computer animation Y.GUO ET AL virtual worlds 00●年◆。eee。ee。ee00。。●。。。.e.e.................0g Most of them exert trivial effect in general.As a result, based on a great deal of training examples which include 3D face models and non-face models,AdaBoost learns the weak classifiers and combines the most effective ones to form a strong classifier. AdaBoost is an iterative process.During every iteration,the classifiers are trained and the one with the lowest classification error is picked out.Thereafter,those (a) (b) difficult examples of classification will be emphasized in the following iterations by assigning them higher training weights.The selected classifiers are finally integrated to form a strong classifier with the learned weighted coefficients. Weak Learning.A fundamental step of AdaBoost iteration is the weak learning algorithm,which is devised +d=0 to select the weak classifier that can best separate the 3D face and non-face models.Since the geometric context (c) (d) is N-dimensional,the weak classifier is built upon the distribution of the 3D face models'corresponding Figure 4.Symmetry detection.(a)The 3D model;(b)sampled geometric contexts. feature points.(c)Quads space for those potentially symmetric As a preprocessing,we first align the examples of feature points.Some quads in the circle cluster together.(d)The 3D face models together with the non-face models extracted symmetric plane with equation aox+boy coz+ into consistent reference frame.Then we voxelize their d6=0,d6=01. surfaces into uniform voxel representation and compute the geometric contexts separately. to construct geometric context.We take the one that For the corresponding surfacial voxels at the same is perpendicular to the symmetric plane and passes position on 3D face examples,we assume that every through the first eigenvector produced with principle element3(n)of their geometric contexts with consistent component analysis on sampled vertices as the base edge length follows Gaussian distribution.Hence the N- plane.The algorithm for reflective symmetric detection dimensional geometric contexts S satisfy N-dimensional Gaussian distribution, is rather fast and effective. Gi(S)=as e-(s-msi)cs'(s-ms1) (8) AdaBoost Learning-Based 3D Face Model Detection where irepresents the index of the current voxel position. ms;is the mean vector,Cs;for the covariance matrix,and Through the above reflective symmetry detection asi for a constant which gives unit normalization. algorithm,asymmetric regions are eliminated and With the above distribution,the weak classifier h is only those reflective symmetric and nearly reflective defined as symmetric parts are viewed as the candidates for the 3D face.We then examine those parts and determine hi(S)=sign(Gi(S)-0:) (9) whether the face part exists using AdaBoost learning algorithm. During the weak learning,0 is chosen as a threshold such With voxel representation,the geometric contexts with that the least examples are misclassified. variant edge length R over surface of the symmetric part construct a large feature space.Each geometric context AdaBoost Learning Algorithm.Given the set M of can perform simple classification of 3D face and non-face, training examples which consist of K models and their that is,it yields a weak classifier.Nevertheless,no single labels as following, classifier can achieve high detection rate reliably and accurately.In addition,not all the classifiers are crucial. M=(xj.yj)j=1...K (10) Copyright 2007 John Wiley Sons,Ltd. 488 Comp.Anim.Virtual Worlds 2007;18:483-492 DoL:10.1002/cavY. GUO ET AL. ........................................................................................... Figure 4. Symmetry detection. (a) The 3D model; (b)sampled feature points. (c) Quads space for those potentially symmetric feature points. Some quads in the circle cluster together. (d) The extracted symmetric plane with equation a0x + b0y + c0z + d0 = 0, d0 = 0, 1. to construct geometric context. We take the one that is perpendicular to the symmetric plane and passes through the first eigenvector produced with principle component analysis on sampled vertices as the base plane. The algorithm for reflective symmetric detection is rather fast and effective. AdaBoost Learning-Based 3D Face Model Detection Through the above reflective symmetry detection algorithm, asymmetric regions are eliminated and only those reflective symmetric and nearly reflective symmetric parts are viewed as the candidates for the 3D face. We then examine those parts and determine whether the face part exists using AdaBoost learning algorithm. With voxel representation, the geometric contexts with variant edge length R over surface of the symmetric part construct a large feature space. Each geometric context can perform simple classification of 3D face and non-face, that is, it yields a weak classifier. Nevertheless, no single classifier can achieve high detection rate reliably and accurately. In addition, not all the classifiers are crucial. Most of them exert trivial effect in general. As a result, based on a great deal of training examples which include 3D face models and non-face models, AdaBoost learns the weak classifiers and combines the most effective ones to form a strong classifier. AdaBoost is an iterative process. During every iteration, the classifiers are trained and the one with the lowest classification error is picked out. Thereafter, those difficult examples of classification will be emphasized in the following iterations by assigning them higher training weights. The selected classifiers are finally integrated to form a strong classifier with the learned weighted coefficients. Weak Learning. A fundamental step of AdaBoost iteration is the weak learning algorithm, which is devised to select the weak classifier that can best separate the 3D face and non-face models. Since the geometric context is N-dimensional, the weak classifier is built upon the distribution of the 3D face models’ corresponding geometric contexts. As a preprocessing, we first align the examples of 3D face models together with the non-face models into consistent reference frame. Then we voxelize their surfaces into uniform voxel representation and compute the geometric contexts separately. For the corresponding surfacial voxels at the same position on 3D face examples, we assume that every element
si(n) of their geometric contexts with consistent edge length follows Gaussian distribution. Hence the N￾dimensional geometric contexts S satisfy N-dimensional Gaussian distribution, Gi(S) = αSi e− 1 2 (S−mS i) T CS −1 i (S−mS i) (8) where irepresents the index of the current voxel position. mS i is the mean vector, CS i for the covariance matrix, and αS i for a constant which gives unit normalization. With the above distribution, the weak classifier hi is defined as hi(S) = sign(Gi(S) − θi) (9) During the weak learning, θi is chosen as a threshold such that the least examples are misclassified. AdaBoost Learning Algorithm. Given the set M of training examples which consist of K models and their labels as following, M = (xj , yj )j=1,...,K (10) ............................................................................................ Copyright © 2007 John Wiley & Sons, Ltd. 488 Comp. Anim. Virtual Worlds 2007; 18: 483–492 DOI: 10.1002/cav