Figure 15. Start-End points, Bounding-Edges and Depth-Map for 3 models named bunny, female and dinosaur which have been extracted through 18 silhouettes from different views using hierarchical algorithm.

In figure 15 we have demonstrated the start-end points of edges, bounding edges and depth map of reconstructed 3d models for synthetically objects named horse, bunny, female and dinosaur from 18 silhouette by hierarchical method has been shown.

Implementation and Exprimental Results

To evaluate the efficiency of our approach to 3D model reconstruction by tracking the space curves using the perpendicular double stereo rigs, there are

two questions that we need to consider. First, how good is the estimation of motion parameters? Second, how efficient is our method in practice or how robust is our method against the disturbing effects like noise? In this section, we present the results of experiments designed to address these two concerns. Experimental results are conducted with both synthetic and real sequences that contain different textures of objects. Two synthetic objects named Helga and Cow were captured by perpendicular double stereo cameras in 3D-StudioMax environment. The objects were moved with arbitrary motion parameters and 36 sequences were captured by each camera. Figure 16 shows the camera setup to capture Helga and Cow models along with their extracted space curves. It is clear that the number of outlier points is significantly less than the number of valid unique points. Figure 17 illustrates the curve tracking and motion estimation process by minimizing the geometric distance of curves in the camera images. Two sets of space curves, which have been extracted by two distinct stereo rigs, are projected to the camera planes in the next sequence and motion parameters are adjusted in which the projection of space curves in the related camera planes to be as close as possible to nearby curves. Figures 18 and 19 tend to demonstrate how the projection of unique points becomes closer and closer to edge curves, in each iteration, to minimize the geometric distance error. To evaluate the estimation error of motion process, variation of the six motion parameters across time has been plotted in figures 20 and 21 for Cow and Helga sequences. Comparing diagrams of true motion with diagrams of estimated motion by single stereo and perpendicular double stereo setup reveals the superiority of the perpendicular double stereo to the single stereo. The assessment is also given numerically in table 4 and table 5. Figures 22 and 23 show the temporal sequence of Helga and result of implementation for virtual camera alignment across time. In addition, this figure illustrates how the silhouette cones of virtual cameras are intersected to construct the object visual hull. Figure 24 compares the quality of reconstructed Helga model using true motion information and estimated motion information by single and perpendicular double stereo setups. Figures 25 to 27 demonstrate the result of implementation for Cow sequences. To evaluate the robustness of motion estimation against the noise and color maladjustment, comparison between single stereo and perpendicular double stereo are given both quantitatively and qualitatively in table 6 and figure 28. To get the qualified edge curves in noisy image, it will be necessary to smooth the image before edge detection process. At the same time, smoothing the image makes small perturbation in the position of edge points. The perpendicular double stereo setup appears to be more robust against the perturbation of edge points. Figure 29 to 34 demonstrate the result of

implementation for real objects sequences named Buda, Cactus and Head by perpendicular double stereo setup. All objects have been captured through the turntable sequences. Notice the small perturbations from circular path in the alignment of virtual cameras for Head sequence. These perturbations are caused by the none-rigid motion of body in the neck region. In this experiment, the process of motion estimation has been accomplished only based on the head (not body) region information. Both synthetic and real experimental results demonstrate the honored performance of the presented method for the variety of motions, object shapes and textures.

Figure 16. Reconstructed spaced curves on the surface of synthetic models Helga and Cow (captured in 3D Studio Max).

Figure 17. Motion estimation by tracking the projections of space curves in perpendecular double stereo images.

Figure 18. Motion estimation by minimizing the geometric distance of space curves from adjacent curves in four camera images. Convergence of algorithm is shown for different number of iterations.

Figure 19. Motion estimation by minimizing the geometric distance of space curves from adjacent curves in the projected camera images. Convergence of algorithm is shown for different number of iterations for (a) Cow and (b) Helga.

Figure 21. True and estimated motion parameters for Helga sequences by single and perpendicular double stereo setup.

Table 5. Estimation error of motion parameters for Helga sequences by single and perpendicular double stereo setup

Figure 22. Different views of Helga in 36 sequence of its motion.

Figure 23. Three-dimensional model reconstruction from multivies for Helga sequences. (top) Trajectory of estimated virtual cameras by perpendicular double stereo rigs along with two silhouette cones intersection, (middle) extraction of visual hull by all silhouette cones intersection and, (down) color mapping from visible cameras.

Figure 24. Reconstructed model of Helga including the bonding edges visual hull, depthmap and texture mapped 3D model: (a) estimated-motion with single stereo, (b) estimatedmotion with perpendicular double stereo, and (c) true-motion.

Figure 25. Different views of Cow in 36 sequence of its motion.

Figure 26. Trajectory of estimated virtual cameras by perpendicular double stereo rigs and extraction of visual hull by silhouette cones intersection (Cow sequences).

Figure 27. Reconstructed model of Cow including the bonding edges visual hull, depthmap and texture mapped 3D model: (a) estimated-motion with single stereo, (b) estimatedmotion with perpendicular double stereo and (c) true-motion.

Table 6. Estimation error of motion parameters for noisy sequences of Helga by single and perpendicular double stereo setup (σn2 = 0.1)

Figure 28. Effect of noise and color unbalance in 3D reconstruction: (a) noisy images of different cameras with σ2 = 0.1, (b) reconstructed model with single stereo rig and (c) reconstructed model with perpendicular double stereo rigs.

Figure 29. Different views of Buda in 36 sequence of its motion.

Figure 31. Different views of Cactus in 36 sequence of its motion.

Figure 33. Different views of Head (the picture of author) in 36 sequence of its motion.