Morphological scale-space using generalized zero-crossings as feature points

We prove that the scale-space of a 1D gray-scale signal based on morphological filtering satisfies the causality (no new feature points are created as scale gets larger). We refine the standard definition of zero-crossing so as to allow signals with certain discontinuity, 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. The previous work by Chen-Yan (1989) used an inappropriate definition of feature points, which led them to an incorrect proof of the causality. Our causality results does not apply to a more general 2D gray scale image. Causality results on the standard morphological blurring, obtained as byproduct, are also included