Abstract
The map-seeking circuit (MSC) is an explicit biologically-motivated computational mechanism which provides practical solution of problems in object recognition, image registration and stabilization, limb inverse-kinematics and other inverse problems which involve transformation discovery. We formulate this algorithm as discrete dynamical system on a set Δ=Π L ℓ=1 Δ(ℓ), where each Δ(ℓ) is a compact subset of a nonnegative orthant of ℝn, and show that for an open and dense set of initial conditions in Δ the corresponding solutions converge to either a vector with unique nonzero element in each Δ(ℓ) or to a zero vector. The first result implies that the circuit finds a unique best mapping which relates reference pattern to a target pattern; the second result is interpreted as “no match found”. These results verify numerically observed behavior in numerous practical applications.
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References
Arathorn, D.: Map-seeking: recognition under transformation using a superposition ordering property. Electron. Lett. 37(3), 164–165 (2001)
Arathorn, D.: Map-Seeking Circuits in Visual Cognition. Stanford University Press, Palo Alto (2002)
Arathorn, D.: Memory-driven visual attention: an emergent behavior of map-seeking circuits. In: Itti, L., Rees, G., Tsotsos, J. (eds.) Neurobiology of Attention. Academic/Elsevier, Dordrecht (2004)
Arathorn, D.: A cortically-plausible inverse problem solving method applied to recognizing static and kinematic 3D objects. Neural Inf. Process. Syst. (NIPS) 18 (2005)
Arathorn, D.: Computation in the higher visual cortices: map-seeking circuit theory and application to machine vision. In: Proceedings of IEEE AIPR 2004, October 2004
Arathorn, D.: Cortically plausible inverse problem method applied to complex perceptual and planning tasks. In: Proceedings SPIE Defense and Security Symposium, April 2006
Arathorn, D.: From wolves hunting elk to rubik’s cubes: are the cortices composition/decomposition engines? In: Proceedings AAAI Symposium on Connectionist Compositionality, October 2004
Arathorn, D.: The map-seeking method: from cortical theory to inverse-problem method. In: 2005–2006 IMA/MCIM Industrial Problems Seminar. http://www.ima.umn.edu/industrial/2005-2006/abstracts.html#arathorn
Harker, S., Gedeon, T., Vogel, C.: Analysis of constrained optimization variants of the map-seeking circuit algorithm. J. Math. Imaging Vis. (2007). doi: 10.1007/s1085-007-0024-7
Hale, J.K., Kocak, H.: Dynamics and Bifurcations. Springer, New York (1991)
La Salle, J.: The extent of asymptotic stability. Proc. Natl. Acad. Sci. USA 46, 363–365 (1960)
Vogel, C., Arathorn, D., Parker, A., Roorda, A.: Retinal motion tracking in adaptive optics scanning laser ophthalmoscopy. In: Proceedings of OSA Conference on Signal Recovery and Synthesis, Charlotte, NC, June 2005
Vogel, C., Arathorn, D., Roorda, A., Parker, A.: Retinal motion estimation and image dewarping in adaptive optics scanning laser ophthalmoscopy. Opt. Express 14, 487–497 (2006)
Snider, R., Arathorn, D.: Terrain discovery and navigation of a multi-articulated linear robot using map-seeking circuits. In: Proceedings SPIE Defense and Security Symposium, April 2006
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Gedeon, T., Arathorn, D. Convergence of Map Seeking Circuits. J Math Imaging Vis 29, 235–248 (2007). https://doi.org/10.1007/s10851-007-0028-3
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DOI: https://doi.org/10.1007/s10851-007-0028-3