The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.12 - Dec. (2011 vol.17)
pp: 2555-2562
Jan-Henrik Haunert , University of Würzburg
Leon Sering , University of Würzburg
ABSTRACT
Mobile users of maps typically need detailed information about their surroundings plus some context information about remote places. In order to avoid that the map partly gets too dense, cartographers have designed mapping functions that enlarge a user-defined focus region - such functions are sometimes called fish-eye projections. The extra map space occupied by the enlarged focus region is compensated by distorting other parts of the map. We argue that, in a map showing a network of roads relevant to the user, distortion should preferably take place in those areas where the network is sparse. Therefore, we do not apply a predefined mapping function. Instead, we consider the road network as a graph whose edges are the road segments. We compute a new spatial mapping with a graph-based optimization approach, minimizing the square sum of distortions at edges. Our optimization method is based on a convex quadratic program (CQP); CQPs can be solved in polynomial time. Important requirements on the output map are expressed as linear inequalities. In particular, we show how to forbid edge crossings. We have implemented our method in a prototype tool. For instances of different sizes, our method generated output maps that were far less distorted than those generated with a predefined fish-eye projection. Future work is needed to automate the selection of roads relevant to the user. Furthermore, we aim at fast heuristics for application in real-time systems.
INDEX TERMS
Cartography, schematic maps, fish-eye view, graph drawing, optimization, quadratic programming.
CITATION
Jan-Henrik Haunert, Leon Sering, "Drawing Road Networks with Focus Regions", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 12, pp. 2555-2562, Dec. 2011, doi:10.1109/TVCG.2011.191
REFERENCES
[1] M. Agrawala and C. Stolte, Rendering effective route maps: Improving usability through generalization. In Proc. 28th Annu. Conf. Comput. Graphics (SIGGRAPH'01), pages 241–249, 2001.
[2] J. Böttger, U. Brandes, O. Deussen, and H. Ziezold, Map warping for the annotation of metro maps. IEEE Comput. Graph., 28 (5): 56–65, 2008.
[3] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, Cambridge, UK, 2004.
[4] M. A. A. Cox and T. F. Cox, Multidimensional scaling. In C. Chen, W. Härdle, and A. Unwin editors, , Handbook of Data Visualization, pages 315–347. Springer, Berlin, Germany, 2008.
[5] D. Fairbairn and G. Taylor, Developing a variable-scale map projection for urban areas. Comput. Geosci., 21 (9): 1053–1064, 1995.
[6] E. R. Gansner, Y. Koren, and S. C. North, Topological fisheye views for visualizing large graphs. IEEE T. Vis Comput. Gr., 11 (4): 457–468, 2005.
[7] M. R. Garey and D. S. Johnson, Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1990.
[8] F. Guerra, and C. Boutoura, An electronic lens on digital tourist city-maps. In Proc. 20th Int. Cartogr. Association Conf, pages 1151–1157, 2001.
[9] L. Harrie, L. T. Sarjakoski, and L. Lehto, A mapping function for variable-scale maps in small-display cartography. J. Geospatial Eng., 4 (2): 111–123, 2002.
[10] L. Harrie and T. Sarjakoski, Simultaneous graphic generalization of vector data sets. GeoInformatica, 6 (3): 233–261, 2002.
[11] S.-H. Hong, D. Merrick, and H. A. D. do Nascimento, The metro map layout problem. In Proc. Australasian Symp. Inform. Visualization - Volume 35, (APVis '04), pages 91–100. Australian Computer Society, 2004.
[12] B. Jenny, Geometric distortion of schematic network maps. Bulletin of the Society of Cartographers, 40: 15–18, 2006.
[13] N. Kadmon and E. Shlomi, A polyfocal projection for statistical surfaces. Cartogr. J., 15 (1): 36–41, 1978.
[14] C. Kaiser, F. Walsh, C. J. Q. Farmer, and A. Pozdnoukhov, User-centric time-distance representation of road networks. In Proc. 6th Int. Conf. Geographic Inform. Sci., (GIScience'10), pages 85–99. Springer, Berlin, Germany, 2010.
[15] N. Karmarkar, A new polynomial-time algorithm for linear programming. In Proc. 16th Annu. ACM Symp. Theory Comput., (STOC '84), pages 302–311. ACM, 1984.
[16] A. Klippel, K.-F. Richter, T. Barkowsky, and C. Freksa, The cognitive reality of schematic maps. In Map-based Mobile Services – Theories, Methods and Implementations, pages 57–74. Springer, Berlin, Germany, 2005.
[17] J. Kopf, M. Agrawala, D. Bargeron, D. Salesin, and M. Cohen, Automatic generation of destination maps. ACM T., 29 (6): 158:1–158:12, 2010.
[18] Y. K. Leung and M. D. Apperley, A review and taxonomy of distortion-oriented presentation techniques. ACM T. Comput.-Hum. Int., 1: 126–160, June 1994.
[19] Q. Li, Variable-scale representation of road networks on small mobile devices. Comput. Geosci., 35 (11): 2185–2190, 2009.
[20] H. Lintzöft, 50 Jahre Falkplan. In Patent-Plan Hamburg (Reprint anläslich 50 Jahre Falkplan 1945–1995). Falk-Verlag, Hamburg, Germany, 1995.
[21] D. Merrick and J. Gudmundsson, Increasing the readability of graph drawings with centrality-based scaling. In Proc. 2006 Asia-Pacific Symp. Inform. Visualization - Volume 60, APVis '06, pages 67–76. Australian Computer Society, 2006.
[22] K.-R. Müller, S. Mika, G. Rätsch, K. Tsuda, and B. Schölkopf, An introduction to kernel-based learning algorithms. IEEE T. Neural Networ., 12 (2), 2001.
[23] K. G. Murty, Linear complementarity, linear and nonlinear programming. Heldermann, Berlin, 1988.
[24] M. Nöllenburg and A. Wolff, Drawing and labeling high-quality metro maps by mixed-integer programming. IEEE T. Vis Comput. Gr., 17 (5): 626–641, 2011.
[25] K.-F. Richter, D. Peters, G. Kuhnmünch, and F. Schmid, What do focus maps focus on? In Proc. Int. Conf. Spatial Cognition VI: Learning, Reasoning, and Talking about Space, volume 5248 of Lecture Notes In Artificial Intelligence, pages 154–170. Springer, Berlin, Germany, 2008.
[26] M. Sarkar and M. H. Brown, Graphical fisheye views. Commun. ACM, 37: 73–83, December 1994.
[27] F. Schmid, Knowledge-based wayfinding maps for small display cartography. J. Location Based Services, 2 (1): 51–83, 2008.
[28] M. Sester, Optimization approaches for generalization and data abstraction. Int. J. Geogr Inf. Sci., 19 (8–9): 871–897, 2005.
[29] E. Shimizu and R. Inoue, Time-distance mapping: visualization of transportation level of service. In Proc. Symp. on Environmental Issues Related to Infrastructure Development, pages 221–230, 2003.
[30] J. Stott, P. Rodgers, J. C. Martinez-Ovando, and S. G. Walker, Automatic metro map layout using multicriteria optimization. IEEE T. Vis Comput. Gr., 17: 101–114, January 2011.
[31] Y.-S. Wang, T.-Y. Lee, and C.-L. Tai, Focus+context visualization with distortion minimization. IEEE T. Vis Comput. Gr., 14 (6): 1731–1738, 2008.
[32] D. Yamamoto, S. Ozeki, and N. Takahashi, Focus+glue+context: an improved fisheye approach for web map services. In Proc. 17th Annu. ACM Symp. Advances Geogr. Inform. Syst. (ACM-GIS'09), pages 101– 110. ACM, 2009.
[33] A. Zipf and K.-F. Richter, Using focus maps to ease map reading: Developing smart applications for mobile devices. Künstliche Intelligenz (KI), 02 (4): 35–37, 2002.
21 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool