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Discrete derivative approximations with scale-space properties: A basis for low-level feature extraction

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Abstract

This article shows how discrete derivative approximations can be defined so thatscale-space properties hold exactly also in the discrete domain. Starting from a set of natural requirements on the first processing stages of a visual system,the visual front end, it gives an axiomatic derivation of how a multiscale representation of derivative approximations can be constructed from a discrete signal, so that it possesses analgebraic structure similar to that possessed by the derivatives of the traditional scale-space representation in the continuous domain. A family of kernels is derived that constitutediscrete analogues to the continuous Gaussian derivatives.

The representation has theoretical advantages over other discretizations of the scale-space theory in the sense that operators that commute before discretizationcommute after discretization. Some computational implications of this are that derivative approximations can be computeddirectly from smoothed data and that this will giveexactly the same result as convolution with the corresponding derivative approximation kernel. Moreover, a number ofnormalization conditions are automatically satisfied.

The proposed methodology leads to a scheme of computations of multiscale low-level feature extraction that is conceptually very simple and consists of four basic steps: (i)large support convolution smoothing, (ii)small support difference computations, (iii)point operations for computing differential geometric entities, and (iv)nearest-neighbour operations for feature detection.

Applications demonstrate how the proposed scheme can be used for edge detection and junction detection based on derivatives up to order three.

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  1. Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Computational Vision and Active Perception Laboratory, S-100 44, Stockholm, Sweden

    Tony Lindeberg

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Lindeberg, T. Discrete derivative approximations with scale-space properties: A basis for low-level feature extraction. J Math Imaging Vis 3, 349–376 (1993). https://doi.org/10.1007/BF01664794

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