Principles for Automatic Scale Selection

An inherent property of objects in the world is that they only exist as meaningful entities over certain ranges of scale. If one aims at describing the structure of unknown real-world signals, then a multiscale representation of data is of crucial importance. Whereas conventional scale-space theory provides a well-founded framework for dealing with image structures at di erent scales, this theory does not directly address the problem of how to select appropriate scales for further analysis. This chapter outlines a systematic methodology of how mechanisms for automatic scale selection can be formulated in the problem domains of feature detection and image matching ( ow estimation), respectively. For feature detectors expressed in terms of Gaussian derivatives, hypotheses about interesting scale levels can be generated from scales at which normalized measures of feature strength assume local maxima with respect to scale. It is shown how the notion of -normalized derivatives arises by necessity given the requirement that the scale selection mechanism should commute with rescalings of the image pattern. Speci cally, it is worked out in detail how feature detection algorithms with automatic scale selection can be formulated for the problems of edge detection, blob detection, junction detection, ridge detection and frequency estimation. A general property of this scheme is that the selected scale levels re ect the size of the image structures. When estimating image deformations, such as in image matching and optic ow computations, scale levels with associated deformation estimates can be selected from the scales at which normalized measures of uncertainty assume local minima with respect to scales. It is shown how an integrated scale selection and ow estimation algorithm has the qualitative properties of leading to the selection of coarser scales for larger size image structures and increasing noise level, whereas it leads to the selection of ner scales in the neighbourhood of ow eld discontinuities.