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

研究者对Mumford-Shahmodel进行了扩展，以处理图像分析中模型不能完全令人满意的某些问题。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、计算神经科学和具有聚类和分类的一般数据挖掘等许多重要领域取得了成功，但它在对更一般类别的图像和视觉信号进行建模和计算方面逐渐暴露出不足。这里的方法是在随机、功能和几何建模和分析三个不同的方向上推进Mumford-Shah模型。新模型可以有效和忠实地分析和计算更复杂的信号模式，这些模式需要固有的随机性，振荡或高阶几何规律，并且超出了原始mumford - shah模型的范围。该项目所必需的主要数学工具包括随机分析、泛函分析、微分几何、非线性偏微分方程、非凸优化和科学计算。在许多涉及视觉和图像的国家重要领域，一个共同的基本问题是确定是否存在某种特定的感兴趣的模式，无论是敌人伪装的坦克，乘客随身携带的长刀，还是脑部扫描中的肿瘤块。在人类智能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。在物体识别和提取中揭示和模拟这种卓越的人类智能是分割问题的主要目标，在当今信息时代，当到处都遇到大量的图像/视觉数据集时，这一点比以往任何时候都更加紧迫。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品质。在Mumford-Shah的经典方法的基础上，通过结合几种现代数学工具，研究者开发了新的模型，这些模型在检测真正复杂的模式时更加忠实，更广泛地适用和灵活，并且在计算上更加高效和易于处理。该项目有助于培养这些关键领域的研究生。该结果对于模式分类和数据分析方法广泛使用的科学、工程、医学和工业应用具有重要价值。

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.