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

摘要：本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的三组问题：(1)图像分割与视频中目标跟踪的应用；(2)像素域和变换域的图像绘制问题；(3)基于变分pde的模型在流形图像处理中的应用。该方法利用了计算数学中的强大概念，如对偶性、凸化、非光滑优化、快速组合优化、数值PDE技术、谐波分析和计算微分几何。研究的重点是计算效率、特征提取、目标跟踪、全局分析和优化以及几何和纹理信息的保存等关键问题，这些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疗成像等许多领域。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很敏感。此外，数值算法可能陷入局部非最优极小值。为了获得更好的分割算法，我们将先前为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连接）扩展到其他分割模型，例如受纹理合成技术启发的非局部分割模型。对于图像绘制，最近的进展包括PDE和几何技术，纹理合成，以及用于图像缺失变换的混合框架（例如小波或傅里叶）。这项研究正在检验这些方法之间的协同作用，并推导出结合每种方法最佳特征的模型和算法。对于流形上的图像处理，我们正在使用保角映射技术和基于pde的图像处理模型来推导针对一般曲面的有效算法。新的应用包括一般表面上的地标的自动跟踪和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.