The geometry and matching of lines and curves over multiple views

This paper describes the geometry of imaged curves in two and three views. Multi-view relationships are developed for lines, conics and non-algebraic curves. The new relationships focus on determining the plane of the curve i n a projective reconstruction, and in particular using the homography induc ed by this plane for transfer from one image to another. It is shown that g iven the fundamental matrix between two views, and images of the curve in e ach view, then the plane of a conic may be determined up to a two fold ambi guity, but local curvature of a curve uniquely determines the plane. It is then shown that given the trifocal tensor between three views, this plane d efines a homography map which may be used to transfer a conic or the curvat ure from two views to a third. Simple expressions are developed for the pla ne and homography in each case. A set of algorithms are then described for automatically matching individua l line segments and curves between images. The algorithms use both photomet ric information and the multiple view geometric relationships. For image pa irs the homography facilitates the computation of a neighbourhood cross-cor relation based matching score for putative line/curve correspondences. For image triplets cross-correlation matching scores are used in conjunction wi th line/curve transfer based on the trifocal geometry to disambiguate match es. Algorithms are developed for both short and wide baselines. The algorit hms are robust to deficiencies in the segment extraction and partial occlus ion. Experimental results are given for image pairs and triplets, for varying mo tions between views, and for different scene types. The methods are applica ble to line/curve matching in stereo and trinocular rigs, and as a starting point for line/curve matching through monocular image sequences.