Gradient watersheds in morphological scale-space

After introducing several results relating to the modification of the homotopy of gradient functions based on extrema in the base image and building on earlier results in morphological scale-space, we introduce a scale-space monotonicity theorem for regions of an image defined by watersheds of a gradient function modified to retain only the local minima or maxima of its smoothed parent image. We then illustrate the theorem with an example of the scale-space extraction of texture features from the nuclei of cervical cells.