ANNUAL REPORTPortable Digital Mouth andO lusion Reprodu er(KSTC-144-401-07-015)PI: Fuhua (Frank) ChengDepartment of Computer S ien eCollege of EngineeringUniversity of Kentu ky

Tel: (859) 268-5823Email: heng� s.uky.eduhttp://www. s.engr.uky.edu/~ heng/4/25/2008

SummaryThis is a summary of our work during the past ten months (June 1, 2007 - Mar h 31, 2008).Te hni al tasks �nished during this period in lude: the onstru tion of a standard model,the onstru tion of a o lusion model, the design of a mouth s an system, mesh simpli� a-tion te hnique, mesh interpolation te hnique, mesh urvature omputation te hnique, meshsegmentation te hnique, onstraint based s aling, and o�set surfa e generation te hnique.Te hni al tasks to be �nished in lude: last step of the onstru tion of the mouth s an sys-tem, and feature based mat hing te hnique. We expe t our �rst prototype to be ready inthree to four months.We onsider this grant, up to this point, a su ess. We not only have rea hed mostof our resear h goals, i.e., developing the ne essary imaging system and required geometri algorithms to support the reprodu tion of a patient's mouth and o lusion, but also produ edan MS (Ds. Jiaxi Wang, graduated in Mar h, 2008, urrently working for a ompany inLexington, KY), 3 journal papers and 2 onferen e paper. We anti ipate another MS (Mr.Conglin Huang) to be produ ed at the end of this year and a PhD (Mr. Fengtao Fan) to beprodu ed at the end of next year.

Table of Contents1 Introdu tion 11.1 Ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Current Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Obje tive 43 Te hni al Tasks 74 Experimental Method 85 Results and Dis ussion 206 Con lusions 27Referen es29

1 Introdu tion1.1 Ba kgroundChairside CAD/CAM systems have been used by dentists for treatment of patients for morethan 20 years. The �rst dental CAD/CAM system, CEREC 1 [123℄, was introdu ed to thedental ommunity on Sept. 19, 1985 (see Figure 1(a) for su h a system). The idea ame outof one person: Dr. Werner H. M�ormann at the Dental S hool of the University of Zuri h.Today, we have CEREC 3 (Figure 1(b)) and CEREC inLab [136℄. Several other ommer ialFigure 1: (left) CEREC-1; (right) CEREC-3.dental CAD/CAM systems su h as Cer on [84℄, De im [75℄, etkon [88℄, Everest [106℄, GN-1[97℄, DigiDent, DentaCAD [87℄, Lava [143℄, Medifa turing [68℄, Pre ident DCS [83℄, Pro era[133℄, Pro 5O [80℄, Wol-Ceram [153℄, and ZENO Te [151℄ are also available. All an do3D design of veneers, inlays/onlays, rowns/Cores, bridges/frameworks [138℄. The CERECsystem, used by more than 17,000 dentists and in 28 dental s hools in this ountry alone,remains the most popular one. It is possible now for a patient to go to a dentistry to getdental restoration servi e su h as the ones mentioned above in only one trip [138℄. However,one wouldn't be so lu ky if a (removable) partial denture (Figure 2(a)) is needed. The patientwill have to visit the dentistry at least three times and the total pro ess will take more than3-4 weeks. This is be ause the dentist still needs to take an impression of the patient's mouth(Figure 2(b)), send it to a dental lab for an experien ed dental te hni ian to make a partialdenture (Figure 2( )) out of it, wait for the partial denture to be sent ba k, try it on thepatient, make ne essary adjustment, and try on the patient again. More time will be needed

if the partial denture has to be sent ba k to the dental lab for stru tural modi� ation.(a) (b) ( )Figure 2: (a) A partial denture, (b) a PVS-impression taken from a patient, and ( ) makinga partial denture.One would ask: Current CAM te hnology should be able to handle the partial denturemanufa turing pro ess. So what is the problem here? Indeed, the required te hnology andmaterials are both there [96℄[123℄. The problem is: we don't have a CAD representation ofthe patient's mouth required by a CAM system for the manufa turing pro ess.

(a)(b) ( )Figure 3: Brontes' produ t: (a) the 3D imaging system; (b) bolt-on amera; ( ) systemillustration.

1.2 Current Situation[Intraoral Data A quisition℄ Current intraoral data a quiring devi es an be lassi�edinto three ategories: video amera based, X-ray based, and a tive wavefront sampling based.In the �rst ase, the hand-held devi e, mainly used as a visualization or inspe tion tool, anshow live video of the teeth. The �eld of view is usually small, overing only one or twoteeth (Figure 1). These ameras produ e no 3D information that is riti al for CAD/CAMdesign. Besides, for ea h tooth, they an only see part of it, not the entirety. Produ ts inthis ategory in lude Gendex's A uCam series [95℄ and SUNI's SuniCam USB amera [140℄.The Sirona's CEREC 3 amera [136℄ uses a tive infra-red light to a quire a depth map ofthe s ene (Figure 1). A spe ial painting is required and the re onstru tion pro ess takesabout 45 se onds for just one tooth. It is diÆ ult to extend it to build a 3D representationfor the entire mouth. In the se ond ase, by doing a full X-ray s an (200 X-rays or more),the devi e an get quite good 3D information of all the teeth and the jaws. Produ ts su has Sirona's Galileos 3D [136℄, AFP Digital's Digital X-Ray [95℄, and Kodak's RVG DigitalRadiographiy System [109℄ are in this ategory. But like all the X-ray images, this approa hdoes not provide good information on soft tissues su h as the gums whi h are riti al in there onstru tion of the patient's mouth for partial denture design. In the third ase, by usinga tive proje tion to determine 3D surfa e data from 2D images [142℄[94℄, the single- ameradevi e an a tually see in 3D. Brontes' produ t is the only produ t in this ategory [74℄ (seeFigure 3 for the imaging system, the amera and the system illustration of this produ t).However, like all the imaging systems, it an only provide information on visible portions ofan obje t, it an not provide informtation on the portions of an obje t that it an not see.Therefore, with urrent intraoral data a quiring devi es, it is not easy to a quire enoughdata for the onstru tion of a omplete CAD representation of a patient's mouth and theo lusion. That is why dentists are still using the impressing-taking approa h to reprodu e

a patient's mouth even though this approa h has problems su h as:� the need of taking multiple impressions;� remakes and multiple try-ins of the partial dentures due to poor quality of the impres-sions;� over extended or under extended borders of the partial dentures; and� dimension instability due to alginate shrinkage/expansion.[Shape Representation℄ Another problem with the urrent hairside CAD/CAM systemsis the la k of design support for ompa t, one-pie e representation of the mouth. CAD rep-resentations supported by urrent hairside CAD/CAM systems are either mesh based orNURBS-based [145℄. Mesh-based representations are expensive to maintain and pro ess be- ause usually ex essively large amount of verti es and fa es are needed in the representationto rea h a required pre ision. On average, 20,000 verti es and fa es are needed in a singletooth representation in this approa h [145℄. The NURBS-based approa h, on the other hand,limited by the re tangular grid topology of its parameter spa e, an not represent ompli- ated shape with only one surfa e. Therefore, urrent hairside CAD/CAM systems arehindered not only by insuÆ ient data for the re onstru tion of a mouth, but also ineÆ ientCAD modeling te hniques in representing the mouth.2 Obje tiveThe goal of this proje t is to develop a produ t alled Portable Digital Mouth andO lusion Reprodu er (PDMOR) that is apable of reprodu ing the mouth and o lusionof a patient.

Figure 4: Con eptual deisgn of an MSS.A PDMOR is omposed of aMouth S an System (MSS), a at-panel liquid rystal display(LCD) monitor, a PC, and a set of re onstru tion, modeling and rendering programs (seeFigure 4 for the design).These omponents will be mounted on a small wheeled table to ensure easy mobility ina dental lini or a dental lab. A dentist puts the MSS into the patient's mouth, instead ofusing the traditional impression taking approa h, to get (multiple-view) image data of thevisible portions of the teeth and gums of the patient. The image data then go through a tri-angulation, a feature dete tion and a registration pro esses to get an as omplete as possiblerepresentation of the teeth and gums of the patient. This still in omplete representation isthen ombined with a standard model to re onstru t all the existing teeth and gums of thepatient. The re onstru ted 3D omputer model an be used as a diagnosti aid for treatmentplanning or as a blue print for the design and manufa turing of dental applian es, su h aspartial dentures. It an also be used for patient edu ation and identi� ation purpose.A PDMOR an re onstru t missing teeth of the patient a ording to appropriate param-eters, and trim the mouth model in anyway a user desires. Therefore, this produ t will notonly make the hair side job of a dentist easier, but also make the job of a dental te hni ianin designing and manufa turing dental applian es, su h as dental partials, easier and more

pre ise. Other advantages of this produ t in lude: no all-ba k is ne essary for a patient,data an be sent to other pla es easier and data an be used for mu h longer time.[Te hni al Innovation℄ The te hni al innovation of the proje t is two-fold: the produ tand the te hnologies. First, the development of PDMOR itself is revolutionary. No produ tshave ever been developed (or, su essfully developed) to re onstru t the omplete mouth ofa patient. This is the �rst time a produ t in that dire tion will be developed and marketed.The produ t will have signi� ant impa t on both dental pra ti e and the dental industry.A tually, if a entralized national dental database an be built, this produ t will also haveimpa t on dental re ord keeping, dental data transmission, remote treatment planning, andidenti� ation/se urity issue.Se ond, the underneath idea of a PDMOR is also safe and innovative. With the amerabeing able to move axially, we have an imaging system that an see more points of the mouththan anybody else.However, be ause of obstru tion and onstrained working spa e, no dental imaging sys-tems an see all the points in the mouth of a patient, in luding ours. To use in ompletedata to re onstru t the mouth, one needs to ompare the a quired data of the patient witha standard model to determine how the standard model should be morphed and deformedto get a representation for ea h tooth and gum of the patient. This requires both novelmodeling and novel representation te hnologies.The re onstru tion of a patient's o lusion is also ne essary and important. A dentalapplian e, su h as a partial denture, designed based on a mouth model without a propero lusion pattern would not work be ause it an not perform hewing fun tion properly (seeFigure 5(b) for su h a mouth model). Only a mouth model with an appropriate o lusionsu h as the one shown in Figure 5(a) an ensure proper fun tion of the resulting partialdenture.

(a) (b)Figure 5: (a) Mouth with a working o lusion, (b) without a working o lusion.3 Te hni al TasksBe ause the intraoral data a quiring pro ess an not get omplete data of the mouth, astandard model has to be used with the a quired data to build a patient's mouth ando lusion. The idea: for ea h tooth (gum) of the patient, we ompare the a quired (partial)data of that tooth (gum) with the orresponding tooth (gum) in the standard model todetermine how the representation of the standard model should be morphed and deformedto get a representation of that tooth (gum) for the patient.This pro ess involves several major te hni al steps. First, a standard model has to bebuilt. This model should be able to provide information eÆ iently and a urately enough forall subsequent omparison, mat hing and morphing pro esses. Se ond, the data set re eivedfrom the intraoral data a quiring pro ess has to be segmented into groups so that the pointsof ea h group are either from an individual tooth or a gum.For ea h resulting new data point group, we need to perform a oarse-grained mat h-ing [69℄ pro ess to identify the orresponding tooth (gum) in the standard model. Thispro ess should be done using a feature-based approa h to make it eÆ ient. So we need toknow the features of ea h segmented data group, as well as the features of ea h tooth andgum of the standard model [115℄. We then perform a �ne-grained mat hing to identify theexa t lo ation of ea h segmented data set on the representation of the standard model.The next step is to modify the standard model so we an get a representation for ea htooth and gum of the patient. This pro ess involves rigid motions, s aling and deformation.

Everything up to this point is point-based. After this step, we need to perform a surfa e�tting pro ess so a parametri representation an be obtained for ea h tooth and gum of thepatient.Te hniqeus that have to be developed here in lude feature dete tion, feature based mat h-ing, subdivision surfa e based interpolation te hniques, point based data segmentation, pointbased 3D re onstru tion, standard model onstru tion, oarse-grained mat hing, �ne-grainedmat hing, morphing of subdivision surfa es, onstraint based deformation, o�set and blendingsubdivision surfa e generation, and subdivision surfa es interse tion. The standard model,the mouth model, and the o lusion of the patient will be represented by Catmull-Clarksubdivision surfa es.Subdivision surfa e based modeling and representation te hniques are more powerful thanNURBS or Splines based te hniques in that� the new te hniques require only one surfa e to represent the �nal result no matter how ompli ated the shape and topology. This will not only make representation easier tohandle, but make shape manipulation easier to perform as well;� a voxel-based representation for the mouth model an also be reated. This repre-sentation is good for visualization and virtual environment appli ations. But mostimportantly, it provides all the digital information required for subsequent design andmanufa turing of the dental applian es and, hen e, making the design and manufa -turing pro ess easier and more pre ise.4 Experimental MethodIn this se tion we explain how we realize our obje tives.[Pointwise 3D Re onstru tion℄ Pointwise 3D re onstru tion starts with the intraoral

data a quisition pro ess, to be performed by the Mouth S an System (MSS). An MSS is omposed of an Axial Stereo Vision unit (see Figure 4 for the design of an Axial StereoVision unit). The on ept of axial stereo vision has been studied for a while [130℄[81℄[139℄.But usually only an oridnary lens is used in the system. Our design of the MSS is novelbe ause it is the �rst time a ylindri al mirror is used in su h a system. The parts of ourAxial Stereo Vision unit are des ribed as follows (see Figure 4(a) for the labels):1. IEEE-1394 amera from Point Gray Resear h, the resolution is 1280� 9602. External light to amera, an xenon lamp old light from LOEN In .3. Cylindri al Mirror4. Camera Mount5. Position en oder from Mi rote h Laboratory In .6. Translation stage7. Gear box8. DC motor. The motor will stop when the amera tou hes either end of the TranslationStage.9. Motor power.In our ase, the ylindri al mirror ould be very lose to the target (teeth) when the systemmoves deeper into the mouth. In su h a ase, the radial distortion ould be signi� ant. Onthe other hand, the fo al length of the mirror is relatively large in our system whi h helpsredu e the signi� an e of this error.The solution for this distortion is to orre t the image measurement to those that wouldhave been obtained under an ideal pinhole lens. This pro ess is illustrated in Figure 6 [139℄.A thorough error analysis will be performed for this part.

Figure 6: The image of a square with radial distortion and the orre ted image.In order to retrieve 3D oordinates from 2D images, the images need to be alibrated.The famous Tsai te hnique [146℄ will be used here. The remaining part of this task is wellstudied and developed in the literature.[Depth Re onstru tion℄ It is possible to al ulate the depth information using an te h-nique alled light attenuation sterro (LAS). However, this new te hnique ould fail if dire tillumination is not strong enough to rea h the ba k side of the teeth. With our novel ap-proa h of the MSS stru ture, we an use a te hnique alled multiview radial atadioptri imaging to apture the teeth from just one shot [127℄.[Standard Model Constru tion℄ sin e point based representation is too expensive forstorage and pro essing, and information on individual teeth and gums is needed for there onstru tion pro ess of teeth and gums of the patient, the best hoi e to build a standardmodel is to reate a parametri representation for ea h individual tooth and gum. Thiswill be done by s anning individual teeth and gums, forming a 2-manifold mesh with adesired ombinatorial stru ture through triangulation of the unorganized point loud, doinga simpli� ation pro ess to redu e the omplexity of the 2-manifoldmesh, and then performinga surfa e �tting pro ess to get a parametri representation for ea h tooth and gum.We don't anit ipate any te hni al problems for the �rst three steps. A tually the s an-ning pro ess and the 2-manifold mesh generation pro ess have already been done for all the

(a) (b) ( )Figure 7: (a) Arti� ial erami teeth s anned for the standard model; (b) upper jaw teethof the standard model; ( ) lower jaw teeth of the standard model.teeth (Figure 7). The third step, mesh simpli� ation, is ne essary be ause the number ofpoints generated by the s anning pro ess is very large (more than 20,000 points/fa es gen-erated for ea h tooth), way beyond the apability of any of the urrent surfa e interpolationte hnologies. A shape-preserving simpli� ation te hnique [79℄ will be used for this step, al-though a te hnique published earlier an a hieve the same goal as well [104℄. The fourthstep, however, requires spe ial e�ort. Catmull-Clark subdivision surfa es (CCSSs) will beused for the �tting pro ess [76℄. We address the te hni al issues of this pro ess next.[Subdivision Surfa e based One-Surfa e Fitting℄ B-spline and NURBS surfa es havebeen used extensively in �tting 3D data points [90℄[91℄. Be ause of the re tangular natureof their parameter spa es, they an only �t data points from a re tangular grid. They annot �t data points of arbitrary topology unless the data point set is segmented into groupsea h homeomorphi to a re tangular grid. But smoothness between the B-spline or NURBSsurfa es that �t adja ent data point groups is not guaranteed. A better approa h is to usesubdivision surfa es in the �tting por ess be ause it is possible to �t any data points withonly one subidivion surfa e and, onsequently, no segmentation of the data set is requiredin the shape re onstru tion pro ess. Subdivision surfa es in lude uniform B-spline surfa es,pie ewise B�ezier surfa es, non-uniform B-spline surfa es and NURBS surfa es as spe ial ases

[134℄. So they are the most general surfa e representation s heme so far. See Figure 8(d)for the representation of a ventilation ontrol omponent with a single subdivision surfa e.The initial ontrol mesh and the ontrol meshes after one re�nement and two re�nementsare shown in (a), (b) and ( ), respe tively. The ventilation ontrol omponent has seventeenholes (handles). Therefore, it an not be represented by a single B-spline or NURBS surfa e.(a) (b) ( ) (d)Figure 8: (a) Initial ontrol mesh, (b) ontrol mesh after one re�nement, ( ) after twore�nements, and (d) limit surfa e of a ventilation ontrol omponent.The �tting pro ess will follow the following path. First, we perform one or more levelsof Catmull-Clark subdivision on the given data points P to get a �ner ontrol mesh G. Gsatis�es the following property: ea h fa e of G is a quadrilateral and ea h fa e of G hasat most one extra-ordinary vertex. The verti es of G are divided into two ategories. Avertex of G is alled a Type I vertex if it orresponds to a vertex of P . Otherwise it is alleda Type II vertex. Q is then de�ned as a ontrol mesh with the same number of verti esand the same topolgy as G. We assume Q has m verti es Q = fQ1;Q2; � � � ;Qmg, m > n,and the �rst n verti es orrespond to the n Type I verti es of G (and, onsequently, the nverti es of P ). These n verti es of Q will also be alled Type I verti es and the remainingm� n verti es Type II verti es. This way of setting up Q is to ensure the parametri formdeveloped for a CCSS pat h [20℄ an be used for the limit surfa e of Q, denoted S(Q), andwe have enough degree of freedom in our subsequent work. Note that m is usually mu hbigger than n. The remaining job then is to determine the position of ea h vertex of Q. Thedi�eren e from previous work here is we will ignore the similarity onstraint, fo us on the

properties of the surfa e itself only. Figure 9 is an example from our work in this dire tion [?℄.

(a) (b) ( )Figure 9: (a) Given data meshM , (b) limit surfa e ofM , ( ) subdivision surfa e intgerpolatesM .[Point based Data Segmentation℄ In this resear h, segmentation of 3D points sampled bythe intraoral data a quisition devi e will be done using a ombination of edge-based approa h[115℄ and region-based approa h [148℄. First, this is possible be ause topologi al informationon the data sets is known, therefore seed surfa es an be de�ned for a region-based approa h.Se ond, this is ne essary be ause the 3D data set re eived from the data a quisition pro essdoes not ontain many of the natural boundaries between adja ent teeth (the amera annot see them), therefore an edge-based approa h an not do the work ompletely by itself.An edge dete tor designed as a onvolution mask will be reated �rst. The edge dete tor ombines the fun tions of Gaussian smoothing and di�erential property estimation. Edgedete tion is then a zero rossing sear h in the onvoluted signal [77℄. The information re- eived from the edge dete tor will be ombined with all the boundaries of the gaps in thedata set to form the path in doing the segmentation.[Coarse-Grained Mat hing℄ A natural attempt here is to use the outline of ea h seg-mented data group to ompare with the outlines of the teeth and gums in the standard

model to determine the orresponden e. This naive approa h works for the front teeth,maybe even the anine teeth, but not the other teeth be ause in most of the ases the am-era an not see the entire tooth, therefore, the outline of the segmented data group does notrepresent the real outline of a tooth. Geometri features of the segmented data group willhave to be used as well.For ea h segmented data group, we will identify geometri features using a point-basedapproa h [115℄, fo using on orners and ridges. For teeth and gums in the standard model,the features will be identi�ed by examining the urvature of the subdivision surfa e repre-sentation. Note that parametrization te hniques for Catmull-Clark subdivision surfa es areavailable [20℄[159℄[51℄. Therefore, identifying features for obje ts represented by Catmull-Clark subdivision surfa es is possible.[Fine-Grained Mat hing℄ The job here is, for ea h tooth and gum of the standard model,�nd its best position respe t to the segmented, partial representation of the orrespondingtooth (if there is one) or gum of the patient. This on eptually simple task is the most riti alstep of this resear h be ause the orre tness of the re onstru tion result and, onsequently,the �tness of the partial denture(s) to be designed subsequently are largely determined bythe result of this step. This step starts with the positioning of the gums, and then the frontteeth, the molar teeth and the remaining teeth (in that order). The mat hing will be doneby minimizing the distan es between riti al points of the orresponding teeth and gumswith the lo ation of the teeth with respe t to the gums as onstraints.The distan e will be de�ned by a lo al quadrati approximation to the squared distan efun tion (SDF) [132℄. Given a surfa e S, the SDF assigns every point P of the embeddingspa e the squared distan e from P to S. SDF has been applied to solve a number of shape�tting problems [131℄[78℄[149℄. The lo al properties of the SDF for a manifold are presentedin [63℄. Its geometry is studied in detail in [132℄. The advantages of SDF in lude (1) one an

ompletely avoids the parametrization problem, (2) the approximation pro edure is fasterand more stable. Our approximation is de�ned as follows.Let S be an oriented surfa e S(u; v) with a unit normal ve tor �eld n(u; v) = e3(u; v). Atea h point S(u0; v0), we de�ne a lo al right-handed frame whose �rst two ve tors e1 and e2determine by the prin ipal urvature dire tions. These ve tors are not uniquely determinedat an umbili al point, but in that ase we an take any two orthogonal tangent ve tors e1and e2. The de�ned frame is referred as the prin ipal frame M . The two prin ipal urvature enters at the onsidered surfa e point S(u; v) an be expressed in M as (0; 0; pi). Thequadrati approximant Fd of d2 at (0; 0; d) is given by:Fd(x1; x2; x3) = (d=(d� p1))x21 + (d=(d� p2))x22 + x23[Morphing of Standard Model℄ In general, due to hange of urvature distribution aftera s aling pro ess, it is not possible for the new surfa e �S to have exa tly the same shape anddimension as the un onstrainedly s aled surfa e while arrying all the original features. Anapproximation method has to be used to onstru t �S. In this work, the new surfa e will be onstru ted following the �x-and-stret h based approa h [55℄.The main idea of this approa h is to �x regions of the given subdivision surfa e that ontain the features while s aling and stret hing the remaining part of the surfa e untilsome onditions are rea hed (see Figure 10). The surfa e is divided into three parts, thefeatures (region I of the smaller ellipse in Figure 10), neighboring regions of the features(region II in Figure 10), and the remaining part (region III in Figure 10). The featuresthat need to be �xed during the s aling and stret hing pro ess have to be transformed toappropriate lo ations �rst (see region I of Figure 10). Region III is simply s aled using thegiven s aling fa tors. Region II is stret hed to provide a smooth onne tion between the

I

IIIII

Scaling

S

Stretching Relocation(a) (b)Figure 10: (a) Basi idea of the �x-and-stret hing approa h, (b) onstrained s aling: before(left) and after (right).relo ated features and the s aled region III.The stret hing pro ess ensures that the shape and urvature distribution of �S are as lose to those of the un onstrainedly s aled version of the given CCSS, S, as possible, while arrying all the original features Ci. This is a hieved by minimizing a shape-preserving ob-je tive fun tion de�ned on the di�eren e of these two surfa es on neighboring regions of thefeatures. EÆ ient te hniques have been developed for the energy evaluation pro ess and theoptimization pro ess. The stret hing pro ess does not hange the topologi al stru ture of thesubdivision surfa e. Therefore, the resulting surfa e �S is again a CCSS. Figure 10(b) showsan example of this on ept where the hole of the ro ker arm is the feature held hangedduring the stret hing pro ess.[Constraint Based Deformation℄ We need an automati shape shrinking/expandingmethod for subdivision surfa es that would stop the shrinking/expanding pro ess on e somepre-set onditions are met. In this ase, the pre-set onditions are lo ations of adja entteeth, teeth on the opposite jaw, and information we re eived from the segmented results ofthe in omplete representation of the teeth and gums.The di�eren e between this te hnique and the te hnique presented in [55℄ is that in this

Figure 11: onstraint based deformation: (a) before deformation; (b) after deformation. ase, the entire surfa e is s aled while, in the latter ase, only ertain portions of the surfa eare s aled (see Figure 10 for details). Another di�eren e is, in this ase, the s aling is notuniform even the entire surfa e is s aled (see Figure 11).A yet third di�eren e is, to ensure the partial denture is lo ked tight enough by its adja- ent teeth, the above onstraint should be slightly over-satis�ed, i.e., the shrinking/expandingpro ess does not stop immediately on e the surfa e tou hes adja ent obje ts, but ontinuesthe shrinking/expanding pro ess until an o�set boundary ondition is rea hed. One wayto satisfy this ondition is to s ale down the size of the adja ent teeth slightly, and usethe new shapes as the onstraints. Another issue is, if the shrinking/expanding pro esstou hes one side �rst, one should modify the orientation of the subdivision surfa e insteadof its lo ation. The physi al meaning of this is, the tooth should be sheared instead of moved.[O�set and Blending Subdivision Surfa e Generation℄ For a given subdivision surfa eS, the work we need to do here is to reate an o�set surfa e S� for a spe i�ed portion of thegiven surfa e. Sin e subdivision surfa e is almost everywhere regular ex ept at a few extra-ordinary points, an intuitive approa h is to partition the spe i�ed region into sub-regionsof re tangular topology and use the well-studied o�set surfa e generation for onventionalparametri surfa es (see, e.g., [67℄) to generate an o�set surfa e for ea h sub-region and thenstit h these o�set surfa es up to form an o�set surfa e for the entire region. This seeminglyreasonable approa h gives us a surfa e that is smooth only in the interior of the partitioned

sub-regions, not at the boundary urves. This is not a eptable in our ase be ause a partialdenture manufa tured using this model would have sharp edges at the sub-region boundariesand hurt the patient. Our approa h here is to use a ombination of onstrained s aling and onstrained translation to get a general o�set surfa e generated and then use surfa e-surfa einterse tion to remove undesired portion of the surfa e. This approa h is independent of thetopology of the base surfa e and, onsequently, an be used for surfa es who parameters arenot re tangular su h as subdivision surfa es.A blending surfa e will be generated by mixing several base surfa es to form a new surfa e.Ea h base surfa e ontributes ertain share in the formulation of the new surfa e. Theweight of ea h base surfa e is determined by a real-valued fun tion alled "weightfun tion"or "blendingfun tion". Blending surfa e generation for B-spline and B'ezier surfa es hasbeen well studied [73℄. As in the o�set surfa e generation ase, those te hniques an not beuses dire tly be ause the presen e of extra-ordinary points.The basi idea is to onstru t a rail urve on both surfa e, using the surfa e-surfa einterse tion te hnique to be developed next, then build a general blending model. This isbe ause the onstru tion of a smoothing surfa e for two interse ting surfa es requires the omputation of the interse tion urve in ertain ases only. The omputed interse tion urvedoes not have to be exa t; a good approximation would usually be enough. After all, this urve is only used in generating two appropriate rail urves and an appropriate blendingarea. Note that the onstru tion of the rail urve and the blending area should be performedin parameter spa e to avoid unne essary adjustment pro ess. Another important issue inthis dire tion is the smoothing of sharp edges and sharp orners. A blending te hnique forthe smoothing of a sharp orner shared by three fa es will be developed here too.[Subdivision Surfa es Interse tion℄ The interse tion operation will be performed in theparameter spa es of the subdivision surfa es, not in obje t spa e. A ubi frame bu�er will be

reated for ea h losed subdivision surfa e (a solid: a tooth or a gum). The representation ofea h tooth (gum) will be voxelized �rst and then a volume ooding is performed to mark allthe voxels that are inside the given tooth (gum). Therefore, voxels in ea h ubi frame bu�er an be lassi�ed into three ategories: (1) IN voxels, (2) ON voxels and (3) OUT voxels. Let��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

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(b)Figure 12: Performing interse tion operation in 2D parameter spa e.S(u; v) be a pat h of the CCSS representation of a tooth (gum) A. For ea h subpat h ofS(u; v) resulted from the voxelization pro ess, we voxelize it one more time using the methoddis ussed in [50℄. However, this time we do not write voxels into A's ubi frame bu�er, butlook up the voxel values in both tooth A and tooth B's ubi frame bu�ers. If voxel valuesof this subpat h in both ubi frame bu�ers are either IN or ON, then this is a subpat h tokeep. Subpat hes of this type will be alled K-subpat hes. If voxel values of this subpat h areall OUT in both ubi frame bu�ers, then this is a subpat h to dis ard. Subpat hes of thistype are alled D-subpat hes. Otherwise, i.e., if some of the voxel values are IN or ON andsome of the voxel values are OUT, then this is a pat h with some portion to keep and someportion to dis ard. Subpat hes of this type are alled I-subpat hes (interse ting subpat hes).For example, the re tangles shown in Fig. 12 (a) are the parameter spa es of the resultingsubpat hes when the re ursive voxelization pro ess stops and the dashed polyline is part ofthe interse tion urve of the two given teeth in this pat h's 2D parameter spa e. We an seethat subpat h A1A2A4A3 in Fig. 12(a) is an I-subpat h. The interse tion urve dete tingpro ess then basi ally is pro ess to tra e the I-subpat hes.

[Out ome Assessment℄ We will use only one metri in assessing the out ome of the in-novation resear h des ribed above. We will onsider the out ome a good one if the relativeerror in ea h ase is smaller than or equal to 3% of the dimemsion of a tooth. Measuringabsolute error does not make mu h sense here be ause the dimension of a tooth is alreadyrelatively small. The reason for using 3% for the relative error bound is be ause it orre-sponds to half a pixel in a resolution of 1280� 960. This bound will be used both for DMRand shape representation.5 Results and Dis ussion[Mouth S an System℄ We have performed omparison of two di�erent designs. In the�rst ase, the amera looks through the ylindri al mirror (the axis of the mirror and the amera's opti al axis are oin ident). The amera images s ene points both dire tly andafter re e tion by the mirror. As a result, s ene points are imaged from di�erent viewpointswithin a single image.The imaging system in this ase aptures the s ene from the real viewpoint of the ameraas well as a ir ular lo us of virtual viewpoints produ ed by the mirror. To see this onsidera radial sli e of the imaging system that passes through the opti al axis of the amera. Thereal viewpoint of the amera is lo ated at O. The mirrors m1 and m2 (that are straightlines in a radial sli e) produ e the two virtual viewpoints V1 and V2, respe tively, whi h arere e tions of the real viewpoint O. Therefore, ea h radial sli e of the system has two virtualviewpoints that are symmetri with respe t to the opti al axis. Sin e the omplete imagingsystem in ludes a ontinuum of radial sli es, it has a ir ular lo us of virtual viewpointswhose enter lies on the amera's opti al axis.The three viewpoints O, V1, and V2 in a radial sli e proje t the s ene onto a radial linein the image, whi h is the interse tion of the image plane with that parti ular sli e. This

Figure 13: First design drawing of a ylindri al mirror.radial image line has three segments - JK, KL, and LM. The real viewpoint O of the ameraproje ts the s ene onto the entral part KL of the radial line, while the virtual viewpointsV1 and V2 proje t the s ene onto JK and LM, respe tively. The three viewpoints (real andvirtual) apture only s ene points that lie on that parti ular radial sli e. If P is su h a s enepoint, it is imaged thri e (if visible to all three viewpoints) along the orresponding radialimage line at lo ations p, p1, and p2. Sin e this is true for every radial sli e, the epipolarlines of su h a system are radial. Sin e all radial image lines have three segments (JK, KL,and LM) and the lengths of these segments are independent of the hosen radial image line,the aptured image has the form of a donut. The amera's real viewpoint aptures the s enedire tly in the inner ir le, while the annulus orresponds to re e tion of the s ene - thes ene as seen from the ir ular lo us of virtual viewpoints.To determine the depth of a parti ular s ene point, its proje tions in the image, i.e.,

orresponding points, have to be identi�ed via stereo mat hing. As the epipolar lines areradial, the sear h for orresponding points needs to be restri ted to a radial line in the image.However, most stereo mat hing te hniques reported in literature deal with image pairs withhorizontal epipolar lines. Therefore, it would be desirable to onvert the information apturedin the image into a form where the epipolar lines are horizontal. Re all that a radial line inthe image has three parts - JK, KL, and LM, one for ea h viewpoint in the orrespondingradial sli e. We reate a new image alled the entral view image by sta king the KL partsof su essive radial lines. This view image orresponds to the entral viewpoint in the radialsli es. We reate similar view images for the virtual viewpoints in the radial sli es - the leftview image by sta king the LM parts of su essive radial lines and the right view image bysta king the JK parts. To a ount for the re e tion of the s ene by the mirror the ontentsof ea h JK and LM lines are ipped. Observe that the epipolar lines are now horizontal.Thus, traditional stereo mat hing algorithms an now be dire tly applied.In the se ond ase, the obje t is pla ed inside the ylindri al mirror, and the amera usesa �sh-eye lens. In this ase, the light rays are re e ted more than on e. We do not need to onvert it into 3 or more images like what we did in the �rst ase. Sin e in this ase �ndingthe angle of in iden e an solve the depth problem without �nding the image plane of ea h amera.The on lusion is, the �rst approa h is better than the se ond ase. Hen e, an MSS basedon the �rst approa h is being built now. The ylindri al mirror is urrently being built by a ompany in entral Europe (Cze h Republi ). The University has made the �rst payment.The se ond payment will be sent to that ompany when the mirror is delivered to our labat the beginning of May. The Initial design the mirror is shown in Figure 13. An improveddesign is shown in Figure 14.[Mesh Simpli� ation℄ A mesh simpli� ation te hnique by onstri ting triangles has beendeveloped [113℄. Constri ting error is de�ned by a ombination of squared volume error

Figure 14: Se ond design drawing of a ylindri al mirror.variation with onstraint (SVEC), shape fa tor and normal onstraint fa tor of triangles.Gaussian urvature fa tor of ea h onstri ted triangle is used to distinguish sharp featuretriangles from blunter feature triangles. The triangle with the minimum onstri ting erroris onstri ted �rst. New vertex lo ations of triangles with weaker feature are determined byminimizing the onstri tion error of the mesh model. For a sharp feature triangle, the vertexwith maximal absolute Gaussian urvature of the three verti es is used as the new vertex.This new te hnique is simple, steady and an preserve mesh features well. An example ofthis te hnique is shown in Figure 17.[Constru tion of the gum model℄ Our �rst attempt was performed at the Viz Center.The results were not a eptable (see Figures 15 and 15).We then swit h to a di�erent apprao h. Instead of s anning gums we obtained from Dr.Burt, we s anned gums made of plaster (see Figure 16(a)). The result was quite good (see

(a) Gum model (b) S anned bottomgumFigure 15: (a) S anned gum model and (b) software pro essed s anned result.Figure 16(b)).

(a) Plaster model (b) S anned resultFigure 16: (a) Plaster model used to build the gum model; (b) software pro essed result ofthe s anned upper gum.The s anned results were then simpli�ed using the result presented in [113℄. The simpli-�ed results are shown in Figure 17).The simpli�ed meshes are then interpolated using our te hniques to generate parametri representations (see Figure 18).The parametrized representations are then extended to make the holes smaller so that

(a) Simpli�ed gum I (b) Simpli�ed gum IIFigure 17: Simpli�ed gums.

(a) Interpolated top gum (b) Interpolated bottom gumFigure 18: Simpli�ed gums are interpolated to reate parametri representations.when we put teeth into the holes, we will not get gaps anywhere in our model. The extendedresults are shown in Figure 19).[Standard Model℄ With the extended gums and the simpli�ed teeth, a standard model isthen built by putting the teeth into the holes of the extended gums. Di�erent views of thestandard model are shown in Figure 20. We believe this is the best standard model we haveseen so far.[O lusion Model℄ An o lusion model based on the standard model has been develope.Examples of the o lusion model are shown in Figure 21.

(a) extended top gum (b) Extended bottom gumFigure 19: Extended gums.[Teeth and Gum Mat hing℄ A urvature omputing program has been built and testedduring this time. With this program, features of a patient's teeth an be identi�ed and ompared with the standard model to identify the orresponden e between the input teethand teeth in our database. Teeth mat hing based on urvature distribution is urrently beendeveloped.[Teeth and Gum Segmentation℄ A segmentation program has been developed. The teethand the gums of the standard model are segmented using this te hnique and the results areshown in Figure 22. The original mesh is shown in Figure 20. Corre ted We use di�erent olors for di�erent teeth and gums to show the orre tness of our results.[O�set Surfa e Generation℄ An o�set surfa e generation te hnique for Loop subdivisionsurfa e has been developed.[Out ome Assessment℄ We will use only one metri in assessing the out ome of the in-novation resear h des ribed above. We will onsider the out ome a good one if the relativeerror in ea h ase is smaller than or equal to 3% of the dimemsion of a tooth. Measuringabsolute error does not make mu h sense here be ause the dimension of a tooth is alreadyrelatively small. The reason for using 3% for the relative error bound is be ause it orre-sponds to half a pixel in a resolution of 1280� 960. This bound will be used both for DMR

(a) Top-1 (b) Top-2

( ) Bottom-1 (d) Bottom-2Figure 20: Standard model built by our team.and shape representation.6 Con lusionsTe hni al tasks �nished during this 10-month period (6/1/07 - 3/31/08) in lude: the on-stru tion of a standard model, the onstru tion of a o lusion model, the design of a mouths an system, mesh simpli� ation te hnique, mesh interpolation te hnique, mesh urvature omputation te hnique, mesh segmentation te hnique, onstraint based s aling, and o�setsurfa e generation te hnique. Te hni al tasks to be �nished in lude: last step of the on-stru tion of the mouth s an system, and feature based mat hing te hnique.

(a) Open-1 (b) Open-2

( ) Closed-1 (d) Closed-2Figure 21: O lusion model built by our team.We onsider this grant, up to this point, a su ess. We not only have rea hed mostof our resear h goals, i.e., developing the ne essary imaging system and required geometri algorithms to support the reprodu tion of a patient's mouth and o lusion, but also produ edan MS (Ds. Jiaxi Wang, graduated in Mar h, 2008, urrently working for a ompany inLexington, KY), 3 journal papers and 2 onferen e paper. We anti ipate another MS (Mr.Conglin Huang) to be produ ed at the end of this year and a PhD (Mr. Fengtao Fan) to beprodu ed at the end of next year.

(a) Top-1 (b) Top-2Figure 22: Results of segmenting the bottom teeth: (a) with gum; (b) without gum.A knowledgementResear h work presented in this report is supported by the Kentu ky S ien e and Te hnologyCorporation (grant number: KSTC-144-401-07-015). Teeth data and gum data used in thisproje t are provided by Prof. H.T. Yao and his resear h group at the National Chung-ChengUniversity of Taiwan.

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