computer animation Y.GUO ET AL virtual worlds 00●年◆。ee。ee。ee00。。●。。。.e.e.................0g An alternative surrounding entity of the reference point is the ball,which can be divided with respect to its radius,spherical elevation as well as rotation angles in the framework of spherical coordinate.Nevertheless,in such way,the computation of geometric context is time consuming.As can be seen in the next subsection,the geometric context described in Subsection Definition of Geometric Context'can be computed quickly by using integral volume,a new volume-encoding form. Calculation of Geometric Context (a) (b) Our 3D face detection approach needs to compute Figure 1.Geometric context.(a)The zoomed-in side view.(b) geometric context for each surfacial voxel of the A 2D illustration.Each 3:(n)of S:is valued with the ratio symmetric model or its symmetric part.Although of sub-cube;(n)'s gray voxel number to sub-cube (n)'s volume current voxelization algorithm19 has achieved nearly of voxel. real-time performance for complex models,computation of geometric context still undergoes much redundant processing.We introduce here integral volume,a new encode geometric features.Figure 2 shows an example volume-encoding form to improve the efficiency,which of such division on the nose of 3D face. is computed in the voxelization process. The principle of integral volume resembles integral Property of Geometric Context image that has been used in 2D face detection.2 We first give its explanation on 2D space,then extend it to 3D. Similar to other geometric descriptors successfully In Figure 3(a),the 2D shape contour is surrounded by used in 2D shape matching'6 and geometric data one rectangle,which is divided into many 2D voxels. registration,17.18 geometric context is a quantity com- Every voxel is set a value V,and if the voxel is on the puted for each surfacial voxel on the model,based on contour,Vis set1,otherwise 0.In Figure3(b),every voxel the local shape around the reference vertex.As the local is set a integral volume value IV that equals the number surrounding box is divided and the shape ingredient is of all voxels lying on the contour in its upper-left area, computed in each subdivision,the geometric context is that is, not only rich and compact,but also highly discriminative to characterize facial features. IV(xo,yo)= ∑Vx,) (4) x≤0，JyS30 where V(x,y)is the value of voxel with discrete coordinate (x,y)in Figure 3(a). Symmetric plane xX/ 0000 0000 0000 0000 000 000 0008222 00 00 001/ 350 666 00 0 0 00 0 0 46888 0 0 T■T 十257912134 0000 12579121314 Volume Integral Volume (a) (b) (a) (b) Figure 2.Division of surrounding box.(a)The frontal view of Figure 3.2D case of integral volume.(a)The shape contour surrounding box.(b)A zoomed-in slant view of the division. voxel isset 1,while the rest0.(b)Forevery position,the integral v:is enveloped in the central sub-cube. volume stores the sum of its upper-left voxels'value. Copyright2007 John Wiley Sons,Ltd. 486 Comp.Anim.Virtual Worlds 2007;18:483-492 DoL:10.1002/cavY. GUO ET AL. ........................................................................................... Figure 1. Geometric context. (a) The zoomed-in side view. (b) A 2D illustration. Each
si(n) of Si is valued with the ratio of sub-cubei(n)’s gray voxel number to sub-cubei(n)’s volume of voxel. encode geometric features. Figure 2 shows an example of such division on the nose of 3D face. Property of Geometric Context Similar to other geometric descriptors successfully used in 2D shape matching16 and geometric data registration,17,18 geometric context is a quantity com￾puted for each surfacial voxel on the model, based on the local shape around the reference vertex. As the local surrounding box is divided and the shape ingredient is computed in each subdivision, the geometric context is not only rich and compact, but also highly discriminative to characterize facial features. Figure 2. Division of surrounding box. (a) The frontal view of surrounding box. (b) A zoomed-in slant view of the division. vi is enveloped in the central sub-cube. An alternative surrounding entity of the reference point is the ball, which can be divided with respect to its radius, spherical elevation as well as rotation angles in the framework of spherical coordinate. Nevertheless, in such way, the computation of geometric context is time consuming. As can be seen in the next subsection, the geometric context described in Subsection ‘Definition of Geometric Context’ can be computed quickly by using integral volume, a new volume-encoding form. Calculation of Geometric Context Our 3D face detection approach needs to compute geometric context for each surfacial voxel of the symmetric model or its symmetric part. Although current voxelization algorithm19 has achieved nearly real-time performance for complex models, computation of geometric context still undergoes much redundant processing. We introduce here integral volume, a new volume-encoding form to improve the efficiency, which is computed in the voxelization process. The principle of integral volume resembles integral image that has been used in 2D face detection.2 We first give its explanation on 2D space, then extend it to 3D. In Figure 3(a), the 2D shape contour is surrounded by one rectangle, which is divided into many 2D voxels. Every voxel is set a value V, and if the voxel is on the contour, V is set 1, otherwise 0. In Figure 3(b), every voxel is set a integral volume value IV that equals the number of all voxels lying on the contour in its upper-left area, that is, IV(x0, y0) = x≤x0,y≤y0 V(x, y) (4) where V(x, y) is the value of voxel with discrete coordinate (x, y) in Figure 3(a). Figure 3. 2D case of integral volume. (a) The shape contour voxel is set 1, while the rest 0. (b) For every position, the integral volume stores the sum of its upper-left voxels’ value. ............................................................................................ Copyright © 2007 John Wiley & Sons, Ltd. 486 Comp. Anim. Virtual Worlds 2007; 18: 483–492 DOI: 10.1002/cav