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# Principles for Automatic Scale Selection

## Tony Lindeberg

Technical report ISRN KTH NA/P--98/14--SE. Department of Numerical Analysis and Computing Science, KTH (Royal Institute of Technology), S-100 44 Stockholm, Sweden, August 1998.

In: B. Jähne (et al., eds.), Handbook on Computer Vision and Applications,, volume 2, pp 239--274, Academic Press, Boston, USA, 1999.

## Abstract

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 multi-scale representation of data is of crucial importance. Whereas conventional scale-space theory provides a well-founded framework for dealing with image structures at different scales, this theory does not directly address the problem of how to select appropriate scales for further analysis. This article outlines a systematic methodology of how mechanisms for automatic scale selection can be formulated in the problem domains of feature detection and image matching (flow 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 $\gamma$-normalized derivatives arises by necessity given the requirement that the scale selection mechanism should commute with rescalings of the image pattern. Specifically, 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 reflect the size of the image structures.

When estimating image deformations, such as in image matching and optic flow 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 flow 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 finer scales in the neighbourhood of flow field discontinuities.

Keywords: scale, scale-space, scale selection, normalized derivative, feature detection, blob detection, corner detection, frequency estimation, Gaussian derivative, scale-space, multi-scale representation, computer vision

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