Functional, Stochastic and Geometric New Advancements of the Mumford-Shah Model

ShenDMS-0604510 调查人员开发扩展Mumford-Shah模型来处理图像分析中的某些问题，该模型并不完全令人满意。 尽管经典的Mumford-Shah模型在计算机视觉、图像处理、生物医学成像、认知感知、计算神经科学以及基于聚类和分类的一般数据挖掘等重要领域得到了成功的应用，但在对更一般的图像和视觉信号进行建模和计算时，它逐渐暴露出其不足。 这里的方法是推进Mumford-Shah模型在随机，功能和几何建模和分析三个不同的方向。 新的模型可以有效地和忠实地分析和计算更复杂的信号模式，需要固有的随机性，振荡，或高阶几何和超出了原来的Mumford-Shah模型的范围。 主要的数学工具，必不可少的项目涉及随机分析，功能分析，微分几何，非线性偏微分方程，非凸优化，科学计算。 在许多涉及视觉和图像的国家重要领域中，一个共同的基本问题是确定是否存在某种特定的感兴趣模式，无论是敌人伪装的坦克，乘客随身携带的长刀，还是大脑扫描中的肿瘤或块。 在人类智慧瞬间做出决定的背后，是复杂（但通常是潜意识的）和聪明的决策规则、平衡原则和信息块的集合。 在对象识别和提取中揭示和模仿人类如此卓越的智能是分割问题的主要目标，并且它从来没有比今天的信息时代更紧迫，因为到处都遇到大量的图像/视觉数据集。 自动化、准确性、通用性、适应性和速度是过去几十年来推动所有研究工作的关键素质。 在经典的Mumford-Shah方法的基础上，通过结合几种现代数学工具，该算法开发了新的模型，这些模型在检测真实复杂模式时更忠实，适用范围更广，更灵活，计算效率更高，更易于处理。 该项目帮助研究生在这些关键领域。 结果是有价值的科学，工程，医学和工业应用模式分类和数据分析方法被广泛使用。

ShenDMS-0604510 The investigator develops extensions to the Mumford-Shahmodel to deal with certain problems in image analysis for whichthe model is not entirely satisfactory. Despite its successfuluse in a variety of important areas including computer vision,image processing, biomedical imaging, cognitive perception,computational neural science, and general data mining withclustering and categorization, the classical Mumford-Shah modelhas gradually revealed its insufficiency in modeling andcomputing more general classes of image and visual signals. Theapproach here is to advance the Mumford-Shah model in threedistinct directions of stochastic, functional, and geometricmodeling and analysis. The new models can effectively andfaithfully analyze and compute more complex signal patterns thatdemand innate randomness, oscillations, or high-order geometricregularities and are beyond the scope of the originalMumford-Shah model. The main mathematical tools essential forthe project involve stochastic analysis, functional analysis,differential geometry, nonlinear partial differential equations,non-convex optimizations, and scientific computing. A fundamental problem common in many nationally importantareas involving vision and images is to determine whether somespecific pattern of interest is present, be it an enemy'sdisguised tank, a long knife in a passenger's carry-on, or atumor block in a brain scan. Behind the split-second decision byhuman intelligence is an ensemble of sophisticated (but oftensubconscious) and clever decision rules, balance principles, andinformation blocks. To reveal and emulate such remarkable humanintelligence in object identification and extraction is the majorgoal of the segmentation problem, and it has never been moreurgent than in today's information age when massive sets ofimage/visual data are encountered everywhere. Automation,accuracy, genericity, adaptivity, and speed are key qualitiesthat have been driving all the research efforts in the past fewdecades. Based on the classic method of Mumford-Shah, byincorporating several modern mathematical tools the investigatordevelops newer models that are more faithful in detecting realcomplex patterns, more broadly applicable and flexible, andcomputationally more efficient and tractable. The project helpstrain graduate students in these crucial areas. Results arevaluable for scientific, engineering, medical, and industrialapplications where pattern classification and data analysismethods are widely used.