Navigating in a Shape Space of Registered Models

Transcription

1 Navigaing in a Shape Space of Regisered Models Randall C. Smih, Member, IEEE, Richard Pawlicki, Isván Kókai, Jörg Finger and Thomas Veer, Member, IEEE Absrac New produc developmen involves people wih differen backgrounds. Designers, engineers, and consumers all have differen design crieria, and hese crieria inerac. Early conceps evolve in his kind of collaboraive conex, and here is a need for dynamic visualizaion of he ineracion beween design shape and oher shape-relaed design crieria. In his paper, a Morphable Model is defined from simplified represenaions of suiably chosen real cars, providing a coninuous shape space o navigae, manipulae and visualize. Physical properies and consumer-provided scores for he real cars (such as weigh and sporiness ) are esimaed for new designs across he shape space. This coupling allows one o manipulae he shape direcly while reviewing he impac on esimaed crieria, or conversely, o manipulae he crierial values of he curren design o produce a new shape wih more desirable aribues. Index Terms Morphable model, shape space, barycenric coordinaes, design space. INTRODUCTION This paper describes echniques for navigaing in a shape-space of regisered models creaed from examples. Models are defined as vecors of geomeric feaures wih which o draw he model, and oher associaed model aribues. A se of regisered models are all described by he same feaures in he same order, and only differ by he feaure values. Our examples are cars, bu he navigaion echniques described are more widely applicable. They enable dynamic visual exploraion of he shape-space, guided by esimaes of physical properies for new shapes, and predicions for appearance-based semanic labels (e.g. spory ). In any new produc developmen effor, here is a planning phase in which coarse goals and crieria mus firs be raionalized and balanced. We do no prejudge he relaive imporance of any crieria aesheic, engineering, or marke-based and herefore emphasize exploraion and visualizaion raher han opimizaion. We believe many requiremens are negoiable, and rade-offs will be made if alernaives are visually eviden. In he following we develop inegraed echniques and associaed GUIs o suppor hese aciviies which we hink can grealy aid in design raionalizaion: Exploraion of he inerplay of crieria wih each oher and resuling design shapes. Navigaion oward suiable regions of shape space. Direc adjusmen of design geomery. Resricion o design subspaces by consrains. The abiliy o creae represenaive conceps simply and rapidly a a very early sage in meeings, or wih cusomers has significan poenial. I should be noed, however, ha he described research is no a fielded applicaion, nor are he GUIs described currenly under usabiliy esing.. Relaed Work This effor builds on he original work on Morphable Models [][2], and he performance-based mehods for ailoring human face models during synhesis [3]. Barycenric coordinaes [4], principal Randall C Smih and Richard Pawlicki are wih GM R&D, randall.c.smih@gm.com. Thomas Veer, Isván Kókai,and Jörg Finger are wih Universiy of Basel, Thomas.Veer@unibas.ch. Manuscrip received 3 March 2007; acceped Augus 2007; posed online 2 November componens [5], and eigensysems [6] are sandard ools in his lieraure. We augmen hese wih addiional mahemaical ools for consraining and guiding navigaion hrough he infinie variaions of he design space. Specifically, by inegraing linear consrains and linear regression ino his mix, we make i easier o suppor engineering and consumer-based crieria as funcions of shape. These echniques depend upon regisered models in conras wih more general shape similariy measures [7][8] based on aggregaed properies used in shape rerieval. The boom-up design-from-examples approach also conrass wih, bu may complemen, op-down design sysems using shape grammars [9]. The laer embeds knowledge abou appropriae feaure combinaions in generaive rules; he former derives i from saisics of well-chosen examples. Boh mus decide which feaures are relevan, and a hybrid soluion may ulimaely be mos powerful. The change from faces o cars may seem like a small sep, bu isn'. Regisraion algorihms for faces [3], and human bodies [0] do no adap well o 3D car models whose surfaces are criically defined by a few major flow curves. Prior regisraion work wih car surfaces [] produced resuls no smooh enough for our purposes, and we hypohesize he lack of his consrain is one reason. Similarly, poin-o-poin correspondence ediors [2] do no suppor hese specialized needs. Mos likely, car surface meshes need o be derived from hese criical flow curves [3] prior o regisraion, or he curves hemselves used o define he Morphable Model wih he surfaces generaed from hem secondarily [4]. We have no solved he auomaic regisraion problem eiher, bu uilize flow curves in our represenaion and hand-fiing of models o emplaes in order o produce usable resuls for esing (see Secion 2)..2 Organizaion of he Paper Secion 2 provides background on he represenaions used (boh 2D and 3D examples). Secion 3 reviews and inegraes he basic mahemaical seps for visually exploring he design space guided by differen kinds of crieria. In Secion 3. barycenric coordinaes are used o browse hrough he variey of shapes, using GUIs based on he simplex. The GUI also acs as a color-coded parameer map of he shape region indicaing direcions o increase, decrease, or mainain he value of a seleced scalar crierion. Secion 3.2 follows wih a recursive weighed leas squares mehod ha enables direc manipulaion of shape feaures coupled wih a parameer map showing how deviaions from he curren feaure value affec any oher seleced crierion. Secion 3.3 ransforms he linear consrain sysem o principal componens of he shape-space, making compuaions more racable in he lower dimensional space. The

2 relaion of barycenric coordinaes o PC space is esablished o inegrae hese effors. In Secion 3.4 linear regression is added as anoher way o relae aribues o shape variables. Physical aribues of vehicles and consumer feedback on shapes may no be obviously relaed o he shape geomery, bu may indeed show a relaion hrough linear regression on he basis vehicles, or more generally on he shape principal componens. Any esablished relaionship can hen be used as a linear consrain or color-coded in a parameer map o guide navigaion. In addiion, regression on shape PCs enables predicion of aribues of new shapes no derivable from he basis se of exemplar shapes, and opens he possibiliy for explanaion of hese aribues in erms of shape feaures. In Secion 4 a simple scenario ies hese echniques (and GUIs) ogeher. In Secion 5 a number of open quesions are discussed and Secion 6 summarizes and concludes. 2 REPRESENTATION In order o explore uses of morphable car models, we have creaed a succession of 2D and 3D hand-regisered daabases fi o sandardized emplaes (Figure ). The figures hroughou he paper will use one or anoher represenaion, bu he echniques are applicable o any of hem. The old curve emplae (op) is used for models in Figures 7 and 8. The newer emplaes conain separae meshes for he body, roof, and wheel openings among ohers, since hese sub-surfaces can move relaive o one anoher by large amouns across a variey of cars which would disor a single mesh. The meshes represen "base" surfaces heoreical shapes defined primarily by a few curves. The "fille" curves for he roof and body 3D emplaes are shown in yellow. They are insrumenal by hemselves in defining he size, proporions, and shape of he roof and body segmens, in he same sense ha a skech aris defines a concep wih a few srokes. They are responsible for conrolling he major highlighs in he vehicle appearance, which can be seen for 4 of he fied 3D vehicles in Figure. The res of he emplae mesh is designed around hese curves wih higher mesh densiy in areas ha vary subsanially from car o car. Given he fully-deailed arge vehicle meshes, he arge's characerisic curves mus be idenified and mached o he emplae curves. The heoreical arge "base" surfaces mus be arfully buil, passing smoohly hrough exraneous geomeric deail, wih he emplae verices disribued o capure he shape, above, below, or on he arge surfaces. The difficulies inheren in dealing wih segmenaion of he arge ino base shapes, idenifying and ignoring exraneous deail, and generalizing he arge shapes o an ideal parially defined by some exising curves is beyond he abiliy of curren auomaic regisraion algorihms. 2. Shape-Space Definiion A shape-space is defined by n exemplars models chosen for some paricular qualiies. Each is represened by a d-dimensional vecor x he defining geomeric feaures for he vehicles which ogeher wih some fixed opological informaion used in drawing, form he emplae. Typically, d >> n. An exemplar is named (e.g. x 0 ), idenifying i as a paricular insance of x wih paricular feaure values. A monage of hiry 2D exemplars is shown in Figure. A subscrip ( x i ), or no subscrip, will represen any design; a superscrip plus ( ) indicaes an updaed design. The exemplars are grouped as columns in a marix. The deviaion marix addiionally has he average ( x ) of he column vecors subraced. D ~ D d, n d, n [ x0, x, K, xn ] [ x0 x, x x, K, xn x] () Fig.. Versions of 2D and 3D emplaes. Designs ( x ) are affine combinaions of he exemplars, wih he affine consrain on he componens of he blending vecor : 3 NAVIGATION ~ x = D = x D [ i] =. (2) The following secions develop echniques for navigaion in he design space while simulaneously viewing and manipulaing he design shape and crieria of ineres. For navigaion, i is concepually convenien o view any creaed design o be he saring poin for he nex evaluaion in a sequence by adding combinaions of he (original) shape deviaions o i: ~ x x = x D x = x = i i i 0. (3) All designs, including x, lie in he same hyperplane. The deviaion vecors span he hyperplane, so any linear combinaion of hem will say in he subspace. The addiional resricion on in (2) is only necessary for equaliy wih he lef-hand side of he equaion. The vecor in (3) is unresriced, bu can be adjused for differen purposes wihou affecing he resul. Since he sum of deviaions from he mean is zero, ~ ~ 0 D = D ( α) = (4) any vecor wih he same value in each componen can be added o and is mapped o 0 by (3). Two such adjusmens are useful. The vecor componens can be shifed by he minimum componen value o a new minimum value of zero useful for seing sliders ha have minimum zero values:

3 sliders =. (5) The oher adjusmen ses so is componens sum o one again: = min [ i] n. (6) Equaion (3) can be rewrien, if desired, as a single sep adjusmen o he mean, or an affine combinaion of he exemplars as before: x = x D = D ~. (7) 3. Barycenric Coordinaes In his secion, he vecor in (2) will be manipulaed hrough a user inerface, enabling exploraion of he design space calculaion of a mulilinear esimae of any vehicle s properies and visualizaion of he resulan inerpolaed shape. The mos direc implemenaion o specify uses n sliders ranging from zero o one wih one slider for each exemplar. To save space, hese were arranged in a circle, as shown on he lef in Figure 4. This ype of inerface is inconvenien, bu is a way o fully specify any vehicle in he space. If n poins are affinely independen hey form he verices of an n-simplex he convex hull of he poins embedded in a n R or greaer dimension [5]. The 2-simplex is a hyperplane in riangle, embedded in he plane conaining is hree verices. Any oher poin in he hyperplane can be referred o he simplex as an affine combinaion of is verices. In (2), is he barycenric coordinae of he poin x wih respec o he n-simplex defined by he poins (columns) of D. If any componens of are negaive or greaer han one he poin is ouside he simplex bu sill in he conaining hyperplane. Barycenric coordinaes are coordinae frame free ; given a paricular poin p, each barycenric coordinae for i is calculaed by a funcion per verex of he simplex [4]. There is a mapping for any poin p of one n-simplex o is corresponding poin in any oher n- simplex hrough he simplex verices (columns of V and D ) and p s barycenric coordinae: Deviaions from he aribue mean are blended and added o he aribue value a for he curren design. I follows from (8) ha if he same aribues are associaed wih he verices of boh simplexes, he barycenric coordinae of a poin in one can be used o esimae he aribue a he corresponding poin in he oher. Wih ha raionale, he res of he secion will describe he use of simplexrelaed GUIs for parially specifying. Generalized barycenric coordinaes [4][6] exend barycenric coordinae calculaions from he 2-simplex o planar n-sided convex polygons. We use i in a 2D GUI for blending hree (or n) vecors (Figure 2). The barycenric coordinae for he cursor poin inside he figure is calculaed [4] and blends he deviaion shape vecors assigned o he verices by (3). The resul is added o he curren design shape and he vehicle is hen drawn. A cursor poin on a verex regeneraes he associaed exemplar, or on an edge inerpolaes wo exemplars, assuming he deviaions are being added o he mean design. A cursor poin in he middle he barycener for a sandard simplex averages he deviaions (which equals 0). Oherwise an inerior poin blends all deviaions hose associaed wih nearby verices have he mos weigh. Any oher scalar propery of he models (e.g., wheelbase ) can be inerpolaed across he inerior of he polygon, and is value color-coded (hrough five 20% ranges, from minimum o maximum values in he Figure). While a GUI could be made using a 3-simplex (erahedron) and a 3D cursor, his approach canno be direcly applied in higher dimensions. However, oher han he riangle, he polygon is no a simplex, and only a small fracion of he values of available in he n-simplex can be generaed. For example, i is no possible for widely separaed verices o have srong affecs on he blended resul; so he ordering of properies on he verices is imporan. In Figure 3, exemplars were sored on wheelbase, and associaed wih he n-gon verices in a zigzag paern across he n-gon, from boom o op, o creae low and high clusers on opposie poles. Deviaions were blended for long (op) and shor (boom) wheelbase and added o he average car shape (middle). Noe ha he average car characer is generally mainained. Whichever soluion is being added o he deviaions will appear when he cursor is in he middle of he parameer map in ha case all he deviaions are being weighed equally and sum o zero. If he aribues disribued over he basis vehicles have significan ouliers, linear inerpolaion may no be sufficien. Appropriaeness mus be deermined in any case. p = V a x = (8) p D p The imporan propery of barycenric coordinaes is heir wellknown abiliy o inerpolae vecors on he simplex verices smoohly over he inerior commonly used o inerpolae colors on riangle verices across inerior pixels or o inerpolae physical properies across finie elemens. The inerpolaed scalar propery a a poin is compued using he barycenric coordinae of ha poin, and a vecor a of aribue values (one value per simplex verex) p ( a a * ) p a = a. (9) Fig. 2. Mousing around in a parameer value map. Fig. 3. Three wheelbase variaions on he average 2D car (middle).

4 weigh. C is iniially approximaed by he sample covariance of he exemplar daa and R is user-adjusable. Any common scale facor in C and R cancels ou of () and can be divided ou of he final resul (2). So unless a probabiliy is needed, (3) will be used for convenience. I should be clear ha in a sequence H, z, and R are poenially differen each sep, and ha he updaed x becomes x in he nex sep. To enable navigaing he design space freely wihou limiing he space o smaller and smaller subspaces, equaion () will be uilized wihou updaing C each ime. x = x CH [ HCH C = C CH [ HCH C iniial S x R] R] ~ ~ [ DD ] d d ( z Hx), HC () (2) (3) Fig. 4. Changing he wheelbase updaes oher parameer maps. Figure 4 shows a ypical arrangemen wih he design window, sliders, and muliple parameers maps. One maximized slider is an indicaion ha he curren design is an exemplar (X22 in Figure ), and no somehing else. The inse shows how he parameer map for curb weigh changes as he cursor is moved wihin he wheelbase n-gon from A o B o C while holding down a mouse buon. Afer he firs change is made (A) he esimaed curb weigh for he new design is shown, and he color map indicaes i is in he 60%-80% of maximum (red) range. While creaing a smaller wheelbase (B), all oher parameer maps coninuously change o cener on he evolving curren design and esimaed curb weigh has decreased. I increases again as he wheelbase is increased (C), so he wo variables are posiively correlaed. The user could have kep he wheelbase nearly consan insead by moving he cursor along a color conour while observing he oher parameers, and he changing design shape. The whole subspace of vehicles wih a consan wheelbase, however, is no reachable wih his one GUI alone, which can access only a porion of he n-simplex and is inerior. The shapes reached will be influenced srongly by he curren design. To ge o oher porions of he space, seps involving oher GUIs need o be performed. 3.2 Recursive Weighed Leas-Squares We also need o manipulae he geomeric feaures of he curren design direcly. The design can be consrained by a linear relaion H (a consan marix) on he design (for example, geomeric poin offses): In Figure 5 (righ side) H is a 3xd marix of zeros wih a single in each row; i selecs a paricular 3D poin from he design. Dragging ha poin wih he 3D cursor o a new posiion ( z ) creaes a posiion difference used in () o adjus he enire model, finding he soluion closes o he saring design ha saisfies he consrain wihin a olerance. In a probabilisic conex, i is he minimum variance soluion for Gaussian variables [8]. I is considerably easier o use a 2D cursor and manipulae a 2D secion e.g., he mid-plane exraced from he 3D model. In Figure 5 (lef side), dragging a poin on he secion updaes he secion dynamically (as a morphable sub-model) in is window, and he 3D morphable model updaes simulaneously as well. A range of deviaed posiions of any paricular 2D poin in he curren design can be encoded in a grid, and he minimum disance design soluions () calculaed wih he same saring design value for each grid poin deviaion. Any oher scalar propery or linear funcion of he design variables can hen be calculaed per grid poin soluion, and he grid poin assigned a color based on is value. In Figure 6, he red curve in he op secion is being manipulaed by one of is endpoins, and wo insances of he design and he ouline difference is shown. Associaed wih he wo designs are wo parameer maps color-coded hrough five 20% ranges, from minimum o maximum values for he parameer across he exemplar se. On op of ha is a scaer plo of he deviaions from he mean (cener grid poin) of he designaed drag poin for he 30 exemplars. The lef-mos grid shows a cursor square a he iniial drag poin value for he design; in he cener image he cursor has moved o he lef, eiher from dragging he 2D poin in he model window, or moving he cursor on he parameer map. In eiher case, i is clear how o move he poin o increase, decrease, or mainain he curren parameer value while viewing he resuling and ransiional shapes. A map for he deviaion of any wo parameers from he curren design values can be similar consruced by combining he wo scalar linear funcions. z = Hx. (0) Differen H funcions can be pre-deermined and sored. For any one of hem, he curren design value for z can simply be calculaed and displayed. If we provide a arge (consrain) z value ha is differen, we wan he overall design o adjus, bu we also allow a olerance for he arge. How much he design bends o he consrain, and how much he consrain accommodaes he curren design, are handled by a weighed average described below. This form is usually seen in a probabilisic filer conex [7]. The iniial design poin is he mean. The covariance marices C and R ac as (inverse) weigh marices for he curren design and he consrain respecively he larger he variance, he smaller he Fig. 5. Dragging a seleced 2D or 3D poin o reshape he 3D car.

5 Fig. 6. Dragging a 2D poin guided by a parameer value map. Applying Equaion wihou applying Equaion 2 allows he user o jump around in he whole design space freely; bu previous consrains are no enforced. Tha s plausible while browsing, o undersand he landscape and shape ineracions wih various crieria shown in parameer maps. Equaion 2 is applied o commi o he consrain. In Figure 7 a vehicle wheelbase is exended and locked a a paricular value. The res of he vehicle reshapes when a second poin is subsequenly moved (up, and hen down) bu he wheelbase is fixed (wihin a specified olerance). The second poin can be locked down, and so on. Significanly, he updaed covariance marix remembers hese los degrees of freedom and implemens locking in new vehicle shape calculaions (2). 3.3 Principal Componens C (Equaion 2) can be large, even hough i is sparse. Transforming he problem ino is principal componens avoids creaion and use of C alogeher, reducing he dimensionaliy of he problem space and enhancing real-ime performance. PCs are a se of uncorrelaed parameers ha may be used o generalize he idea of shape beyond he basis se, and allow predicions and explanaions of properies for oher vehicles. In his secion he recursive weighed leas squares formulae are ransformed ino he PC space, allowing direc manipulaion of meaningful design feaures economically. Singular Value Decomposiion is applied in (4) o rewrie he exemplar deviaion marix ino a produc of marices: U is column orhogonal; V is square and orhogonal; and Σ is a diagonal marix. Fig. 7. Locking using linear consrains o limi he design space. ~ D U Σ V d,n = d, n n, n n, n U U = I, V V = I (4) Since d>n, and D ~ is composed of mean-deviaion column vecors, he rank of D ~ is a mos n. A leas one Singular Value will appear as a zero diagonal elemen in Σ. Using (4), he scaled covariance marix in (3) becomes ~ ~ [ D D ] = UΣV VΣ U = UΛU d, d. (5) The columns of U are he eigenvecors of (3) and he diagonal elemens of Λ are he corresponding eigenvalues. In Principal Componen Analysis, he diagonal elemens of Λ (and Σ ) are sored in descending order placing any zeros in he las diagonal elemens. The columns of U and V are sored correspondingly. Jolliffe [5] defines he (cenered) principal componens using he sored eigenvecors as which, using (4) gives u U ~ x (6)

6 Finally, using he definiion ~ u = U D = ΣV. (7) G HU (8) we ge he recursive forms of () and (2) in he lower-dimensional PC space, and can move back and forh o feaure space where he variables have more obvious meanings for user manipulaion. u = u ΛG [ GΛG R] u = 0 Λ = Λ ΛG i= 0 ( z Hx) [ GΛ R] GΛ (9) G (20) Noe ha Λ and Σ are no longer diagonal. Barycenric coordinaes can now be ied o PC space by solving = VΣ 0 U ( x x) = VΣ 0 u (2) using (4) and (6). Noe ha locking (use of equaion 20) resrics subsequen new designs u o a subspace, which resrics in urn via (2) using he un-updaed Σ from (4). Corresponding cursor poins in he n-gon for hese resriced vecors are compued as before (see [4]). Alernaively, an unresriced vecor can be compued from he cursor posiion in he n-gon [4], and used in (7) wih updaed Σ o generae new designs in he resriced subspace. 3.4 Linear Regression In [3], a subjecive score (like facial boniness ) for each exemplar was used o calculae a boniness shape deviaion vecor ha could be scaled and added o anoher face model for effec: ~ x x a = ( ) = x ~ x a i ( ) i s * ~ x a (22) a he elemens of an aribue vecor (he scores), and s he wih i scale facor. The echnique associaes a semanic label wih a direcion in shape-space. The aribue scores can be garnered from expers, or drawn from he populaion. In Figure 8, we selfscored a number of vehicles (no he same se as Figure ) for he word spory. No surprisingly, cars aain characerisics of spors car ye he ransiions are ineresing. Alernaively, (23) uses scores o deermine a linear funcion on he principal componens of he shape-space. For scores on all he exemplars, = ei rivially gives back he already-supplied aribue for he designaed exemplar, or oherwise inerpolaes he values. a = a a = a( VV ) = β ( V ) (23) Subsiuing he parenheical erm which is equal o he Ideniy marix and regrouping, ransforms he idenify funcion ino a linear funcion wih coefficiens β on he sandardized PCs. Since PCs are orhogonal, eliminaing small PCs o produce a linear relaion wih only a few imporan parameers is simple: zero ou heir coefficiens. Figure 9 shows a linear fi of six of he larges PCs from a 2D sample se of 30 cars o heir acual weigh, wih approximaely 90% of he weigh variance 'explained' by he linear model. The larges PCs are relaed o size and proporion, bu i is sill no obvious ha a 2D ouline should be a good firs-order predicor for real vehicle mass. Neverheless, he example is only illusraive of he poenial no inended o be regarded as a validaed resul. Fig.8. Adding increasing amouns of spory o an iniial vehicle. Fig. 9. Real and fi curb weigh for hiry cars from six 2D shape PCs

7 4 USAGE SCENARIO An example of use is shown in Figure 0. An iniial design has already been creaed (op frame), as indicaed by he paern in he sliders below i. The depiced vehicle was iniially exemplar X4 (from Figure ) and has had is wheelbase grealy reduced using he n-gon labelled wheelbase. The curb weigh was also reduced as a resul. The characer of he original exemplar is sill plainly eviden. In he middle frame, he design has been furher modified by dragging a poin on he rear glass of he model (highlighed in red) in he spory direcion dicaed by he spory GUI. The cursor (large whie do) in ha GUI indicaes he movemen of he dragged poin from middle o lef (in he op frame o he middle frame), making he wheelbase smaller sill. The resuling design is shown in he boom panel wih he original X4 ouline in red. 5 DISCUSSION There are many quesions ha have no been addressed in his paper: linear vs. non-linear mehods; he number of exemplars needed; wheher o rea size and proporion informaion separaely from shape; and how o handle deails hose aspecs of individual vehicles ha are no amenable o capure hrough common feaure vecors and summarizaion hrough saisics. We have been pleasanly surprised a how well linear models have worked boh in he 2D models ses, and in a small (8 model) se of 3D models. As menioned in he inroducion, iniial emplae definiion and regisraion of models are difficul problems. We have ieraed over a number of emplaes, and arfully hand-regisered models o hese emplaes, giving us good resuls. One obvious place o es he validiy of linear inerpolaion is shown in Figure 7. Each car was modelled wih an occupan in design posiion (heel poin, hip poin), and he occupan was linearly inerpolaed wih he vehicle. The occupan was no a kinemaic model is limbs were jus drawn beween he reposiioned joins (poins). One would expec ha wih oo much shape variaion he figure s limbs would srech, raher han roae ino posiion, and indeed his happens. Over a large range (from ruck o low-slung spors car) his change was only a small proporion of limb lengh, amouning o a few millimeers perfecly valid for illusraion and discussion. Since he 2D emplae has been evolving, we have no been able o accumulae models wihou rework from sage o sage limiing us a any paricular sage o less han one hundred models a any ime in he basis se. Mixing vehicles across segmen rucks, SUVs, sedans can produce very ineresing cross-over resuls, bu is going o ake many vehicles o span he large shape variaions. For many pracical purposes, i is more desirable o model wihin segmen (Figure has no SUVs or rucks in i). The more homogenous he model se, he more likely i is ha linear regression will work well. Projecing a new vehicle ino he shape-space, and regeneraing i does no produce a near-replica of he shape; more models will be needed in he se, bu we can predic he number. On he oher hand, smaller model ses like he ones wih which we are experimening have been used o predic properies of new vehicles o firs order, and heir inerpolaions do produce an ineresing variey of pleasing shapes. The exemplars have no been normalized o separae shape characerisics from he dominaing influence of size and proporion his will be a fuure invesigaion. The firs few PCs describe mos of he se variance, since hey represen he large variaions in size and configuraion (cab forward or rear). Wih a sample of 30 2D cars, here are sill 29 PCs (he maximum number) achieving numerical significance based on a common cu-off hreshold bu heir value afer he firs few is dubious. However, if no PCs are eliminaed, hey by definiion mus reconsruc he exemplars, leading o some ineresing cases where a small PC maers. Even hough an exemplar may be regisered o he shape emplae, i can sill conain an idiosyncraic sub-shape highly unique o i wihin he se. In one insance, a unique bumper was idenified wih a small PC, in he sense ha he PC value changed dramaically in he one vehicle only. We were able o graf he bumper ono anoher car by subsiuing he unique PC value for his ino he second vehicle s PC descripion. To define a design emplae, or achieve model regisraion, we have o focus on common characerisics. Two-door and four-door vehicles are boh represened in Figure by hining a door placemen wih a single door cu. Oherwise we would segregae hese wo ypes of vehicles we don wan doors parially emerging, or popping in and ou of he design as a whole during morphing. However, some level of deail seems required o gain accepance. I would be ineresing o undersand how effecive (in erms of informaion conen) simple line models are vs. more realisic models. We have added pseudo-3d shading as a exure o he 2D models, conrolled by he underlying mesh so ha i reshapes wih he vehicle. Shading informaion is no conained in he morphable model, and addiional care needs o be aken for decisions based on appearance. Fig. 0. Two sep modificaion of X4.

8 Fig.. Monage of hiry 2D vehicles and four of he eigh 3D vehicles fi o he emplaes 6 CONCLUSION Concepual design involves many people wih differen backgrounds. Ariss and designers, engineers, and consumers all have differen crieria. The echniques developed in he previous secions can be used o explore he inerplay of hese crieria wih shape, and sugges ways o move he design in a posiive direcion. Iniial discussions wih prospecive users have generaed many ideas in design, engineering, and markeing where an effecive shape-space ool could be of use. To make our resuls more effecive, we need o build subsanially larger daabases o be able o capure more suble variaions; es and validae he linear predicion models vs. higher order models; and look ino he effecs of removing size and proporion informaion and reaing i separaely from shape. Finally, a good auomaic 3D model regisraion algorihm would make large 3D daabases possible, and 3D applicaions viable. REFERENCES [] M. Jones, T. Poggio, Hierarchical Morphable Models, CVPR, pp , 998. [2] M. Jones, T. Poggio, Mulidimensional Morphable Models, Proceedings Inernaional Conference on Compuer Vision, 998. [3] V. Blanz, T. Veer, A Morphable Model for Synhesis of 3D Faces, Proceedings SIGGRAPH, 999. [4] J. Warren, S. Schaefer, A. Hirani, M. Desbrun, Barycenric Coordinaes for Convex Ses, Technical Repor, Rice Universiy, [5] I. T. Jolliffe, Principal Componen Analysis, New York, Springer- Verlag, 986. [6] M. Turk, A. Penland, Eigenfaces for Recogniion, Journal of Cogniive Neuroscience, Vol. 3, No., pp. 7-86, 99. [7] T. Funkhouser, M. Kazhdan, P. Shilane, P. Min, W. Kiefer, A. Tal, S. Rusinkiewics, D. Dobkin, Modeling by Example, Proceedings SIGGRAPH, [8] M. Kazhdan, T. Funkhouser, S. Rusinkiewics, Shape Maching and Anisoropy, Proceedings SIGGRAPH, [9] S. Orsborn, J. Cagan, R. Pawlicki, R. Smih, Pushing he Limis of Vehicle Design: Uilizing a Parameric Shape Grammar o Explore Cross-Over Vehicle Conceps, Proceedings DETC, [0] B. Allen, B. Curless, Z. Popovic, The Space of Human Body Shapes: Reconsrucion and Parameerizaion from Range Scans, Proceedings SIGGRAPH, [] C. Shelon, "Morphable Surface Models", Inernaional Journal of Compuer Vision, Vol. 38, No., pp. 75-9, [2] D. Wiley, N. Amena, D. Alcanara, D. Ghosh, Y. Kil, E. Delson, W. Harcour-Smih, F. Rohlf, K. S. John, B. Hamann, Evoluionary Morphing, Proceedings IEEE Visualizaion, VIS 05, pp [3] P. Alliez, D. Cohen-Seiner, O.Devillers, B. Levy, M. Desbrun, Anisoropic Polygonal Remeshing,SIGGRAPH, 2003/ACM TOG. [4] I. Kokai, J. Finger, R. Smih, R. Pawlicki, T. Veer, "Example-Based Concepual Syling Framework for Auomobile Shapes, Proceedings, Fourh Eurographics Workshop on Skech-Based Inerfaces and Modeling, (Acceped for publicaion) [5] H.S.M. Coxeer, Regular Polyopes, Third Ediion, Dover Ediion 973. [6] M. Meyers, H. Lee, A. Barr, M. Desbrun, Generalized Barycenric Coordinaes on Irregular Polygons, Journal of Graphics Tools, Nov [7] A. Gelb, (Ed.), Applied Opimal Esimaion, M.I.T. Press, 984. [8] R. Smih, R. Pawlicki, Concepual Shape Design Using Opimal Esimaion Techniques GM R&D Collaboraion Repor, 2003.