Mathematical Sciences: Statistical Image Models

拟议的研究重点是统计方面的两类图像模型：吉布斯随机场（GRF）的离散晶格（第一部分）和变形模板连续域（第二部分）。 对于第一类，目的是适合GRF，其生成具有类似于自然纹理的分析和视觉特征的图像。 这包括参数估计和模型选择：前者是一个推断问题，估计未知参数（可能是高维的）中包含的能量函数，导致GRF;后者是一个多重决策问题，选择一个能量函数从一个有限的候选人。 还将考虑一些复杂的问题，如可能的长期依赖性和间接观测。 第二类是在激光雷达传感器的目标检测的背景下研究的。 所考虑的特定对象是无约束环境中的人脸。 人脸检测被表述为贝叶斯推理问题，它包括先验（形状模型），似然（数据模型）和算法（从后验的模拟和基于后验的推理）。特别地，模拟将通过跳跃扩散过程进行，并且推断是识别与最大后验概率对应的对象的数量。 纹理是各种图像的主要特征。 第一部分的研究尝试从统计的角度渲染纹理。 例如，木纹显示出很强的方向性，而沙纹显示出各向同性和随机性。这样的差异可以在由计算机图形学使用相同模型的不同参数创建的图像中表示。 第二部分的研究将试图回答的问题"有没有任何具体的对象在图片？”“如果有，有多少？"这里采用了一种基于模型的方法，不仅根据观察到的图像提供了对这些问题的答案，而且还量化了与每个答案相关的机会。 因此，我们应该寻找更有可能（有更大机会）是正确的答案。

The proposed research focuses on statistical aspects of two classes of image models: Gibbs random fields (GRFs) on discrete lattices (Part I) and deformable templates on continuous domains (Part II). For the first class, the purpose is to fit a GRF which generates images with analytical and visual characteristics similar to those of natural textures. This includes parameter estimation and model selection: the former is an inference problem of estimating the unknown parameter (maybe high dimensional) contained in the energy function that induces the GRF; the latter is a multiple decision problem of choosing an energy function from a finite set of candidates. Some complicated issues, such as possible long-range dependence and indirect observations, will also be considered. The second class is studied in the context of object detection from laser radar sensors. The particular objects considered are human faces in an unconstrained environment. The face detection is formulated as a Bayesian inference problem, which consists of the prior (shape models), the likelihood (data models) and the algorithms (simulation from the posterior and inference based on the posterior). In particular, the simulation will be carried out by jump-diffusion processes and the inference is to identify the number of object(s) corresponding to the maximal posterior probability. Texture is a dominant feature in representation of various kinds of images. The first part of the proposed research attempts to render textures from statistical point of view. For instance, wood grain shows strong directional tendency but sand patterns appear isotropic and random. Such a difference can be represented in images created by computer graphics using different parameters for the same model. The second part of the proposed research will try to answer the questions "Is there any specific object in the picture?" "If yes, how many?" Here a model-based approach is taken which not only provides an swers to those questions based on the observed images but also quantifies the chance associated with each answer. Therefore, we should seek answers that are more likely (with greater chance) to be correct.