Fractional-Order Total Variation Based Image Denoising, Deconvolution, and CT Reconstruction Under Poisson Statistics

Image processing and analysis have gained increasing popularity nowadays for its applications in medical imaging, astronomy, astrophysics, surveillance, image compression, and transmission. In this dissertation, we work on three types of image restoration: image denoising, deconvolution, and computer tomography (CT) reconstruction. In many photonlimited imaging systems, acquired data is usually corrupted by Poisson noise and blurring artifacts. Different from Gaussian noise that is commonly used in the scientific community, Poisson noise depends on the image intensity, which makes image restoration very challenging. Moreover, the underlying images often contain complex geometries, and hence it is desirable to impose a regularization to preserve piecewise smoothness. For this purpose, we propose to use the fractional-order total variation (FOTV) regularization. Specifically, for image denoising, we can establish the existence and uniqueness of a solution to our proposed model. To solve the problem efficiently, we adopt three numerical algorithms based on the Chambolle-Pock primal-dual method, a forward-backward splitting scheme, and the alternating direction method of multipliers (ADMM), each with guaranteed convergence. Various experimental results demonstrate the effectiveness and efficiency of our proposed methods over the state-of-the-art in Poisson denoising. Blurring is always inevitable, as the data recorded by a digital device is an average over neighboring pixels, leading to a blurred image. The blurring process can be modeled as a convolution of an underlying image with a point spread function (PSF). We consider both non-blind and blind image deblurring models, in which blind refers to the case of an unknown PSF. In the pursuit of the high-order smoothness of a restored image, we use the FOTV regularization to remove the blur and Poisson noise simultaneously. We develop an ADMM-based algorithm for non-blind deblurring and an expectation-maximization (EM) algorithm in the blind case. A variety of numerical experiments demonstrate that the proposed algorithms can efficiently reconstruct piecewise-smooth images degraded by Poisson noise and various types of blurring, including Gaussian and motion blurs. Specifically for blind image deblurring, we obtain significant improvements over the state-of-the-art. Lastly, we consider a CT reconstruction problem, where the noise is traditionally modeled by Gaussian distribution. We propose using the FOTV regularization and a data fidelity term for Poisson noise to reconstruct the CT image. We show in experiments that the Gaussian noise is indeed suitable when the highest intensity value (or called peak value) is large. But the Poisson distribution is a more appropriate distribution for relatively lower peak values. Furthermore, we demonstrate that FOTV-based regularization outperforms the classic methods in the CT reconstruction, especially when the data has a small peak value, or the signal-to-noise ratio is low.