Scale Space Properties of the Multiscale Morphological Closing-opening Filter

\Scale-space" is an important recent concept used in image and signal processing and pattern recognition. Traditional scale-space is generated by a linear Gaus-sian smoothing operation. We present here a nonlinear type of smoother corresponding to the multiscale opening and closing operations of mathematical morphology which also generates a \scale-space." We show that a parabolic structuring element possesses desirable properties and we demonstrate the necessary monotonic scale-space property for this structuring element.