Investigación

Geometric associative memories applied to pattern restoration

B. Cruz, R. Barrón, and H. Sossa

Centro de Investigación en Computación – Instituto Politécnico Nacional, Av. Juan de Dios Bátiz and M. Othón de Mendizabal México, D.F. 07738. MÉXICO, Tel. 5729 6000 ext, 56512. Fax 5729 6000 ext. 56607. E–mail: benji@helgrind.net, tbarron@cic.ipn.mx, hsossa@cic.ipn.mx

Recibido el 5 de octubre de 2009.
Aceptado el 8 de febrero de 2010.

Abstract

Two main research areas in Pattern Recognition are pattern classification and pattern restoration. In the literature, many models have been developed to solve many of the problems related to these areas. Among these models, Associative Memories (AMs) can be highlighted. An AM can be seen as a one–layer Neural Network. Recently, a Geometric Algebra based AM model was developed for pattern classification, the so–called Geometric Associative Memories (GAMs). In general, AMs are very efficient for restoring patterns affected BY either additive or subtractive noise, but in the case of mixed noise their efficiency is very poor. In this work, modified GAMs are used to solve the problem of pattern restoration. This new modification makes use of Conformal Geometric Algebra principles and optimization techniques to completely and directly restore patterns affected by (mixed) noise. Numerical and real examples are presented to test whether the modification can be efficiently used for pattern restoration. The proposal is compared with other reported approaches in the literature. Formal conditions are also given to ensure the correct functioning of the proposal.

Keywords: Associative memories; pattern restoration; mixed noise; conformal geometric algebra.

Resumen

Dos áreas de investigación muy importantes en reconocimiento de patrones son la clasificación y la restauración de patrones. En la literatura, se han propuesto muchos modelos para resolver varios de los problemas relacionados con estas dos áreas. Entre estos modelos, hay que resaltar a las memorias asociativas (MA). Una MA puede ser vista como red neuronal de una sola capa. Recientemente, un nuevo modelo de MA basado en la llamada álgebra geométrica fue desarrollado para la clasificación de patrones: las llamadas memorias asociativas geométricas (MAG). En general, las MA son muy eficientes en la restauración de patrones afectados por ruido ya sea aditivo o substractivo, pero en el caso de ruido mezclado su eficiencia es muy pobre. En este trabajo se utilizan MAGS modificadas para resolver el problema de la restauración de patrones. Esta nueva modificación hace uso de principios del álgebra geométrica conforme y de técnicas de optimización para restaurar patrones afectados con ruido mezclado en forma directa y completa. Se presentan, además, ejemplos numéricos y con datos reales para probar la propuesta. Finalmente, se presenta una comparación con otras reportadas en la literatura. También se proporcionan algunas condiciones que garantizan el funcionamiento de la propuesta.

Descriptores: Memorias asociativas; restauración de patrones; ruido mixto; álgebra geométrica conforme.

PACS: 89.20.Ff; 87.57.Nk; 87.80.Xa

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Acknowledgements

The authors thank the National Polytechnic Institute of Mexico (SIP–IPN) under grants 20090620 and 20091421. Humberto Sossa thanks CINVESTAV–GDL for the support to do a sabbatical stay from December 1, 2009 to May 31, 2010. Authors also thank the European Union, the European Commission and CONACYT for the economic support. This paper has been prepared by economic support of the European Commission under grant FONCICYT 93829. The content of this paper is the exclusive responsibility of the CIC–IPN and it cannot be considered that it reflects the position of the European Union. We thank also the reviewers for their comments for the improvement of this paper.

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