New PDE Based Models and Numerical Techniques in Level Set Surface Processing, Imaging Science and Materials Science

Stanley J. Osher和Luminita A. Vese研究人员与初级教师，博士后研究员和学生一起开发成像科学中四阶偏微分方程的新颖有效的计算技术，包括医学成像，图像处理，计算机视觉，图形学以及材料科学。他们从Yves Meyer对之前的电视模型所做的分析开始。这给了一个独特的非线性偏微分方程和泛函分析的混合，以及新的和非常有用的模型图像分解。这导致研究人员分析成像科学中使用的更一般的四阶方程，具有几何应用和解释，以及新的有效的计算和数值方法来近似这些方程和其他相关的偏微分方程。考虑的问题包括图像分解为卡通加纹理，晶体生长中形状的伍尔夫进化，Eulerelastica模型的图像复原，通过水平集的几何图像恢复，以及从属于同一未知表面的斑块形式的无组织数据点（如LIDAR， LADAR， 3D云或地球科学数据）重建表面。这项研究极大地推动了材料科学（如微芯片设计）、图像分析、计算机视觉和图形学的发展。这些领域在美国信息技术产业、纳米科学、国土安全和医学成像领域具有重要的战略价值。研究人员和合作者开发的水平集方法和应用程序已经影响了许多技术领域——例如，b谷歌网站上的“水平集方法”有近5000个点击量，从好莱坞图形到水下爆炸，到客机控制，再到开发新的微芯片设计等等，应用范围非常广泛。研究人员的新模型减轻了处理大规模数据集的繁琐人工操作的负担，提高了领域科学家以及普通用户的效率，促进了建模和渲染任务，并简化了信息和材料处理的管道。

Stanley J. Osher and Luminita A. Vese The investigators together with junior faculty, postdoctoralfellows, and students develop novel and efficient computationaltechniques for partial differential equations of fourth orderarising in imaging science, including medical imaging, imageprocessing, computer vision, and graphics, as well as materialscience. They begin with analysis done on their previous TV modelby Yves Meyer. This gives a unique blend of nonlinear partialdifferential equation and functional analysis as well as new andvery useful models for image decomposition. This leads theinvestigators to the analysis of more general fourth orderequations used in imaging science, with geometric applicationsand interpretations, as well as new efficient computational andnumerical methods to approximate these and other related partialdifferential equations. Problems considered include imagedecomposition into cartoon plus texture, Wulff evolution ofshapes in crystal growth, image disocclusion by the Eulerelastica model, image restoration via geometry of level sets, aswell as reconstruction of surfaces from unorganized data points(such as LIDAR, LADAR, 3D cloud or geoscience data) in the formof patches belonging to the same unknown surface. This investigation dramatically advances thestate-of-the-art in material science (e.g., microchip design),image analysis, computer vision, and graphics. These fields areof great strategic value in the US information technologyindustry, in nanoscience, in homeland security, and in medicalimaging. The level set method and applications, developed by theinvestigators and collaborators, has impacted numerous areas oftechnology -- see, e.g., the Google website for close to 5,000hits on "Level Set Methods" with a spectacular range ofapplications from Hollywood graphics to underwater explosions tocontrol of passenger aircraft to developing new microchip designsand beyond. The investigators' new models reduce the burden oflaborious human operations for the processing of large-scale datasets, enhance the efficiency of domain scientists as well asordinary users, facilitate modeling and rendering tasks, andstreamline the pipeline of information and materials processing.