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Planar Numerical Signature Theory Applied to Object Recognition

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Abstract

Geometric invariants play a crucial role in the field of object recognition where the objects of interest are affected by a group of transformations. However, designing robust algorithms that are tolerant to noise and image occlusion remains an open problem. In particular, numerical signature-invariants in terms of joint invariants, as an approximation to the differential signature-invariants, suffer instability, bias, noise and indeterminacy in the resulting signatures. This paper addresses some of these issues in respect of planar signatures. To improve the stability in the Euclidean case, we replace Heron’s formula by the “accurate area” and then we demonstrate that the proposed algorithm is, not only numerically stable but is also, in terms of mean square error, a closer approximation (by at least a factor of three) compared with the original formulation of Calabi. To reduce noise in the resulting curves “the n-difference technique” and “the m-mean signature method” are introduced and we show that these methods are capable of minimizing noise by more than 90 %. The n-difference technique can also be applied to eliminate indeterminacy in the outputs. For the equiaffine case, we improve and extend the required formulation for the implementation of Signature theory for any planar meshes with a general position property. Moreover, we introduce a general formulation for the full conic sections to determine an equiaffine-invariant numerical approximation to the equiaffine arc length, measured along the given curve between any two points of the mesh. Finally, we demonstrate the discriminative power of the concept of discrete signature analysis for distinguishing normal and abnormal regions in the medical imaging domain.

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Acknowledgement

We would like to thank Peter J Olver for his advice and comments.

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Authors and Affiliations

  1. Department of Basic Sciences, Eslamshahr Branch, Islamic Azad University, Tehran, Iran

    Reza Aghayan

  2. Quantitative Medical Imaging International Institute (QMI3), Faculty of SEC, Kingston University London, Penrhyn Road, Kingston upon Thames, Surrey, KT1 2EE, UK

    Reza Aghayan, Tim Ellis & Jamshid Dehmeshki

Authors
  1. Reza Aghayan
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  2. Tim Ellis
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  3. Jamshid Dehmeshki
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Correspondence to Reza Aghayan.

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Aghayan, R., Ellis, T. & Dehmeshki, J. Planar Numerical Signature Theory Applied to Object Recognition. J Math Imaging Vis 48, 583–605 (2014). https://doi.org/10.1007/s10851-013-0427-6

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  • DOI: https://doi.org/10.1007/s10851-013-0427-6

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