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Theory of Computing Systems

Erschienen in:

02.07.2021

Generating Visual Invariants −a New Approach to Invariant Recognition

verfasst von: Reza Aghayan

Erschienen in: Theory of Computing Systems | Ausgabe 1/2022

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Abstract

This paper is devoted to a new paradigm for the invariant recognition of visual objects through introducing ‘the generating invariant’ of the underlying visual geometry which allows to numerically calculate differential signature curves in a fully group-invariant manner. Then, we utilize the results to work on the unsolved problem of approximating similarity differential invariant signatures based on suitable combination of joint invariants of the underlying group action. We also illustrate that, compared to the traditional and current schemes, the new paradigm is reliable, stable, significantly minimizes the effects of noise and indeterminacy, and Signature-inverse Theorem is correct in terms of it. Besides we demonstrate that the ratio of elapsed time and inaccuracy of the outcomes resulted from the Calabi et al.’s scheme to our formulation follow an exponential decay model. And finally, application to real images are discussed.

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Calabi et al. would have preferred they be known ‘joint differential invariants’, [9].

In their approach, a higher order differential invariant is approximated by some joint differential invariant which depends on lower order derivatives evaluated at several points on the curve. In Calabi et al.’s view, this was only a partial resolution of the difficulty, since to compute such a semi-differential invariant, each derivative appeared in it must be evaluated by a discrete approximation, therefore, the original high order differential invariant is itself approximated in the end by a fully discrete finite difference version. Besides, to maintain invariance, we must use a finite difference approximation to the semi-differential invariant by joint invariants. So it always ends up approximating differential invariants by joint invariants anyway, [9].

leading to the complicated versions of this theorem and the loss of uniqueness.

Of writing and approximating similarity DISCs by suitable combination of the underlying joint invariants.

γ is said to be s-smooth if it is s times differentiable. Also, γ is regular if its first derivative never vanishes.

2d-cross product v ×w refers to the determinant |vw| for any pair of vectors v and w.

From now on, to avoid the ambiguity caused by sign and without loss of generality, we assume κ is positive.

As mentioned earlier, the post-processing n-difference and m-mean signature techniques are able minimize the effect of noise and indeterminacy,[2].

To avoid inflection points one way is to consider convex ordinary meshes. However, this idea doesn’t work where the curve includes a line segment.

Note that, identity (27) presents the ‘full’ Euclidean signature of γ, [3].

the number of points of the sampling.

Since the similarity curvature \(\mathrm {\kappa _{S\mathbb {I}}}\) is a third order differential invariant, two smooth convex curves passing through a point p have the same similarity curvature at p iff they have a third order contact at p, i.e. one needs at least four points to approximate a similarity differential invariants.

The angle 𝜗 is calculated in terms of the third derivatives of γ and the osculating circle C at the point of interest.

A computed tomography technique by which arterial and venous vessels are visualized through the body.

The same reasons demonstrate that our formulation has also the same advantages compared to the current formulation.

Aghayan, R.: Numerical joint invariant level set formulation with unique image segmentation result, Machine Vision and Applications, https://doi.org/10.1007/s00138-020-01134-w (2021)

Aghayan, R., Ellis, T., Dehmeshki, J.: Planar numerical signature theory applied to object recognition. J. Math. Imaging Vision 48(3), 583–605 (2014)MathSciNetCrossRef

Aghayan, R.: Orientation-invariant numerically invariant joint signatures in curve analysis. Int. J. Comput. Math. 3(1), 13–30 (2018)MathSciNet

Aghayan, R.: Signature-inverse Theorem in Mesh Group-planes, arXiv:2006.03759 (2020)

Boutin, M.: Numerically invariant signature curves. Int. J. Comput. Vision 40(3), 235–248 (2000)CrossRef

Bruckstein, A.M., Katzir, N., Lindenbaum, M., Porat, M.: Similarity invariant signatures and partially occluded planar shapes. Int. J. Comput. Vision 7(3), 271–285 (1992)CrossRef

Bruckstein, A.M., Halt, R.J., Netravali, A.N., Richardson, T.J.: Invariant signatures for planar shape recognition under partial occlusion. CVGIP: Image Understanding 58, 49–65 (1993)CrossRef

Bruckstein, A.M., Netravali, A.N.: On differential invariants of planar curves and recogn-izing partially occluded planar shapes. Ann. Math. Artificial Intel. 13, 227–250 (1995)CrossRef

Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numeri-cally invariant signature curves applied to object recognition. Int. J. Comput. Vision 26, 107–135 (1998)CrossRef

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Cartan, É.: La méthode du repére mobile, la théorie des groupes continus et les espaces général-isés Exposés de Géométrie, 5 Paris Hermann et cie (1935)

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Fels, M., Olver, P.J.: Moving coframes II. Regularization and theoretical foundations. Acta Appl. Math. 55, 127–208 (1999)MathSciNetCrossRef

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Olver, P.J.: Classical Invariant Theory. Cambridge Univ Press, New York (1999)CrossRef

13.

Pauwels, E., Moons, T., Van Gool, L. J., Kempenaers, P., Oosterlinck, A.: Recognition of planar shapes under affine distortion. Int. J. Comput. Vision 14, 49–65 (1995)CrossRef

14.

Moons, T., Pauwels, E., Van Gool, L., Oosterlinck, A.: Foundations of semi-differential invariants. Int. J. Comput. Vision 14, 25–48 (1995)CrossRef

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Weiss, I.: Geometric invariants and object recognition. Int. J. Comput. Vision, vol. 10, 3 (1993)

Titel: Generating Visual Invariants −a New Approach to Invariant Recognition
verfasst von: Reza Aghayan
Publikationsdatum: 02.07.2021
Verlag: Springer US
Erschienen in: Theory of Computing Systems / Ausgabe 1/2022
Print ISSN: 1432-4350
Elektronische ISSN: 1433-0490
DOI: https://doi.org/10.1007/s00224-021-10042-z

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