Balanced Binary Tree Schemes for Computing Zernike Radial Polynomials

Zernike radial polynomials 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 Zernike radial polynomials 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 \textit{balanced binary tree} (BBT) schemes for computing Zernike radial polynomials: firstly we established an elegant formulae for computation; secondly we proposed the recursive and iterative algorithms based-on BBT; thirdly we analyzed the computational complexity of the algorithms rigorously; finally we verified and validated the performance of BBT schemes by testing the running time. Theoretic analysis shows that the computational complexity of BBT recursive algorithm 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 BBT iterative algorithm (BBTIA), which is stable and efficient for realtime applications.