CAREER: Mathematical Foundations of Computer Graphics

本研究的目的是加强计算机图形学的理论基础，通过仔细制定其数学基础，应用计算机科学的形式化方法，并探索与其他学科的基本联系。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)形式化方法。在每个类别中都描述了几个特定的项目。在这笔赠款下所做的工作将在所使用的工具和所调查的领域中不同于以往的工作。应用的新工具包括功能分析中的数学方法（如测度论）、基于信息的复杂性（如信息半径）和计算机科学的形式化方法（如精算）；这些工具将应用于计算机图形学的基本问题，例如推导和澄清放射学原理，对基于图像的渲染的准确性施加先验限制，以及区分非朗伯场景的图像。这些项目牢牢扎根于作者以前所做的工作。要研究的新领域包括证明呈现算法的正确性，“反转”呈现算法以响应用户查询，以及形式化人机交互中默认假设、歧义和矛盾的使用。这项工作的基本教育目标是为计算机图形学注入适当的数学结构，培养未来图形学研究人员构建严格证明和正式可验证算法的艺术，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。这些目标不能通过引入单一课程来实现，而是需要让学生在许多层面上接触必要的概念。为此，我们将在现有的二年级计算理论课程中引入计算机图形学的元素，并设计一门高级研究生课程，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化验证的作用。

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.