From the Superderivative to a Relative-Gradient Edge Operator

Alejandro Garnung Menéndez

First-derivative edge detectors such as Sobel, Prewitt, Roberts and Scharr measure absolute grey-level differences. We revisit the log-gradient magnitude, a relative-gradient edge operator, and motivate it by analogy with the superderivative, a multiplicative derivative in which the independent variable is perturbed multiplicatively rather than additively. In one and two dimensions the logarithm of the superderivative is a radial logarithmic derivative; its local, origin-free, isotropic analogue is the magnitude of the logarithmic gradient. The resulting operator measures relative rather than absolute change, giving an edge-strength map that is invariant under global intensity rescaling. We present the theory, show how the operator can be implemented in two steps---a classical gradient kernel plus a pointwise division---and compare it with standard first-derivative filters on synthetic images with ground-truth edges and on the official BSDS500 benchmark. We also study a parametric family that interpolates between the absolute gradient and the log-gradient; empirically, the best members are small intermediate exponents, which edge the classical kernels by a small but statistically…