An Optimal Matching Algorithm Based on Rough Localization and Exact Adjustment

Aiming at the problems on estimate of the initial transformation, matching precision and global optimization in matching, this paper presents a matching method, which is based on rough localization and exact adjustment. By means of rotation, translation and coincidence of the minimum bounding boxes of surface and measured points, rough localization is realized, which produces a good estimate for the follow-up iterative algorithm. The closest points that were calculated via the normal projection of the sample points then establish the correspondence between two objects. An iterative process is used in the exact adjustment to ensure the global optimization of match. For reducing the effect of bad points or local distortion, a maximum distance criterion is adopted to refine the transformation between objects. A computer simulation is given to demonstrate that the algorithm is steady and effective. Introduction In manufacturing industries, surface matching is used widely, such as the optimization of distribution of stock allowance, localization of blank parts, and precision measurement of free-form surface etc. [4,5,13]. So, extensive research has been developed and many methods were proposed during two decades recently. Besl and McKay [1] proposed the iterative closest point (ICP) algorithm for registration of 3D shapes by establishing the correspondence between the objects based on the closest distance at each iteration. Sharp et al. [2] combined Euclidean invariants into ICP algorithm to improve its convergence to the global minimum. Menq et al. [5] developed an iterative approach together with nonlinear optimization methods for precision measurement of sculptured surface in CAD-directed inspection. Li et al. [13] presented a unified geometry theory for localization of three types of workpieces and proposed an effective algorithm. Approaches above-mentioned are to solve the square objective function to be minimized to find the best transformation for localization, but they require a good initial estimate to ensure convergence to the global minimum. Otherwise, correspondence can be also established by calculating the same feature between two objects, in [6] K.H. Ko constructed surface-intrinsic-wireframe for given surfaces and calculated the umbilical points to match surfaces, in [3] he used intersection of iso-curvature lines of the Gaussian and the mean curvatures to establish correspondence to find the transformation between objects. But it is difficult to achieve the global optimization, as the calculation of umbilical points or curvatures that is restricted by measurement data sometimes is not accurate. In this paper, we propose surface rough localization by minimum bounding box to produce a good initial estimate for the follow-up exact adjustment in which an iterative process is used to ensure the global optimization of match, and construct a maximum distance criterion to refine the transformation between objects. Finally an example illustrates the optimal-match algorithm. Surface rough localization Measurement without reference is employed in modern industry results in the difference between Key Engineering Materials Vols. 291-292 (2005) pp 661-666 online at http://www.scientific.net © (2005) Trans Tech Publications, Switzerland Online available since 2005/Aug/15 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.34-17/04/08,11:56:34) the measuring coordinate system and the design coordinate system. Therefore, it is necessary that the measured data must be aligned with CAD model again to ensure the identification between two coordinate systems for succeeding machining. By this processing, a good initial transformation is gained for the follow-up exact adjustment. This procedure is called as surface rough localization. In this paper, the rough localization is realized by matching between the minimum bounding boxes of the measured data and surface. Minimum Bounding Box. The minimum bounding box (MBB) is defined as a cuboid with minimum volume, which envelops the dataset or the surface. The approaches of determination of MBB of arbitrary objects are referred to [8,9]. Each face of the MBB by Chan’s method [9] is parallel to the coordinate plane, which is advantageous to match the MBB. The procedures of the determination of the MBB are summarized as follows: 1. Find the axis aligned bounding box (AABB) of the model as shown in Fig. 1. 2. Construct three axes x R , y R and z R (they are normal to the coordinate planes YZ , ZX , XY , respectively and pass through the box center) as shown in Fig. 1. 3. Then the model is iteratively rotated about the three axes respectively until its projected bounding box area along the rotary axis on the coordinate plane achieves a minimum. 4. The model is then oriented at a position which gives a minimum bounding box volume. Rotary angles are recorded asα , β andγ , then the transformationT is calculated. Rough Localization. The minimum bounding boxes of the surface and the point set are shown in Fig. 2. The procedures of the rough localization between two objects are described as follows: 1. Translate the center of MBBP and that of MBBS to the origin, respectively. 2. MBBP and MBBS are rotated about the coordinate axis X ,Y , Z until two boxes are coincident (four instances). The transformation of point set and surface are p T ′ and s T ′ , respectively. 3. Judge whether the point set matches the surface correctly or not. If it is not, go to step 2 and choose other coincident instance. 4. Calculate the transformation to complete the rough localization. Let } { i p P = be measured data set to be aligned with surface ) , ( v u S S = . p T and s T are the transformation of the point set and the surface during calculating the MBB, respectively. We can gain the following equation: Fig. 1. Aligned bounding box and three rotary axes a) Minimum bounding box of surface (MBBS) b) Minimum bounding box of point set (MBBP) Fig. 2. Minimum bounding box (MBB) 662 Advances in Abrasive Technology VIII