Quantization of the Nonlinear Sigma Model Revisited

We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomaly-free under additional assumptions on the target homogeneous space, in agreement with the work of Friedan. We carry out some explicit computations for the $O(N)$-model. Finally, we show how a suitable notion of the renormalization group establishes the Ricci flow as the one loop renormalization group flow of the nonlinear sigma model.Comment: 51 page

Mathematical approaches to digital color image denoising

Many mathematical models have been designed to remove noise from images. Most of them focus on grey value images with additive artificial noise. Only very few specifically target natural color photos taken by a digital camera with real noise. Noise in natural color photos have special characteristics that are substantially different from those that have been added artificially. In this thesis previous denoising models are reviewed. We analyze the strengths and weakness of existing denoising models by showing where they perform well and where they don't. We put special focus on two models: The steering kernel regression model and the non-local model. For Kernel Regression model, an adaptive bilateral filter is introduced as complementary to enhance it. Also a non-local bilateral filter is proposed as an application of the idea of non-local means filter. Then the idea of cross-channel denoising is proposed in this thesis. It is effective in denoising monochromatic images by understanding the characteristics of digital noise in natural color images. A non-traditional color space is also introduced specifically for this purpose. The cross-channel paradigm can be applied to most of the exisiting models to greatly improve their performance for denoising natural color images.Ph.D.Committee Chair: Haomin Zhou; Committee Member: Luca Dieci; Committee Member: Ronghua Pan; Committee Member: Sung Ha Kang; Committee Member: Yang Wan

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Adaptive Edge-guided Block-matching and 3D filtering (BM3D) Image Denoising Algorithm

Image denoising is a well studied field, yet reducing noise from images is still a valid challenge. Recently proposed Block-matching and 3D filtering (BM3D) is the current state of the art algorithm for denoising images corrupted by Additive White Gaussian noise (AWGN). Though BM3D outperforms all existing methods for AWGN denoising, still its performance decreases as the noise level increases in images, since it is harder to find proper match for reference blocks in the presence of highly corrupted pixel values. It also blurs sharp edges and textures. To overcome these problems we proposed an edge guided BM3D with selective pixel restoration. For higher noise levels it is possible to detect noisy pixels form its neighborhoods gray level statistics. We exploited this property to reduce noise as much as possible by applying a pre-filter. We also introduced an edge guided pixel restoration process in the hard-thresholding step of BM3D to restore the sharpness of edges and textures. Experimental results confirm that our proposed method is competitive and outperforms the state of the art BM3D in all considered subjective and objective quality measurements, particularly in preserving edges, textures and image contrast