, Statistical Modeling And Applications

The thesis presents an new image decomposition for feature extraction, which is called the pyramidal dual-tree directional filter bank (PDTDFB). The image representation has an overcomplete ratio of less than 8/3 and uses a separable filter bank implementation structure. We discuss how to utilize both magnitude and phase information obtained from the PDTDFB for the purpose of texture image retrieval. The relative phase, which is the difference of phases between two adjacent complex coefficients, has a linear relationship with the angle of dominant orientation within a subband. This information is incorporated to form a new feature vector called CDFB-RP. Another application of PDTDFB is texture segmentation. A new feature extraction method is proposed for texture segmentation. The approach is based on incorporating the phase information obtained from complex filter banks. The PDTDFB is used to decompose a texture image in order to provide complex subband coefficients. The local mean direction, extracted from the phases of the coefficients, is defined as additional features for classification and segmentation.We proposed a modified version of the PDTDFB for image denoising. Unlike the previous approach, the new FB provides an approximately tight-frame decomposition. Then we proposed the complex Gaussian scale mixture (CGSM) for modeling the distribution of complex directional wavelet coefficients. The statistical model is then used to obtain the denoised coefficients from the noisy image decomposition by Bayes least squares estimator. Performance of the denoised images using the PDTDFB is compared with the conventional transforms including the orthogonal wavelet, the contourlet and the steerable pyramid.A new approach which exploits the probabilistic properties from the phase information of two-dimensional complex wavelet coefficients for the image modeling is developed. Definition, property and statistics of relative phase of the complex coefficients are studied in detail. We proposed von Mises and wrapped Cauchy for the probability density function (pdf) of the relative phase in the complex wavelet domain. The von Mises and wrapped Cauchy models are compared, and the simulation results show that the wrapped Cauchy fits well with the peaky and heavy-tailed pdf of the relative phase and the von Mises fits well with the pdf which is in Gaussian shape. For most of the test images, the wrapped Cauchy model is more accurate than the von Mises, when images are decomposed by different complex wavelet transforms including the DTCWT, the PDTDFB and a modified version of curvelet. With the assumptions of the Gaussian image model as well as the Gaussian scale mixture (GSM), the marginal and joint distributions for the phase of the complex wavelet coefficients are studied in detail. From these hypotheses, we then derive the probability density function of the relative phase (RP-PDF) in complex wavelet domain. We propose the maximum-likelihood method to estimate two RP-PDF parameters. The RP-PDF fits well with behaviors of the relative phase from various real images including texture images as well as natural images. The RP-PDF model is compared with the von Mises and wrapped Cauchy distributions. The experimental results, in which the real images are decomposed by various complex wavelets such as the DTCWT, the PDTDFB and the curvelet, show that the RP-PDF model for relative phase is more accurate than the others.