A Bayesan Mxture Model for Mult-vew Face Algnent Y Zhou, We Zhang, Xaoou Tang, and Harry Shu Mcrosoft Research Asa Bejng, P. R. Chna {t-yzhou, xtang, hshu}@crosoft.co DCST, Tsnghua Unversty Bejng, P. R. Chna w-z0@als.tsnghua.edu.cn Abstract For ult-vew face algnent, we have to deal wth two ajor probles:. the proble of ult-odalty caused by dverse shape varaton when the vew changes draatcally;. the varyng nuber of feature ponts caused by self-occluson. Prevous wors have used nonlnear odels or vew based ethods for ult-vew face algnent. However, they ether assue all feature ponts are vsble or apply a set of dscrete odels separately wthout a unfor crteron. In ths paper, we propose a unfed fraewor to solve the proble of ult-vew face algnent, n whch both the ult-odalty and varable feature ponts are odeled by a Bayesan xture odel. We frst develop a xture odel to descrbe the shape dstrbuton and the feature pont vsblty, and then use an effcent EM algorth to estate the odel paraeters and the regularzed shape. We use a set of experents on several datasets to deonstrate the proveent of our ethod over tradtonal ethods.. Proble descrpton Paraetrc deforable odels [,, 3, 4, 5, 6] have been wdely used n such vson tass as age segentaton and object localzaton. Aong dfferent odels, statstcal shape odels [3, 6] have shown state of the art perforance for shape regstraton by usng statstcal technques to descrbe shape dstrbuton n order to regularze shapes n shape regstraton. However, the shape odels used n these ethods are generally Gaussan lnear odels and are only capable of descrbng faces wth lted vew changes. When vew changes draatcally, the Gaussan lnear odels often fal to odel shapes properly. To solve the ultodal shape regstraton proble, two an approaches have been proposed. One s the vew based ethod whch uses a set of dfferent odels to represent shapes fro dfferent vews [4], and the other s the non-lnear odel ethod whch uses nonlnear odels to represent shape varatons [5]. The vew-based ethods treat a set of dscrete odels separately wthout a unfor crteron to regularze shape n a ultodal fraewor. It s very dffcult to cover unlted vew change possbltes wth a sall set of dscrete odels. Therefore, the ethod has hgh requreents on the vews of the tranng data, whch has to be close to the testng data. Snce every vew has to be coputed, the coputaton cost sgnfcantly ncreases. On the other hand, the non-lnear odel ethods generally assue all the feature ponts are vsble. But ths assupton ght not hold for ult-vew faces. In fact, an portant ssue for large vew varaton s that soe feature ponts ght becoe nvsble at soe vews because of self-occluson. So ths requres the odel to be able to handle varyng feature pont sets. However, the varyng densonal odel optzaton tself s an ll posed proble of exponental coplexty. Instead of solvng such a varyng denson optzaton proble, we wll treat the pont vsblty as a rando varable and then nfer the probablty of the vsblty of each pont. In ths paper, we propose a ultodal Bayesan fraewor for ult-vew face algnent. Frst, the probles of ult-odalty and varable feature ponts are forulated n a unfed Bayesan fraewor. Specally, we use a xture odel to descrbe the shape dstrbuton and pont vsblty, and then derve the posteror of the odel paraeters gven an observaton of unnown vald feature ponts. Second, an EM algorth s gven to estate the odel paraeters, the regularzed shape, and the vsblty of ts ponts. Extensve experents on several datasets clearly show the effcacy of the new algorth.. Proble forulaton Our probablstc forulaton ncludes a pror xture odel of the shape and pont vsblty and a lelhood odel about the observaton of varable feature ponts n the age space. At the end of ths secton, we derve the posteror about the odel paraeters and forulate a herarchcal hdden varable odel for the proble... Pror odel A shape wth N landar ponts s labeled by a 3Ndensonal vector (x,x,v,,x N,x N,v N ), n whch the par (x, x ) s the coordnates of the th pont and v s a 0- varable ndcatng the vsblty status of the th pont. Usng x=(x,x,,x N,x N ) T to denote all pont coordnates and usng v=(v,,v N ) T as the vsblty ndcator for all ponts, we represent a shape as (x, v). The pror odel ncludes a xture shape odel of x and a xture vsblty odel of v. A xture shape odel s learned after 0-7695-37-/05/$0.00 (c) 005 IEEE
algnng all the shapes to a coon coordnate frae. Coputng the coon coordnate frae and algnng all the shapes s a typcal Generalzed Procrustes Analyss (GPA) [3, 5, 6]. In ths paper, we learn a -cluster xture PPCA odel [9], p( x b) = π f ( x b ) =, N = π ( πσ ) exp x µ Φ b = σ where s the cluster nuber, π s the cluster weght and f ( b ) s a noral densty functon; b=(b,,b ) s called the shape paraeter and each b follows a Gaussan dstrbuton p(b )=N(0, Λ ) n whch Λ s a dagonal atrx; µ s the center of the th cluster and Φ s the prncpal atrx whose coluns are the egenvectors. The pror landar pont vsblty odel s defned as a xture Bernoull odel, N v ( ) ( ) ( ) v p v = π q q, = = where the cluster weght π s the sae as Eq. and q s a value between 0 and whch defnes the probablty of the th pont to be vsble for the shapes n the th cluster. Each q s learned after the xture shape odel s obtaned. For a tranng shape x, ts probablty for belongng to the th cluster s denoted as f (x). We copute f (x) by frst projectng x nto the prncple subspace Φ to get ts shape paraeter b and then ultplyng the probablty p(b ) wth f (x b ). In other words, we copute the Mahanalobs dstance of x to the cluster center µ as the suaton of the M-dstance n the prncpal subspace and the M-dstance fro the prncpal subspace [9]. So far, wth the tranng set {(x,v ),,(x (L),v (L) )}, the pror vsble probablty q of pont n cluster s L L l l l q = v f ( x ) f ( x ). (3) l= l= Fgure shows the two cluster centers of the xture shape odel wth the vsble probablty of each landar pont llustrated as the degree of the darness. The darer the pont s, the ore lely t ght be occluded... Lelhood odel For the current searched age, the observaton for the shape regularzaton s the updated shape after local texture atchng [3, 6]. We call ths updated shape as the observed shape [6] and denote t as y. The observed shape y s n the age space and there s a slarty transforaton dfference between y and the shapes n the underlyng shape space. We use a vector θ=(c,c,s,θ) T to represent ths set of transforaton paraeters, n whch c=(c,c ) T s the translaton, s s the scale and θ s the rotaton. We use T θ ( ) to denote the slarty transforaton ncurred by ths set of pose paraeter,.e. for a shape vector x, T θ (x)=s(i N U θ )x+ N c. Here s the Kronecer product, N s a N-densonal colun vector whose eleents are all one, I N s the N-densonal Fgure : Centers of the two clusters n the xture shape odel. Fgure : The herarchcal hdden varable odel. dentty atrx and U θ s the rotaton atrx wth the angle θ. We denote T - θ ( ) as the nverse transforaton of T θ ( ). The observed shape y s nown to be nosy and soe of ts feature ponts ght be occluded. So gven the underlyng shape x and ts pont vsblty v, y s assued to follow a dstrbuton n Eq. (4), v p( y x, v, θ) = ( πρ ) exp ( v ) T θ ( ) ρ y x,(4) where the notaton s array ultplcaton whch s the entry-by-entry product of two vectors and the notaton s the L nor. The varance ρ s estated fro the update of the shape n the local texture atchng step [6]. The vsblty varable v ndcates the vald feature ponts of the observed shape y..3. Herarchcal hdden varable odel The paraeters to be estated here are the shape paraeter b and the pose θ. Based on the pror odels,, and the lelhood odel (4), the posteror of the paraeters s obtaned by ntegratng the hdden varables, p( b, θ y) p( y x, v, θ) p( x b) p( v) p( b) dxdv. (5) The varables x and v are the hdden varables n our forulaton. In addton, the cluster ndcator varable s denoted as w=(w,,w ) T, whch only taes value le (0,,0,,0, 0) T and ts dstrbuton functon s, w p( w = ) = π or p ( w ) = π. (6) = And then gven the cluster varable w, the dstrbuton of x and v can be rewrtten as w (, ) ( ) p x w b = f x b, = ( ) ( ) N v v p( v w ) = q q. = = Wth x, v, and w, the whole fraewor can be descrbed by a herarchcal hdden varable odel n Fgure. 3. Shape regularzaton by EM algorth w
In ths secton, we descrbe an EM algorth to copute the paraeter MAP estaton n the herarchcal hdden varable odel n Fgure. 3.. EM algorth for MAP of paraeters Instead of drectly optzng the posteror functon n Eq. (5), we apply the EM algorth to fnd the MAP estaton of paraeters. The E step coputes the expectaton of the log-posteror. It s done through the coputaton of the expectaton of several suffcent varables. The dervaton of these suffcent statstcs s presented n Appendx A and B. And we suarze the E step as follows. E) Estaton of the cluster weght: wˆ = E[ w y ]. E) Estaton of the pont vsblty: vˆ = E[ v y, w = ]. E3) Estaton of the shape: xˆ [, ˆ = E x y v, w = ] = (( vˆ ) ) Φ b ( ˆ + v ) ( pφ b + ( p) T θ ( y )). In the above steps, refer to Eq. n Appendx B for the coputaton of w and Eq. for the coputaton of v. Each eleent of v s a value between 0 and, and t deternes how uch we wll use the nforaton of each pont fro the observaton when we estate the underlyng shape x. The for of the estaton of the shape x tells us: ) f soe eleent of v s sall, that s, the correspondng pont s supposed to be very lely to be occluded, we wll put less confdence on that pont; ) when the observed shape s used for the estaton of x, ts weght depends on the rato between ts nose and the nose n the shape space. Please refer to Eq. (0) n Appendx B for the dervaton of x. Wth the expectaton of the suffcent statstcs coputed n the E step, the M step tres to axze the expected log-posteror. It s done separately for the shape and pose paraeters, M) update of shape: wˆ Λ T b = ( Φ xˆ ), wˆ Λ + σ M) update of pose: n ˆ ( ˆ ) ( ( ˆ )) θ = w v y T θ x. = Fro Eq. M), we can see when projectng the shape x to the PCA subspace Φ, t s regularzed by shrnng ts weght along each prncpal drecton. And the ntensty of shrnage depends on the energy Λ at the prncpal drecton. If the energy Λ s sall, that eans the dstrbuton along the prncpal drectons s copact, then the shrnage wll be sgnfcant; otherwse, the shrnage wll be nor. In addton, the cluster weght w s also used to copute the shrnng factor. If the cluster weght s sall,.e. t s unlely that the shape coes fro ths cluster, the shrnage wll be uch sgnfcant for the shape paraeter of ths cluster. Eq. M) shows that the cluster weght also contrbutes to the weghted least square process to update the pose paraeter θ. And the pont vsblty v deternes the contrbuton of each feature pont n the update of pose. 3.. The regularzed shape and ts relablty Once we get the MAP paraeter (b, θ), we can regularze the observed shape y as the transfored expected underlyng shape x wth the pose paraeter θ. And the expected tangent shape x s E x y, b, θ = wˆˆ x. (7) = So the regularzed shape s a weghted average of shapes n dfferent clusters whch are the best atched to the observed shape. It s weghed by the weght of the cluster, whch tells us how lely the current shape belongs to ths cluster. And we can also get the expected pont vsblty v of the regularzed shape x as, E ( v y, b, θ ) = wˆˆ v. (8) = The pont vsblty estaton s a value between 0 and, and t actually gves us a easureent of the relablty of each feature pont of the regularzed shape. In fact, although the shape x and the pont vsblty v are assued ndependent n the pror xture odel, ths ndependence s not tenable any ore after the observaton y s ncorporated. Ths s llustrated by the forulaton n Fgure. The estaton of shape x s based on ts vsblty probablty v. And the pont vsblty probablty easures how consstent one pont s n the shape odel, or n other words, to all other ponts. 4. Experental analyss We apply the herarchcal xture odel to ult-vew face algnent. In ths secton, we descrbe the tranng procedure n detals and evaluate the perforance of the algorth n ters of accuracy and stablty. We deonstrate that, wth a set of dscrete vew odels, we can handle face ages wthn a range of vews. The xture odel can not only copute postons of the feature ponts, but also estate the relablty of each pont. An nterestng byproduct of the new algorth s that we can also use the odel to estate the vew drecton of a face. 4.. Tranng the xture odel The database we use for tranng and testng ncludes both the real and synthetc ages. In order to better evaluate the algorth, we dvde our database nto three parts: Dataset I, Dataset II, and Dataset III. Dataset I s coposed of the frontal, 40 and 50 degree vewed face ages, whch wll be used to tran the pror xture odel. There are 30 frontal face ages n Dataset I whch coe fro the FERET database [7] and the AR database [8]. And there are 00 face ages for the 40 and 50 degree vews respectvely, whch are generated fro the USF Huan ID 3D database [0]. Dataset II also contans vewed face ages generated fro these 3D data. The faces n Dataset II are of 0, 0, and 30 degree
Fgure 3: Faces wth labeled landars. vews and there are 00 ages for each vew. The ages n Dataset II are well labeled, and snce they are dfferent fro those n Dataset I, they wll be used as the ground truth for testng. The ages n Dataset III are all synthetc ages. We use 490 ages n the FERET and AR databases and apply the technque n [] to generate the 3D odel of the face n each age. Then, usng the 3D odel, we can synthesze vewed face ages at arbtrary vews. In Dataset III, we have totally 490x5 synthetc face ages, so there are 490 ages at each vew drecton fro 0 to 50 degrees. The faces n Dataset I and II are labeled wth 83 feature ponts. Fgure 3 shows labeled faces at 0 and 50 degree vews. We draw the ponts wth three dfferent shapes to represent dfferent eanngs. The crcle ponts (whte),.e. the ponts on eyes, eyebrows, and outh, are those of texture sgnfcance n the age and have the sae geoetrcal eanngs at each vew. These ponts are chosen to correspond to the sae ponts on the 3D odel. More precsely, the locatons of these ponts at dfferent vews all have correspondng ponts on the frontal vew face at whch all the ponts are vsble. Unle the crcle ponts, the daond ones (blac) do not have the sae geoetrcal eanng for all the vews. In fact, they are selected because they have strong texture sgnfcance on the age, for exaple the edges, and have portant functons n age searchng. Ths nd of ponts ncludes those on the nose and the left part of slhouette (boundary wth the bacground). All the other ponts (gray) are drawn wth squares for those wthout texture eanngs on the ages. They are chosen to correspond to the sae ponts on the 3D face odel. These ponts are on the rght part of the slhouette. After we fnsh labelng all the locatons of these feature ponts, ther vsbltes are autoatcally coputed wth the 3D odel of each face. Tranng the herarchcal odel ncludes learnng a xture shape odel for all the 83 feature ponts and a xture vsblty odel for each pont. In the pror odel, we learn a xture PPCA odel for the shape and a xture Bernoull odel about the vsblty of each pont as descrbed n Secton.. Tranng s done on the ages n Dataset I and two clusters are learned for the xture odel. Fgure shows the two cluster centers. One cluster s the faces of nearly the frontal vews and the other s the faces of 40 and 50 degree vews. Gven a shape x, the probablty p(w = x) descrbes the lelhood that x Fgure 4: Local updatng step. coes fro the cluster and how close t s to ths cluster center, so ths probablty ay gve us soe nforaton on the vew of x. Based on ths consderaton, we learn a regresson functon of the vew over the cluster weght estaton fro Dataset I and II. The probablty p(w = x) s proportonal to p(w =)f (x). Snce the vew dstrbuton n Dataset I and II s very sparse, whch only taes values as 0, 0, 0, 30, 40 and 50, the regresson functon we learned s a very sple nd-order polynoal. Thus, wth ths regresson functon and the estated weght we can roughly estate the vew drecton of an nput face. 4.. Local texture update Slar to ASM and BTSM, the process of our ethod s coposed of two steps: local texture atchng and global shape regularzaton. However, because of the ultodalty of the ages we odel, we use a slghtly dfferent local texture atchng strategy. The local texture odel s learned for each cluster separately. Specfcally, for each cluster, the local texture odel s traned on the faces whose shapes are wthn the three standard devatons of the PPCA odel of each cluster. Our local texture odel s the sae as ASM and BTSM, whch s based on gray gradents at the feature pont. Also, our ethod s pleented n a ult-resoluton fraewor and a threelayer Gaussan age pyrad s fored on each age by down-saplng ts pxels to obtan ages of coarser resolutons. And the local odels are traned for each layer. Therefore, for each pont, there are two local odels at each layer. When we use the two odels to do local updatng, we frst calculate the update of the pont wth each local odel and then wegh the results to get a fnal one. And the weght s just the weght estaton of the cluster. Fgure 4 shows ths local updatng process. After the local updatng, n the shape regularzaton step, we use the EM algorth n Secton 3 to estate the shape and pose paraeters, and fnally get the estaton of the regularzed shape and the relablty of each pont. 4.3. Algnent accuracy The accuracy of our algorth s tested on Dataset II whch has labeled ground truth. It should be noted that the faces n Dataset II are of 0~30 degree vews, whch do not appear n our tranng data n Dataset I. We ntend to show that usng only a set of dscrete vew odels, we
(a) (b) Fgure 5: Statstcs of algnent errors. (a) Denstes of error dfferences. (b) Denstes of errors of top 0 worst ponts. (a) Fgure 6: Results on testng ages. can handle face ages n a range of vews. To deonstrate the accuracy, we copare our ethod wth another two exstng ethods: the vew based odel and the nonlnear odel. The algnent error s calculated as the average pont-to-pont dstance between the algned ponts and the labeled ground truth. For the vew based odel, we tran two lnear BTSM odels [6] for frontal vew and 40-50 vew respectvely wth the face ages n Dataset I and present the best result of the ndvdual vewed BTSM odels. For other nonlnear odels, we copare ours wth that usng a xture Gaussan odel for representng shape varatons n ASM [5]. The three algorths use the sae local texture descrpton and start wth the sae ntalzaton. The algnent accuracy of the three ethods s copared n Fgure 5. Fgure 5 (a) shows the overall proveent of our odel over the other two. It shows that our ethod s better than the xture ASM on 8% ages and outperfors the best vewed BTSM on 9% ages. The proveent can be seen ore clearly f we only concentrate on the worst algned ponts. Fgure 5 (b) shows the average errors of top 0 worst algned ponts: whle 8% ages algned by the xture ASM and 74 % ages algned by the vewed BTSM have ths error above 0 pxels, our odel has an error below 0 pxels on 75% ages. Soe results of the three ethods are also shown n Fgure 6. These results show two aspects of our algorth. Frst, copared wth vew based ethods, our ethod can reasonably wegh each xture coponent odel. Snce the (b) (c) Fgure 7: (a) Algned results. (b) Vew estaton. (c) Estaton of the vsblty probablty of the left eye nner corner pont. weghted cobnaton of xture coponents ples a uch bgger saple space, our ethod can also wor on the cases even f slar vewed saples are not n the tranng data. Second, copared wth other nonlnear odels, our odel consders the vsblty of the feature ponts and uses ths vsblty easure to prevent the ssng or sleadng features to run the overall result. 4.4. Vew and pont vsblty estaton One nterestng result of our ethod s that we can estate the face vew drecton usng only the D odel. We do ths through the estaton of the cluster weghts. In our odel, ths weght estaton represents how far the face s fro the cluster center n ters of the Mahalanobs dstance. We have one cluster of nearly frontal faces and the other of about 40~50 degree vewed faces. It s reasonable to assue that the faces between 0 and 50 degrees have dfferent dstances to the two cluster centers. And ths conjecture s confred by our experental results. The dotted lne n Fgure 7 (b) shows vew estaton results when the vew of a face changes fro 0 to 50 degrees and Fgure 7 (a) shows the algnent results. We can see the estaton s consstent for the contnuous change of vews. It shows the weght estaton s rather stable. Although the results see not ftted well between 40 to 50 degrees, ths s reasonable snce 40 and 50 degree vewed faces are n the sae cluster n our odel so that the vews between the cannot be classfed well. Another nterestng outcoe s the estaton of pont vsblty and ths quantty s a probablty between 0 and whch can gve us a relablty easure for each pont. Fgure 7 (c) shows the curve of the estated vsble probablty of the left-eye nner corner pont wth the change of vews. The gray dot ponts exactly correspond to the results vsualzed n Fgure 7 (a).
(a) (b) Fgure 8: Statstcs of vew estaton errors. (a) Denstes. (b) Eprcal CDFs. Fnally, we test the behavor of our vew estaton ethod on Dataset III. Dataset III s coposed of synthetc vewed face ages fro 0 to 50 degree vews and there are 490 ages at each vew. Snce the weght wll becoe undscrnatng between 40 and 50 degree vews, we present the results n these vews separately. Fgure 8 (a) shows the densty of the vew estaton errors. The x-axs s the dfference of the estated vew and the true vew. We can see that for the vews under 40 degrees, the vew estaton s roughly a Gaussan dstrbuton around the true vew wth a standard devaton of about 0 degrees. Fgure 8 (b) further plots the eprcal c.d.f. of the absolute errors. It shows that for the face vews bellow 40 degrees, 80% of the have the vew estaton errors under 8 degrees and 90% under 0 degrees. 5. Concluson and dscusson Ths paper has presented a new nonlnear odel for ultodal shape regstraton proble. By odelng shape and pont vsblty n a xture probablstc odel, we have handled the large shape varaton and varable feature ponts n a unfed fraewor. Wthn ths fraewor, an effcent EM algorth for shape regularzaton s developed. When leveragng all the odaltes n our odel, we reasonably wegh ther contrbutons n a sense of goodness-of-ft. Snce ths nd of contnuous weght values are ore accurate than the dscrete values used by tradtonal vew based ethods, our ethod s ore stable when the vew changes. That partally explans why our odel can wor on a range of vews wth the pror odel learned only on several vews. On the other hand, by ntroducng a hdden varable to ndcate the vsblty status of each pont, we handle the self-occluson stochastcally. 6. References [] A. Blae and M. Isard. Actve Contours. Sprnger, 998. [] M. Kass, A. Wtn, and D. Terzopoulos. Snaes: Actve contour odels. IJCV, :3--33, 987. [3] T. F. Cootes, C. Taylor, D. Cooper, and J. Graha. Actve shape odels ther tranng and ther applcatons. Coputer Vson and Iage Understandng, 6:38-59, January 995. [4] T. F. Cootes, G. V. Wheeler, K. N. Waler, and C. J. Taylor. Vew-based Actve Appearance Models. Iage and Vson Coputng, 00. [5] T. F. Cootes and C. Taylor. A xture odel for representng shape varaton. BMVC, 997. [6] Y. Zhou, L. Gu and H. J. Zhang. Bayesan Tangent Shape Model: Estatng Shape and Pose va Bayesan Inference. IEEE Conf. on CVPR, 003. [7] P. J. Phllps, H. Moon, S. A. Rzv, and P. J. Rauss. The FERET evaluaton ethodology for face recognton algorths. IEEE Trans. on PAMI, (0):090--04, 000. [8] A.M. Martnez and R. Benavente. The AR Face Database. CVC Techncal Report #4, June 998. [9] M. E. Tppng and C. M. Bshop. Mxtures of Probablstc Prncpal Coponent Analysers. Neural Coputaton, July 998. [0] V. Blanz and T. Vetter. A orphable odel for the synthess of 3D-faces. ACM SIGGRAPH, 999. [] Y.X. Hu, D. L. Jang, S. C. Yan, and H. J. Zhang. Autoatc 3D Reconstructon for Face Recognton. Intl. Conf, on AFGR, 004. Appendx A. Expectaton of log-posteror The expectaton of the log-posteror of the coplete varable (y,x,v,w) s coputed by the followng functon, { T Q( b, θ b, θ ) =,,, E we E wy vw y xvw y b Λ b = (9) + ρ ( v ) ( y Tθ ( x) ) + σ x Φ b + const So the coputaton of Q-functon s to copute the wˆ E vˆ = E v y, w =, and suffcent statstcs = [ w y ], [ ] xˆ = E[ x y, v ˆ, w = ]. B. Soe condtonal probabltes ) Posteror of the shape x n cluster wth vsblty v: p( x y, v, w = ) f( x b ) p( y x, v, θ) (0) exp x Φ b ( v ) T θ ( ) σ s ρ x y So the expectaton of x s that of Eq. E3) n Secton 3.. ) Posteror of pont vsblty v for each cluster: p( v y, w = ) p( v w = ) p( y x, v, θ) f ( x b ) dx v N v v ( πδ ) ( q ) ( q ) = ( πρ )( πσ ) ( v ) z' ( v ) z ( v ) z exp + + δ σ ρ s 3) Posteror of vew class: p( w = y) p( w = ) p( y w = ) N q δ z z z π ( q ) exp + + + πσ ρ δ σ ρ s = where δ ( σ s ρ ) = +, p = ( s ρ ) ( σ + s ρ ), p = p, z = Φ b, z = Tθ ( y),and z = pz+ pz, z s the th pont of z.