Constant-Time Calculation of Zernike Moments for Detection with Rotational Invariance

We construct a set of special complex-valued integral images and an algorithm that allows to calculate Zernike moments fast, namely in constant time. The technique is suitable for dense detection procedures, where the image is scanned by a sliding window at multiple scales, and where rotational invariance is required at the level of each window. We assume no preliminary image segmentation. Owing to the proposed integral images and binomial expansions, the extraction of each feature does not depend on the number of pixels in the window and thereby is an <inline-formula><tex-math notation="LaTeX">$O(1)$</tex-math><alternatives><inline-graphic xlink:href="bera-ieq1-2803828.gif"/></alternatives></inline-formula> calculation. We analyze algorithmic properties of the proposition, such as: number of needed integral images, complex-conjugacy of integral images, number of operations involved in feature extraction, speed-up possibilities based on lookup tables. We also point out connections between Zernike and orthogonal Fourier–Mellin moments in the context of computations backed with integral images. Finally, we demonstrate three examples of detection tasks of varying difficulty. Detectors are trained on the proposed features by the RealBoost algorithm. When learning, the classifiers get acquainted only with examples of target objects in their upright position or rotated within a limited range. At the testing stage, generalization onto the full <inline-formula><tex-math notation="LaTeX">$360^\circ$</tex-math><alternatives><inline-graphic xlink:href="bera-ieq2-2803828.gif"/></alternatives></inline-formula> angle takes place automatically.