This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Spatial Reasoning with Incomplete Information on Relative Positioning
September 2005 (vol. 27 no. 9)
pp. 1473-1484
This paper describes a probabilistic method of inferring the position of a point with respect to a reference point knowing their relative spatial position to a third point. We address this problem in the case of incomplete information where only the angular spatial relationships are known. The use of probabilistic representations allows us to model prior knowledge. We derive exact formulae expressing the conditional probability of the position given the two known angles, in typical cases: uniform or Gaussian random prior distributions within rectangular or circular regions. This result is illustrated with respect to two different simulations: The first is devoted to the localization of a mobile phone using only angular relationships, the second, to geopositioning within a city. This last example uses angular relationships and some additional knowledge about the position.

[1] B. Landau and R. Jackendoff, “‘What’ and ‘Where’ in Spatial Language and Spatial Cognition,” Behavioral and Brain Sciences, vol. 16, pp. 217-265, 1993.
[2] H.F. Durrant-Whyte, “Uncertain Geometry in Robotics,” IEEE J. Robotics and Automation, vol. 4, no. 1, pp. 23-31, Feb. 1988.
[3] J. Borenstein, H.R. Everett, and L. Feng, Where am I? Sensor and Methods for Mobile Robot Positioning. Univ. of Michigan, 1996.
[4] R. Smith, M. Self, and P. Cheeseman, Estimating Uncertain Spatial Relationships in Robotics, chapter 3, pp. 167-193, Springer-Verlag, 1990.
[5] R. Chatila and J.-P. Laumond, “Position Referencing and Consistent World Modeling for Mobile Robots,” Proc. IEEE Int'l Conf. Robotics and Automation, pp. 138-145, 1985.
[6] R. Smith and P. Cheeseman, “On the Representation and Estimation of Spatial Uncertainty,” The Int'l J. Robotics Research, vol. 5, no. 4, pp. 56-68, 1987.
[7] H.P. Moravec and A. Elfes, “High Resolution Maps from Wide Angle Sonar,” Proc. IEEE Int'l Conf. Robotics and Automation, pp. 116-121, 1985.
[8] W. Burgard, D. Fox, D. Hennig, and T. Schmidt, “Estimating the Absolute Position of a Mobile Robot Using Position Probability Grids,” Proc. 14th Nat'l Conf. Artificial Intelligence (AAAI), vol. 2, pp. 896-901, 1996.
[9] D. Fox, W. Burgard, and S. Thrun, “Active Markov Localization for Mobile Robots,” Robotics and Autonomous Systems, vol. 25, pp. 195-207, 1998.
[10] J.J. Leonard and H.F. Durrant-Whyte, “Mobile Robot Localization by Tracking Geometric Beacons,” IEEE Trans. Robotics and Automation, vol. 7, no. 3, pp. 376-382, June 1991.
[11] M. Betke and L. Gurvits, “Mobile Robot Localization Using Landmarks,” IEEE Trans. Robotics and Automation, vol. 13, no. 2, pp. 251-263, 1997.
[12] J.R. Spletzer and C.J. Taylor, “A Bounded Uncertainty Approach to Multi-Robot Localization,” Proc. Int'l Conf. Intelligent Robots and Systems, pp. 1258-1265, 2003.
[13] L. Doherty, K.S.J. Pister, and L. El Ghaoui, “Convex Position Estimation in Wireless Sensor Networks,” Proc. IEEE Infocom, vol. 3, pp. 1655-1663, 2001.
[14] L. Vieu, “Spatial Representation and Reasoning in Artificial Intelligence,” Spatial and Temporal Reasoning, pp. 5-41. Dordrecht, Kluwer, 1997.
[15] A.G. Cohn, “Qualitative Spatial Representations,” Proc. IJCAI99 Workshop Adaptive Spatial Representations of Dynamic Environments, pp. 33-52, 1999.
[16] G. Ligozat, “Reasoning about Cardinal Directions,” J. Visual Languages and Computing, vol. 9, pp. 23-44, 1998.
[17] J.F. Allen, “Maintaining Knowledge about Temporal Intervals,” Comm. ACM, vol. 26, no. 11, pp. 832-843, 1983.
[18] S.K. Chang, Q.Y. Shi, and C.W. Yan, “Iconic Indexing by 2D Strings,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 9, no. 3, pp. 413-428, May 1987.
[19] C. Freksa and K. Zimmermann, “On the Utilization of Spatial Structures for Cognitively Plausible and Efficient Reasoning,” Proc. IEEE Int'l Conf. Systems, Man, and Cybernetics, pp. 261-266, Oct. 1992.
[20] J. Freeman, “The Modeling of Spatial Relations,” Computer Graphics and Image Processing, vol. 4, pp. 156-171, 1975.
[21] K. Miyajima and A. Ralescu, “Spatial Organization in 2D Segmented Images: Representation and Recognition of Primitive Spatial Relations,” Fuzzy Sets and Systems, vol. 65, pp. 225-236, 1994.
[22] J.M. Keller and X. Wang, “Comparison of Spatial Relation Definitions in Computer Vision,” Proc. Third Int'l Symp. Uncertainty Modeling and Analysis and Ann. Conf. North Am. Fuzzy Information Processing Soc., pp. 679-684, Sept. 1995.
[23] P. Matsakis and L. Wendling, “A New Way to Represent the Relative Position between Areal Objects,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 7, pp. 634-642, July 1999.
[24] J.M. Keller and X. Wang, “Learning Spatial Relationships in Computer Vision,” Proc. Int'l Conf. Fuzzy Systems, pp. 118-124, 1996.
[25] L.T. Koczy, “On the Description of Relative Position of Fuzzy Patterns,” Pattern Recognition Letters, vol. 8, pp. 21-28, 1988.
[26] I. Bloch, “Fuzzy Relative Position between Objects in Image Processing: A Morphological Approach,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 7, pp. 657-664, july 1999.
[27] I. Bloch and A. Ralescu, “Directional Relative Position between Objects in Image Processing: A Comparison between Fuzzy Approaches,” Pattern Recognition, vol. 36, pp. 1563-1582, 2003.
[28] S. Dutta, “Approximate Spatial Reasoning: Integrating Qualitative and Quantitative Constraints,” Int'l J. Approximate Reasoning, vol. 5, pp. 307-331, 1991.
[29] Fuzzy Logic Techniques for Autonomous Vehicle Navigation, Studies in Fuzziness and Soft Computing, D. Driankov and A. Saffiotti, eds. Springer-Physica Verlag, 2001.
[30] L. Talmy, “How Language Structures Space,” Spatial Orientation: Theory, Research and Application, H.L. Pick and L.P. Acredolo, eds. New York: Plenum Press, 1983.
[31] E.T. Jaynes, “Information Theory and Statistical Mechanics,” Physical Rev., vol. 106, no. 4, pp. 620-630, 1957.
[32] R. Dehak, “Inférence Quantitative des Relations Spatiales Directionnelles,” PhD thesis, Ecole Nationale Supérieure des Télécommunications, Paris, France, 2002E043, 2002.
[33] R. Simmons and S. Koenig, “Probabilistic Robot Navigation in Partially Observable Environments,” Proc. Int'l Joint Conf. Artificial Intelligence, pp. 1080-1087, 1995.
[34] P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection. New York: John Wiley, 1987.
[35] E. Grosicki, K. Abed-Meraim, and R. Dehak, “A Novel Method to Fight the Non-Line-of-Sight Error in AOA Measurements for Mobile Location,” Proc. IEEE Int'l Conf. Comm., vol. 5, pp. 2794-2798, 2004.

Index Terms:
Index Terms- Probabilistic geometry, spatial reasoning, geometrical inference.
Citation:
Sidi Mohammed R?da Dehak, Isabelle Bloch, Henri Ma?tre, "Spatial Reasoning with Incomplete Information on Relative Positioning," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 9, pp. 1473-1484, Sept. 2005, doi:10.1109/TPAMI.2005.186
Usage of this product signifies your acceptance of the Terms of Use.