Reflective Symmetry Detection by Rectifying Randomized Correspondences

Images often contain multiple reflective symmetries. We propose a method to detect multiple reflective symmetries at different scales and viewpoints based on the J-linkage framework [5], which combines ideas from RANSAC (robustness against outliers) and the Hough transform (multiple models detection through voting schemes). How J-linkage works in symmetry detection ? Given N reflective matches M = {mi}i=1,··· ,N in one image, we estimate K symmetries by randomly sampling K valid minimal seed sets from M. We will see that two matches are sufficient to define a minimal seed set. We thus obtain K symmetries, each with an associated consensus set (the subset of compatible with each symmetry). A binary N×K matrix is thus built, where the entry (i, j) is 1 if the i-th match is in the consensus set of the j-th symmetry, and 0 otherwise. Each row of this matrix indicates which symmetries are preferred by each match and is considered as a binary feature vector for that match. Using these features, agglomerative hierarchical clustering based on the Jaccard distance is used to cluster the matches. Finally, each, large enough, cluster corresponds to a local symmetry. How to rectify the distorted symmetry from two matches ? In practice, the symmetry can be observed in a slanted view and thus undergoes some perspective distortion. In this more general case, m1 = {p1,p1} and m2 = {p2,p2} intersecting at the vanishing point v are necessary to determine the symmetric axis (see Fig. 1(a)). We decompose the homography, which rectifies the distorted symmetry, into three parts: