用于计算泽尼克径向多项式的平衡二叉树方案
Balanced Binary Tree Schemes for Computing Zernike Radial Polynomials
AI总结：
针对泽尼克径向多项式现有计算方案存在溢出、不稳定或复杂度高的问题，提出基于平衡二叉树的递推与迭代算法，经理论分析与实验验证，迭代算法稳定高效，适用于实时场景，兼具教育价值。
摘要：
泽尼克径向多项式（Zernike radial polynomials, ZRP）在光学设计、成像系统、图像处理系统等应用领域发挥着重要作用。目前，通过计算机程序自动计算ZRP的数值方案主要有两类：一类基于定义式，其中的阶乘运算可能导致溢出问题，且高阶导数处理难度大；另一类基于递推，要么存在不稳定性，要么计算复杂度较高。本文重点研究用于计算ZRP的平衡二叉树（balanced binary tree, BBT）方案：首先建立了简洁的计算公式；其次提出了基于BBT的递推算法与迭代算法；随后严格分析了这些算法的计算复杂度；最后通过测试运行时间对BBT方案的性能进行了验证与确认。理论分析表明，平衡二叉树递推算法（balanced binary tree recursive algorithm, BBRTA）的计算复杂度为指数级，迭代算法的计算复杂度为二次级，这一结果与运行时间测试高度吻合。实验显示，在不同计算平台上，平衡二叉树迭代算法（balanced binary tree iterative algorithm, BBTIA）的耗时约为1~10微秒，具备稳定性与高效性，适用于实时应用。从STEM教育的角度来看，BBT与ZRP的关联展现了工程问题背后离散数学结构的美感与应用价值，值得向大学生、计算机程序员和光学工程师介绍。
英文摘要：
Zernike radial polynomials (ZRP) play a significant role in application areas such as optics design, imaging systems, and image processing systems. Currently, there are two kinds of numerical schemes for computing the ZRP automatically with computer programs: one is based on the definition in which the factorial operations may lead to the overflow problem and the high order derivatives are troublesome, and the other is based on recursion which is either unstable or with high computational complexity. In this paper, our emphasis is focused on exploring the balanced binary tree (BBT) schemes for computing the ZRP: firstly an elegant formulae for computation is established; secondly the recursive and iterative algorithms based-on BBT are proposed; thirdly the computational complexity of the algorithms are analyzed rigorously; finally the performance of BBT schemes by testing the running time is verified and validated. Theoretical analysis shows that the computational complexity of balanced binary tree recursive algorithm (BBRTA) and iterative algorithm are exponential and quadratic respectively, which coincides with the running time test very well. Experiments show that the time consumption is about $1\sim 10$ microseconds with different computation platforms for the balanced binary tree iterative algorithm (BBTIA), which is stable and efficient for real-time applications. In the sense of STEM education, the connection of the BBT and ZRP exhibits the beauty and applications of discrete mathematical structure behind the engineering problem, which is worthy of introducing to the college students, computer programmers and optics engineers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
