Mathematical Sciences: Statistical Image Models

本文主要研究两类图像模型的统计特性：离散格子上的Gibbs随机场(第一部分)和连续区域上的可变形模板(第二部分)。对于第一类，其目的是适应生成具有与自然纹理相似的分析和视觉特征的图像的GRF。这包括参数估计和模型选择：前者是估计导致GRF的能量函数中包含的未知参数(可能是高维)的推理问题；后者是从有限的候选集合中选择能量函数的多决策问题。一些复杂的问题，如可能的长期依赖和间接观察，也将被考虑。第二类是在激光雷达传感器目标检测的背景下研究的。所考虑的特定对象是不受约束的环境中的人脸。人脸检测被描述为一个贝叶斯推理问题，它由先验(形状模型)、似然(数据模型)和算法(后验模拟和基于后验推理)组成。特别地，该模拟将通过跳跃扩散过程来进行，其推论是识别与最大后验概率相对应的对象数目(S)。纹理是表示各种图像的主要特征。本研究的第一部分试图从统计学的角度对纹理进行渲染。例如，木材纹理表现出强烈的方向性，而沙纹则表现出各向同性和随机性。这种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。这项拟议研究的第二部分将试图回答这样的问题：“图片中有什么特定的物体吗？”“如果是，有多少？”这里采用了一种基于模型的方法，它不仅根据观察到的图像为这些问题提供了一个SWERS，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(更有可能)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.