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Figure 18.8 Topological shapes for model description. (a) A model with clockwise rotation of the starting junction; (b) a model with counter-clockwise direction for the starting junction.
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Figure 18.8(a) and 18.8(b), respectively. In each extraction, there are enough line groups to guide a hypothesis for 3D object recognition. Table 18.2 presents the matching results in Figure 18.9 corresponding to the models of Figure 18.8. All extractions are enumerated for each model. The above experimental results demonstrate that the proposed framework can find similar shapes even from a cluttered and distorted line set. By the local comparison-based matching criterion permitting a shape distortion for the reference model shape, reasonable line grouping and shape matching are possible, without increasing the time complexity. The grouping and the junction detection are all performed in 1 s, on a Pentium II-600 desktop machine. In all test sets, any length ratio of the model or scene lines has not been used, because the scene lines are easily broken in outdoor images. Angle relations and line ordering between two neighboring junctions are well preserved, even in broken and distorted line sets.
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Figure 18.9 A topological shape extraction for 3D object recognition. (a) Original image; (b) line extraction; and (c), (d) found topological shapes.
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Fast Object Recognition Using DP
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Figure 18.9 (continued)
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Table 18.2 The topological matching results. J1 Model 1 (39, 80) (39, 79) (39, 86) (118, (40, (106, (94, (116, 41) 41) 41) 41) 41) J2 (80, 41) (86, 41) (86, 41) (41, (41, (41, (41, (41, 80) 80) 80) 80) 80) J3 (41, 70) (41, 70) (41, 70) (80, (80, (80, (80, (80, 38) 38) 38) 38) 38) J4 (70, 69) (70, 69) (70, 69) (38, (38, (38, (38, (38, 56) 56) 56) 56) 56)
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8.2 Collinearity Tests for Random Lines
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We tested the stability of the two collinear functions proposed in Section 6 by changing the standard deviation as a threshold for four end points of the two constituting lines. The noise was modeled as 2 Gaussian random noise having mean 0 and variance 0 . A total of 70 lines were randomly generated in a rectangular region of size 100 100 in Figure 18.10, hence the number of possible line pairs was 70 C2 = 2415. Finally, only two line pairs were selected as satisfying the two collinear conditions of Equation (18.19) and Equation (18.25) under 0 = 0 1. When we reduced the variation value, there were a few collinear sets. By a simple definition of collinear functions and control of the variance 0 as Gaussian perturbation, we could systematically obtain the collinear line set without referring to heuristic parameters.
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100 90 80 70 60 y 50 40 30 20 10 0 0 10 20 30 40 50 x (a) 60 70 80 90 100
Figure 18.10 Collinear lines according to the change of standard deviation 0 as a threshold. (a) The randomly generated original line set; (b) line pairs detected for 0 = 0 4; (c) for 0 = 0 3; and (d) for 0 = 0 1.
Fast Object Recognition Using DP
Figure 18.10 (continued)
9. Conclusions
In this chapter, a fast and reliable matching and grouping method using dynamic programming is proposed to extract collections of salient line segments. We have considered the classical dynamic programming as an optimization technique for geometric matching and grouping problems. First, the importance of grouping to object recognition was emphasized. By grouping together line features that are likely to have been produced by a single object, it has long been known that significant speed-ups in a recognition system can be achieved, compared to performing a random search. This general fact was used as a motive to develop a new feature grouping method. We introduced a general way of representing line patterns and of using the patterns to consistently match 2D and 3D objects. The main element in this chapter is a DP-based formulation for matching and grouping of line patterns by introducing a robust and stable geometric representation that is based on the perceptual organizations. The end point proximity and collinearity comprised of image lines are introduced as two main perceptual organizing groups to start the object matching or recognition. We detect the junctions as the end point proximity for the grouping of line segments. Then, we search a junction group again, in which each junction is combined by the collinear constraint between them. These local primitives, by including Lowe s perceptual organizations and acting as the search node, are consistently in a linked form in the DP-based search structure. We could also impose several constraints, such as parallelism, the same line condition and rotational direction, to increasingly narrow down the search space for possible objects and their poses. The model description is predefined for comparison with the scene relation. The collinear constraint acts to combine the two junctions as a neighborhood for each other. The DP-based search algorithm reduces the time complexity for the search of the model chain in the scene. Through experiments using images from cluttered scenes, including outdoor environments, we have demonstrated that the method can be applied to the optimal matching, grouping of line segments and 2D/3D recognition problems, with a simple shape description sequentially represented.