Algorithms for fast computation of Zernike moments and their numerical stability

Accuracy, speed and numerical stability are among the major factors restricting the use of Zernike moments (ZMs) in numerous commercial applications where they are a tool of significant utility. Often these factors are conflicting in nature. The direct formulation of ZMs is prone to numerical integration error while in the recent past many fast algorithms are developed for its computation. On the other hand, the relationship between geometric moments (GMs) and ZMs reduces numerical integration error but it is observed to be computation intensive. We propose fast algorithms for both the formulations. In the proposed method, the order of time complexity for GMs-to-ZMs formulation is reduced and further enhancement in speed is achieved by using quasi-symmetry property of GMs. The existing q-recursive method for direct formulation is further modified by incorporating the recursive steps for the computation of trigonometric functions. We also observe that q-recursive method provides numerical stability caused by finite precision arithmetic at high orders of moment which is hitherto not reported in the literature. Experimental results on images of different sizes support our claim.

Graphical AbstractOrders of moment vs execution time for ZMs for algorithms A(existing method of ZMs using GMs), B(proposed fast method) and C(proposed fast method with quasi-symmetry).Orders of moment vs execution time for ZMs for algorithms C, D(q-recursive) and E(qθ-recursive method).Figure optionsDownload high-quality image (216 K)Download as PowerPoint slideOrders of moment vs reconstruction error (∈L) for image size 512 × 512 pixels using algorithm D.Research Highlights
► Two fast algorithms for the calculation of Zernike moments (ZMs) are developed.
► Fast algorithm (qθ-recursive) uses recurrence relation for the computation of trigonometric functions.
► Numerical stability of q-recursive method with symmetry/quasi-symmetry for very large orders of moment established.
► Fast Algorithm for computing ZMs from Geometric Moments (GMs) proposed.