3D Face recognition by ICP-based shape matching

Transcription

1 3D Face recognton by ICP-based shape matchng Boulbaba Ben Amor, Karma Ouj, Mohsen Ardablan, Lmng Chen LIRIS Lab, Lyon Research Center for Images and Intellgent Informaton Systems, UMR 505 CRS Centrale Lyon, France Abstract In ths paper, we propose a novel face recognton approach based on.5d/3d shape matchng. Whle most of exstng methods use facal ntensty mage, we am to develop a method usng three-dmensonal nformaton of the human face. Ths s the man nnovaton of our technology. In our approach, the 3D dmensonal nformaton s ntroduced n order to overcome classcal face recognton problems whch are pose, llumnaton and facal expresson varatons. The paradgm s to buld a 3D face gallery usng a laser-based scanner: the off-lne stage. In the on-lne stage, the recognton, we capture one.5d face model at any vew pont and wth any facal expressons. Our processng allows the dentfcaton of the presented person by performng the captured model wth all faces from the database. Here, the Iteratve Closest Pont-based matchng algorthm provdes the pose of the probe whereas the regon-based metrc provdes a spatal devaton between the probe and each face from the gallery. In ths metrc, we calculate the global recognton score as a weghted sum of regon-based dstances already labelled as mmc or statc regons. For automatc 3D face segmentaton, we use an mmerson verson of watershed segmentaton algorthm. Ths paper also presents some experments n order to shown llumnaton, pose and facal expresson compensatons. I. ITRODUCTIO AD MOTIVATIO Over the past few decades, bometrcs and partcularly face detecton [], analyss, measurement and descrpton have been appled wdely n several applcatons such as recognton, vdeo survellance, access control, producton of personal documents (.e. passport, natonal dentty card [], etc). However, as descrbed n the Face Recognton Vendor Test (FRVT) [3] and other face recognton evaluatons reports, most commercal face recognton technologes (FRT) suffer from two knds of problems. The frst one concerns nter-class smlarty such as twns classes, fathers and sons classes. Here, the people have smlar appearances whch make ther dscrmnaton dffcult. The second and the more mportant s the ntraclass varatons causes by sgnfcant changes n lghtng condtons, pose varatons (.e. three-dmensonal head orentaton), and facal expressons (see fgure ). In fact, lghtng condtons dramatcally change the face appearance, so approaches based on ntensty mages are not suffcent for overcomng ths problem. Pose varatons also present mmense problem for recognton and handcap for performng comparsons between frontal face mages (from dataset) and changed vewpont mages (the probe). Furthermore, compensaton of the facal expressons s also a dffcult task n D-based approaches, because t sgnfcantly changes the texture mage of the face. Current state-of-the-art n face recognton s rch n works whch am to resolve problems regardng the challenge. The majorty of these researches use ntensty mages of the face, called D model-based technques, for recognton. In contrast, the second paradgm to recognton s the 3D model-based technques n whch researches explot, n addton to textural and slhouette nformaton, the three-dmensonal shape of the face n order to mtgate some varatons. It s n ths category of technques that the proposed approach belongs. Typcally, the 3D faces of nterest are saved n a lbrary durng an offlne phase. Durng the onlne recognton phase, a sngle captured.5d model of the face present n the scene s matched wth the models n the lbrary n order to fnd the dentty and the pose of the person. Inter-class smlartes Class Class j Intra-class varatons Pose Illumnaton Facal expressons Fgure. Inter-class smlartes and ntra-class varatons: face recognton problems The remander of the paper s organzed as follows: Secton () presents a state-of-the-art n 3D face recognton technques partcularly model-based ones. Secton (3) descrbes an overvew of the proposed approach. In Sectons (4) we focus on developed works for.5d and 3D face acquston for recognton. The secton (5) descrbes the database collecton procedure. In sectons (6) and (7), we emphasze our recognton algorthms usng the.5d/3d face matchng coupled wth a regon-based metrc. Fnally, a concluson and future works are presented n secton (8).

2 II. RELATED WORK O 3D FACE RECOGITIO Most face recognton algorthms belong to one of the two man famles: appearance-based and model-based. The appearance-based methods, also called vew-based, use statstcal technques to analyze the dstrbuton of the mage vectors n the vector space, and derve an effcent and effectve representaton (feature sub-space) accordng to dfferent applcatons. In fact, the face mage s consequently consdered as a hgh-dmensonal vector (.e. a pont n a hgh-dmensonal vector) and the dmensonal reducton mathematcal tools (lnear and non-lnear) are appled to these vectors. In contrast, the frst model-based approaches are based on D feature ponts whch are located on the face. Indeed, they compute some dstances between these features, consdered as a bometrc sgnature. These systems are enhanced, thereafter, by the ntroducton of the Actve Appearance Model [4] whch ams to cancel the facal expresson on a gven face and synthesze novel expressons n the same face. A more advanced 3D morphable face model presented n [7, 6] s explored to capture the true 3D structure of human face surface. Both morphable model methods come under the framework of nterpretaton through synthess. These technques use a generc 3D face model obtaned by PCA as the mddle step for fttng the 3D morphable model to the presented probe mage. The man advantage of model-based methods s that the model, whch ntegrates pror human knowledge, has ntrnsc physcal relatonshp wth real faces. As a recent sub-category of the model-based approaches, we fnd the 3D face matchng methods. The uses of 3D mages allow the overcomng of lmtatons due to pose and llumnaton varatons. The appearance of the face s more senstve than ts 3D shape to facal expressons. In addton, the depth mage presents several mportant topologcal descrptors, especally curvature whch s very nterestng for feature face localzaton. All these advantages, as mentoned n [5], encourage the followng of these methods n the face recognton challenges. We can group these approaches nto two famles: statstcal-based methods and shape matchngbased methods. The works, whch belong to the frst famly, apply the classcal lnear and non-lnear dmensonal reducton (PCA, LDA ) technques to the range mages [9,, 3] from the data collecton n order to buld the sub-space. The second knd of approaches uses classcal shape matchng algorthms n order to compute the spatal devatons between the probe and the 3D mages from the gallery [8]. The test mage s typcally a.5d face model wth unknown orentaton or a complete 3D face model. These technques requre good ntalzaton for good matchng and provde correspondng ponts, spatal devaton between them and pose parameters of the test mage. One of the most mportant works s descrbed n [,7,8] n whch the authors combne a surface matchng algorthm wth an appearance synthess approach for a best recognton rate. The fnal recognton result s a weghted sum of the two scores. In [4], the authors present a novel dea whch represents the facal surface as an sometrc surface (length preservng). Usng a global transformaton based on geodesc dstance, the obtaned forms are ndependent of facal expressons. After ths transformaton, they perform one classcal rgd surface matchng and PCA for sub-space buldng and face matchng. A good revew and comparson of these technques s gven n [0]. Another nterestng study whch compares ICP and PCA based approaches s gven n [6]. Here, the authors show a baselne performance between these approaches and conclude that ICP-based method performs better than a PCA-based method. Ther challenge s expresson changes partcularly, eye lp open/closed and mouth open/closed. In the present work, we propose a novel approach for face matchng usng.5d partal model n any vewpont and wth any facal expressons. Already, the 3D face database s bult n an offlne phase. The shape matchng s performed by the well-known Iteratve Closet pont matchng algorthm and the recognton s done by a weghted sum of spatal devaton n dfferent regons of the reference face. III. PROPOSED APPROACH: OVERVIEW Our people dentfcaton approach s based on the face, the most mportant nformaton to recognze someone. Snce 960, researchers have worked on ths popular challenge n order to provde an effcent soluton for many applcatons such as vdeo survellance, mmgraton control, access control, etc. The proposed soluton can be used for authentcaton or for enrollment processes (fgure ). In an offlne phase, we buld our 3D face database wth neutral expressons, the faces nsde are composed of the 3D meshes and the assocated texture mages both saved n the same VRML(Vrtual Realty Modelng Language) fle. In the onlne step, we frst capture one.5d face and conserve only the skn regon. Then we match the gven partal model wth all faces n the dataset. The core of our algorthm conssts of the algnment step then the matchng step of the gven surfaces. For the frst task, approxmatng the transformatons between the vews, we apply a coarse algnment, and then we perform a fne algnment by ICP algorthm. Ths algorthm s an teratve procedure mnmzng the mean square error (MSE) between ponts n one vew and the closest ponts, respectvely, n the other. 3D face database Authentcaton Threshold.5D Face Model (probe) Fne algnment Coarse algnment 3D shape Matchng.5D Face Model (probe) Fgure. Proposed approaches for face recognton (enrollement) and authentcaton based on.5d/3d face shape matchng Recognton

3 The results are two matched sets of ponts n the.5d probe model and the 3D face model from the database. Furthermore, global spatal devaton between each par of ponts s provded. The recognton process s based on the obtaned dstrbuton of ths dstance produced by regons. IV..5D AD 3D FACE PHOTOGRAPHY FOR RECOGITIO For recognton, we have already developed a complete.5d and 3D human face acquston framework based on a stereo sensor coupled wth a structured lghtng source. In [5], we propose an accurate and, at the same tme, a low-cost soluton dedcated to the 3D model-based face recognton technques (3D-FRT). In our approach, we frst calbrate the stereo sensor n order to extract ts optcal characterstcs and geometrcal parameters. Second, eppolar geometry coupled wth a projecton of specal structured lght on a face, mproves the resoluton of the stereo matchng problem, by transformng t nto a one-dmensonal search problem and a sub-pxel features matchng. ext, we apply our adapted and optmzed dynamc programmng algorthm to pars of features whch are already located n each scanlne. Fnally, 3D nformaton s found by computng the ntersecton of optcal rays comng from the par of matched features. The fnal face model s produced by a ppelne of four steps: (a) Splne-based nterpolaton, (b) Partal models algnment then ntegraton, (c) Mesh generaton, and (d) Texture mappng. Fgure 3 llustrates some stages of ths approach. (a) Three partal models for the regstraton: left, frontal, and rght 3D textured model (VRML format) (b) Texture to shape assocaton procedure Fgure 4. 3D face database collecton For our prmary experments, our database contans 0 3D faces and 80.5D test models. For each person the test dataset contans 9 partal models ( frontal, profles and 6 wth expressons), as llustrated by fgure 5. Both 3D and.5d faces have about 7000 vertces and trangles whch s a approprate resoluton for representng 3D shape and performng n short tme the algnment steps. 3D Shape Texture 3D nterpolated ponts Trangulaton 3D Surface (a) rght, face, left (b) neutral, exclamaton, unhappy, smlng, content, very happy Fgure 5. (a)face pose varatons (b) facal expresson varatons Fgure 3..5D face acquston usng stereo sensor asssted structured lght V. DATABASE COLLECTIO As descrbed on the system overvew secton, we must have a 3D face database for performng recognton from any vewpont. The complete 3D face s obtaned by mergng three partal models as shown n fgure 4 (a). Frst, we photograph the subject n three drectons then we regster the.5d partal meshes and merge the texture mages. All reconstructed models must have neutral facal expressons as shown by fgure 4(b). They have also the same number of vertces n ther mesh. The man goal of ths database collecton process s to evaluate the robustness of our developed technques and others to llumnaton, pose and facal expresson varatons. VI. ICP-BASED FACE MATCHIG STRATEGY The man contrbuton of our approach s the use of dmensonal nformaton whch s lost by projecton n the twodmensonal photos. The ntutve way for recognton s the shape matchng process. Many solutons are developed for ths task especally for range mage regstraton. The basc algorthm s Iteratve Closest Pont developed by Bessel and al. and publshed n [9]. In our approach we consder one

4 coarse algnment step, whch approxmates the rgd transformaton between models, then we perform our optmzed ICP algorthm varant whch rapdly converges to a global mnma resultng from ths ntal soluton. A. Coarse algnement Currently we are workng on full-automatc coarse algnment procedure based on 3D face features located by curvature analyss. However, we present here a manual step n whch the user must select more than two corresponded 3D ponts n the probe model and the 3D face model from the gallery. Rgd transformaton (R, T, s) ncludng rotaton R, translaton T and scale s s computed usng the selected ponts. Ths procedure presents a good ntalzaton stage before the fne algnment stage usng ICP. Fgure 3 llustrates ths process and shows the result of the rgd transformaton appled to one of the gven models. There are two advantages of the coarse algnment: good ntalzaton whch guarantees the convergence to global mnma and reduces the convergence tme of the ICP algorthm. Selected features teraton k of the algorthm, R=R k R and t=t+t k. The crteron to be mnmzed n the teraton k becomes (4): e(r k,t k ) = (Rk (Rp + t) + tk y (4) = The ICP algorthm presented above always converges monotoncally to a local mnmum [9]. However, we can hope for a convergence to a global mnmum f ntalzaton s good. For ths reason, we perform the above coarse algnment procedure before the fne one. Fgure 7 llustrates zooms of some regons n algned models; here the 3D model s one neutral 3D model from the gallery whereas the.5d model s a scan of the same face wth facal expressons (opened mouth and eyes). It s clear that fne algnment contrbutes to mnmzng the dstance between the ponts. In our algorthm we use, for performng ICP, a set of features selected based on the tolerance level of spatal devaton. Ths allows a rapd convergence of the algorthm whch processes only these ponts and cancels ponts whch presents spatal devaton value superor to the tolerance value. In contrast, correspondence concerns all ntersectng ponts. The steps of the algorthm are gven as follows: Algorthm: ICP-based matchng 3D face model (From database).5d face model (Captured) Fgure 6. The coarse algnment step (ntlsaton for ICP) B. Fne algnment Our fne algnment algorthm s based on the well-known Iteratve Closet Pont (ICP) algorthm [9] whch s an teratve procedure mnmzng the mean square error (MSE) between ponts n one vew and the closest ponts, respectvely, n the other. At each teraton of the algorthm, the geometrc transformaton that best algns the probe model and the database model s calculated. Intutvely, startng from the two sets of ponts P = {p }, as a reference data, and X = {y }, as a test data, the goal s to fnd the rgd transformaton (R,t) whch mnmzes the dstance between these two sets of ponts. The prncple of ICP conssts of determnng for each pont p of the reference set P the nearest pont n the second set X wthn the meanng of the Eucldean dstance. The rgd transformaton, mnmzng a least square crteron (3), s calculated and appled to the each pont of P: e(r, t) = (Rp + t) y (3) = Coarse algnment result Ths procedure s alternated and terated untl convergence (.e. stablty of the mnmal error). Indeed, total transformaton (R,t) s updated n an ncremental way as follows: for each Inputs: P = {p} (model from database), X = {y} (scan model) Outputs: (R,T) whch mnmse MSE, matched ponts, spatal devaton - Fnd closest pont pars (nt.: coarse algnment soluton) - Compute best transform whch mnmzes the MSE, - Apply transform to the scan model, - Iterate untl numercal convergence, compute at each teraton (R k,t k ), Fgure 7. Iteratve Closest Pont for fne algnement VII. FACE RECOGITIO METRIC Performng rgd matchng based on the ICP algnment algorthm provdes good recognton results. However, t s senstve to sgnfcant facal expresson changes. For ths reason and based on some emprcal observatons of face anthropometry, we partton the face model nto regons labeled as mmc and statc. The mmc regons show manly varatons n the face shape (especally chn, mouth and eyes) whereas

5 statc areas present small deformatons (the rest of the face). In the recognton process we take nto account ths classfcaton n order to assocate dfferent weghts wth statc and mmc regons. The segmentaton process s done automatcally by the watershed algorthm whch outputs 3D homogenous areas based on the gradent of the range mage. In the sub-sectons below we descrbe, frstly, the global dstance for recognton. Secondly, we propose an anthropometrc study of the face and an automatc segmentaton step. Fnally, we propose a novel regon-based metrc whch bears n mnd deformable regons and statc regons and consequently more robust to facal expressons. A. Watershed-based face segmentaton 3D surface segmentaton nvolves parttonng the surface nto groups of subsets of meshes whch are homogeneous wth respect to some crtera. To understand watershed-based segmentaton we consder shape of face as a topographc surface. If we flood ths surface from ts mnma and, f we prevent the mergng of the waters comng from dfferent sources, we partton the mage nto two dfferent sets: the catchment basns and the watershed lnes. Ths s closely the prncpal of the mmerson verson of ths algorthm proposed by L. Vncent and P. Solle n [0] whch s used n our technque. The man problem of ths approach s the oversegmentaton phenomena due to the small varatons whch exst n the surface of the face. Our soluton s to apply a Gaussan then a medan flter on the range mage n order to cancel the small varatons whch appear n the 3D surface and to elmnate the undesrable pcks. As llustrated by fgure 8, ths pretreatment allows us to overcome ths well-known problem n watershed based segmentaton. segmentaton process. We also present results of Gaussan 8(e) and mean curvature 8(f) to dscrmnate some mportant features whch wll be used to automate the coarse algnment stage. In (5) K s the Gaussan curvature and H s the mean curvature. eg f K = EG F, (5) eg + Eg ff H = EF G where E, F, and G are coeffcents of the frst fundamental form and e, f, and g are coeffcents of the second fundamental form []. B. Global dstance Our frst approach for recognton usng.5d/3d shape matchng s based on spatal devaton dstrbuton between matched ponts n probe and the 3D faces from the gallery. The global dstance s measured for each par of correspondng ponts obtaned by ICP. Fgure 6 shows two hstograms of dstrbuton and a color map whch presented by colors the dstance under the request (the.5d model). For the frst request, wth changed facal expressons, as shown n fgure 9 that the hgher parts of the models are better matched than the lower parts. Ths s because the hgher parts are more statc than the lower ones. Ths s the object of the second approach presented n sub-secton C, n whch a regon-based dstance s proposed n order to take n account these varatons. In the second gven example, the request (left profle) s perfectly matched to the 3D face and the pose and the dentty of the person s gven correctly. (a) (b) (c) (d) (e) (f) Fgure 8. (a) 3D face (b) watershed segmentaton wthout flterng (c) watershed wth gaussan flter (d) watershed wth medan flter (e) Gaussan curvature (f) mean curvature It s clear that the number of regons n Fg. 8(b) and 8(c) s more sgnfcant than n fg. 8(d). Ths s done by flterng pretratement appled to the surface of the shape before the Fgure 9. Examples of spatal devatons and colormaps between the.5d scan of face and the 3D face from the database

6 C. Regon-based dstance As llustrated n fgure 9, the rgd matchng process presents some lmts to facal expresson varatons. Our soluton s to partton the face model to two knds of regons: the mmc and the statc regons. Ths human face segmentaton model s the result of emprcal and anthropometrc studes. For the statc regons, the rgd matchng s more sgnfcant for computng recognton score. After surface segmentaton we attrbute to the obtaned regons, dfferent weghts and the global dstance takes nto account the label of each regon. Ths metrc s more robust than the global dstance calculaton. Equaton (6) gves the global recognton score functon of regon-based scores. Here λ represent the weghts, ψ s the ndvdual regon score and ψ s the recognton score. ψ = λ ψ (6) Ths defnes a novel face matchng metrc whch s robust to facal expressons by segmentng the face model to statc and mmc regons and concludes wth a regon-based matchng score. Here the.5d/3d matchng allows the pose problem compensaton and the regon-based score mtgates the facal expresson varatons. Texture nformaton s not ntegrated n ths work so llumnaton varatons are also compensated. Moreover, t wll be ntegrated n order to enhance the recognton/authentcaton decson. After probe/gallery models matchng, we can extract corresponded texture and compute smlarty based on one of the tradtonal methods (correlaton, PCA, etc.) snce the pose s already dentfed. VIII. COCLUSIO AD FUTURE WORK In ths paper we have presented a novel face recognton method whch s based on the Iteratve Closest Pont matchng algorthm and a novel regon-based metrc. As mentoned n secton 5, we are currently concentratng on developng a full automatc procedure for the coarse algnment stage. Ths s based on curvature analyss of the 3D face surface. Our future work concerns the ntegraton of ntensty nformaton n our recognton process after the 3D matchng. Indeed, the texture mage presents the appearance of the human face and provdes complementary nformaton for recognton. After ICP algnment we have correspondng 3D ponts n the probe model and each face belongng to the dataset. In addton, face models are composed of textural and dmensonal nformaton, so after ICP processng we can match texture nformaton n probe and gallery models. Ths can enhance the recognton results and ensure the automatc decson process. Moreover, the presented spatal devaton s the result of pont-to-pont dstance so we wll use pont-to-plane dstance whch s more precse for computng spatal devaton between models. REFERECES [] Tsshkou D., Hammam M., Chen L., Face Detecton n Vdeo Usng Combned Data-mnng and Hstogram based Skn-color Model, IEEE Thrd Internatonal Symposum on Image and Sgnal Processng and Analyss, ISB , Rome, Italy, September 003. pp [] K. Peng, L. Chen and S. Ruan, Face Image Encrypton and Reconstructon for Smart Cards Usng Fourrer-Melln Transform, Proc. of MEDIAET 04, Tozeur, Tunse, 004, pp. [3] P.J. Phllps, P. Grother, R.J Mcheals, D.M. Blackburn, E Tabass, and J.M. Bone, FRVT 00: Evaluaton Report,. March 003 [4] T.F. Cootes, G.J. Edwards, and C.J. Taylor, Actve appearance models, Proc. ECCV, vol., pp , 998. [5] B. Ben Amor, M. Ardablan, L. Chen, 3D Face Modelng Based on Structured-lght Asssted Stereo Sensor. Proceedng of ICIAP 005, Caglar, Itala, 6-8 September 005. [6] Kyong I. Chang, Kevn W. Bowyer, and Patrck J. Flynn. Effects on facal expresson n 3D face recognton. In SPIE Conference on Bometrc Technology for Human Identfcaton, volume 5779 of SPIE Proceedngs, Orlando, FL, 005. [7] Xaoguang Lu and Anl K. Jan. Integratng range and texture nformaton for 3D face recognton. In Proc. 7th IEEE Workshop on Applcatons of Computer Vson, pages 56-63, Breckenrdge, CO, 005. [8] Xaoguang Lu and Anl K. Jan. Deformaton analyss for 3D face matchng. In Proc. 7th IEEE Workshop on Applcatons of Computer Vson, pages 99-04, Breckenrdge, CO, 005. [9] Gang Pan and Zhaohu Wu. 3D face recognton from range data. Internatonal Journal of Image and Graphcs, 5(3):-, 005. [0] Kyong I. Chang, Kevn W. Bowyer, and Patrck J. Flynn. An evaluaton of mult-modal D+3D face bometrcs. IEEE Transactons on Pattern Analyss and Machne Intellgence, 7(4):69-64, 005. [] Xaoguang Lu, Ren-Len Hsu, Anl K. Jan, Behrooz Kamgar-Pars, and Behzad Kamgar-Pars. Face recognton wth 3D model-based synthess. In Proc. Internatonal Conference on Bometrc Authentcaton, volume 307 of Lecture otes n Computer Scence, pages 39-46, Hong Kong, 004. Sprnger-Verlag. [] Thomas Heseltne, ck Pears, and Jm Austn. Three-dmensonal face recognton: An egensurface approach. In Proc. IEEE Internatonal Conference on Image Processng, pages 4-44, Sngapore, 004. [3] Sotrs Malassots and Mchael G. Strntzs. Pose and llumnaton compensaton for 3D face recognton. In Proc. IEEE Internatonal Conference on Image Processng, pages 9-94, Sngapore, 004. [4] Alexander M. Bronsten, Mchael M. Bronsten, and Ron Kmmel. Expresson-nvarant 3D face recognton. In Proc. Internatonal Conference on Audo- and Vdeo-based Bometrc Person Authentcaton, volume 688 of Lecture otes n Computer Scence, pages 6-70, Guldford, UK, 003. [5] Chenghua Xu, Yunhong Wang, Tenu Tan, and Long Quan. Depth vs. ntensty: Whch s more mportant for face recognton? In Proc. 7th Internatonal Conference on Pattern Recognton, Cambrdge, UK, 004. [6] Mun Wa Lee and Surendra Ranganath. Pose-nvarant face recognton usng a 3D deformable model. Pattern Recognton, 36(8): , 003. [7] Volker Blanz and Thomas Vetter. Face recognton based on fttng a 3D morphable model. IEEE Transactons on Pattern Analyss and Machne Intellgence, 5(9): , 003. [8] Charles Beumer and Marc Acheroy. Face verfcaton from 3D and grey level clues. Pattern Recognton Letters, :3-39, 00. [9] P. Besel and. Mckay: A method for regstraton of 3D-shapes. IEEE trans. Pattern analyss and Machne ntellgence, 4():39-56, 99. [0] L. Vncent and P. Solle, "Watersheds n Dgtal Spaces: An Effcent Algorthm Based on Immerson Smulatons", IEEE Transactons on PAMI, Vol. 3, o. 6, June 99, pp [] Gray, A. "The Gaussan and Mean Curvatures" and "Surfaces of Constant Gaussan Curvature." 6.5 and Ch. n Modern Dfferental Geometry of Curves and Surfaces wth Mathematca, nd ed. Boca Raton, FL: CRC Press, pp and , 997.