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 O(1)O(1) 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 360_360^\circ angle takes place automatically.

中文翻译：

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用于旋转不变性检测的 Zernike 矩的恒定时间计算


我们构建了一组特殊的复值积分图像和一种算法，可以快速（即在恒定时间内）计算 Zernike 矩。该技术适用于密集检测程序，其中图像由多个尺度的滑动窗口扫描，并且每个窗口级别都需要旋转不变性。我们假设没有初步图像分割。由于所提出的积分图像和二项式展开，每个特征的提取不依赖于窗口中的像素数量，因此是 O(1)O(1) 计算。我们分析该命题的算法属性，例如：所需积分图像的数量、积分图像的复共轭性、特征提取中涉及的操作数量、基于查找表的加速可能性。我们还指出了在积分图像支持的计算背景下 Zernike 和正交 Fourier_Mellin 矩之间的联系。最后，我们演示了不同难度的检测任务的三个示例。检测器通过 RealBoost 算法针对所提出的特征进行训练。在学习时，分类器仅熟悉处于直立位置或在有限范围内旋转的目标对象的示例。在测试阶段，会自动泛化到完整的 360_360^\circ 角度。