Local Appropriate Scale in Morphological Scale-Space.pdf

Local Appropriate Scale in MorphologicalScale-SpaceUllrich KotheFraunhofer Institute for Computer Graphics RostockJoachim-Jungius-Str. 9, 18059 Rostock, GermanyEmail: koethe@egd.igd.fhg.deAbstract. This paper discusses the problem of selecting appropriatescales for region detection prior to feature localization. We argue thatan approach in morphological opening-closing scale-space is better thanone in Gaussian scale-space. The proposed operator is based on a newshape decomposition method called morphological band-pass lter thatdecomposes an image into structures of dierent size and dierent cur-vature polarity. Local appropriate scale is then dened as the scale thatmaximizes the response of the band-pass lter at each point. This op-erator gives constant scale values in a region of constant width, and itszero-crossings coincide with local maxima of the gradient magnitudes.Its usefulness is demonstrated by some examples.1 IntroductionSince their introduction by Witkin [14] scale-space representations have becomea universal approach to a wide variety of computer vision tasks. They are basedon the observation that real world objects and their projections onto imagesexist as meaningful entities only over certain ranges of scale. By making scale aparameter, an image can be transformed into a family of gradually simplyedversions of itself. The scale parameter controls the amount of smoothing, thusthe greater it is the more ne scale information is suppressed.The most common implementation of this idea is the Gaussian scale-spacewhich is dened by a convolution of the image f(x) with a Gaussian kernelwhere the scale parameter determines the width of the Gaussian. The propertiesof this scale-space have been studied intensively by several researchers, see e.g.Lindeberg [8]. Since it is a pure scale-space, i.e. it does not require any priorknowledge about the image content and treets all scales equally, an importantquestion arises: If no scale is special in any way, how