ABSTRACT
A fundamental problem in model-based computer vision is that of identifying which of a given set of geometric models is present in an image. Considering a “probe” to be an oracle that tells us whether or not a model is present at a given point, we study the problem of computing efficient strategies (“decision trees”) for probing an image, with the goal to minimize the number of probes necessary (in the worst case) to determine which single model is present. We show that a ⌈lg k ⌉ height binary decision tree always exists for k polygonal models (in fixed position), provided (1) they are non-degenerate (do not share boundaries) and (2) they share a common point of intersection. Further, we give an efficient algorithm for constructing such decision trees when the models are given as a set of polygons in the plane. We show that constructing a minimum height tree is NP-complete if either of the two assumptions is omitted. We provide an efficient greedy heuristic strategy and show that, in the general case, it yields a decision tree whose height is at most ⌈lg n ⌉ times that of an optimal tree. Finally, we discuss some restricted cases whose special structure allows for improved results.
- 1.E. Arkin and J. Mitchell. Applications of combinatorics and computational geometry to pattern recognition. Technical report, Gornen University, 1990.]]Google Scholar
- 2.P. Belleville and T. Shermer. Probing polygons minimially is hard. Technical Report 91-12, Dept. of Computing Science, Simon Fraser University, Burnaby, B. C. Canada, December 1991.]]Google Scholar
- 3.H. Bernstein. Determining the shape of a convex n-sided polygon by using 2n + k tactile probes, information Processing Letters, 22:255-260, 1986.]] Google ScholarDigital Library
- 4.E. Bienenstock, D. Geman, and S. Geman. A relational approach in object recognition. Technical report, Brown University, 1988.]]Google Scholar
- 5.E. Bienenstock, D. Geman, S. Geman, and D.E. Mc- Clure. Phase II technical report, development of laser radar ATR algorithms. Technical Report Contract No. DAAL02-89-C-0081, CECOM Center for Night Vision and Electro-Optics, October 1990.]]Google Scholar
- 6.B. ChazeUe and H. Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. J. A CM, 39:1-54, 1992.]] Google ScholarDigital Library
- 7.R. Chin and C. Dyer. Model-based recognition in robot vision. Computing Surveys, pages 67-108, 1986.]] Google ScholarDigital Library
- 8.H. De Fraysseix, J. Pach, and R. Pollack. How to draw a planar graph on a grid. Combinatorica, 10:41-51, 1990.]]Google Scholar
- 9.I-I. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, Heidelberg, 1987.]] Google ScholarDigital Library
- 10.A. Fiat and A. Shamir. How to find a battleship. Networks, 19:361-371, 1989.]]Google Scholar
- 11.R. J. Fowler, M. S. Paterson, and L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12:133-137, 1981.]]Google Scholar
- 12.K. Y. Fung, T. M. Nicholl, R. E. Tarjan, and C. J. Van Wyk. Simplified linear-time Jordan sorting and polygon clipping. Inform. Process. Lett., 35:85-92, 1990.]] Google ScholarDigital Library
- 13.M. Garey. Optimal binary identification procedures. SIAM J. Appl. Math., 23:173-186, 1972.]]Google ScholarDigital Library
- 14.P. Gaston and T. Lozano-Perez. Tactile recognition and localization using object models: The case of polyhedra on a plane. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6:257-266, May 1984.]]Google ScholarDigital Library
- 15.M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, NY, 1980.]] Google ScholarDigital Library
- 16.W. E. Grimson. The combinatorics of local constraints in model-based recognition and localization from sparse dat~. J. A CM, 33:658-686, 1986.]] Google ScholarDigital Library
- 17.W. E. Grimson and T. Lozano-Perez. Localizing overlapping parts by searching the interpretation tree. IEEE Transactions on Pattern Recognition and Machine Intelligence, PAMI-9:469-482, July 1987.]] Google ScholarDigital Library
- 18.K. Hoffmann, K. Mehlhorn, P. Rosenstiehl, and R. E. Tarjan. Sorting Jordan sequences in linear time using level-linked search trees. Inform. Control, 68:170-184, 1986.]] Google ScholarDigital Library
- 19.L. Hyafil and R. Rivest. Constructing optimal binary decision trees is NP-complete. Information Processing Letters, 5:15-17, 1976.]]Google ScholarCross Ref
- 20.D. S. Johnson. Approximation algorithms for combinatorial problems. J. Comp. Sys. Sciences, 9:256-278, 1974.]]Google ScholarDigital Library
- 21.E. Joseph and S. Skiena. Model-based probing strategies for convex polygons. Computational Geometry: Theory and Applications, 2, 1992.]] Google ScholarDigital Library
- 22.D. Lichtenstein. Planar formulae and their uses. SIAM J. Comput., 11:329-343, 1982.]]Google ScholarDigital Library
- 23.K. Lyons and D. Rappaport. An efficienlL algorithm for identifying objects using robot probes. The Visual Computer, to appear.]]Google Scholar
- 24.B. Motet. Decision trees and diagrams. Computing Surveys, pages 593-623, 1982.]] Google ScholarDigital Library
- 25.C. H. Papadimitriou. On certain problems in algorithmic vision. Technical report, Computer Science, UCSD, 1991.]]Google Scholar
- 26.S. Skiena. Interactive reconstruction via probing. IEEE Proceedings, 80:1364-1383, 1992.]]Google ScholarCross Ref
Index Terms
- Decision trees for geometric models
Recommendations
Rank-Balanced Trees
Since the invention of AVL trees in 1962, many kinds of binary search trees have been proposed. Notable are red-black trees, in which bottom-up rebalancing after an insertion or deletion takes O(1) amortized time and O(1) rotations worst-case. But the ...
Geometric Decision Tree
In this paper, we present a new algorithm for learning oblique decision trees. Most of the current decision tree algorithms rely on impurity measures to assess the goodness of hyperplanes at each node while learning a decision tree in top-down fashion. ...
Spanning Trees in Multipartite Geometric Graphs
Let R and B be two disjoint sets of points in the plane where the points of R are colored red and the points of B are colored blue, and let $$n=|R\cup B|$$n=|RýB|. A bichromatic spanning tree is a spanning tree in the complete bipartite geometric graph ...
Comments