Rotation invariance in gradient and higher order derivative detectors

Abstract

This paper addresses the problem on how to evaluate various operators used for estimation of derivatives in images. Such operators are extremely commonly used, for instance to detect edges. For bandlimited correctly sampled signals ideal derivative operators are easy to define. For 2D signals the first derivative operators take the form of a rotation invariant pair. Rotation invariance is also a natural requirement for the non-ideal practically implementable operators of which the Sobel operator is one example. Second degree derivators do not form a set of rotation invariant operators by itself. Instead, certain linear combinations of these derivators display this property. The Fourier domain is used extensively in the analysis, partly because the angular variation in the form of circular harmonics is preserved over the Fourier transform. Circular harmonics expansion is used for evaluation of practically implementable operator kernels. A quantity called total harmonic distortion (THD) is defined to capture the overall deviation from rotation invariance. In a comparison with two other simple kernels it seems that the Sobel operator does fairly well. Errors in magnitude and orientation estimation follow the THD-values quite closely. For all pairs of operators (convolution kernels), which are to be employed for orientation estimation, the paper presents a method to evaluate their quality. Besides, the paper brings about new insights in the interconnected concepts of edge detectors, line detectors, derivative operators, rotation invariance, and rotational symmetry.