Matching of Objects Nodal Points Improvement Using Optimization

The main objective of this work was to improve a previously developed object matching methodology. This overall methodology includes: a modeling phase; followed by a modal analysis; the construction of a matrix that relates both sets of objects points; and the matching phase. The previously implemented matching phase is based on a local search; with this solution the relation between objects nodes (points) are not considered. To overcome this, we implemented a new matching solution, using optimization techniques, based on a global search. This solution is compared with the previous one allowing the verification of the results improvement. The local and the global search methods used in this work allow only matches of the usual type “one to one”. However, there are situations when this type of matches is not the most adequate, since it can imply loss of information. To avoid this problem, we developed a new algorithm applicable to contour objects that finds satisfactory matches of type “one to many” or vice-versa. INTRODUCTION Establishing correspondences between objects is an essential step of some computational vision domains, as for example, analyzing image objects movement/deformation. The resolution methods for the matching problem usually include restrictions that avoid incorrect matches according to the considered criteria. Examples of those restrictions are [1]: the order [2, 3], epipolar restrictions (rigidity restrictions) [2, 3], uniqueness [4], visibility [5], and proximity. To determine the matches it can be used, for example, image correlation (i.e. images similarity is assumed) [1, 6, 7], the points’ proximity [7, 8], or the disparity fields smoothness [2]. The matching problem can also be interpreted as an optimization problem, where the objective function can depend, for example, of any image relationship mentioned in the last paragraph, and the restrictions have to form a non-empty space of possible solutions for the optimization problem. This solution can use: dynamic programming [2], graph search [3], bipartite graph matching [9], convex minimization [7], etc. Non-optimal approaches include: greedy algorithms [10], simulated annealing [11], relaxation [4], etc. In this work, the matching process is based on the construction of a matrix (the affinity matrix) that is obtained by one of the two different methodologies previously implemented [12, 13]: (1) using the shape modal analysis [14]; or (2) using the finite elements method and modal analysis [15]. Both methods establish the correspondences from the displacement analysis of each point in the respective modal space. These methodologies can be used to satisfactorily match 2D or 3D objects. However, such as they were initially proposed, the matching phase has two considerable disadvantages: 1) the matches are established using a local approach, i.e. the relationship between the points of a same object is not considered; 2) when the considered objects have a different number of elements, the excess points are not matched, since in this approach the contemplated correspondence type is the usual “one to one”. Here, the first disadvantage is overtaken [16] using global optimization methods in the matching process, after adjusting this problem to Inverse Problems, Design and Optimization Symposium Rio de Janeiro, Brazil, 2004 a classical assignment problem [17]. This new methodology improved the results in several aspects (number of matches, robustness, etc.). The problem of the excess points is solved with the adaptation of matches of type “one to many” and vice versa, using neighborhood criterions applicable to contour objects. The developed algorithm matches successfully the excess points in the examples considered. BASE METHODOLOGIES In this section are presented the two adopted modeling methods (geometrical and physical) that can be used in the construction of the affinity matrix. After the model derivation, the construction of the affinity matrix and the local approach, previously used for the determination of the matches, is explained.