Graph-based image restoration and subpixel-based image scaling : analysis in the continuous domain

Abstract

To process discrete digital images, a straightforward approach is to view them as arrays of numbers and then directly operate on the arrays. An alternative paradigm, however, is to regard them as samples from continuous signals. Hence, we can apply a continuous model to analyze and resolve discrete image processing problems. By virtue of this continuous-domain paradigm, more mathematical tools become available and formal continuous analysis can be performed, providing profound insights into the problem being considered. Moreover, for problems involving entities that are inherently continuous, it is more accurate to do analysis in the continuous domain. In this thesis, we focus on two image processing problems: graph-based image restoration and subpixel-based image scaling. Unlike existing works applying the continuous-domain paradigm, it is non-trivial to analyze these two problems in the continuous domain. However, we manage to do so and answer/resolve several fundamental aspects of the two problems. We first consider the problem of image restoration with graph Laplacian regularization. A graph Laplacian regularizer is a recent, popular prior assuming the target pixel patch to be smooth with respect to an appropriately chosen graph. However, the mechanisms and implications of imposing the graph Laplacian regularizer on the original inverse problem are not well understood. In this work, we interpret neighborhood graphs of pixel patches as discrete counterparts of Riemannian manifolds and perform analysis in the continuous domain, bringing novel understandings to several key problems of graph Laplacian regularization. Specifically, we show the convergence of the graph Laplacian regularizer to a continuous-domain functional and derive the optimal graph Laplacian regularizer for the problem of image denoising. Then with the notion of anisotropic diffusion, we explain the behavior of graph Laplacian regularization during iterations, e.g., its tendency to promote piecewise smooth signals under certain settings. To verify the analysis, an iterative image denoising algorithm is developed. Experimental results show that our algorithm performs competitively with state-of-the-art denoising methods for natural images, and outperforms them significantly for piecewise smooth images. In the second problem, we aim at re-scaling a given image for a color display panel by controlling its subpixels individually, so as to improve the luminance resolution of the displayed image. However, improved luminance resolution brings chrominance distortion, making it crucial to suppress color error while maintaining sharpness. Moreover, it is difficult to develop a scheme that is applicable for various subpixel arrangements and for arbitrary scaling factors. By taking into account the low-pass nature of the human visual system (HVS), we address the aforementioned issues with a generalized continuous-domain analysis model. Our developed continuous-domain algorithm is accurate. The error of the resulting images is only introduced by the discrete implementation, and a higher computational budget leads to results with higher accuracy. Experiments show that the proposed method provides sharp images with negligible color distortions. Our method is comparable to the state-of-the-art methods for the RGB stripe arrangement, and outperforms them for other subpixel arrangements.

Date of Award	2016
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology