J. Eur. Opt. Society-Rapid Publ. 22, 31( 2026) 317
Figure
4. Accuracy comparisons( a)–( f) at six positions on the unit circle:( x, y)( �0.5000000, �0.8660254),( �0.7071068, �0.7071068),( �0.5000000, 0.8660254),( 0.5000000, 0.8660254).( 0.5000000, �0.8660254), and( �0.8660254, �0.5000000). Each comparison includes three curves: the one above for 5050 Zernike components of orders 0 to 99, calculated using the direct transformation method, as the comparison reference; the middle one for the absolute difference of the component-wise recursive method to the reference; and the curve below for the block-wise recursive method. All difference curves are shown with the same value scale: 4 10 �13.
recursion) were further compared with the direct transformation method( equation( 10)). Unlike that the two recursive methods, which were fully implemented in MATLAB with double precision, the direct transformation method requires additional support from an external library for“ arbitrary-precision-arithmetic” due to the rapidly increasing integer values( i. e., binomial factors) for polynomials of order higher than 50. The comparison results for the six positions with the largest deviations, all located on the unit circle, are shown in Figure 4. It is evident that for both recursive methods, all deviations are smaller than approximately 4 10 �13 compared to the direct transformation method.
It should be noted that this work does not constitute a definitive assessment of the accuracy of one method compared to others, but rather a preliminary evaluation to support the conceptual investigation of the relationship between Zernike polynomials and Pascal’ s triangle. Further evaluations with more detailed application scenarios are to be expected, as the difference curves reveal some orderdependent structural information which, although very small, is not random noise.
4.3 Memory requirement
Memory requirements are often a critical factor in development due to detailed application scenarios. This work offers high flexibility for practical system / algorithm development: Block-wise recursive computation for generating basis functions requires no additional memory; similarly, virtually no additional memory is needed for surface / wavefront analysis, which uses transformation matrices for coefficient operations and derivative computations, since all transformation matrices can be rapidly computed by block-wise recursive algorithms, e. g., equation( 11) for the forward coordinate system transformation in Zernike computations.
5 Conclusion
In this work, the computation of Zernike basis functions was divided into two parts: the part irrelevant to the coordinate values and the part relevant to the coordinate values. The first part comprises the computation of the coordinate transformation along with an orthogonalization process( based on homogeneous xy polynomials). This leads to two cases of block-wise recurrence in the coordinate-independent Zernike computation. The use of these two blockwise recurrences, supported by Pascal triangle, enables a block-wise direct transformation method and a block-wise recursive computation method for Zernike polynomials. The latter is more suitable for computing basis functions, while the former is better suited for operations and analyses