CAREER: Mathematical Foundations of Computer Graphics

这项研究的目的是通过仔细阐述计算机图形学的数学基础，应用计算机科学的正式方法，并探索与其他学科的基本联系，来加强计算机图形学的理论基础。工作计划分为四类：1)数学基础，2)数值方法，3)计算复杂性，4)形式方法。在每个类别中都描述了几个具体的项目。在这笔赠款下完成的工作将有别于以前在所应用的工具和所调查的领域中所做的工作。将应用的新工具包括泛函分析的数学方法(例如测量理论)、基于信息的复杂性(例如信息半径)和计算机科学的形式化方法(例如精化演算)；这些工具将应用于计算机图形学的基本问题，例如推导和澄清辐射测量原理，对基于图像的绘制的精度施加先验限制，以及区分非朗伯场景的图像。这些项目植根于作者之前的工作。在需要研究的新领域中，包括证明渲染算法的正确性，响应用户查询而对渲染算法进行反转，以及形式化使用默认假设、歧义和人机交互中的矛盾。这项工作的基本教育目标是向计算机图形学注入适当的数学结构，培训未来的图形学研究人员构建严格证明和形式可验证的算法的艺术，并将计算机图形学的工具和基本概念整合到核心计算机科学课程中。这些目标不是通过引入一门课程就能实现的，而是需要让学生接触到许多层次上的必要概念。为此，将把计算机图形学的元素引入现有的计算理论二年级课程，并将设计一门研究生级别的高级课程，探索计算机图形学、人机交互和人工智能的相互作用，同时强调数学抽象和形式验证的作用。

The objective of this research is to strengthen the theoretical foundations of computer graphics by carefully formulating its mathematical underpinnings, applying formal methods of computer science, and exploring fundamental connections with other disciplines. The work plan is organized into four categories: 1) mathematical underpinnings, 2) numerical methods, 3) computational complexity, and 4) formal methods. Within each category several specific projects are described.Work done under this grant will depart from previous work in both the tools applied and in the areas investigated. Among the novel tools to be applied are mathematical methods from functional analysis (e.g. measure theory), information-based complexity (e.g. radius of information), and formal methods of computer science (e.g. refinement calculus); these tools will be applied to fundamental problems of computer graphics, such as deriving and clarifying radiometric principles, placing a priori limits on the accuracy of image-based rendering, and differentiating images of non-Lambertian scenes. These projects are firmly rooted in previous work performed by the author.Among the novel areas to be investigated are proving the correctness of rendering algorithms, "inverting" rendering algorithms in response to user queries, and formalizing the use of default assumptions, ambiguity, and contradiction in human-computer interaction. The fundamental educational objectives of this work are to infuse computer graphics with appropriate mathematical structure, to train future graphics researchers in the art of constructing rigorous proofs and formally verifiable algorithms, and to integrate the tools and fundamental concepts of computer graphics into the core computer science curriculum. These goals cannot be attained through the introduction of a single course, but will instead require exposing students to the necessary concepts at many levels. Toward this end, elements of computer graphics will be introduced into an existing sophomore-level course on the theory of computation, and an advanced graduate-level course will be devised that explores the interplay of computer graphics, human-computer interaction, and artificial intelligence, while emphasizing the role of mathematical abstraction and formal verification.