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A cortical based model of perceptual completion in the roto-translation space. (English) Zbl 1088.92008

Birindelli, Isabeau (ed.) et al., Proceedings of the workshop on second order subelliptic equations and applications, Cortona, Italy, June 16–22, 2003. Potenza: Universit√† degli Studi della Basilicata, Dipartimento di Matematica, S.I.M. Lecture Notes of Seminario Interdisciplinare di Matematica 3, 145-161 (2004).
The authors are concerned with mathematical modeling of perceptual completion. The human visual system can recover missing parts of an image, complete objects and create contours. The reconstruction is performed via minimization of the elastica functional \[ \int_{\gamma}\left( 1+k^{2}\right) ds, \] where the integral is computed on the missing part of the boundary and \(k\) is its curvature. The authors suggest a model inspired by the architecture of the visual cortex which can be also considered as a lifting in the roto-translation group of models based on the elastic functional. The completion algorithm for missing boundaries is performed through a diffusion-concentration process and an orientation selectivity process.
Convergence of the diffusion concentration algorithm is proved and a numerical scheme approximating the model equations with finite differences is suggested. Two numerical experiments are considered – a completion of a figure that is only partially lifted in the roto-translation space and a classical cognitive image of Kanizsa fishes.
For the entire collection see [Zbl 1058.00010].

MSC:

92C20 Neural biology
91E30 Psychophysics and psychophysiology; perception
65N06 Finite difference methods for boundary value problems involving PDEs
92C55 Biomedical imaging and signal processing
49J45 Methods involving semicontinuity and convergence; relaxation
65R10 Numerical methods for integral transforms
68U10 Computing methodologies for image processing
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