Local Appropriate Scale in Morphological Scale-Space

This paper presents a novel approach to selecting appropriate scales for region detection prior to feature localization. We develop and formalize a number of requirements that should be fulfilled by such an appropriate scale operator and show by theoretical considerations and experiments that a morphological opening-closing scale-space meets these requirements better than Gaussian scale-space. As a prerequisite for appropriate scale measurements we generalize morphological decomposition methods and introduce a morphological band-pass filter. It decomposes an image into structures of different size and different curvature polarity ("light and dark blobs"). It may thus be seen as a morphological analogy to the important Laplacian of Gaussian operator. The local appropriate scale is than defined as the scale that maximizes the response of the band-pass filter at each point. This operator has a number of interesting properties: Most notably it gives constant scale values in a region of con stant width, and its zero-crossings coincide with local maxima of the gradient magnitudes. Some example applications show that the new operator is very useful to tune subsequent operators towards optimal scales.