three-dimensional distance field metamorphosisdcor/online_papers/papers/tog98.pdf · geometry and...

Three-Dimensional Distance FieldMetamorphosis

DANIEL COHEN-ORDAVID LEVINandAMIRA SOLOMOVICITel-Aviv University

Given two or more objects of general topology, intermediate objects are constructed by adistance field metamorphosis. In the presented method the interpolation of the distance fieldis guided by a warp function controlled by a set of corresponding anchor points. Some rules fordefining a smooth least-distorting warp function are given. To reduce the distortion of theintermediate shapes, the warp function is decomposed into a rigid rotational part and anelastic part. The distance field interpolation method is modified so that the interpolation isdone in correlation with the warp function. The method provides the animator with atechnique that can be used to create a set of models forming a smooth transition between pairsof a given sequence of keyframe models. The advantage of the new approach is that it iscapable of morphing between objects having a different topological genus where no correspon-dence between the geometric primitives of the models needs to be established. The desiredcorrespondence is defined by an animator in terms of a relatively small number of anchorpoints.

Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometryand Object Modelling—curve, surface, solid, and object representations; I.3.6 [ComputerGraphics]: Methodology and Techniques—interaction techniques; I.3.7 [Computer Graph-ics]: Three-Dimensional Graphics and Realism—animation

General Terms: Algorithms

Additional Key Words and Phrases: Computer animation, interpolation, morphing, radialbasis function, shape-blending, warping

1. INTRODUCTION

Surface metamorphosis is the continuous evolution of a surface from asource surface, through intermediate surfaces, into a designated surface.This object-space process of generating intermediate 3D models is to be

Authors’ address: School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv 69978,Israel.Permission to make digital / hard copy of part or all of this work for personal or classroom useis granted without fee provided that the copies are not made or distributed for profit orcommercial advantage, the copyright notice, the title of the publication, and its date appear,and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, torepublish, to post on servers, or to redistribute to lists, requires prior specific permissionand / or a fee.© 1998 ACM 0730-0301/98/0400–0116 $5.00

ACM Transactions on Graphics, Vol. 17, No. 2, April 1998, Pages 116–141.

distinguished from image-space metamorphosis [Beier and Neely 1992;Wolberg 1990]. Figure 1 shows a sequence of images from an animationmovie showing a metamorphosis of a source 3D model (on the upper left)into a designated 3D model (on the bottom right). Note that the cameraroams slightly during the animation and the model casts a shadow thatevolves according to the shape of the 3D model. Such view-dependenteffects are impossible with image-space metamorphosis.

This article describes a method that allows the user to create a series ofobjects which forms a smooth transition between keyframe objects. Themethod presented is explicitly designed for surfaces of solid objects, that is,the boundary, in any dimension, between the inside of the solid and theoutside. However, the method is presented for the metamorphosis ofsurfaces of 3D solid objects.

Fig. 1. 3D morphing between a cow and a tiger. Note that the camera roams during theanimation and the model casts a shadow that evolves according to the shape of the 3D model.

Distance Field Metamorphosis • 117

ACM Transactions on Graphics, Vol. 17, No. 2, April 1998.

The problem of blending two surfaces, even polyhedral ones, is notsimple. In order to blend two polyhedral models, most techniques requireestablishing a full correspondence between their structures [Kent et al.1992]. However, a correspondence alone does not guarantee a smoothtransition from the source model to the designated model, and the vertices’paths have to avoid troublesome situations such as self-intersections.Two-dimensional shape-blending techniques that deal with the vertices’path problem are not easy to extend to 3D ones.1 Another class of methods,the image metamorphosis techniques [Beier and Neely 1992; Wolberg1990], has rather straightforward extensions to voxel-space [Hughes 1992;He et al. 1994; Lerios et al. 1995]. However, the exact location of thesurface of the voxel-based intermediate objects is not explicitly defined.

The problem of 2D shape blending can also be considered as one of bodyreconstruction from cross-sections [Sederberg and Greenwood 1992]. Someof the reconstruction methods use bivariate interpolation after finding acorrespondence between the contours of the cross-sections. When the cross-sections are not dense, they may exhibit significant differences in thegeometry and topology of the boundary contours. A method that easilyhandles that issue is the distance field interpolation (DFI) presented andanalyzed in Levin [1986]. DFI is a general set-valued interpolation methodfor the reconstruction of an n-dimensional model from a sequence of its(n 2 1)-dimensional cross-sections. It uses a univariate interpolation of thedistance field, and performs well for cross-sections of different topologicalgenera in any dimension. The DFI method has been adapted successfully inbiomedical reconstruction applications [Herman et al. 1992].

The metamorphosis presented here is based on the DFI method. How-ever, it is extended in the sense that the interpolation, and thus theblended surface, follows a warp transformation guided by means of auser-defined control. The control is defined by a point-to-point correspon-dence between prescribed anchor points on the given intermediate objects.To avoid large deviations of the in-between shapes from the two originalshapes, we decomposed the warp transformation W into a rigid (rotationand translation) transformation and an elastic transformation, which areseparately interpolated. The rigid-elastic decomposition of the warp func-tion, and its particular interpolation are so chosen to minimize the distor-tion of the intermediate surfaces. The rigid transformation is used to rotateand translate the source object to match the coarse features of the desig-nated object, and the finer features of the object are evolved by the elasticpart. In Cohen-Or et al. [1996] we have shown how these principles can beapplied to surface reconstruction from 2D cross-sections.

In Section 2 we present the different terms used in the field of morphing,and survey-related work. Section 3 describes the DFI method. The warpfunction, its decomposition, and parameterization, are introduced in Sec-

1Please see Goldstein and Gotsman [1995], Sederberg et al. [1993], Shapira and Rappoport[1995], and Sun et al. [1995].

118 • D. Cohen-Or et al.


tion 4. In Sections 5 and 6 we describe some implementation details anddiscuss the results in Section 7. We conclude with a summary in Section 8.

2. BACKGROUND—FROM WARPS TO METAMORPHOSIS

The problem of transition from one geometrical model to another can beformally stated as follows: given two models, a source S and a target T,construct a set of transformations {Wt u t [ [0, 1]}, such that W0 5 I is theidentity transformation; that is, W0(S) 5 S, and W1(S) 5 T. Intermediatemodels or in-between “blends” of S and T are defined by Wt(S), t [ (0, 1).

It is important to distinguish between methods operating on a discreterepresentation, where the transformation is applied to the pixels of animage [Beier and Neely 1992; Lee et al. 1995; Wolberg 1990] (image-basedmethods) or to a voxel representation [He et al. 1994; Hughes 1992; Lerios1995], and methods operating on a combinatorial representation (e.g.,polygons and polyhedra).2

The term morphing is used for all methods that gradually and continu-ously deform S into T, while producing the in-between models. Themorphing process is usually composed of warping and interpolation. Sincemorphing involves two objects, we perform the warping operation first, toobtain an approximated alignment so that the two shapes will be relativelysimilar, and then we blend the two warped objects into one. This interpola-tion process in image-based techniques is often referred to as cross-dissolving, where the values of corresponding pixels are blended, yieldingthe effect of fading-in of the target image with a fade-out of the sourceimage. Figure 2 illustrates the image-based morphing, where cross-dissolv-ing is used to blend the gray-level values of corresponding pixels.

Warping can be generally described as a mapping or a transformationthat can be applied either in image space [Arad and Reisfeld 1995;Ruprecht and Muller 1995; Wolberg 1990] or object space [Sederberg andParry 1986; True and Hughes 1992; Carmel and Cohen-Or 1997]. Thismapping essentially distorts the object it is applied to, according to somepredetermined constraints, which are usually specified by the user. Suchuser control over the warping is crucial in obtaining satisfactory results,and can be achieved, for instance, through specifying a predeterminedmapping for a set of points (point-to-point correspondence).

It is important to note that, in practice, warping (and hence morphing)requires a good level of user specification to achieve the desired results. Allforms of specification (e.g., point-to-point correspondence) are based on theidea of defining the transformation by manipulating only a finite and, it ishoped, small number of parameters. From these specifications, mathemat-ical techniques should be used to compute the transformation for any pointin the warping domain.

Shape-blending algorithms usually refer to techniques operating on acombinatorial representation. Most of the work done here has dealt with

2Please see Goldstein and Gotsman [1995], Kent et al. [1992], Sederberg et al. [1993], Shapiraand Rappoport [1995], Sun et al. [1995], and Kanai et al. [1997].



polygonal or polyhedral objects.3 The general method of these algorithms isto displace the vertices, edges, and faces of S over time to coincide in

3Please see Kent et al. [1992], Sederberg et al. [1993], Shapira and Rappoport [1995], Sun etal. [1995], Parent [1992], Lazarus and Verroust [1994], DeCarlo and Gallier [1996], Decaudin[1996], Carmel and Cohen-Or [1997], and Kanai et al. [1997].

Fig. 2. Image-based morphing process.



position with the vertices, edges, and faces of T. This usually requiressolving the vertex correspondence problem, which determines which vertextravels to which vertex during the deformation process. However, establish-ing suitable correspondence is difficult, and does not guarantee goodbehavior of the intermediate models. In addition, a shape-blending tech-nique needs to control the vertex paths, that is, the paths along which thevertices travel between their corresponding positions, to avoid self-intersec-tions or major changes in intrinsic geometric properties, such as angles andlengths.4

Research effort in the metamorphosis field initially focused on the 2Dproblem. For 2D polygonal shapes, Sederberg and Greenwood [1992] pro-posed a solution to the vertex correspondence problem, and the vertex pathproblem was dealt with in Sederberg et al. [1993]. Other 2D algorithmsthat deal with the vertex path of polygonal shapes were proposed byShapira and Rappoport [1995] who used a connecting skeleton to controlthe interior of the shape as well as its boundary, and by Goldstein andGotsman [1995] who used a multiresolution representation of simple poly-gons. As for image space techniques, Beier and Neely [1992] suggested analgorithm based on fields of influence surrounding 2D control primitives. Atreatise of image warping methods can be found in Wolberg’s [1990] book.

Less work has been done in 3D morphing. Chen and Parent [1989] brieflyaddressed an extension of a transformation for piecewise linear 2D con-tours to 3D objects represented by a set of planar contours. Another shapetransformation algorithm was proposed by Kent et al. [1992], where thecorrespondence problem for polyhedral objects was solved by merging theirtopological structures (see also Kanai et al. [1997]). Kaul and Rossignac[1991] computed the Minkowski sum of scaled versions of the models, andthen used it to obtain a smooth transition between the models. Lerios et al.[1995] extended the work of Beier and Neely [1992] where fields ofinfluence of 3D primitives were used to warp a voxel space. The voxel-basedmodel was then rendered by volume-rendering techniques.

Recently, Sun et al. [1995] presented a solution to the vertex pathproblem of polyhedral models, which is an extension of the intrinsic shapetransformation scheme suggested in Sederberg et al. [1993] for 2D poly-gons. Rather than considering a polyhedron as a set of independent verticesor faces, their algorithm treats a polyhedron as a graph representing theinterrelation between vertices, where intrinsic shape parameters, such asdihedral angles and edge lengths, are used for interpolation.

Other 3D metamorphosis algorithms deal with sampled or volumetricrepresentation of the objects. Hughes [1992] proposed transforming theobject into the Fourier domain, and then interpolating the low frequenciesover time while the high frequencies are slowly added in, thus minimizingthe object distortion caused by its high-frequency components. A method

4Please see Goldstein and Gotsman [1995], Kent et al. [1992], Sederberg et al. [1993], Shapiraand Rappoport [1995], and Sun et al. [1995].



using the wavelet transform instead of the Fourier transform was proposedby He et al. [1994]. This method enables a more accurate transition of theisosurfaces, due to the better localization of the wavelet transform in thespatial domain.

Payne and Toga [1992] gave a distance-field volumetric representation of3D biomedical objects. Their technique consists of cross-dissolving thedistance values of each voxel and then reconstructing the intermediatesurfaces out of the intermediate distance-field. However, no warping orcorrespondence was established between the source and the target object.This lack of control over the morphing process often yields poor results. Wepresent a method that uses the distance-field volumetric representation of3D objects, but in addition, establishes a point-to-point warp functionbetween the two models that can be seen as a way of enforcing topologicalcorrespondence and geometrical properties.

3. DISTANCE FIELD INTERPOLATION

Reconstruction of 3D objects from cross-sections is of major importance inmany applications, particularly in biomedical imaging. One class of recon-struction methods builds a polygonal surface by assuming the cross-sections consist of closed polygonal contours [Barequet and Sharir 1996].Another class of methods uses a volumetric approach. The entire volume(3D voxel array) is reconstructed from the gray levels of the cross-sections.The missing data are reconstructed by some interpolation of the grayvalues of the cross-sections. The surface is then defined and reconstructedby isosurfacing methods. These methods do not yield smooth spatial trans-formations, as can be seen in Figure 3. Figure 3(a) shows that sections ofthe source object dissolve instead of spatially transforming into the targetobject, as in Figure 3(c).

The method of DFI presented in Levin [1986] achieves better reconstruc-tion of the surface by interpolating distance values rather than gray values.This method has been successfully adapted in biomedical reconstructionapplications (see e.g., Herman et al. [1992] and Payne and Toga [1992]).

The DFI method can be described in terms of n-dimensional surfacereconstruction from (n 2 1)-dimensional cross-sections (n . 1). Consider asolid object V , Rn, discretized or not. Given a finite set of (n 21)-dimensional cross-sections of the object,

V tj 5 $ x 5 ~ x1, . . . , xn21! u~ x1, . . . , xn21, tj! [ V%, t0 , t1. . . , tM ,(1)

we would like to reconstruct V. Throughout the article we use lower indicesfor numbering entities and upper indices to denote the components of avector. Let us define the signed distance fields at the levels t0, t1, . . . , tM.



Fig. 3. Comparing gray-level interpolation with warping to the DFI with warping: (a) resultsof the gray-level interpolation before thresholding; (b) after thresholding; (c) results of the DFItechnique. Note the discontinuity between the third and the fourth objects in (b).



For x 5 ( x1, . . . , xn21),

Dtj~ x! 5 H2dist~ x, V tj!

dist~ x, V tj!

if x [ V tj

otherwise,(2)

where Vtjdenotes the boundary of Vtj

, and dist denotes the Euclideandistance in Rn21. Now, using univariate interpolation (with respect to theparameter t) we interpolate, between the cross-sections, the distancevalues of points having the same first n 2 1 coordinates. The resultinginterpolant approximates the n 2 1 Euclidean distance between y 5( x1, . . . , xn21, t) and Vt 5 V uyn5t.

Once this approximated distance field is available, the surface of theobject can be determined by the zero points of the distance field or, in thediscretized version, by the boundary between the positive and negativevalued lattice points, whereas the volume itself (its interior) is defined asthe set of all negative valued points.

Formally, the DFI method defines an approximated domain V ' V asfollows: for y 5 ( x1, . . . , xn21, t) we first define the interpolant dx(t) ( x 5( x1, . . . , xn21)) by univariate interpolation of the values {Dtj

( x)} j50M . Now,

V 5 $ y udx~t! # 0%. (3)

For more details on the choice of the interpolation method, and forapproximation rate analysis, see Levin [1986].

Clearly, this method can be used for morphing, where the problem ofmatching and interpolating the deformations of (n 2 1)-dimensionalobjects can be viewed as an n-dimensional shape reconstruction. The DFImethod works well provided that “corresponding parts” of the two objectsare properly aligned. Otherwise, some parts may unexpectedly disappearand reappear. The approach taken in this work to overcome this restrictionis the combination of DFI with a proper warp transformation.

4. WARP-GUIDED DFI MORPHING

Let us consider for simplicity the blending of two objects, S 5 V0 and T 5V1. DFI gives us a method of blending by interpolating (linearly) thedistance functions D0 and D1, as described in Section 3. We call thisblending process no-warp DFI blending. No-warp DFI blending is quiterestricted, and may produce unsatisfactory results (see the example inFigure 4). To demonstrate this, consider the case when T is just a rotationof S, and S is just a thin rod. No-warp DFI blending at t 5 1⁄2 gives eithera very small object, or an empty object, which is not the naturally expectedresult. On the other hand, no-warp DFI blending works nicely if S is a thinrod and T is just the rod S without its middle third.

Consider now the possibility that T is obtained by a deformation (warp)of S; that is, T 5 W1(S). Furthermore, let us assume that T graduallyevolves from S, through a continuum of objects Wt(S), where {Wt}t[[0,1] are



smooth transformations, smoothly changing with t, and W0 [ I is theidentity transformation. In such a case the blending can be defined by thewarp itself; this is pure-warp blending. Pure-warp blending is quite re-stricted, as it does not allow changes in the genus of the objects.

We now hybridize the two different methods into a more powerful tool[Beier and Neely 1992; Lee et al. 1995]. The first step of the hybrid methodis to find a smooth warp {Wt}t[[0,1] such that W1(S) ' T in some chosensense. Then the DFI method is applied, now guided by the warp {Wt}. Note

Fig. 4. Comparing the morphing results using the DFI method: (a) without any warp, and (b)using a proper warp. The objects are defined by the white contours.



that the warp operates in Rn21 and the interpolation of the distance field isperformed along the nth dimension. As stated previously, we first find asmooth warp {Wt}t[[0,1] such that W1(V0) approximates V1 as well aspossible. In addition, we use the two signed distance functions D0 and D1

as defined by Eq. (2). For x 5 ( x1, . . . , xn21) we then interpolate thevalues D0(W0( x)) and D1(W1( x)) with respect to the parameter t, denotingthe interpolant dx(t). The approximated domain V defined by the hybridwarped DFI procedure is

V 5 $ y 5 ~Wt~ x!, t! udx~t! # 0%. (4)

4.1 Obtaining the Approximated Warp Wt

We define the warp Wt so that N points { p0,i}i51N in the source domain are

mapped to corresponding points { p1,i}i51N in the designated domain. Later

we refer to these points as anchor points. The warp is to be determined sothat

W1~ p0,i! 5 p1,i , 1 # i # N. (5)

The warp function maps the source (level zero) anchor points to theircorresponding designated (level one) points, and other points in the sourcedomain comply with those constraints. Defining the warp by anchor pointsis attractive because it is a naturally applicable concept in any dimension.From the user’s point of view, this method is intuitive and very easy tohandle. It also has the advantage of being robust and very stable in thesense that the results are not very sensitive to small perturbations in theanchor points.

We now define the warp transformation based on anchor points, startingwith the case n 5 3; that is, the cross-sections are in R2. To achieve leastdistortion of the in-between objects, the warp transformation Wt is decom-posed into a rigid part and an elastic part. To explain the motivation here,let us view a very simple case: consider the blending of two objects in R2,S 5 V0 and T 5 V1, where T 5 RuS and Ru is a rotation in angle u. Thereare two obvious options for defining a warp Wt. One is the linear interpola-tion warp Wt 5 (1 2 t)I 1 tRu. The other is the linear rotation warp Wt 5Rtu. Both warps vary smoothly from the identity I at t 5 0 to Ru at t 5 1.However, the second option is preferable since it is an isometry for any t(i.e., nondistorting). Now consider the case where T is obtained from S by arotation Ru, a translation c, and an elastic transformation E; that is, T 5E(RuS 1 c). In the following we choose the rotation Ru and the translationc so that E is as close as possible to I in some sense (see Figure 5). Thewarp Wt is then defined by

Wt~ x! 5 ~~1 2 t!I 1 tE!~Rtux 1 tc!, x [ R2. (6)



We describe how to obtain the rotation, the translation, and the elastictransformation, and consider the main case n 5 4, namely, the morphing of3D objects.

4.2 The Rigid Transformation in R3

To start with, we are given two ordered lists of points that, taken together,define pairs of anchor points in R3, { p0,i} i51

N and { p1,i} i51N . The rigid part of

the transformation is defined by the rotation R and the translation c, whichminimize

Q 5 Oi51

N

iRp0,i 1 c 2 p1,ii2, (7)

Fig. 5. The rigid part minimizes the work of the elastic part: (a) blending with the rigidtransformation; (b) without it.



where i z i is the Euclidean norm on R3. Imagine that the transformedpoints {Rp0,i 1 c} are connected by identical elastic springs to thecorresponding points { p1,i}. The form Q represents the elastic energy ofthis system, and the rigid transformation that minimizes Q brings V0 to anequilibrium position in the springs’ system.

The preceding least squares fitting problem of the two sets of anchorpoints in R3 is solved using an explicit algorithm, which involves thesingular value decomposition (SVD) of a 3 3 3 matrix [Arun et al. 1987]. Asa result, we obtain a 3 3 3 rotation matrix R and a translation vector c. Inorder to interpolate the rotation defined by the matrix R, we use theQuaternion approach [Shoemake 1985], thus obtaining the intermediaterotation matrices Rt, for t [ (0, 1).

Finally, the warp transformation in the 3D case is defined by

Wt~ x! 5 ~~1 2 t!I 1 tE!~Rtx 1 tc!, x [ R3. (8)

4.3 The Elastic Warp in R3

Once we have computed the rigid transformation, we move on to calculat-ing the elastic transformation. In accordance with Equations (6) and (9), welook for a transformation E, in general, a nonlinear transformation, E:R3 3 R3, such that

E~Rp0,i 1 c! 5 p1,i , 1 # i # N. (9)

This is a multivariate scattered data interpolation problem, which wesuggest solving by using radial basis functions, abbreviated RBF [Dyn1987; Arad and Reisfeld 1995]. In R3 this means solving three interpolationproblems in R3 for each component of the vector equation (9).

Radial basis functions have proven to be an effective tool in multivariateinterpolation problems of scattered data. Given scattered points { xi} , Rd

and corresponding data values {Fi} , R, we look for an interpolatoryfunction S( x) of the form

S~ x! 5 Oi51

N

aig~i x 2 xii!, (10)

such that S( xi) 5 Fi, 1 # i # N. Here i z i denotes the usual Euclideannorm on Rd and g : R1 3 R. A function of this form is usually referred to asa pure radial sum. Using radial functions reflects the fact that thescattered data have no preferred orientation [Dyn 1987]. Other multivari-ate approximations, generalizing the univariate splines, use augmentedradial sums, where the sum in Eq. (10) is augmented by a low-degreepolynomial. Our discussion is mainly concerned with the 3D case (d 5 3).We define the elastic transformation E : R3 3 R3 as

E~q! 5 Z~q! 1 L~q!, (11)



where L is an affine transformation and Z is of the form

Z~q! 5 ~S1~q!, S2~q!, S3~q!!, (12)

where Sk(q) 5 ( i51N ai

kg(iq 2 qii), for 1 # k # 3, q 5 (q1, q2, q3)T [ R3,g : R1 3 R is a univariate function, and {qi} for i 5 1, 2, . . . , N are thesource anchor points after the rigid transformation has been applied to them.

The preceding definition of the transformation E includes a linear part L.This linear part is added because pure radial sums may yield poor approx-imation of the transformation for points away from the anchor points.Moreover, the linear part is natural here since we would like to exactlyreconstruct those transformations E that are affine, and especially theidentity transformation.

A proper choice of g is also important. Choosing g as a function with afast decay, for example, the Gaussian g(r) 5 exp(2r2/s2) with a small s,results in a finer local influence of the radial part, which can be used toobtain different effects for various parts of the object. Other choices areg(r) 5 r2 log(r) in R2 and g(r) 5 r3 in R3, which have global effect [Dyn1987; Ruprecht and Muller 1995]. The different effects in R2 are demon-strated in Figure 6. In the following we describe the computation of theelastic warp in R3.

After computing the rigid part, namely, the rotation R and the transla-tion c, the elastic mapping E is defined by N interpolation conditions:

E~Rp0,i 1 c! ; E~qi! 5 p1,i ; yi , i 5 1, . . . , N, (13)

where qi 5 (qi1, qi

2, qi3)T and yi 5 ( yi

1, yi2, yi

3)T. This interpolation problemis always solvable if we use an augmented radial approximation (with aproperly chosen g) of the form

E~q! 5 Oi51

N

aig~iq 2 qii! 1 Aq 1 a4 , (14)

where A 5 (a1, a2, a3)T, ai [ R3, 1 # i # N, a, 5 (a,1, a,

2, a,3)T [ R3,

1 # , # 4.Thus E is determined by N 1 4 coefficients in R3. The computation of

those coefficients involves the solution of three square linear systems ofsize N 1 4 each, where N conditions are derived by the interpolationrequirements, as defined in (13), and the additional compatibility condi-tions are

Oi51

N

aik 5 O

i51

N

aikqi

1 5 Oi51

N

aikqi

2 5 Oi51

N

aikqi

3 5 0, k 5 1, 2, 3.

These conditions guarantee that the transformation is affine reducible;that is, the transformation is purely affine whenever possible. The system



of equations for the vectors of unknowns uk 5 (a1k, . . . , aN

k )T and vk 5 (a1k,

a2k, a3

k, a4k)T, k 5 1, 2, 3, is

HGuk 1 Hvk 5 bk

HTuk 5 0, (15)

where bk 5 ( y1k, . . . , yN

k )T, G 5 { g(iqi 2 qji)}i, j51N , and H is an N 3 4

matrix with an ith row {qi1, qi

2, qi3, 1}, 1 # i # N.

Now that the warp transformation Wt is fully defined, the warped DFIapproximation can be obtained by applying the definition of Eq. (4). In thefollowing section we describe the application of the method in a discretevoxel-based environment.

5. THE SIGNED 3D DISTANCE TRANSFORM

Given a geometric object, its signed 3D distance field is generated by firstrasterizing the object into a binary discrete volumetric representation. The

Fig. 6. Different effects when using a g function (c) with fast decay and (d) with a globaleffect. The source and target anchor points are shown in (a) and (b), respectively.



binary volume, which represents the solid object, consists of binary-valuedvoxels, which are classified either as feature or nonfeature voxels. Featurevoxels are defined as those that either intersect the surface of the object orare contained within it. Assume now that the feature voxels are containedin a discrete bounding box partitioned into voxels at a certain resolution.

Given a binary 3D discrete volume, the 3D distance transformationconverts the volume into a distance field as defined in Eq. (2). We extendthe 2D discrete distance transform [Borgefors 1986] to 3D. The basic idea isthat the global operation of calculating the distances can be approximatedby propagating local distances. These methods are most efficient, and areeasily extended to 3D.

The basic process of computing the distances of the nonfeature voxels tothe nearest feature voxel, is done as follows. The original volume isinitialized such that feature voxels are assigned a zero value and nonfea-ture voxels are assigned infinity. We use a discrete distance called Chamferdistance [Borgefors 1986] which approximates the Euclidean distance. Ituses a 3 3 3 3 3 cubic mask (illustrated in Figure 7), where the localdistances are scaled to avoid floating point arithmetic.

The distance transform process consists of two successive scannings ofthe volume, a forward scan and a backward scan. The 3D Chamfer distancemask is partitioned into two half-masks. The first half, denoted forwardmask (FM), contains the distances of the voxels that are positioned in themask “before” the central voxel in the scanning direction. The second half,denoted backward mask (BM), contains the other voxels in the mask. Bothhalves contain the central voxel.

At each step the mask is placed over a voxel v0 5 ( x0, y0, z0) and itsvalue V(v0) is replaced by the following computation.

Forward Scan: top to bottom, back to front, left to right,

V~v0! 5 min~i,j,k![FM

$V~ x0 1 i, y0 1 j, z0 1 k! 1 d~i, j, k!%.

Backward Scan: bottom to top, front to back, right to left,

V~v0! 5 min~i,j,k![BM

$V~ x0 1 i, y0 1 j, z0 1 k! 1 d~i, j, k!%.

Fig. 7. The three layers of the cubic Chamfer distance mask.



Here (i, j, k) is the position in the mask (the center being (0, 0, 0)), andd(i, j, k) is the local distance from the mask center. The process describedassigns to each nonfeature voxel the distance to its nearest feature voxel,and the feature voxels are assigned the value zero.

However, for our application, this basic computation of the distancetransform needs to be extended to evaluate the distances of the featurevoxels as well as the nonfeature voxels, as defined by Eq. (2). To match thisdefinition, we switch the roles of the feature and the nonfeature voxels, andrepeat the distance transform to compute a second distance field volume,containing the distances of the feature voxels to the nearest nonfeaturevoxel. All the positive distance values in the second distance field arenegated, and the two discrete distance fields are combined (by discardingthe zero-valued distances in both volumes), to yield the representation ofEq. (2).

The source and the target objects to be morphed are represented asdiscrete distance field volumes (DF-volumes). Constructing the intermedi-ate object essentially requires generating its discrete DF-volume, out ofwhich its surface can be extracted [Lorensen and Cline 1987].

6. THE DISCRETE MORPHING PROCEDURE

In this section we present the morphing procedure used for generating theintermediate DF-volume in a voxel-based representation. The formal defi-nition of the warped DFI method is given in Eq. (4). However, theimplementation of this formula in a discrete environment is not straight-forward, and involves an additional discretization approximation. Forclarity, let us consider the procedure for the case of two 3D objects, S 5 V0and T 5 V1.

As explained in Section 2, the morphing procedure combines two simul-taneous processes: warping and cross-dissolving. The warp mapping isdetermined by the mapping of the anchor points, whereas the cross-dissolving is performed by interpolating the distance values of correspond-ing voxels (see Figure 8). We combine two backward mappings, as describedin the following with respect to Figure 9, in order to perform this process.

Let D0 and D1 be the two DF-volumes of the source and the designatedobjects, respectively, and let us denote the DF-volume of the desiredintermediate object Dt. We assume that all these DF-volumes and theanchor points are contained within a finite set X of R3.

Constructing the intermediate DF-volume Dt requires the evaluation ofthe correct distance value to be associated with each of the voxels at level t.The first step of this process is to apply the warp transformation Wt, asdefined in Eq. (11), to the source anchor points in order to find the locationof the anchor points in the intermediate level. These points are denoted{mi [ Wt( p0,i)}i51

N . In the next step, for any voxel at level t we would liketo find the corresponding voxels at levels 0 and 1. This is done using thetwo backward mappings B1 and B2, such that B1, B2 : X 3 R3, and

B1~mi! 5 p0,i , B2~mi! 5 p1,i , i 5 1, . . . , N. (16)



It is important to note here that although B1 is the inverse of Wt at theanchor points, this does not necessarily hold for other points. Finding B1and B2 is done by the process described in the previous section.

Fig. 8. Warp-Guided DFI: (a) source distance field warped towards the target; (b) intermedi-ate interpolated Distance Field; (c) designated distance field warped towards the source; thecontours are highlighted in white.



Once B1 and B2 are available, Dt is generated as follows: for each v [ X,the distance value Dt(v), is evaluated by

Dt~v! 5 ~1 2 t! D0~B1~v!! 1 tD1~B2~v!!, (17)

where the discrete distance functions D0 and D1 are extended to X bytrilinear interpolation. Hence, the two distance values of the correspondingvoxels at levels 0 and 1 are interpolated linearly. For some voxels v [ X, itmay occur that either B1(v) or B2(v) is not in X, in which case we assign tothis point an approximated distance value by adding the distance from thispoint to the distance value of its nearest voxel in X. In order to speed upthe computation, the trilinear interpolation can be applied only to distancevalues that are close to zero, since this is where the surface is located, andavoided elsewhere.

7. RESULTS

We have tested the algorithm on several 3D examples, as shown in Figures1, 12, and 13. Two choices of radial basis functions for computing theelastic warp were examined. The Gaussian function g(r) 5 exp(2r2/s2) isused in Figures 12 and 13, whereas the g(r) 5 r3 basis function is used forthe morphing example given in Figure 1. For more details on the choice ofthe radial functions, see Arad and Reisfeld [1995]. For a comparison ofrun-times using radial basis function transformations, see Ruprecht andMuller [1995].

Figure 10 shows two results of the voxelization and the surface recon-struction of 3D models. Note that the large number of triangles in thereconstructed surfaces depends directly on the resolution used, regardlessof the complexity of the original object representation, and as the resolutiongrows the main problem becomes the storage space consumption. However,a large number of redundant polygons in regions of low curvature can beremoved by decimation and simplification processes (e.g., Schroeder et al.[1992]). Once the voxelized DF-volume is computed, the time cost for

Fig. 9. Transformations involved in the generation of Dt. Wt maps the anchor points and B1

and B2 backmap the distance values.



creating an intermediate volume is a function of the resolution, the numberof anchor points, and the choice of the radial basis functions.

We have implemented the 3D algorithm on an SGI R4400 machine. Usingan unoptimized C code, it took approximately 40 minutes to create anintermediate 2003 volume, with 20 anchor points and global supportedradial functions. All the models seen in the examples were originallyrepresented as polyhedra, and the voxelization was performed in a 2003

resolution. The RayShade raytracer was used to render the images.Figure 1 shows a 3D morphing sequence between the surfaces of a cow

and a tiger. We used g(r) 5 r3 as the radial function, and controlled themorphing with 18 anchor points, located on the legs, head, tail, and alongthe sides of the animals.

The morphing in Figure 12, between the surfaces of a Triceratops and aniron, is controlled by 16 anchor points using the Gaussian as the radialfunction with 3s 5 20. Note that only the iron object has a hole.

Figure 13 shows a 3D morphing sequence between two key objects thathave a different number of connectivity components. The source is com-posed of two objects, a mushroom and an iron, and the target is a teapot.The morphing is controlled using 11 anchor points, located on the handle of

Fig. 10. Surface reconstruction results: (a) original donkey (874 triangles); (b) model voxel-ized at a resolution of 1003 (26,764 triangles); (c) original iron (2,600 triangles); (d) modelvoxelized at a resolution of 1003 (42,482 triangles).



Fig. 11. Comparative results of morphing algorithms: (a) Sederberg et al.; (b) Shapira andRappoport; (c) Goldstein and Gotsman; (d) DF morphing. The anchor points are marked on thetwo key figures.



the teapot and the iron, on the spout, and along the sides of the teapot. Weused the Gaussian as the radial function with 3s 5 20. It should beemphasized that the relative positions of these three models are identicalto that shown in the sequence, so that the rotation is performed by the rigidtransformation.

We visually compared the results of our algorithm with the results ofSederberg et al. [1993], Shapira and Rappoport [1995], and Goldstein andGotsman [1995] on the same input. (The first three results were scannedfrom Goldstein and Gotsman [1995].) The results are given in Figure 11,where the anchor points used to create our morphing are marked on thekey figures. The thin-plate spline function g(r) 5 r2 log r has been used asthe radial function. It should be pointed out that the first three methodstreat the objects as polygons and are thus forced to create a propercorrespondence between their geometric features (e.g., vertices), whereas

Fig. 12. 3D morphing between a Triceratops and an iron. Note that only the iron object has ahole.



our method treats the objects as (discretized) contours. That means that inour technique the representation of the two objects may differ and need notbe restricted to polygons, since no explicit correspondence between therepresentation’s features needs to be established.

8. CONCLUSIONS

The article presents an object-space metamorphosis method between two(or more) surfaces of solid objects with general topology, using an extendeddistance field interpolation. The metamorphosis of the surface is guided bycorresponding control points that define a warp function. The warp func-tion is decomposed into a rigid part and an elastic part in order to reducethe distortion of intermediate models. The rigid transformation serves asan approximation to the overall warp transformation, which yields better

Fig. 13. 3D morphing between a mushroom, an iron, and a teapot. Note that the source andthe designated objects have different topological genera.



control over the path of the in-between shape throughout the metamorpho-sis.

The method presented here allows an intuitive and simple manualintervention of the animator who can control the animation process byprescribing a small number of anchor points. The advantage of thisapproach is that no correspondence between the models’ geometric primi-tives (e.g., vertices) needs to be established. Moreover, the input objects donot necessarily need to have the same representation or the same topolog-ical genus, and can have a different number of connectivity components(see DeCarlo and Gallier [1996]).

Although there is no explicit definition of what the in-between objectsshould look like, the sequence of shapes should look “natural” to the viewer.There is a variety of ideas of what the sequence of in-between objectsshould satisfy to appear natural. However, it is rather easy to agree that amorph sequence should satisfy the following criteria to yield a pleasingmorph.

—The volume and the boundary surface area of the objects should changesmoothly and monotonically.

—The boundary surface of the objects should retain the smoothness of theoriginal objects.

—Features common to both source and target objects (e.g., head or legs)should be preserved during the process.

Generally speaking, these criteria aim at treating the objects as rigidlyas possible and at avoiding redundant global and local deformations. Amorph sequence satisfying the preceding criteria probably yields a goodmetamorphosis, but it still remains difficult to obtain an objective compar-ison between different sequences, resulting either from different algo-rithms or from the same algorithm with different parameters.

However, the quality of the morph is very much subject to a proper warp.That is, it works well provided that “corresponding parts” of the two objectsare properly aligned by the warp. Otherwise, some parts may unexpectedlydisappear and reappear, exactly as can happen in pure DFI (with no warp).The fact that the warp is guided by the user is probably the weak part ofthe technique. However, a high-fidelity, eye-pleasing, automatic alignmentof arbitrary objects is still a major challenge.

ACKNOWLEDGMENTS

The authors would like to thank Pixel Inc., especially Haggai Goldfarb andBoaz Tal, for providing us with the 3D models. Special thanks to Yael Milofor her moral support and encouragement. Oded Cohen and Eran Rosen-berg helped us to produce the video movies. Gershon Elber and Nur Aradprovided us with many helpful comments on early drafts.

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Received July 1996; revised November 1997; accepted December 1997



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