This Article 
 Bibliographic References 
 Add to: 
Visualization of Spatiotemporal Behavior of Discrete Maps via Generation of Recursive Median Elements
February 2010 (vol. 32 no. 2)
pp. 378-384
B.S. Daya Sagar, Indian Statistical Institute-Bangalore Centre, Bangalore
Spatial interpolation is one of the demanding techniques in Geographic Information Science (GISci) to generate interpolated maps in a continuous manner by using two discrete spatial and/or temporal data sets. Noise-free data (thematic layers) depicting a specific theme at varied spatial or temporal resolutions consist of connected components either in aggregated or in disaggregated forms. This short paper provides a simple framework: 1) to categorize the connected components of layered sets of two different time instants through their spatial relationships and the Hausdorff distances between the companion-connected components and 2) to generate sequential maps (interpolations) between the discrete thematic maps. Development of the median set, using Hausdorff erosion and dilation distances to interpolate between temporal frames, is demonstrated on lake geometries mapped at two different times and also on the bubonic plague epidemic spread data available for 11 consecutive years. We documented the significantly fair quality of the median sets generated for epidemic data between alternative years by visually comparing the interpolated maps with actual maps. They can be used to visualize (animate) the spatiotemporal behavior of a specific theme in a continuous sequence.

[1] M. Worboys and M. Duckam, GIS: A Computing Perspective. CRC Press, 2004.
[2] A.U. Frank, “GIS for Politics,” Proc. Ann. Conf. GIS Planet '98 1998.
[3] R.T. Snodgrass, “Temporal Databases,” Theories and Methods of Spatiotemporal Reasoning in Geographic Space, A.U. Frank, I. Campari, and U.Formentini, eds., pp. 22-64, Springer-Verlag, 1992.
[4] C.D. Tomlin, Geographic Information Systems and Cartographic Modeling. Prentice Hall, 1990.
[5] S.M. Lane and G. Birkhoff, Algebra. Macmillan, 1967.
[6] A.U. Frank, “Map Algebra with Functors for Temporal Data,” Proc. ER Workshop Conceptual Modeling for Geographic Information Systems, 2005.
[7] N.C. Cressie, Statistics for Spatial Data. John Wiley and Sons, 1993.
[8] G.T. Herman, J. Zheng, and C.A. Bucholtz, “Shape-Based Interpolation,” IEEE Computer Graphics and Applications, vol. 12, no. 3, pp. 69-79, May 1992.
[9] S.P. Raya and J.K. Udupa, “Shape Based Interpolation of Multidimensional Objects,” IEEE Trans. Medical Imaging, vol. 9, no. 1, pp. 32-42, Mar. 1990.
[10] S. Beucher, “Interpolation d'Ensembles, de Partitions et de Fonctions,” Technical Report N-18/94/MM, Ecole des Mines de Paris, 1994.
[11] F. Meyer, “Interpolations,” Technical Report N-16/94/MM, Ecole des Mines de Paris, 1994.
[12] J. Serra, “Interpolations et Distance de Hausdorff,” Technical Report N-15/94/MM, Ecole des Mines de Paris, 1994.
[13] J. Serra, “Hausdorff Distances and Interpolations,” Math. Morphology and Its Applications to Images and Signal Processing, H.J.A.M. Heijmans and J.B.T.M. Roerdink, eds., Kluwer Academic Publishers, 1998.
[14] M. Iwanowski, “Application of Mathematical Morphology to Image Interpolation,” PhD thesis, School of Mines Paris—Warsaw Univ. of Tech nology, 2000.
[15] J. Vidal, J. Crespo, and V. Maojo, “Recursive Interpolation Technique for Binary Images Based on Morphological Median Sets,” Proc. Int'l Symp. Math. Morphology: 40 Years On, C. Ronse, L. Najman, and E. Decenciere, eds., pp. 53-62, 2005.
[16] J. Serra, Image Analysis and Mathematical Morphology. Academic Press, 1982.
[17] D.J. Burr, “Elastic Matching of Line Drawings,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 3, no. 6, pp. 708-713, Nov. 1981.
[18] S.Y. Chen, W.C. Lin, C.C. Liang, and C.T. Chen, “Improvement on Dynamic Elastic Interpolation Technique for Reconstructing 3-D Objects from Serial Cross Sections,” IEEE Trans. Medical Imaging, vol. 9, no. 1, pp. 71-83, Mar. 1990.
[19] P.N. Werahera, G.J. Miller, G.D. Taylor, T. Brubaker, F. Danesghari, and E.D. Crawford, “A 3-D Reconstruction Algorithm for Interpolation and Extrapolation of Planar Cross Sectional Data,” IEEE Trans. Medical Imaging, vol. 14, no. 4, pp. 765-771, Dec. 1995.
[20] P. Maragos, “Pattern Spectrum and Multiscale Shape Representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 7, pp. 701-716, July 1989.
[21] P. Maragos, “Morphological Filtering for Image Enhancement and Feature Detection,” The Image and Video Processing Handbook, A.C. Bovik, ed., pp.135-156, Elsevier Academic Press, 2005.
[22] F. Hausdorff, Grundzuge der Mengenlchre. Viet and Co. (Gekurzte) Auft., 1914.
[23] M. Iwanowski and J. Serra, “The Morphological-Affine Object Deformation,” Mathematical Morphology and Its Applications to Image and Signal Processing, J. Goutsias, L. Vincent, D.S. Bloomberg, eds., pp. 81-90, Kluwer Academic Publishers, 2000.
[24] H.-L. Yu and G. Christakos, “Spatio-Temporal Modeling and Mapping of the Bubonic Plague Epidemics in India,” Int'l J. Health Geographics, vol. 5, no. 12, 2006, doi: 10.1186/1476-072X-5-12.
[25] P. Maragos and R.D. Ziff, “Threshold Superposition in Morphological Image Analysis Systems,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 5, pp. 498-504, May 1990.

Index Terms:
GISci, spatial interpolation, mathematical morphology, thematic maps, dilation, erosion, interpolation formulas, spatial databases and GIS, cartography, morphological image representation, visualization techniques and methodologies, geometrical problems and computations, set theory.
B.S. Daya Sagar, "Visualization of Spatiotemporal Behavior of Discrete Maps via Generation of Recursive Median Elements," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 32, no. 2, pp. 378-384, Feb. 2010, doi:10.1109/TPAMI.2009.163
Usage of this product signifies your acceptance of the Terms of Use.