with silhouette of the object instead of sparse local features. Besides, there exist objects, which cannot be represented adequately by primitive object features as points, lines, or circles. Moreover, pose estimations of global object descriptions are, statistically, more accurate and robust than those from a sparse set of local features. Whereas point features reveal little about surface topology, space curves provide such geometrical cues. Therefore, we focus on the space curves to develop an inverse problem approach.

Robert and Faugeras presented an edge-based trinocular stereo algorithm using geometric matching principles [26]. They showed that given the image curvature of corresponding curve points in two views, it is possible to predict the curvature in the third one and use that as a matching criterion. Schmid and Zisserman offered an extension to Robert method by fusing the photometric information and edge information to reduce the outlier matches [27]. Both these approaches apply many heuristics. Han and Park developed a curve-matching algorithm based on geometric constraints [28]. They apply epipolar constraint between two sets of curves and compute corresponding points on the curves. From the initial epipolar constraints obtained from corner point matching, candidate curves are selected according to the epipolar geometry, curve-end constraints, and curve distance measures. Assuming that the corresponding curves in stereo images are rather similar, they apply curve distance measure as a constraint of curve matching. In general, this assumption is not true, as it will be discussed in section 2.2. Kahl and August developed an inverse method to extract space curves from multiple view images [29]. Instead of first seeking a correspondence of image structure and then computing 3D structure, they seek the space curve that is consistent with the observed image curves. By minimizing the potential associated to prior knowledge of space curves, i.e. average curvature, and potential associated to the image formation model, they look for the candidate space curves. The main deficiency of this method is that the relative motion of the cameras is assumed to be known.

2.1. Differential Geometry of Space Curves

Let P =(X ,Y ,Z)be a point whose position in space is given by the equations X = f (s), Y =g(s) and Z =h(s) where f, g, and h are differentiable functions of s.

As s varies continuously, P traces a curve in space. The differential geometry of curves traditionally begins with a vector R(s)=X (s)i+Y (s)j+Z (s)k that describes

the curve parametrically as a function of s that is at least thrice differentiable.

Figure 1. Space curve geometry in Frenet frame.

2.2. Inverse Problem Formulation

Given a sequence of edge curves of a moving object in the calibrated stereo rig, the problem is to extract the space curves on the surface of the object. It is obvious that the plane curve is established by projection of the space curve to camera plane. As it is shown in figure 2, projections of space curve in two different camera planes do not necessarily have the same shape. In the small base line stereo setup, the assumption of shape similarity for correspondent curves will be reasonable because of the small variation of viewpoint. However, in general, for any curve pair in two camera planes we can find one space curve by intersecting the projected rays from plane curves into space through the camera centers. Therefore, the inverse problem of determining the space curve from plane curves is an ill posed problem. To find a way out to this problem, we consider the fundamental theorem of space curves as:

Theorem 1. If two single-valued continuous functions κ(s) and τ(s) are

given for s >0 , then there will exist exactly one space curve determined except for orientation and position in space, i.e. up to a Euclidean motion, where s is the arc length, κ is the curvature, and τ is the torsion [30].

The fundamental theorem illustrates that the parameters κ(s)and τ(s) are intrinsic characteristics of space curve that do not change when the curve moves

Figure 3. Reconstructed 3D points from different pairs of points in left and right camera images in rectified stereo configuration. Horizantal scan line is the epipolar line.

Applying the constraint of horizontal scan line, as the epipolar line, the following equations can be easily derived: