Mathematical Models & Computational Algorithms for Image Processing, computer Vision & Computer Graphics

该方案解决了图像处理、计算机图形学和计算机视觉交叉领域中的三组问题：(1)图像分割及其在视频目标跟踪中的应用；(2)像素域和变换域的修复问题；(3)基于变分偏微分方程模型在流形上的图像处理应用。该方法利用了计算数学中强大的概念，如对偶、凸化、非光滑优化、快速组合优化、数值偏微分方程组技术、调和分析和计算微分几何。研究的重点是关键问题，如计算效率、特征提取、目标跟踪、全局分析和优化，以及保留几何和纹理信息，这些都是进一步发展的关键。这项研究将影响从娱乐到国土安全和医学图像的许多领域。对于图像分割，典型的模型通常是非凸的，并且允许许多局部最优解，使得它们对初始猜测敏感。此外，数值算法可能陷入局部非最优极小。为了获得更好的分割算法，我们将早期为Chan-Vese分割模型开发的一种新的凸化技术(通过TVL1模型和水平集之间的深度联系)扩展到其他分割模型，如受纹理合成技术启发的非局部分割模型。在修复方面，最近的进展包括偏微分方程和几何技术，纹理合成，以及用于修复缺失变换的混合框架(例如，小波或傅立叶)。这项研究正在检验这些方法之间的协同作用，并推导出结合了每种方法的最佳特征的模型和算法。对于流形上的图像处理，我们使用保角映射技术和基于偏微分方程的图像处理模型来推导出适用于一般曲面的高效算法。新的应用包括自动跟踪一般表面上的地标，以及在地标匹配中结合形状信息。

Abstract - ThieleThis proposal addresses three sets of problems in the intersection of image processing, computer graphics and computer vision: (1) image segmentation with applications to object tracking in videos, (2) inpainting problems, both in the pixel domain and the transform domain, and (3) applications of variational PDE-based models to image processing on manifolds. The approach makes use of powerful concepts from computational mathematics, such as duality, convexification, non-smooth optimization, fast combinatorial optimization, numerical PDE techniques, harmonic analysis, and computational differential geometry. The research is focusing on key issues, such as computational efficiency, feature extraction, object tracking, global analysis, and optimization, and preserving both geometric and texture information, which are keys to further advances. The research will impact many areas ranging from entertainment through homeland security and medical imaging.For image segmentation, typical models are generally nonconvex and admit many local optimal solutions, making them sensitive to initial guesses. Moreover, numerical algorithms can be trapped in local non-optimal minima. To obtain better segmentation algorithms, we are extending a novel convexification technique (through the deep connection between TVL1 models and level sets) developed earlier for the Chan-Vese segmentation model to other segmentation models, such as non-local segmentation models inspired by texture synthesis techniques. For inpainting, recent advances include PDE and geometry techniques, texture synthesis, and a hybrid framework for inpainting missing transform (e.g. wavelets or Fourier). This research is examining the synergy between these methods and deriving models and algorithms that combine the best features of each. For image processing on manifolds, we are using conformal mapping techniques and PDE-based image processing models to derive efficient algorithms for general surfaces. New applications include the automatic tracking of landmarks on general surfaces and the incorporation of shape information in landmark matching.