Scale-space using mathematical morphology

In this paper, we prove that the scale-space of a one-dimensional gray-scale signal based on morphological filterings satisfies causality (no new feature points are created as scale gets larger). For this we refine the standard definition of zero-crossing so as to allow signals with certain singularity, and use them to define feature points. This new definition of zero-crossing agrees with the standard one in the case of functions with second order derivative. In particular, the scale-space based on the Gaussian kernel G does not need this concept because a filtered signal G * f is always infinitely differentiable. Using this generalized concept of zero-crossing, we show that a morphological filtering based on opening (and, hence, also closing by duality) satisfies causality. We note that some previous works have mistakes which are corrected in this paper. Our causality results do not apply to more general two-dimensional gray scale images. Causality results on alternating sequential filter, obtained as byproduct, are also included.