Abstract
A crucial issue when applying topographic maps for clustering purposes is how to select the map's overall degree of smoothness. In this paper, we develop a new strategy for optimally smoothing, by a common scale factor, the density estimates generated by Gaussian kernel-based topographic maps. We also introduce a new representation structure for images of shapes, and a new metric for clustering them. These elements are incorporated into a hierarchical, density-based clustering procedure. As an application, we consider the clustering of shapes of marine animals taken from the SQUID image database. The results are compared to those obtained with the CSS retrieval system developed by Mokhtarian and co-workers, and with the more familiar Euclidean distance-based clustering metric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G.J. Deboeck and T. Kohonen, Visual Explorations in Finance with Self-Organizing Maps, Heidelberg: Springer, 1998.
M. Cottrell, P. Gaubert, P. Letremy, and P. Rousset, “Analyzing and Representing Multidimensional Quantitative and Qualitative Data: Demographic Study of the Rhône Valley. The Domestic Consumption of the Canadian Families,” in Kohonen Maps, Proc. WSOM99, E. Oja and S. Kaski (Eds.), Helsinki, 1999, pp. 1-14.
K. Lagus and S. Kaski, “Keyword Selection Method for Characterizing Text Document Maps,” in Proc. ICANN99, 9th Int. Conf. on Artificial Neural Networks, vol. 1, London: IEE, 1999, pp. 371-376.
J. Vesanto, “SOM-Based Data Visualization Methods,” Intelligent Data Analysis, vol. 3, no. 2, 1999, pp. 111-126.
J. Vesanto and E. Alhoniemi, “Clustering of the Self-Organizing Map,” IEEE Trans. Neural Networks, vol 11, no. 3, 2000, pp. 586-600.
J. Himberg, J. Ahola, E. Alhoniemi, J. Vesanto, and O. Simula, “The Self-Organizing Map as a Tool in Knowledge Engineering,” in Pattern Recognition in Soft Computing Paradigm, Nikhil R. Pal (Ed.), Singapore: World Scientific Publishing, 2001, pp. 38-65
T. Kohonen, “Self-Organized Formation of Topologically Correct Feature Maps,” Biol. Cybern., vol. 43, 1982, pp. 59-69.
T. Kohonen, Self-Organization and Associative Memory, Heidelberg: Springer, 1984.
T. Kohonen, Self-Organizing Maps, Heidelberg: Springer, 1995.
H. Ritter, “Asymptotic Level Density for a Class of Vector Quantization Processes,” IEEE Trans. Neural Networks, vol. 2, no. 1, 1991, pp. 173-175.
D.R. Dersch and P. Tavan, “Asymptotic Level Density in Topological Feature Maps,” IEEE Trans. Neural Networks, vol. 6, 1995, pp. 230-236.
M.M. Van Hulle, Faithful Representations and Topographic Maps. From Distortion-to Information-Based Self-Organization, Haykin, S. (Ed.), New York: Wiley, 2000.
M.M. Van Hulle, “Kernel-Based Topographic Map Formation by Local Density Modeling,” Neural Computation, vol. 14, 2002, pp. 1561-1573.
M.M. Van Hulle, “Monitoring the Formation of Kernel-Based Topographic Maps,” in IEEE Neural Network for Signal Processing Workshop 2000, Sydney, 2000, pp. 241-250.
M.M. Van Hulle and T. Gautama, “Monitoring the Formation of Kernel-Based Topographic Maps with Application to Hierarchical Clustering of Music Signals,” J. VLSI Signal Processing Systems for Signal, Image, and Video Technology, vol. 32, 2002, pp. 119-134.
M.M. Van Hulle, “Kernel-Based Equiprobabilistic Topographic Map Formation,” Neural Computation, vol. 10, 1998, pp. 1847-1871.
F. Mokhtarian, S. Abbasi, and J. Kittler, “Efficient and Robust Retrieval by Shape Content Through Curvature Scale Space,” in Proc. International Workshop on Image DataBases and MultiMedia Search, Amsterdam, The Netherlands, 1996, pp. 35-42.
L. Devroye, “Universal Smoothing Factor Selection in Density Estimation: Theory and Practice (with discussion),” Test, vol. 6, 1997, pp. 223-320.
S.R. Sain, K.A. Baggerly, and D.W. Scott, “Cross-Validation of Multivariate Densities,” Journal of the American Statistical Association, vol. 89, no. 427, 1994, pp. 807-817.
P. Hall and M.P. Wand, “On the Accuracy of Binned Kernel Density Estimators,” Journal of Multivariate Analysis, vol. 56, 1996, pp. 165-184.
E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, London: Chapman and Hall, 1999.
R.L. Graham, D.E. Knuth, and O. Patashnik, Answer to Problem 9.60 in Concrete Mathematics: A Foundation for Computer Science, Reading, MA: Addison-Wesley, 1994.
T. Gautama and M.M. Van Hulle, “Hierarchical Density-Based Clustering in High-Dimensional Spaces Using Topographic Maps,” in IEEE Neural Network for Signal Processing Workshop 2000, Sydney, 2000, pp. 251-260.
M.P. Dubuission Jolly, S. Lakshmanan, and A.K. Jain, “Vehicle Segmentation and Classification Using Deformable Templates,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 3, 1996, pp. 293-308.
K.-M. Lee and N. Street, “Automatic Image Segmentation and Classification Using On-Line Shape Learning,” in Fifth IEEE Workshop on the Application of Computer Vision, CA, USA: Palm Springs, 2000.
F. Mokhtarian, “Silhouette-Based Isolated Object Recognition Through Curvature Scale Space,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 5, 1995, pp. 539-544.
W. Niblack, R. Barber, W. Equitz, M. Flickner, E. Glasman, D. Petkovic, and P. Yanker, “The QBIC Project: Querying Images By Content Using Color, Texture, and Shape,” SPIE, vol. 1908, 1993, pp. 173-187.
J. MacQueen, “Some Methods for Classification and Analysis of Multivariate Observations,” in Proc. Fifth Berkeley Symp. on Math. Stat. and Prob., vol. 1, 1967, pp. 281-296.
P.R. Krishnaiah and L.N. Kanal, Classification, Pattern Recognition, and Reduction of Dimensionality, Handbook of Statistics, vol. 2, Amsterdam: North Holland, 1982.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Van Hulle, M.M., Gautama, T. Optimal Smoothing of Kernel-Based Topographic Maps with Application to Density-Based Clustering of Shapes. The Journal of VLSI Signal Processing-Systems for Signal, Image, and Video Technology 37, 211–222 (2004). https://doi.org/10.1023/B:VLSI.0000027486.56120.e7
Published:
Issue Date:
DOI: https://doi.org/10.1023/B:VLSI.0000027486.56120.e7