SciELO - Scientific Electronic Library Online

vol.56 issue2Análisis cuantitativo de interacciones moleculares proteína-proteína mediante la combinación de microarreglos y un lector óptico basado en el fenómeno de resonancia de plasmones superficialesAnisotropic cosmology in Sáez-Ballester theory: classical and quantum solutions author indexsubject indexsearch form
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • Have no similar articlesSimilars in SciELO


Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.56 n.2 México Apr. 2010




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:,,


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



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.



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





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.



1. J.A. Anderson, Mathematical Bioscience 5 (1972) 197.        [ Links ]

2. V. Chinarov and M. Menzinger, Biosystems (Elsevier Science) 68 (2003) 147.        [ Links ]

3. W.K. Clifford, American Journal of Mathematics 1 (1878) 350.        [ Links ]

4. B. Cruz, H. Sossa, and R. Barrón, Neural Process. Lett. 25 (2007) 1.        [ Links ]

5. B. Cruz, R. Barrón, and H. Sossa, "Geometric Assocaitives Memories and their Applications to Pattern Classification." In (to appear in) Geometric Algebra Computing for Computing Science and Engineering, by Bayro Corrochano and G. Sheuermann (London: Springer Verlag, 2009).        [ Links ]

6. B. Cruz, R. Barrón, and H. Sossa, "Geometric Associative Memory Model with Application to Pattern Classification." In Proc. of 3rd Internat. Conf. on Appl. of Geom. Algebras in Comput. Sci. and Eng., AGACSE 2008, 2008.        [ Links ]

7. Fukushima, Kunihiko. "Restoring partly occluded patterns: a neural network model." Neural Networks (Elsevier Science) 18, no. 1 (2005): 33–43.        [ Links ]

8. Gonzalez, Rafael C., and Richard E. Woods. Digital Image Processing. Third Edition. Upper Saddle River, New Jersey: Pearson Prentice Hall, 2008.        [ Links ]

9. Hestenes, David. "Old Wine in New Bottles." In Geometric Algebra: A Geometric Approach to Computer Vision, Quantum and Neural Computing, Robotics, and Engineering, by Eduardo Bayro–Corrochano and Garret Sobczyk, 498–520. Boston: Birkhauser, 2001.        [ Links ]

10. Hestenes, David, and Garret Euguene Sobczyk. Clifford Algebra to Geometric Calculus. Kluwer: Springer Verlag, 1984.        [ Links ]

11. Hestenes, David, Hongbo Li, and Alyn Rockwood. "New Algebraic Tools for Classical Geometry." Geometric Computing with Clifford Algebras, 2001.        [ Links ]

12. Hildenbrand, Dietmar. Geoemtric Computing in Computer Graphics using Confromal Geoemtric Algebra. Tutorial, TU Darmstadt, Germany: Interactive Graphics Systems Group, 2005.        [ Links ]

13. Hitzer, Eckhard. "Euclidean Geometric Objects in the Clifford Geometric Algebra of (Origin, 3–Space, Infinity)." Bulletin of the Belgian Mathematical Society 11, no. 5 (2004): 653–662.        [ Links ]

14. Hopfield, John Joseph. "Neural Networks and physicals systems with emergent collective computational abilities." Proceedings of the National Academy of Sciences C–79 (1982): 2554–2558.        [ Links ]

15. Kohonen, Teuvo. "Correlation Matrix Memories." IEEE Transactions on Computer C–21, no. 4 (1972): 353–359.        [ Links ]

16. Ritter, Gerhard X., Peter Sussner, and Juan Luis Diaz de Leon. "Morphological Associative Memories." IEEE Transactions on Neural Networks, 1998: 281–293.        [ Links ]

17. Sossa, Juan Humberto, and Ricardo Barrón. "New Associative Model for Pattern Recall in the presence of Mixed Noise." IASTED Fith International Conference on Singal and Image Processing (SIP 2003), 2003: 485–490.        [ Links ]

18. Steinbuch, Karl. "die Lernmatrix." Kybernetik 1, no. 1 (1961): 26–45.        [ Links ]

19. Sussner, Peter. "Observations on Morphological Associative Memories and the kernel method." Neurocomputing, no. 31 (2000): 167–183.        [ Links ]

20. Willshaw, David J., O. Peter Buneman, and Hugh Christopher Longuet–Higgins. "Non–holographic associative memory." Nature 222 (1969): 960–962.        [ Links ]

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License