Investigations of Tensor Voting Modeling

Joanna Beltowska†‡ Ken Museth‡ David Breen†

†Computer Science DepartmentDrexel University

Philadelphia, PA USA

‡Science and Technology DepartmentLinkoping UniversityNorrkoping, Sweden

ABSTRACT

Tensor voting (TV) is a method for inferring geometric structures from sparse, irregular and possiblynoisy input. It was initially proposed by Guy and Medioni [GM96] and has been applied to severalcomputer vision applications. TV generates a dense output field in a domain by dispersing informationassociated with sparse input tokens. In 3-D this implies that a surface can be generated from a set ofinput data, giving tensor voting a potential application in surface modeling. We study the tensor votingmethodology in a modeling context by implementing a simple 3-D modeling tool. The user creates asurface from a set of points and normals. The user may interact with these tokens in order to modify thesurface. We describe the results of our investigation.

Keywords: implicit surface modeling, surface reconstruction, tensor voting

1 INTRODUCTION

As the use of implicit 3D representations gains pop-ularity, the need for modeling software that pro-vides implicit model editing capabilities increases.New surface editing approaches are continually be-ing explored and much research is being conductedto find ways to interactively modify implicit models.Towards these ends we investigated tensor voting asa possible technology that could provide a new andinteresting approach for editing implicit models.

Tensor voting is a method for grouping geomet-ric features [MLT00]. It can be used to generatesurfaces from a sparse set of possibly noisy and ir-regular input data, and therefore may provide novelediting capabilities within a 3-D modeling context.The goal of our work was to investigate TV asa technique for interactive surface modeling. Wewere interested in determining if TV can be usedto model simple objects, edit them interactivelyand control their shape via TV “input tokens.” Toachieve this, we developed a TV modeling system(TVMS) based on an already existing TV frame-work and conducted experiments to evalute TV ina 3-D modeling context. Spheres were used to ex-amine the parameters of TVMS and two modelswere sculpted in order to get an evaluate the poten-tial capabilities and limitations of the TV modelingprocess.

1.1 Related Work

We looked to other novel modeling systems for ideasand inspiration. In general, these are systems thatmodel 3-D objects with geometry other than polyg-onal meshes. The Vector Field Based Shape Defor-mations of von Funck et al. [vFTS06, vFTS07] isan interactive method for shape modeling with vol-ume preservation. Guo et al.[GHQ04] use scalar–fields to drive a point-set surface editing process.The point-sample based shape modeling presentedby Pauly, Keiser, Kobbelt and Gross [PKKG02] isa hybrid system which couples the benefits of im-plicit and parametric surface representations, us-ing the implicit surface definition of moving leastsquares projection along with point-based visual-ization. Pointshop 3D is a framework developed byZwicker et al. [ZPKG02] that transfers the func-tionality of 2D image editing operations to 3D, De-waele and Cani [DC04] present a virtual clay mod-eling approach, which implements interactive shapemodeling capabilities which may be based on hapticfeedback. Museth et al. [MBWB02, MBW+05] de-veloped interactive editing operators for a relativelynew type of implicit surface, level set models.

In our approach, as in many of the approaches de-scribed above, the object is represented using con-trol points, which can be modified by the user inorder to edit the surface. By using the TV method-ology in a novel context, we have employed the ideasof local and global editing operators, point-basedsurface representation and a physical analogy formodeling surfaces.

1

2 TENSOR VOTING

This section provides an introduction tothe Tensor Voting methodology. More de-tails about the methodology can be found in[GM97, TM98, MLT00, TMMS01, TTMM04].

TV has its background in early computer visionproblems where the available data often is sparseand noisy, making it difficult to extract relevant in-formation and structures. TV identifies local fea-ture descriptions by spreading the information as-sociated with shape-related input within a neigh-borhood while enforcing a smoothness constraint.This process refines the information and accentu-ates local features. By doing so, coherent, locallysmooth, geometric features are defined and noise isdiscarded. Each data point communicates its infor-mation in a neighborhood through a voting process.The more information that is received at each datapoint, the stronger is the likelihood of a geometricfeature being present at a certain location. Thislikelihood is expressed through a confidence mea-sure, saliency, which is used in the feature extrac-tion process.

TV is based on two elements; data representa-tion, which is obtained by means of tensor calculus,and communication of data through linear voting,a process similar to linear convolution. The inputelements, referred to as input tokens, are encodedinto tensor form and communicate their informa-tion to their neighboring tokens via pre-calculatedtensor voting fields. After this initial voting step,each token has its confidence and surface orienta-tion encoded into a generic second order symmetrictensor. The tokens vote a second time to propagatetheir information throughout a neighborhood. Theresult is a dense tensor field which assigns a mea-sure of confidence and saliency to each point in thedomain. This dense map is decomposed into threedense maps, each representing a geometric feature(junction, curves or surface), which are analyzedduring feature extraction. TV can be generalized toN-D [TMMS01]. The 3-D case, specifically surfacevoting, is sufficient for our needs and will thereforebe the focus of this paper.

2.1 Tensor Representation

Diagonalizing a second order symmetric tensor,which can be represented by a 3 × 3 matrix,produces the associated characteristic equation.Solving this equation leads to a representationbased on the eigenvalues λ1, λ2, λ3 (in descreasingorder) and the associated eigenvectors e1, e2, e3 ofthe tensor. A second order symmetric tensor maybe graphically represented as an ellipse in 2-D or

an ellipsoid in 3-D. The eigenvalues describe thegeneral size and shape of the ellipsoid and theeigenvectors describe its principal directions. Be-cause of the properties of second order symmetrictensors, the eigenvalues are real and positive, orzero, and the eigenvectors form an orthonormalbasis. The tensor can be decomposed into threecomponents defined by

T = (λ1 − λ2)e1e1T + (1)

(λ2 − λ3)(e1e1T + e2e2

T ) +λ3(e1e1

T + e2e2T + e3e3

T ).

The first term corresponds to a 3-D stick tensor,which implies a surface patch with normal e1. Thesecond term corresponds to a plate tensor and im-plies a curve or surface intersection with tangente3 that is perpendicular to the plane defined by e1

and e2. The last term corresponds to a 3-D balltensor and implies a structure with no orientationpreference.

2.2 Tensor Communication

The voting process is similar to convolution with thedifference that convolution produces scalar valuesand tensor voting produces tensors. Voting kernelswhich encode certain constraints such as smooth-ness and proximity are used. These voting kernelsare continuous tensor fields which assign a value toevery point within the domain. Any voting ker-nel, regardless of dimension, can be derived fromthe 2-D stick tensor, which is therefore referred toas the fundamental 2-D stick voting field (VF) andis presented in Figure 1 (left). For its derivation,see [MLT00] and [TMMS01]. Given an input pointand normal at O, the most likely normal at P toa curve passing through O and P is defined by theosculating circle connecting O and P , because itkeeps curvature constant. The 3-D VF is producedby rotating the 2-D VF around N .

The magnitude of the field is described by a decayfunction, which is expressed in spherical coordinatesin Equation 2 and is presented in Figure 1 (right),

DF (γ, ϕ, σ) = e( γ2+cϕ2

σ2 ). (2)

γ is the arc length of the curve OP , ϕ the curva-ture, c the curvature scale factor and σ the scaleof analysis. c provides additional control on the in-fluence of curvature. σ determines the size of thevoting neighborhood and is the only free parameteradjusted by a user. In practice the range of influ-ence of a particular VF is eliminated once its mag-nitude drops below some value, e.g. 1%. Thereforethe value of σ determines the extent of the votingfield.

2

. �2 corresponds to 2D junction saliency, with totaluncertainty in orientation as indicated by the balltensor, or �e1e

T1 � e2e

T2 �.

In N-D, we have similar geometric interpretation: The

eigenvectors effectively encode orientation (un)certainties:

A normal to a hypersurface is described by the stick tensor,

which indicates certainty along a single, N-D direction.

Orientation uncertainty is indicated by the ball tensor, where

many intersecting hypersurfaces are present and, thus, no

single orientation is preferred. The eigenvalues encode the

magnitudes of orientation (un)certainties:

. ��1 ÿ �2� corresponds to N-D hypersurface saliency,with e1e

T1 indicating the normal direction,

. for 2 � i < N , ��i ÿ �i�1�, andPi

j�1 ejeTj correspond

to orientation uncertainty in i directions, whichactually defines a �N ÿ i�-D feature whose direc-tion(s) are given by ei�1; � � � ; eN . For example, givenN � 3; i � 2, ��2 ÿ �3��e1e

T1 � e2e

T2 � defines a plate

tensor in 3D, which describes a 1D feature, a curveelement, with tangent direction given by e3. Here,the normal orientation uncertainty only spans a planeperpendicular to e3, indicated by a plate tensordefined as

P2j�1 eje

Tj .

. �NPn

j�1 ejeTj corresponds to N-D hyperjunction sal-

iency, with the N-D ball tensor specifying totalorientation uncertainty.

2.2.3 Uniform Encoding

Therefore, given an N-D point, with or without orientation,

we can unify the input into a tensor field by the following: If

the input token is a point without any directional information,

it is encoded as a N-D ball tensor (�1 � �2 � � � � � �N � 1)

since initially there is no preferred orientation. �e1 e2 � � � eN �Tis an N �N identity matrix.

If the input token is a curve element in N-D, it is encoded

as a plate tensor (�1 � �2 � � � � � �Nÿ1 � 1; �N � 0), with eNequal to the direction of the curve tangent. Other plates are

encoded accordingly.If the input tokenis aN-Dhypersurface patch element, then

it is encoded as a stick tensor (�1 � 1; �2 � �3 � � � � � �N � 0),

with e1 equal to the direction of the surface normal to the given

hypersurface patch.In implementation, we first encode the input uniformly

into a tensor field. Each input tensor token then commu-

nicates, by a voting algorithm, in order to obtain a generic

tensor, which describes the orientation preference (or no

preference) at that site.

2.3 N-D Tensor Communication: N-D Tensor Voting

Each input token votes, or is made to align, with

precomputed, discrete versions of the basis tensors in a

convolution-like way, propagating preferred direction in a

neighborhood. We call these precomputed basis tensors

voting fields. As a result, preferred orientation information is

propagated and gathered at each input site. This voting

process consists of two phases:

. token refinement (ªtensorizationº): tensor votes arecollected at input sites only and

. dense extrapolation (ªdensificationº): tensor votesfill the volume for feature extraction.

2.3.1 Token Refinement and Dense Extrapolation

Given a set of input tokens, they are first encoded as tensors as

described in Section 2.2.3. These initial tensors communicate

with each other by token refinement and dense extrapolation.In essence, in token refinement, each token collects all the

tensor values cast at its location by all the other tokens in a

neighborhood. The resulting tensor value is the tensor sum

of all the tensor votes cast at the token location. In dense

extrapolation, each token is first decomposed into its

corresponding N elements. By using an appropriate voting

field, each token broadcasts the information in a neighbor-

hood. The size of the neighborhood is given by the size of

the voting field used. As a result, a tensor value is put at

every location in the neighborhood. The resulting dense

information can be used for feature extraction in which first

derivatives are computed.In fact, these two tasks can be implemented by a voting

process, which in essence involves having each input tokenaligned with precomputed, dense, N-D voting fields. Thealignment is simply a translation followed by rotation in theN-D space.

Therefore, it remains to describe the N-D voting fields,which can be derived from the most basic, fundamental2D stick voting field.

2.3.2 Derivation of N-D Voting Fields

The fundamental 2D stick voting field. Voting fields of anydimensions can be derived from the 2D stick tensor and,therefore, it is called the fundamental 2D stick voting field.Fig. 5 shows this fundamental 2D stick voting field.

In 2D, a direction can be defined by either the tangentvector, or the normal vector, which are orthogonal to eachother. We can therefore define two equivalent fundamentalfields, depending on whether we assign a tangent or normalvector at the receiving site.

Here, we describe the normal version of the 2D stickkernel. The tangent version is similar. Given a point at theorigin with a known normal (N), we ask the followingquestion: For a given point P in space, what is the most likelynormal (at P ) to a curve passing through O and P and normal toN? Fig. 6 illustrates this situation. We claim that theosculating circle connecting O and P is the most likely onesince it keeps the curvature constant along the hypothesizedcircular arc. For a detailed theoretical treatment, please referto [12]. The most likely normal is given by the normal to thecircular arc at P . The length of the normal vector at P , whichrepresents the saliency of the vote, is inversely proportional

834 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 23, NO. 8, AUGUST 2001

Fig. 6. The design of fundamental 2D stick voting field.

equivalent 2 � 2 eigensystem, with its two unit eigenvectors

e1 and e2 and the two corresponding eigenvalues �1 � �2:

��1 ÿ �2�S� �2B; �3�where S � e1e

T1 defines a stick tensor, and B � e1e

T1 � e2e

T2

defines a ball tensor, in 2D. Note that these tensors define the

two basis tensors for any 2D ellipse.Analogously in N-D, in hypersurface detection, a point in

the N-D space can either be: on hypersurface (smooth), at a

discontinuity of order between two and N , or is an outlier.

Consider the two extremes: an N-D point on a hypersurface

is very certain about its normal direction, whereas a point at

a junction has absolute orientation uncertainty. As in 2D,

this whole continuum can be abstracted as a second order

symmetric N-D tensor, or equivalently, a hyperellipsoid. This

hyperellipsoid can be equivalently described by the

corresponding eigensystem with its N eigenvectors

e1; e2; � � � ; eN and the N corresponding eigenvalues,

�1 � �2 � � � � � �N . Rearranging the N �N eigensystem,

the N-D ellipsoid is given by:

��1 ÿ �2�S�XNÿ1

i�2

��i ÿ �i�1�Xij�1

ejeTj � �NB: �4�

In particular, S � e1eT1 and B �PN

i�1 eieTi defines an

N-D stick and ball, respectively, among all the N basistensors. We call the rest of N ÿ 2 basis tensors

Pij�1 eje

Tj

plate tensors. Any hyperellipsoid in N-D can be representedby a linear combination of these N basis tensors.

2.2.2 N-D Tensor Interpretation

We now explain the geometric meaning of the eigensystem

we derived in the previous section. Return to the 2D case.

The eigenvectors encode orientation (un)certainties: Tangent

direction is described by the stick tensor, indicating certainty

along a single direction. At point junctions, where more than

two intersecting lines converge, a ball tensor is used since

there is no preferred orientation. The eigenvalues encode the

magnitude of orientation (un)certainties since they indicate

the size of the ellipse. Hence, given a generic ellipse and its

equivalent eigensystem, we have the following geometric

interpretation:

. ��1 ÿ �2� corresponds to 2D curve saliency, with astick tensor e1e

T1 indicating the curve tangent

direction,

TANG ET AL.: N-DIMENSIONAL TENSOR VOTING AND APPLICATION TO EPIPOLAR GEOMETRY ESTIMATION 833

Fig. 4. A second order symmetric 2D tensor.

Fig. 5. The fundamental 2D stick voting field.

Figure 1: (left) The design of fundamental 2D stick voting field. (right) Magnitude (saliency) of the funda-mental 2D stick voting field. From [TMMS01]

The field has an analogy with particle physics,motivating the choice of a Gaussian function forthe energy decay function. The voting kernel showsthe same behavior as an anisotropic energy field,where an emitting particle Ps, radiates energy inall directions and a receiving particle Pr receivesenergy from all directions, but with a preferencefor a straight path. Therefore received energy is in-versely proportional to distance traveled and curva-ture, producing a loss of energy along curved paths.

After voting, the saliency tensor T(i, j, k) at eachlocation (i, j, k) is the tensor sum of the contribu-tions from each voting token within range,

T(i, j, k) =∑

m,n,o

V (S(m,n, o), ((i, j, k)− (m,n, o)))

(m,n, o) ∈ N(i, j, k), (3)

where N(i, j, k) returns the neighboring locations of(i, j, k) and V is a function that returns a tensorfield corresponding to the ball, stick and surfacevoting kernel components of S at (m,n, o) using thevector (i, j, k)− (m,n, o), which is expressed in the(e1, e2, e3) coordinate system of S.

2.3 Feature Extraction

After the voting process, each point in the domainhas been assigned a second order symmetric tensorthat estimates the structure of the feature type(s)and the associated saliency. Thus, information iscollected at each location in the domain, buildinga saliency map for each feature type. Salient fea-tures can be located by finding local extrema in thisdense tensor map, which in 3-D is decomposed intothree dense vector maps, expressing the occuranceof junctions, curves and surfaces in the domain. Foreach feature type, the dense tensor map can bebroken into a scalar field, expressing the saliency(strength) of the tensor field at any given point,and a vector field, expressing the direction of thefeature corresponding to the saliency value. Thus,at any given location there is a scalar and a unitvector < s, n > present. These dense saliency andvector maps are then used for vote interpretation.

Since we are only interested in the surface featuresproduced by tensor voting, we only examine the theSurface Map < s, n > extracted from the tensorfield. For this case,

s = (λ1 − λ2) n = e1, (4)

where s is the surface saliency and n is the normalto the most likely surface at each point in space.

The most likely surface produced by a set of inputtokens (points and normals) is an extremal surfaceembedded in the Surface Map. A point is on theextremal surface when the saliency s is extremal inthe normal direction n at that point, i.e. ds/dn =0. Reformulating this equation, the surface maybe found by identifying zeros of the the scalar fieldn · ∇s, defined as g field. For practical reasons theSurface Map is calculated on a regular grid and thezero crossings of the extremal surface are identifiedalong grid lines.

3 TENSOR VOTINGMODELING SYSTEM

The tensor voting modeling system, TVMS, is basedon the TV3D framework for tensor voting developedby the USC Computer Vision Group1, and uses QS-plat2 [RL00] for point-based visualization. It hasbeen developed on a MacBook Pro with a 2.2 GHzIntel Core 2 Duo processor. Functionality has beenadded to both frameworks and only surface votingis used from the TV3D framework. Since our goalis to extract a surface from a given input, we haveomited the point and curve voting functionality ofTV3D.

TV3D uses an input of oriented or unorientedpoints, curvels and surfels. In TVMS, the in-put is restricted to points with associated normals.While a surface may be constructed from unorientedpoints only, we have chosen to include direction in-formation in the input data set. The reason for this

1 http://www.cs.ust.hk/∼cstws/research/TensorVoting3D2 http://graphics.stanford.edu/software/qsplat

3

Figure 2: Translating the right-most token one unit at a time. Left: σ = 7. Right: σ = 10.

is that it gives more accurate results, allows us toomit the first voting step, which estimates and as-signs a direction to all unoriented input tokens, andfinally, it provides the user with an additional toolfor manipulating the resulting surface.

These points can be translated and their corre-sponding normals rotated, as well as added to andremoved from the scene. Translation is done byfirst rotating the normal of a selected point andthen translating the point along the normal. Thisis a simple yet intuitive interaction that meets ourneeds. The user can either create an entirely newmodel on his or her own, or read an existing modelfrom a file. A new point is inserted by clicking be-tween two existing points; the new point is insertedat the position halfway between the two points clos-est to the user’s mouse click. Each input token hasa position, a normal, a weight and a local votingneighborhood size, which can be modified by theuser.

Once the user is satisfied with the initial configu-ration of input tokens, (s)he selects an initial globalsigma value and runs the tensor voting process toextract a surface. The surface is quickly renderedas points by default, but, given more time, the usercan also render the surface as a mesh.

3.1 Modeling Parameters

The original TV framework only has one free pa-rameter, namely σ, which is a coefficient in thesaliency decay function and sets the scale of analy-sis. σ is a positive scalar which determines the sizeof the voting fields used in the voting process andis set with an initial global value. Having a singleglobal σ value will create a model with a one levelof detail. In TVMS, σ has been divided into twoparameters, a local σ, σL which is assigned individ-ually for each token and a global σ, σG, which isassigned to all tokens which have not had their lo-cal sigma value changed. By default, σL is set to -1so that the system can distinguish modified tokens

from unmodified. Instead of a single global votingfield, up to n, where n is the number of input to-kens, individual voting fields can be used to vote fora surface. This allow for the level of detail of thesurface to be varied by using a smaller σL in thoseregions where the user desires finer details.

A weight has been added as a third modeling pa-rameter. The weight is a scaler value associatedwith each input token. The token’s normal is mul-tiplied by the weight before voting and surface ex-traction. Increasing the weight value strengthensthe influence of the plane defined by the token; thusflattening the resulting surface near the token. Thedefault value is set to 1.0.

4 RESULTS

In order to investigate the geometric modeling po-tential of tensor voting we conducted a number ofexperiments. Our experiments were performed intwo parts. In the first part, we looked at the lo-cal influence of the individual modeling parameterson a simple shape. In the second part, we exam-ined overall effectiveness of the TV modeling pro-cess when attempting to model two specific shapes.

4.1 Influence of ModelingParameters

The scale of analysis parameter σ affects the radiusof the model and the cost of calculating a surface.Because we wish to extract the surfaces as quicklyas possible, the size of the voting fields, and thusthe size of the models, have to be kept small. Wefound that the radius of our models, in units of vox-els, should be approximately the same as σ. Hav-ing a value of σ between 8 and 10 produced resultsin a reasonable amount of time, which implied amodel radius of slightly more than 10 units. Usinglarger σ values would require bigger models, thusa higher resolution computational grid, and would

4

Figure 3: Left: The 18 input tokens needed to define a sphere. The top and bottom tokens have beenhighlighted and the extent of their voting fields (σG = 10) have been displayed. Center: The resulting sphere.Right: The extent of the voting fields for the top and bottom points, after point translation. The two fieldsno longer overlap and a disjoint surface has been formed.

Figure 4: Rotating the normal of the right-most point and voting with σG = 10. Starting at the top left, thenormal is rotated 0◦, 15◦, 30◦, 45◦, 60◦ and 75◦.

require significantly more time in the surface extrac-tion step. Also, σ should not be set too large, sincethis causes votes to cancel each other out [MLT00]and produce surface holes, further motivating theuse of small σ values.

The model used in our parameter influence exper-iments is a sphere with a radius of 12.5 units, thatwas defined by 18 input tokens. See Figure 3 (left).

To demonstrate the effect of translating input to-kens, we translate one single token on the sphere,namely the right-most one, one unit at a time. Theresults are presented in Figure 2, which consists ofspheres produced with σ = 7 and σ = 10. It canbe seen that the surface does begin to bulge out tofit to the surface patch implied by the translatedtoken. But if the token is translated too far thesurface breaks up.

Insight into this unwanted artefact is provided inFigure 3. Here, the extents of two of the sphere’s 18tokens are displayed. As the top token is translatedaway from the others the two displayed extents no

longer overlap. In this situation two disjoint sur-faces are produced. The TV process interprets thetranslated token as an isolated point and tries toinfer a separate plane from it. This clearly showsthat in order to produce a closed mesh, one needsto ensure that there is significant overlap betweenthe voting fields of neighboring tokens.

Figure 2 also highlights the effect of σ on the sur-face. On the left, with σ = 7, the resulting surfacehas flatter, somewhat sharper, features. When σ isincreased to 10, more voting fields overlap becausethe are larger. The increased number of contribut-ing VF’s at any location creates a smoothing effecton the surface. This is most evident in the top-rightand bottom-left surfaces in each of the examples.As the right-most token is translated, the resultingbump has a smoother transition to the sphere whenσ is greater.

To demonstrate the effect of rotating normals,Figure 4 presents the changes to the sphere pro-duced by rotating the normal of the right-most to-

5

Figure 5: To the left: Incrementing the weight while voting with σG = 10. Weight values, from left to right,are 1.0, 1.5, 2.0, 2.5, 3.0 and 3.5. To the right: Incrementing the weight while voting with σG = 7. Weightvalues, from left to right, are 1.0, 2.0, 3.0, 4.0, 5.0 and 6.0.

.

Figure 6: The bottle model created from 55 tokens. Left: Input tokens. Center: Resulting TV surfacedisplayed as points. Right: TV surface displayed as a mesh.

ken. It can be seen that applying small rotations(angles less than 45◦) to a token can be easily ac-commodated by the tensor modeling system. Thesurface smoothly blends in the disoriented surfacepatch implied by the rotated token. As the rotationangles increases past 45◦ the surface breaks up. Us-ing a higher σ gives slightly better results, but thisstill cannot resolve the conflicting input informationthat implies that a piece of surface is directed intothe sphere.

The weight parameter provides another way tomodify the influence of a token. The weight param-eter may either enlongate or shorten the length ofthe normal n associated with a token after the vot-ing process. Given a large weight the plane associ-ated with the token’s normal has greater influenceon the surface; thus locally attracting and flatteningthe surface, as seen in Figure 5. Tokens with smallweights lose some of their influence and nearby re-gions are smoothed by the other tokens. Also, asa weight is lowered, the nearby surface region be-comes less sensitive to changes in the weight. This

is probably due to the increased influence of neigh-boring tokens.

4.2 Creating Complete Models

To explore the overall tensor voting modeling pro-cess, two complete models were created, a bottleand a car. Both are fairly simple objects, but havegeometric properties, such as varying curvature andlevel of detail, that potentially could reveal anyweaknesses or problems with our TV-modeling sys-tem. Both models were created by initially defin-ing the tokens needed to produce the basic shape.The tokens were then modified to add details or fixproblem. To change the curvature of the surfaces, anumber of normals were rotated. To flatten regionsof the surfaces, weights of nearby tokens were in-creased. To fill holes, new tokens were added or σL

of the neighboring tokens were increased. For thecar, the body was first created. The four wheelswere then added as details later.

The bottle model, defined with 55 tokens, is pre-sented in Figure 6. The TV3D system evaluated its

6

Figure 7: The car model created from 149 tokens. Left: Input tokens. Center: Resulting TV surface displayedas points. Right: TV surface displayed as a mesh.

Surface Map on X x X x X grid in FOO cpu-seconds.The resulting surface may be displayed with points.These are the zero-crossings of the computationalgrid used to calculate the associated g field. TheMarching Cubes algorithm [LC87] may also be ap-plied to the g field grid to produce a mesh. Thecar model, defined with 149 tokens, is presented inFigure 7. The TV3D system evaluated its SurfaceMap on X x X x X grid in FOO cpu-seconds. Theresulting surface is displayed as points and a mesh.

5 DISCUSSION

During the modeling process, a significant prob-lem quickly emerged. Small, densely sampled areas,may produce asymmetric results, which we believeis due to the TV3D surface extraction implemen-tation. The algorithm begins at the most salientinput token and grows a surface from it until thesaliency of the neighborhood drops below a certainthreshold. It then goes to the second most salientinput point and so on. For the bottle, the asymme-try can be seen on the neck, which is not perfectlysymmetric. When modeling the car, the asymme-try problem imposed serious problems shaping themore finely detailed areas. After creating the body,one wheel at a time was added. When the secondwheel was added, its points influenced the pointsof the first wheel, changing its shape. We changedour strategy and modeled a full half of the car inone go, which was then duplicated, mirrored andmerged with the first half. The resulting mesh wasnot closed and after repairing one hole, new holesemerged on other parts of the model. The surfaceextraction of TV3D appears to propagate the influ-ence of the tokens forward as they are processed;thus the order in which tokens are processed affectsthe final extracted surface. The current TV3D im-plementation is elegant, but may not be the correct

computational engine for an interactive modelingapplication.

The local influence of an individual input tokenon the final surface is not as strong nor with thetype of properties as had been desired. Changingthe parameters of a particular input token will inmost cases render a change in the extracted surface,but only if the parameters remain within a certainrange. For instance, if a token is translated far awayfrom the other tokens, it will lose its influence onits neighbors and will produce a surface separatedfrom the rest of the model. Also, the influence ofa single token token decreases exponentially withits distance. As a token is moved away from theremaining tokens, the influence of its neighbors in-creases and they smooth out the contribution of themoved token. Determing these acceptable param-eter ranges and including them as constraints willbe a neccessary part of a future TVMS.

Surface extraction and voting is costly. Each timethe model is modified and the surface must be up-dated, TVMS recalculates the entire surface insteadof just the part that will be affected by the change.If the system could re-vote only the modified inputtokens and update just the affected areas of the sur-face, surface regeneration could possibly be done inreal-time. This will require a redesign of the surfaceextraction algorithm.

6 CONCLUSION

This work has examined the tensor voting method,a technique for geometric feature grouping, in a3D-modeling context. The new system, which em-ploys the TV3D framework for tensor voting andthe point-rendering functionality of QSplat for visu-alization, extracts surfaces from sparse input data,consisting of points and normals. These pointsand normals can be interactively edited by theuser to change the resulting surface. The initial

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goal of modeling simple objects using TV has beenachieved and two new modeling parameters, a localσ, σL, and a weight w, have been introduced andimplemented.

Though the approach is promising, several limita-tions have been found in our current system. Thereis an asymmetry in the extracted surfaces, proba-bly due to the surface extraction implementationin TV3D. Modeling parameter values may be easilyset to produce unstable and unwanted results. Vot-ing and surface extraction is slow for large modelsand large scales of analysis. The curve and junc-tion capabilities of tensor voting should be exploredfor modeling sharp features and fine details. Eachof these limitations should be addressed for futureversions of a TV-based modeling system.

Acknowledgements We would like to thankWai-Shun Tong who provided us with the tensorvoting library and much technical assistance, andChristoffer Westberg for programming our initialpoint rendering software.

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