Blended Deformable Models

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1 Blended Deformable Models (In IEEE Trans. Pattern Analysis and Machine Intelligence, April 996, 8:4, pp ) Doglas DeCarlo and Dimitri Metaxas Department of Compter & Information Science University of Pennsylvania Philadelphia PA Abstract This paper develops a new class of parameterized models based on the linear interpolation of two parameterized shapes along their main axes, sing a blending fnction. This blending fnction specifies the relative contribtion of each component shape on the reslting blended shape. The reslting blended shape can have aspects of each of the component shapes. Using a small nmber of additional parameters, blendingextends the coverage of shape primitives while also providingabstraction of shape. In particlar, it offers the ability to constrct shapes whose gens can change. Blended models are incorporated into a physics-based shape estimation framework which ses dynamic deformable models. Finally, we present experiments involving the extraction of complex shapes from range data inclding examples of dynamic gens change. Keywords: Shape Representation, Shape Blending, Shape Abstraction, Shape Estimation, Physics-Based Modeling Introdction Shape models incorporate trade-offs between conciseness of representation and descriptive power which affect their seflness for different applications. For shape estimation, it is important that shape models cover a wide variety of shapes sing a small nmber of intitive parameters. Finding the right balance is a difficlt and important problem. When the ltimate goal is recognition, abstraction of shape is also a significant isse. There are many crrent shape representations that se a small nmber of parameters, sch as generalized cylinders [3,, 4], sperqadrics [, 5, 8], hyperqadrics [6] and geons []. These are sefl for recognition tasks, bt lack the generality to represent a large class of shapes in a single model. Representations with many parameters, sch as srfaces with free-form deformations [] have a wide shape coverage, bt have too many parameters to be sefl in recognition tasks. Advancing front methods [9] and oriented particle systems [9] provide srface connectivity information and can model srfaces of arbitrary topology, This work was spported by NSF grants IRI-93997, MIP and ARPA grant DAAH-49567

2 bt do not provide a compact representation of shape. In fact, no existing model for shape estimation with a compact representation can represent objects of varying topology in a nified way an abrpt change in the model (both geometric and representational) is reqired to perform the topological change. Making sch a drastic decision dring estimation is often difficlt, and is not likely to be robst. Estimation sing implicit polynomial based representions [] has also been investigated. The degree and configration of the algebraic srface to be sed for fitting mst be specified in advance, ths making smooth topological changes difficlt. Models sch as those sed in solid modeling [7, 7] have flexible and intitive representations, bt they were not designed for shape estimation they were designed for hman se. For shape recovery applications sing CAD models, compactness in representation is not often a major concern. Systems which are applicable for both shape reconstrction and shape recognition have been presented [, 6, 3]. In [6] the shape was specified by its deformation modes and extracted sing a closed-form soltion of modal analysis. Shape was represented in [3] sing a wavelet basis and estimated by embedding it in a probabilistic framework. Both of these methods provide a collection of parameters ordered by level of detail. The models in [, ] incorporate global deformations which represent prominent shape featres, and local deformations which captre srface detail. Abstraction and compactness of representation are distinct concepts, bt often both are reqired in recognition systems. Considering the isse of abstraction, the ability to combine together different shapes into a nified model is very important. Algebraic srface blends [7] provide this ability, bt are not easily applied to shape estimation. Blobby models [3] can also combine shapes, bt lack flexibility in the nderlying combined shapes, reslting in large nmbers of components. We propose an extension to the shape representation of [, ] which we call blended deformable models to address the isse of combining shapes together into a single model. Given two shapes that can be defined parametrically on a common material coordinate space, blended shapes are constrcted by the linear interpolation of two shapes sing a blending fnction that specifies the relative contribtion of each shape on the reslting blended shape. For example, a sphere and a cylinder blended together cold prodce a bllet shaped object (see figre ). In addition, this parameterization is able to represent shapes of gens and : blending a sphere and tors together prodces an object in which the presence of the hole depends the nmber of holes in a shape a sphere has gens, a tors has gens

3 on the vale of the blending fnction. In addition, a geometrically smooth transition from sphere to tors is achievable by smoothly changing the blending fnction. Figre 3 shows a variety of shapes that we can create sing blending. In a nified model, blended models compactly and intitively represent a wide variety of shapes, inclding shapes of varying gens. An abstraction of shape is also provided the above example blended shape is clearly composed of a sphere and cylinder, which are components of the representation. The global natre of these models allows an efficient approach to shape estimation and the ability to handle sitations where range data are incomplete or sparse. In this paper, we show how blended models can be incorporated into the previosly developed physicsbased estimation framework presented in [, ]. We conclde after demonstrating or techniqe throgh a series of experiments involving incomplete range data from varios objects. Geometry of blended models. Deformable model geometry As in [, ], the models sed in this paper are 3-D srface shape models. The position of a point on the model is given in world coordinates by x which is the reslt of a translation and rotation of its position p, with respect to a non-inertial reference frame. The material coordinates =(; v) of these shapes are specified over a domain. The position of a point on the world model at time t, with material coordinates, with respect to an inertial frame of reference is x(;t)=c(t) +R(t)p(;t); () where c is the center of the inertial frame, and R is a rotation matrix which specifies the relative orientation of the inertial frame to a fixed reference frame. In the non-inertial (fixed) reference frame, the position of model points p, is the sm of a reference shape s and a local displacement d so that p(;t)=s(;t)+d(;t): () These local displacements, d, allow the representation of fine detail, while the reference shape, s, captres salient shape featres. The reference shape of the model, s, is constrcted by applying a global deformation 3

4 T (sch as bending) with parameters q T to a shape primitive e as follows: s() =T(e; q T ): (3) For a 3-D shape primitive (sch as a sperellipsoid []), we have e() :! IR 3. To represent the geometry of the primitive, a mesh of nodes is sed, where each node is assigned a niqe point in. The edges connecting the nodes represent connectivity of the nodes in space. Nodes can be merged together to form a closed mesh where points in map to the same 3-D model location (sch as for the poles of a sphere). The primitives we will be considering have global shape parameters q e which specify the shape. Inclding these parameters, we represent the geometric primitive as e(; q e ); (4) which is defined parametrically in over and has global shape parameters q e. Even thogh or framework can be applied to any class of parameterized primitives, we will be sing sperellipsoid and spertoroid primitives [] to create a blended model. We will now extend the above definition of the global shape s to inclde blended models.. Shape blending In a method analogos to the linear interpolation of two points, it is possible to blend two fnctions. Given two fnctions, f (x) and g(x), wecan blend them singa thirdfnction, (x) (with range [; ]), so that h(x) =f (x)(x) +g(x)(, (x)): (5) An example of this is shown in figre. Notice how h(x) =f (x) where (x) =, h(x) =g(x) where (x) =, and how h(x) is between f (x) and g(x) everywhere. Using this idea, we can blend parameterized shapes by the following formla: s(; v) =s (; v)()+s (; v)(, ()); (6) where s and s are two shapes parameterized over, as in figres (a) and (b). Figre (c) shows s, the reslt of blending the shapes shown in figres (a) and (b). The blending fnction sed to blend the shapes is shown in figre (d). The blending is performed along, which corresponds to the z-axis in these shapes 4

5 y h(x)=f(x)(x)+ g(x)(, (x)) f (x) (x) g(x) x x Figre : Blending of two fnctions f (x), g(x) given blending fnction (x) (from pole to pole). This particlar blending fnction was chosen to illstrate how different parts of the component shapes are expressed in the reslting shape. Notice how the top of s looks like s (a cylinder) since ( )=, and how the bottom of s looks like s (a sphere) since (, )=. = α() =, π/ π/ (a) (b) (c) (d) Figre : (a) Shape s (b) Shape s (c) Blended shape s (d) Blending fnction () The global parameters of s will inclde the global shape parameters of s and s, those that specify (see section.4), and the global deformation parameters q T. A common deformation T is applied separately to each shape primitive so that s = T(e ; q T ) and s = T(e ; q T ). These reslting deformed shapes are then blended together sing (6). When blending shapes, not all combinations of primitives will achieve desirable reslts. For example, a blend between two spheres where one is rotated 9 degrees from the other will prodce an interpenetrating object. Bt since we are able to choose the models in advance for a vision application, we can simply choose compatible shapes, sch as a sperellipsoid and a spertoroid. For the prposes of this paper, we will only have vary with instead of both and v. This limits the coverage to axially symmetric shapes. This restriction does not limit the applicability of blending to the process of shape abstraction. A variety of shapes prodced sing this restricted form of blending are shown in figre 3 by blending sperellipsoids and spertoroids. While these shapes are expressible 5

6 sing other representations [3, 7, ], blending provides a compact and abstract representation. Algebraic srface blending [7] is a CAD method for connecting shapes together throgh the constrction of blend srfaces which are placed adjacent to the component shapes. While similar in spirit, the nderlying theory is very different from the blending presented here, since the smooth join between shapes is achieved by geometrically inserting blend srfaces, not by interpolation. Figre 3: Examples of blended shapes.3 Spertoroid definition In addition to the sperellipsoid [], we will be sing the following definition for a spertoroid primitive: e tors (; v; a;a;a3;a4;a5;;) = a a4 + (a 4 + C a a5 + (a 5 + C a3s )C v )S v C A (,=;=] v(,; ] ; (7) where a ;a ;a 3 ; ; > anda 4 ;a 5. a, a and a 3 are size parameters in the x, y and z directions respectively. and are sqareness parameters as in a sperellipsoid. a 4 and a 5 are hole size parameters in the x and y directions. The hole is closed when a 4 = a 5 =, and the hole opens for vales greater than. As in a sperellipsoid, we define C = sgn(cos )j cos j and S = sgn(sin )j sin j. This definition is similar to the spertoroid given by Barr []. The addition of a 5, a second hole size parameter allows asymmetric holes. The presence of the scaling factors =(a 4 + ) and =(a 5 + ) separate the effects of the global size parameters (a, a and a 3 ) from the hole size parameters (a 4 and a 5 ) to allow hole size changes that do not affect the global tors size. 6

7 .4 Blending fnction parameterization The blending fnction is implemented as a non-niform qadratic B-spline fnction [5]. Given different types of shape primitives, the domain of may vary. For a sperellipsoid, maps [, ; ] to [; ]. The B-spline fnction is specified sing L+ control vales fc i j i :::Lgand L knots f i j i :::Lg, with and L fixed to be the lower and pper bonds of the domain of. The fnction has the vales ( )=c and ( L )=c L and has a continos first derivative except where two knot vales are eqal. The parameters sed to constrct the blending fnction are the L + control vales and the L, movable knots ( and L are fixed), which yields L, total parameters to specify. We concatenate all these parameters into the vector q b,sothat q b =(c ;:::;c L ; ;:::; L, ) > : (8) 3 Gens changing It is also possible to blend objects having gens (a sphere) with objects having gens (a tors). A hole will appear in the blended object as changes. There is no smooth transition between these two shapes becase they are not homeomorphic no seqence of deformations will change a sphere into a tors. Yet it is possible to have a transition between the two where there is a single discontinos event when the object changes gens. This event affects only the topology of the object, not the geometry of the shape. An example transition is shown in figre 4. Figre 4 is an illstrative seqence showing how a sphere can be transformed into a tors sing a blended shape. The blended reslt is compted sing (6), where s is a tors and s is a sphere. Initially, in figre 4(a), () = (forallvalesof )andtheblendedobjecthasthegeometryandtopologyofa sphere. The blended shapes in (b) and (c) show what happens if we slowly change () from to. In (c), when () =, the shape is a pinched sphere [8] the poles have dimpled inward ntil they toch. This has the same geometry as a tors (with the hole closed), bt is topologically eqivalent to a sphere. At this time, at the location where the poles toch, we change the connectivity of the srface to be that topologically eqivalent a sphere and cylinder are homeomorphic, bt a sphere and tors are not 7

8 of a tors. A discssion of how the node interconnections change is given in section 3.. Once the pinched sphere is changed into a tors, the tors hole can now be opened by increasing the tors hole size parameters (a 4 and a 5 ), shown in (d) and (e) (shown from a slightly different viewpoint to make the hole visible). (a) (b) (c) (d) (e) Figre 4: A blended shape changing from a sphere (a) to a tors (e) There are two constraints on the parameters of a tors-sphere blend that mst be enforced to insre the blended shape remains closed. The tors hole mst remain closed when the object has gens. When the object has gens, the vales (, ) and ( ) mst weight the tors so that the poles of the sphere are not expressed in the blended shape. For figres 4(d) and (e), the constraint wold be (, )=( )=since s is the tors. These constraints can be implemented in or framework by simply fixing the appropriate parameter vales at the times in the estimation process when they are not permitted to change. This entire process of gens change can be easily integrated into the physics-based estimation framework. For a hole to form, the object is deformed by the data forces into the configration shown in (c). This point can be detected by examination of the blending fnction. At this point, the hole can atomatically open de to forces from the data. Using this method, a hole can form in a physics-based way. The ideas presented can be applied to any shape primitives, althogh the actal steps involved may vary for different primitives. 8

9 3. Node interconnections When altering the topology of an object, the mesh of nodes mst be reconnected to conform to the new topology. This is a straightforward bt necessary part of the gens conversion process. Figre 5 shows how is folded p to prodce a sphere or tors. The arrows in these diagrams indicate two nodes being merged together, since the material coordinates of the nodes map to the same 3-D model coordinates. For both the sphere and the tors, a tbe is made first (the dotted lines). For the sphere in figre 5(a), the north and soth poles are created by closing each end of the tbe. For the tors in figre 5(b), the ends of the tbe are connected together. When the gens changes, the node mesh first mst be nfolded, and then re-folded to have the proper configration. v (a) (b) Figre 5: Node interconnection differences between a sphere (a) and tors (b) 4 Dynamics and generalized forces The dynamics framework given in [] can be sed after several alterations. In this framework all the degrees of freedom needed to specify the shape (translation, rotation, global and local parameters) are collected together to form the generalized coordinates of the model, q, q =(q > c ; q> ; q> s ; q> d )> ; (9) where q c = c(t), q is the qaternion sed to specify R(t), q d specifies the local deformations, and q s =(q > s ; q > s ; q > ; b q> T )> are the global parameters (q s and q s are the parameters of each of the component shapes, q b are the parameters that specify the blending fnction, andq T are the parameters of the global parameterized deformations). When fitting the model to data, the goal of shape reconstrction is to recover the parameters in q. The approach sed here performs the fitting in a physics-based way the data apply forces to the srface of the 9

10 model, deforming it into the shape represented by the data []. The model can be made dynamic in q by introdcing mass, damping and stiffness and embedding it into a Lagrangian dynamics framework. The Lagrange eqations of motion are second order differential eqations []. In shape estimation applications, the mass is set to zero (so that the model has no inertia and comes to rest as soon as the applied forces eqilibrate or vanish), reslting in the following simplified dynamic eqation: D _q + Kq = f q =(f > c ; f> ; f> s ; f> d )> ; () where D and K are the damping and stiffness matrices respectively, and where f q are the generalized forces []. These generalized forces can be frther broken down into components each corresponding to a component of q as given in (9) above. Using (), _q can be compted, and an integration method can be sed to pdate q. Performing this process iteratively reslts in a model more closely representing the desired shape. Throghot the fitting process, parameter schedles are sed [4, 4], as in other physics-based fitting frameworks. The fitting is performed initially sing coarse parameters (translation, rotation, and major axis lengths), followed by the fine parameters (blending parameters, sperqadric sqareness vales). This allows improvements in efficiency by initially redcing the dimension of the parameter space. By initially disabling the fine scale parameters, local minimm soltions also can be avoided. We compte the generalized forces f q from the 3-D applied forces. The comptation of f c, f and f d are the same as described in []. The comptation of f s is given by f s =(RJ s ) > f applied : () We compte J s, the Jacobian for the global shape s, as follows: J s s : () The Jacobian of the global shape, J s, converts applied forces into generalized forces, which will deform the global shape. The addition of blending changes the comptation of J s. In particlar, from (6) and (): J s = ()J s (, ())J s J b ; (3) where J s =@q s is the Jacobian for the first shape, J s =@q s is the Jacobian for the second shape, and J b is the Jacobian for the parameters of the blending fnction, and is described below.

11 Intitively, (3) means the Jacobians for the components of a blended shape have a greater or lesser effect at a particlar location depending on the fnction. Considering the sphere/cylinder blending example in figre, if a force was applied to the top of the shape, only the parameters of the cylinder wold be affected. Similarly, if a force was applied to the bottom of the shape, only the parameters of the sphere wold change. Therefore, the blending fnction has the desirable effect of localizing the effect of a force to the appropriate shape component. The Jacobian matrix J b reflects how the global shape s changes with respect to the blending fnction parameters q b. Given (6) above, J b v) =, s (; v), s (; : b Given that is a B-spline, to b, we apply the prodct rle to the de Boor algorithm [5]. The control vale and knot constraints c i [; ] for all i L, and i j for all i j L, are enforced to insre the components of q b have correctly bonded vales. It is throgh J b that the blending fnction can change to reflect the shape of the data. Note that for blending to occr dring shape estimation at a particlar location on a shape, the nderlying shapes mst differ. If this was not the case, the difference of the two shapes (s, s ) wold be zero, making J b zero. 5 Experiments In the following fitting experiments, we show the reslts of sing blended shapes in or shape reconstrction system. Figre 6 shows information on each of the experiments inclding the nmber of data points, the reslting mean sqared error (MSE), the size of the parameter set, L (the nmber of knots sed to specify the blending fnction), the dimensions of the node mesh, and the nmber of iterations taken for the fit. In each of the examples, the initial model configration is shown. Initially, the model has all global shape parameters eqal to, and is centered at the center of mass of the data. The blending fnction is initialized to () =. Initially, only = of the data are sed (selected randomly). All of the models sed are global in natre no local deformations were sed. Figres 7 throgh 9 show the fitting reslts obtained for the five experiments. Each fitting example

12 Data Sorce Points MSE #Parm L Mesh #Iter light blb MSU (blb) 4.7% sphere/cylinder MSU (cylinder+sphere) % tors CAD generated 53.6% MSU: Michigan State University PRIP database (special thanks to Anil Jain and Tim Newman) Figre 6: Experiment data and statistics starts with the initial configration described above. After this, the first rogh fit by varying only a, a and a 3 of s is shown. The rest of the steps follow after this, and are described in detail below for each example. Figre 7 shows the model in the process of fitting to light blb data. A blend of two sperellipsoids is sed as the model. The initial model and range data are shown in (a), and the rogh fit after the initial fit is shown in (b). The blending fnction changes in (c). In figre 7(d), all the data are sed to complete the final step, where all the parameters are permitted to change. The final blending fnction (e) shows two distinct areas where it is, and where it is, connected by a smooth transition. Figre 8 shows the fitting of a sphere/cylinder object. Similar to the fitting process of the light blb, (b) shows the initial fit, (c) shows the model after the blending fnction changes, and (d) shows the final fit sing all the range data. With each step, the blending fnction is given to show how it changes dring the fitting. Since this object has a corner where the sphere and cylinder meet, the blending fnction in (d) has developed a point where it is not differentiable. Figre 9 shows the fitting of tors data sing a blend of a sperellipsoid and a spertoroid as the model. The initial range data are shown in (a), and the initialization is shown in (b). The rogh initial fit is shown in (c). The poles are pinched together in (d), and the gens atomatically changes to. The hole is plled open in (e) and (f) (which are the same object from different viewpoints). A final fit sing all data is shown in (g). Notice how the blending fnction (f) has (, )=( )=, since the hole is present. When fitting an object with a sperellipsoid-spertoroid, it is necessary that there be some range data from the inside of the hole. Otherwise, the hole will not be able to be plled throgh by data forces. Each iteration with a fll data set takes (on average) = second on a 5 MHz SGI R4 sing data sets of this size. An adaptive Eler method is sed to pdate the object state. Initially, iterations have O(n log d) complexity (where n is the nmber of nodes, d is the nmber of data points) de to initial nearest-node comptations (for force assignment). Once the shape acqires its rogh general shape, the complexity

13 α() π/ π/ (a) (b) (c) (d) (e) Figre 7: Fitting of light blb data and blending fnction (e) α() α() (a) π/ π/ (b) π/ π/ α() α() (c) π/ π/ (d) π/ π/ Figre 8: Fitting of sphere/cylinder data showing evoltion of blending fnction approaches O(n + d) since nearest-node information can often be carried across iterations. Since fewer range data can be sed initially, this offers an additional constant factor speed increase. For the experiments presented here, this reslts in fits with drations ranging from 45 to 6 seconds each. 6 Conclsions and Ftre Work We have developed and presented a new approach to shape modeling and estimation based on shape blending. These models we created can compactly and intitively represent a large class of shapes in a single model, inclding shapes of varying gens. What we have presented here is also likely to be sefl for recognition becase blended shapes can be parameterized sing a small nmber of intitive global parameters. Blending provides a mechanism of changing topology withot geometric discontinity 3

14 (a) (b) (c) (d) α() π/ π/ (e) (f) (g) (h) Figre 9: Fitting of tors data with gens change in (d) (over time). While there is a representational change (clearly some change is necessary to alter topology when dealing with global shapes), we avoid the sdden geometric and representational changes that other compact shape estimations frameworks employ. Redcing the intensity of this decision shold lead to greater robstness. We demonstrated the performance of or techniqe in a variety of shape estimation experiments involving the extraction of shapes with incomplete range data. Crrently, the blending fnction has a large nmber of degrees of freedom. If blending is to be sed for abstraction, this nmber can be drastically redced. Considering the blending fnctions shown in figres 7(e) and 8(d), the blending fnctions vary from to, with a transition in between. A redction in the nmber of parameters cold be achieved by simply parameterizing the location and character of this transition. Blending fnctions with transitions sch as these prodce a blended shape which clearly shows parts of each component shape, and a transition region between the two shapes. The abstractive power of blending is certainly the most sefl characteristic. We are crrently investigating how to extend the somewhat restricted form of blending presented here. By allowing blending to occr in arbitrary locations (not jst axially), we hope to provide a general facility for combining together selected portions of different shapes (inclding the addition of holes at any location). 4

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