| Graphics and image processing has been an important subject in computer graphics and com-puter vision. A major challenge in the graphics and image processing is how to process and handle the graphics and image data, while preserving the main features and structures effectively. In the past decade, sparse regularization achieved great success in the graphics and image processing field, due to the fact that many important classes of signals (e.g. images) have naturally sparse representations with respect to fixed bases. Most sparse regularization methods take a l2+l1  framework that employs l2 norm for data-fitting and norm for regularization. In recent years, some researchers have found that directly using l0 norm in the gradient regularization can achieve higher sparsity in the gradient of the resulting signal, compared with using norm. Higher spar-sity in gradient significantly improves the robustness against noises or outliers, and the ability to preserve sharp features.In this dissertation, we study the  gradient regularization problem and its applications. We present a novel approximation algorithm to solve the the  gradient regularization problem, and present several novel methods based on this model to tackle challenging graphics and image pro-cessing problems, including image smoothing, mesh smoothing, video segmentation and point set resampling. The main contributions are:· We propose a fused coordinate descent algorithm to approximately solve the l0 gradient regularization problem. The main idea of our solver is that only one variable is optimized at a time, while all others are fixed; neighboring variables are fused together once their values are equal, and act as a single variable in next iteration of optimization. Through the fusion of equal neighboring variables, we add the constraint of gradient sparsity implicitly. Compared with the alternative minimization solver proposed by Xu et al., our solver can achieve higher sparsity in the gradient. To verify the effectiveness of our solver, we apply  gradient regularization using our solver in two applications, edge-preserving image smoothing and feature-preserving mesh smoothing. The comparisons with other methods demonstrate that our application methods can produce better results.· We propose a video segmentation method with  gradient regularization. The measure of gradient sparsity essentially encodes the segmentation information:the neighboring el-ements that have zero gradients among them form a group naturally, while the non-zero gradients separate different groups. The spatio-temporal coherence in video segmentation is enforced through a sparsity pursuit manner. As far as we know, this is the first explo-ration to introduce sparsity analysis into video segmentation. To solve the the l0 gradient regularization problem, we extend fused coordinate descent algorithm from 2D image to 3D video volume. The experimental results in the LIBS VX benchmark demonstrate our superior performance to state-of-the-arts in segmentation accuracy and undersegmentation error.· We propose a efficient point set resampling method with  gradient regularization. Our framework can produce a set of clean, uniformly distributed, geometry-maintaining and feature-preserving oriented points. Thanks to the higher sparsity caused by l0 norm, our method outperform current point set resampling methods in robustness to noises or outliers and the ability to keep sharp features. After finding the reasons for efficiency degradation when applying l0 gradient regularization in point set processing, we propose two acceler-ating algorithms including optimized-based local half-sampling and interleaved regulariza-tion. As demonstrated by the experimental results, the accelerated method is about an order of magnitude faster than the original, while achieves nearly the same point set consolidation performance.
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