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 dlg ke 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 tress 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 ...

A fundamental problem in model-based computer vision is that of identifying to which of a given set of concept classes of geometric models an observed model belongs. Considering a "probe" to be an oracle that tells whether or not the observed model is present at a given point in an image, 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 in which class the observed model belongs. We prove a hardness result and give strategies that obtain decision trees whose height is within a log factor of optimal. These results grew out of discussions that began in a series of workshops on Geometric Probing in Computer Vision, sponsored by the Center for Night Vision and Electro-Optics, Fort Belvoir, Virginia, and monitored by the U.S. Army Research Office. The views, opinions, and/or findings contained in this report are those of the authors and should not be con...

) Michael Bender Saurabh Sethia Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794-4400 fbender---saurabh---skienag@cs.sunysb.edu April 22, 1999 1 Introduction Each test or feature in a classification system defines a set partition on a class of objects. Adding new features refines the classification, whereas deleting features may result in merging previously distinguished classes. As an illustration, consider the set of automobile types f VW Beetle, Toyota, Lexus, Cadillac g. The feature size partitions the cars into sets of small and large cars, ff VW Beetle, Toyotag, f Lexus, Cadillac gg. The feature domestic-origin partitions the cars into ff VW Beetle, Toyota, Lexus g, f Cadillac gg. The feature ugly-shape distinguishes f VW Beetle, Cadillac g from f Toyota, Lexus g. Incorporating both size and origin induces the refined partition ff VW Beetle, Toyotag, f Lexus g, f Cadillac gg, whereas the union of all three features completely di...

As the terms are used here, a body in R el is a compact convex set with non-empty interior, and a polytope is a body that has only finitely many extreme points. The class of all bodies whose interior includes the origin 0 is denoted by %%. A set X is symmetric if X =-X. The ray-oracle of a body C e "#( { is the function 0c which, accepting as input an arbitrary ray R issuing from 0, produces the point at which R intersects the boundary of C. This paper is concerned with a few central aspects of the following general question: given certain information about C, what additional information can be obtained by questioning the ray-oracle, and how efficiently can it be obtained? It is assumed that infinite-precision real arithmetic and the usual vector operations in U d are available at no cost, so the efficiency of an algorithm is measured solely in terms of its number of calls to the ray-oracle. The paper discusses two main problems, the first of which—the containment problem—arose from a question in abstract numerical analysis. Here the goal is to construct a polytope P (not necessarily in any sense a small one) that contains C, where this requires precise specification of the vertices of P. There are some sharp positive results for the case in which d = 2 and C is known not to be too asymmetric, but the main result on the containment problem is negative. It asserts that when d 2 s 3 and the body is known only to be rotund and symmetric, there is no algorithm for the containment problem. This is the case even when there is available a certain master oracle whose questionanswering power far exceeds that of the ray-oracle. However, it turns out that even when there is no additional information about C, the following relaxation of the containment problem admits an algorithmic solution based solely on the ray-oracle: construct a polytope containing C or conclude that the centred condition number of C exceeds a prescribed bound. In the other main problem—the reconstruction problem — it is known only that C is itself a polytope and the problem is to construct C with the aid of a finite number of calls to the ray-oracle. That is accomplished with a number of calls that depends on the number of faces (and hence on the 'combinatorial complexity') of C.

Suppose that for a set H of n unknown hyperplanes in the Euclidean d-dimensional space, a line probe is available which reports the set of intersection points of a query line with the hyperplanes. Under this model, this paper investigates the complexity to find a generic line for H and further to determine the hyperplanes in H . This problem arises in factoring the u-resultant to solve systems of polynomials (e.g., Renegar [12]). We prove that d+1 line probes are sufficient to determine H . Algorithmically, the time complexity to find a generic line and reconstruct H from O(dn) probed points of intersection is important. It is shown that a generic line can be computed in O(dn log n) time after d line probes, and by an additional d line probes, all the hyperplanes in H are reconstructed in O(dn log n) time. This result can be extended to the d-dimensional complex space. Also, concerning the factorization of the u-resultant using the partial derivatives on a generic line, we touch upon reducing the time complexity to compute the partial derivatives of the u-resultant represented as the determinant of a matrix.