Natural images or photographs contain rich small-scale patterns, reflecting the process of image formation and playing a core role in visual perception. This paper develops a new framework for characterizing small-scale patterns, reflecting the emergence process of small-scale patterns in natural images. We analyze small-scale patterns by formulating as a sequence of variational problems in Sobolev spaces associated with a sequence of nested subsets families of the image domain. The variational problems reduce the problem of modeling small-scale patterns to that of a sequence of eigenvalue problems. Mathematical analysis on the existence, uniqueness of small-scale patterns, the structure of the small-scale pattern and the convergence behavior of the proposed model are explored in detail. Finally, a novel effective, adaptive, and hierarchical image representation by the large-scale pattern and small-scale patterns is obtained, yielding high quality signal reconstruction over scales. Numerically, small-scale patterns are computed by eigenvalue decompositions of sparse, symmetric matrices of fast decreasing orders, leading to an accurate, efficient, and straightforward algorithm. Experiments were conducted on several natural images, and synthesized images using our model faithfully capture details in different scales, demonstrating the satisfying performance of the algorithm and the effectiveness of the proposed model. The model is useful for a wide variety of applications in image analysis and computer vision such as image restoration, image synthesis and computational visual perception. In this thesis, the new model is applied for one fundamental problem in image processing of image denoising and verified for its practical value. This thesis studies the modeling of small-scale visual patterns in natural images in detail, which is a basic problem in computer vision. The main contributions of this thesis include the following issues: 1. This paper attempts to develop an new hierarchical model of small-scale patterns in natural images. The new model has solid mathematical foundations. Instead of modeling small-scale signals at a fixed scale, here they are analyzed sequentially over several scales. The hierarchical expansion of image signals is adaptive which greatly captures the dynamics characteristic of small-scale patterns. 2. Theoretically, this paper introduces a novel energy functional for modeling small-scale patterns. Existence, uniqueness, and con...
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