points to the tracking curves, with more precise motion parameters, can be achieved.

Once the six motion parameters were estimated for two consecutive sequences, the motion matrix can be constructed as:

The new position of each point, in the next frame, can be calculated by multiplying the motion matrix to the position vector.

4. Motion Estimation Using Double Stereo Rigs

In this section, we present a double stereo configuration to get as much accuracy as possible in estimation of motion parameters. The basic idea to achieve this end is to find an arrangement of stereo cameras in which the sensitivity of image pose variation to space pose variation is maximized. At first, the single stereo setup is investigated and then a perpendicular double stereo configuration is presented and its dominance to the single stereo is demonstrated.

4.1. Single Stereo Rig

As mentioned in section 2.3, the base line of stereo rig is adjusted neither small nor wide to compromise between depth uncertainty and occlusion. Moreover, to utilize the linear part of camera lens and to get rid of the complex computations of nonlinear distortion, the view angle is chosen as small as possible. Hence, the size of the object is usually very smaller than its distance from the camera center, i.e. 2r tz (see figure 8).

Figure 8. Single stereo setup with small view angle ( tz >> 2r ).

Now, we would like to answer the question that how much accuracy is achievable in space motion estimation by tracking the projection of points in camera planes. For the sake of simplicity, we assume that the optical axis of camera1 is in the depth direction of world coordinate (i.e. Zw). Projection of any point (X w ,Yw , Zw ) in the image plane of camera1 is computed as:

Therefore, xim1 and yim1 are very sensitive to Xw and Yw compared to Zw. As we explained in section 3, the six motion parameters are adjusted in which the

convergence of motion estimation algorithm. Combination of this assumption with Eq.32 and Eq.33 can be resulted in:

This equation reveals that the inverse problem of 3D motion estimation by tracking the points in camera plain is an ill posed problem and does not have one

inaccurate 3D motion parameters will be estimated.

4.2. Double Stereo Rigs

Due to the limitation of large base line selection in single stereo rig, both stereo cameras have approximately the same effect in motion estimation process. To take the advantages of both small and wide base line stereo cameras, we present a combined double stereo setup. This combination is composed of two single stereo rigs in which they make angleθ from each other (see figure 9).

Figure 9. Structure of double stereo setup: (a) Double stereo setup with angleθ , (b) Perpendecular double stereo setup.

Similar to single stereo setup and considering the rotation angle of θ for camera3, it can be easily shown that:

By choosing a proper amount ofθ , it is possible to increase the sensitivity of xim3 and yim3 to Zw as much as possible. Therefore, we can minimize the 3D motion estimation errors ∆Xw and ∆Yw by minimizing ∆xim1 and ∆yim1, and the estimation

error ∆Zw by ∆xim3 and ∆yim3. It can be verified, by differentiating, that the maximum value of sensitivity is achieved byθ =90 . Forθ =90 , the Eq.36 is simplified as:

perpendicular double stereo setup and more precise motion parameters will be resulted.

5. Shape Reconstruction from Object Silhouettes Across Time

Three-dimensional model reconstruction by extracting the visual hull of an object has been extensively used in recent years [31-34] and it has become a standard and popular method of shape estimation. Visual hull is defined as a rough model of the object surface, which can be calculated from different views