基于变分偏微分方程的自适应压缩感知及其图像重建应用

项目名称： 基于变分偏微分方程的自适应压缩感知及其图像重建应用

项目编号： No.11301414

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 周彬

作者单位： 西南石油大学

项目金额： 22万元

中文摘要： 图像重建是医学成像、地质勘探、工业检测等应用领域的重要问题。变分偏微分方程的相关研究极大地促进了计算机视觉、图像重建等应用学科的的发展。自Candes、Tao、Donoho利用全变差模型建立压缩感知理论以来，已产生众多压缩感知模型，并在图像重建类问题中取得了一定的成果。这些模型大多和类全变差的一些能量泛函及相应变分偏微分方程有紧密联系。耦合形式的能量泛函和非局部形式的能量泛函能够更准确的表征图像能量，相应的变分偏微分方程具有更丰富的图像处理能力。本项目拟在经典的变分偏微分方程图像处理模型基础上，研究耦合能量泛函和非局部能量泛函及相应的变分偏微分方程，建立具有自适应性的压缩感知模型，提高图像重建精度和重建效率，以适应复杂的图像重建需求。分析模型解的存在唯一性，利用L1范、L2范等进行解估计，误差估计，稳定性分析。结合变分原理、非线性最优化方法等设计有效的数值算法，通过仿真实验加以验证。

中文关键词： 变分；偏微分方程；压缩感知；图像重建；自适应

英文摘要： Image reconstruction is an important problem in medical imaging, geological exploration, industrial detection, and so on. The researches on variational partial differential equation have greatly improved some applied subjects such as computer vision, image reconstruction. Since Candes, Tao, Donoho proposed the compressed sensing theory based on the TV model, many related compressed sensing models have been succeeded introduced to image reconstruction. Most of these models is closely linked to some TV-based energy functionals and the related variational partial differential equations. Coupled energy functionals and non-local energy functionals can be used to measue the image energy more exactly. The related variational partial differential equations hold various ability on iamge processing. Base on the conventional partial differential equation models, we will research coupled energy functionals and non-local energy functionals with the related variational partial differential equations, found adaptive compressed sensing models to get more accurate result with higher efficiency for complex image reconstruction. Existence and uniqueness of the solution will be analyzed. L1 norm, L2 norm and TV norm will be applied to estimate the solution and the error, analyze the stability. Efficiency algorithm will be designed

英文关键词： Variational；Partial differential equation；Compressed sensing；Image reconstruction；Adpative