curve on surface of the object, and range of depth-variation across the curves, the representative curve length may be varied.

3. Rigid Motion Estimation by Tracking the Space Curves

The movement of a rigid object in space, can be expressed as the six rotation and translation parameters. Now, suppose that we have extracted a set of points on the surface of an object and the goal is to estimate 3D motion of the object across time. To estimate the motion parameters, an error function, which describes the difference of points before and after motion, should be minimized. To get rid of photometric information, we define the error function as a distance of unique points from the nearby curves after movement. To explain the problem mathematically, suppose that Wi is the ith unique point, Nu is the total number of

unique points, Pk (R Wi + T) is projection of Wi in camera plane k after movement, and contourk(m) is the curve number m in camera plane k. To estimate

the motion matrix, i.e. R and T, the error component for each unique point in each projected camera image is defined as the minimum distance of that point from nearby curves in that camera. The total error is also calculated by summing error components over all unique points and all cameras.

e = ∑iNu=1 ∑Kk =1eik ; R,T = arg min{e}

Where K is the total number of cameras (K=2, for single stereo rig). To find the minimum distance of each point from nearby curve in the camera image, we use a circle based search area with increasing radius (figure 7). Therefore, the minimum distance is determined as the radius of the first osculating circle with

components of translation vector. The total error function defined in Eq.22 can be minimized by an iterative method similar to the Levenberg-Marquardt algorithm [8]: