carving [35,36] and view ray sampling [37]. In the voxel carving method, a discrete number of voxels are constructed around the volume of interest. Then, each voxel is checked for all silhouettes and any voxels that project outside the silhouettes are removed from the volume. Voxel carving can be accelerated using octree representation which employs coarse to fine hierarchy. In view ray sampling, a sampled representation of the visual hull is constructed. The visual hull is sampled in a view-dependent manner. For each viewing ray in some desired view, the intersection points with all surfaces of the visual hull are computed. Moezzi et al. [38] construct the visual hull using voxels in an off-line processing system. Cheung et al. [39, 40] show that the voxel method can achieve interactive reconstruction results. The polyhedral visual hull system developed by Matusik et al. [41] also runs at interactive rate.

In this section, two efficient algorithms are presented to improve the speed of computations in visual hull extraction. The first algorithm accelerates the voxel carving method. This algorithm reduces the number of check-points at intersection test procedure. The octree division method is optimized, by minimizing the number of check-points, to find intersection between cubes and silhouette images. To accomplish this function, the points are checked on the edges of octree cubes rather than the inside of volume. Furthermore, the points are checked hierarchically and their number is changed corresponding to the size of octree cubes. The second algorithm employs the ray sampling method to extract the bounding edge model of visual hull. To find the segments of any ray which lies inside the other silhouette cones, the points of ray are checked hierarchically.

5.2.1. Volume Based Visual Hull

Many algorithms have been developed to construct the volumetric models from a set of silhouette images [35, 36, 37, 39]. Starting from a bounding volume that is known to surround the whole scene, the volume is divided into voxels. The task is finding which voxels belong to the surface of 3D object, corresponding to the intersection of back-projected silhouette cones. The most important step in these algorithms is the intersection test. To make the projection and intersection test more efficient, most methods use an octree representation and test voxels in a course-to-fine hierarchy.

5.2.1.1. Intersection Test in Octree Cubes

The most important and time-consuming part of octree reconstruction is the cubes intersection check with silhouette images. All algorithms use one common

Obviously, the total number of check points will be smaller than Ntot(max), because intersected cubes normally will be recognized in early checking. This number can be used as a measure to compare different methods of intersection check. It is obvious that the computing time is proportional with Ntot. Edge based checking is one approach to decrease the number of checking points to identify the mark of cube without loss of accuracy. Any intersection between cube and silhouette should be occurred through edges in one-piece objects. Of course there is one exception case when the object is small and posed inside the cube so that there is no intersection between object and cube through the edges. For such cases, the edge base method can not be employed to decide if object is inside the cube or intersects with cube through the face. Therefore checking some points inside the volume will be inevitable for this situation. If the size of bounding cube be selected properly, comparable to that of object, the cube will be larger than the object only in first level of division. So the ambiguity of intersection test through the edges will be stay only for the first level. Since the octree division is done at all times without checking the occurrence of intersection in first level, the use of edge based intersection test for one piece object can be applied with certainty. Another approach to decrease the number of check points is to change the number of points dynamically in each level. In fact, the large cubes may intersect with small parts of silhouette and it needs checking of more points to identify the intersection. In small cubes, this situation can not be occurred and there is no need to check more points. By choosing NL=8 (checking only corners of cube in last level) and increasing checked points with the factor of 'k' in lower levels, we can minimize Ntot (max) as below:

The final approach to increase the speed is to check the edge points hierarchically. In this way the chance to find two different color points in early checks could be increased.

5.2.1.2. Synthetic Model Results

To determine the capability of presented algorithm and to quantify its performance, we have tested it on a synthetically generated image named Bunny and Horse. Simulation was run on PC Pentium-III 933Mhz using Matlab and C++ generated Mex files. In this analysis, 18 silhouettes of bunny from equal space