曲面上图像处理的非局部变分模型与算法

Texture image processing and analysis on 3D objects in the areas of 3D maps, 3D cities, 3D digital cultural inheritages, 3D digital medical analysis, 3D cinemas and virtual environments depend on geometry features of surfaces, on which a pixels locate. It is a complicated problem coupling geometry and images, which can be treated via nonlocal variational models on surfaces for texture preserving. But there are a lot of theoretic difficulties to overcome. In this project, we’ll firstly define some intrinsic nonlocal operators, such as nonlocal gradients, nonlocal divergences, nonlocal curvatures, on implicit surfaces denoted by level set functions and binary value functions, to design nonlocal nonlinear variational diffusion models for texture image denoising on with edges, smoothness, corners and texture preserving properties on surfaces. We’ll also investigate fast numerical methods for the proposed models combining Nesterov’s accelerating algorithm with step predictor-correction strategy and ADMM(Alternating Direction Methods of Multipliers) method to improve efficiency and design narrow band schemes for the new algorithms to reduce computation dimension.

3D地图/城市/数字化文物/影视/数字医学/虚拟场景中3D对象上纹理图像处理与分析依赖于所在曲面几何特征，是复杂的曲面几何与纹理图像耦合问题，曲面上非局部变分模型有望成为纹理等特征保持的有效分析方法，但面临诸多理论与计算困难。本项目首先以曲面上的纹理图像扩散为背景，用水平集方法、二值标记函数方法结合表达隐式曲面，进而定义曲面上具有内蕴形式的非局部梯度、散度、曲率等非局部算子，继而定义边缘、光滑、拐点、纹理等特征保持的曲面上图像噪声去除的非局部非线性扩散变分模型。为提高计算效率，本项目拟结合预估-校正迭代步长方法与交替方向乘子方法设计这些模型的快速算法。为降低求解规模，项目拟提出适于隐式曲面窄带内求解的窄带方法。研究成果亦为曲面上图像处理、分析的其他问题研究奠定理论基础。

曲面上的图像处理是图像和所在曲面几何的耦合问题，现有的变分图像恢复模型仅考虑曲面上的像素点对其所在局部曲面的依赖，以往采用内蕴局部算子概念，很难实现图像的纹理、边缘等特征的同时保持。本项目引入曲面上非局部内蕴微分算子概念，建立了曲面上图像恢复的内蕴非局部变分模型，自然地继承了基于水平集的隐式曲面上局部变分模型及平面图像非局部变分模型的结构，达到了曲面上图像恢复的边缘、纹理等特征的同时保持。此外，本项目还设计了所提出模型的内蕴形式的快速交替方向乘子方法-ADMM方法。完成了项目提出的研究内容，并为曲面上纹理图像分割、修复、增强等奠定了理论基础。