Affine invariant detection of perceptually parallel 3D planar curves

The problem of parallelism detection between two curves has been formulated in this paper as a line detection problem within an affine-invariant local similarity matrix computed for the two curves. Each element of this matrix gives an affine invariant measure of local parallelism for any pair of curve segments along the two curves. This approach enables the detection of a pair of parallel 3D planar curves as well as parallel 2D curves under general affine transform. Two descriptors were also used here to provide a multi-resolution representation of a curve. Since these two descriptors provide sufficient local and semi-local shape information at every feature point on the curves, the process of detecting parallelism is thus robust against both noise and deformations. Moreover, the proposed technique allows all significant pairs of parallel segments within any two curves in the scene to be detected. Experiments on detecting randomly affine-transformed curves, which are obtained from natural images or artificially generated images, have demonstrated the effectiveness of the technique.