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Volume xx (200y), Number z, pp. 1–12

Prototype Modeling from Sketched Silhouettes based onConvolution Surfaces

Chiew-Lan Tai1, Hongxin Zhang 2,1 †, Chun-Kin Fong 1

1 Department of Computer Science, Hong Kong University of Science & Technology, Kowloon, Hong Kong2 State Key Laboratory of CAD&CG, Zhejiang University, P.R. China.

AbstractThis paper presents a hybrid method for creating three-dimensional shapes by sketching silhouette curves. Given asilhouette curve, we approximate its medial axis as a set of line segments, and convolve a linearly weighted kernelalong each segment. By summing the fields of all segments, an analytical convolution surface is obtained. Theresulting generic shape has circular cross-section, but can be conveniently modified via sketched profile or shapeparameters of a spatial transform. New components can be similarly designed by sketching on different projectionplanes. The convolution surface model lends itself to smooth merging between the overlapping components. Ourmethod overcomes several limitations of previous sketched-based systems, including designing objects of arbitrarygenus, objects with semi-sharp features, and the ability to easily generate variants of shapes.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometryand Object Modeling; I.3.6 [Computer Graphics]: Interaction techniques

1. Introduction

In this paper, we present a method for designing prototypemodels from sketched silhouette curves based on a new en-hanced convolution surface model. Prototype design aims atfast and intuitive creation of 3D shapes. In the conceptual de-sign stage, prototype is designed to test and consolidate ideasbefore actual models are created for computer graphics andCAD applications. Existing 3D design software packagesmainly provide powerful precise shape controls, but they arenot always suitable for conceptual design.

Despite today’s highly computerized world, design artistsstill love to express their ideas and imagination on paper be-cause the principle of drawing is simple and intuitive. Thepopularity of hand-held devices, such as PDA and TabletPC, also contributes to the shift in the user interface pref-erence. So in recent years, there have been many efforts indeveloping more intuitive interfaces for interactive design,facilitating conceptual stage design11, 12, 15, 16, 19, 20, 22, 27, 33.

† This author is partially supported by the National Natural ScienceFoundations (No. 60021201, 60203014) and the National Key BasicResearch and Development Program (No. 2002CB312102)

Figure 1: A mouse shape designed using our method andrendered by POVRAY.

A popular strategy for providing intuitive user interfaceis to accept input of freeform sketches, drawn by the userusing a mouse or a pen-like device on a graphic tablet. For

submitted to COMPUTER GRAPHICS Forum (12/2003).

2 C.L. Tai et al. / Prototype Modeling from Sketched Silhouettes

instance, Teddy is one such system developed for designingfreeform polygonal shapes16.

Inspired by these previous systems, we developed a sys-tem, named ConvMo, for designing prototype by generat-ing convolution surfaces from silhouette curves. Our goal isto perform free-form implicit surface design by sketchingcurves; but, unlike Teddy system, we use menus and slid-ers as well. To enable more versatile design, we propose anew enhanced convolution model that allows users to designthe cross-section shape by either sketching a closed profilecurve or specifying shape parameter values. By incorporat-ing the parametric curve formulation into the implicit model,our hybrid approach can easily produce interesting variantsof shapes. Compared with the Teddy system, our method al-lows the user to more freely design objects with high topo-logical complexity due to the nature of implicit surfaces.And our system supports the following additional features:shapes with holes, semi-sharp features, and free-form cross-section shapes. The proposed method is very useful for en-tertainment graphics applications, such as fast modeling andanimation in cartoon styles.

Our main contributions are

• a method for creating convolution surface models fromsilhouette curves—it extracts a skeleton comprising ofline-segments and convolves the skeleton with polyno-mial weight distribution;

• an enhanced convolution surface model that supportscross-section design.

2. Related Work

The term sketch methods usually refer to methods that at-tempt to infer or retrieve 3D information from strokes drawnon 2D interfaces. The most direct and popular goal is to re-construct 3D shapes, however sketch methods have also beenadopted to solve other problems. For example, a system byTolba et al.30 is a tool for perspective drawing. It projects2D strokes on the unit sphere centered at the eye point andthen views them in perspective. Bourguignon et al.5 presentan approach that directly uses the user-drawn strokes to in-fer the sketches representing the same scene from differentviewpoints. These methods focus on illustrating 3D effectsrather than actual 3D shapes, which is our area of interest.

Earlier sketch systems aimed at fast design of mechanicalparts. They usually interpret the user’s 2D sketches into ex-act geometric elements (e.g., lines, circular arcs, or B-splinecurves), generate geometric constraints (such as right angles,tangencies, symmetry, and parallelism) possibly aided by theuser’s explicit selection11, 26, and attempt to recover missingdepth information14.

Most of the recent freeform sketch methods can beroughly divided into two categories according to the ob-ject representation: boundary-based and volume-based ap-proaches.

The Teddy16 system is a typical boundary-based method.From a sketched silhouette, a polygonal mesh is generated,which can then be rotated using a virtual track-ball interface,and interactively edited using extrusion, bending, cutting op-erations, all controlled by strokes. Its innovative interface in-spired many later sketch systems. However, typical of polyg-onal representation, it has limitations on surface smoothnessand fairness, as well as design of structures with complextopology.

Michalik et al.22 proposed a sketch- and constraint-basedapproach by using standard parametric curves and surfaces.They introduced auxiliary surfaces that allow for a reliableinterpretation of the user’s pen-strokes and achieved the goalof B-spline surface sculpting. Parametric representations aremore compact and have higher degree of smoothness thanpolygonal meshes. However, it is well-known that paramet-ric surfaces are cumbersome for representing objects witharbitrarily topology, and surface trimming has high compu-tation price. These drawbacks limit the modeling ability ofparametric surface approaches.

For the purpose of sculpture modeling, volume-basedmethods are more natural. The approach of Owada et al.24

employed a similar interface as the Teddy system but usedvolumetric representation. Volumetric representations relaxthe topological limitation of boundary representations, butthey are not as compact; moreover, achieving smooth resultrequires both higher storage and computation price.

Our implicit surface-based approach can be viewed as amixture of surface and volumetric approaches. Implicit sur-face representations are robust and compact for designingtopologically complex objects. An implicit surface S can bedefined in the following general form:

S = x|n

∑i=1

Fi(x)−T = 0

where Fi are the primitive field functions, and T is the thresh-old value of the iso-surface.

Convolution surface is a type of implicit surface obtainedby convolving a kernel along a skeleton, composed of points,lines and polygons. Generally, convolution surface methodsproduce rotund shapes due to the nature of the kernel func-tions. Grimm proposed an approach that can sweep out im-plicit surfaces of different cross sectional shapes13, but theresulting surface is a variation of distance function surfaces.With the same objective of providing more design freedomin mind, we propose an extended convolution surface modelby incorporating a parametric component. Our approach im-plicitly controls the cross-section via space transform, sup-ported by an intuitive user interface. The output is compati-ble to the original convolution surface model.

More recently, implicit surfaces based on radial basisfunction (RBF) are widely used in point cloud data interpo-lation and approximation problems6, 31. These methods are


C.L. Tai et al. / Prototype Modeling from Sketched Silhouettes 3

(a) silhouette &skeleton

(b) convolutionsurface

(c) add component

(d) modify crosssection of ear

(e) soft carving (f) dog shape

Figure 2: Method overview

more efficient and produce better quality shapes. However,due to their isotropic property, they are inflexible for creatingspecial feature shapes, such as darts and sharp edges. Dinhet al.10 proposed a surface reconstruction method by usinganisotropic basis functions, but their approach does not solvethe feature representing problem completely.

Free-form design with sketching interface based on im-plicit surfaces has been investigated by other researchers aswell. Karpenko et al.19 proposed a system based on vari-ational implicit surfaces method. They presented an ele-gant way to obtain 3D constraint points required by theRBF interpolating method. The RBF method, however, hashigh computation cost, both in solving the linear systemof constraints and in the evaluation step. The authors ofTeddy proposed an enhancement to their well-received sys-tem based on quadratic surface fitting15. This two-step ap-proach can obtain polygonal meshes of higher fairness. Butsuch a hybrid representation requires saving extra intermedi-ate meshes. In contrast to these approaches, our convolutionsurface method has a compact representation, avoids solv-ing a large-scale global constrained problem and computesthe surface parameters in a more natural way.

3. Method Overview

The main steps of our convolution surface-based prototypedesign system are as follows (Figure 2):

1. The user draws a silhouette curve on a projection planeand the system automatically extracts the skeleton struc-ture consisting of line segments (Section 5.1).

2. The system determines the parameters for each skeletalline segment, and generates a rotund generic convolutionsurface model that fits the silhouette curve (Section 5.2).The system outputs a polygonal mesh using the marchingtetrahedra polygonization algorithm2.

3. The user modifies the circular cross-section of the generic

shape by either sketching a cross-section profile or byspecifying shape parameter values provided by our en-hanced convolution surface model (Section 4.2).

4. The user may perform further component-based editingoperations, such as soft carving and hole making (Section6).

5. The user rotates the partially designed shape to obtain anew projection plane, and repeats steps 1-4 to design an-other surface component, which can be merged smoothlywith, or simply attached to, an existing component.

Furthermore, steps 3 and 4 can be repeated on any com-ponent at any time.

4. Convolution Surface Model

Convolution surfaces were introduced by Bloomenthal andShoemake4 as a natural extension to point-based field sur-faces to include higher-dimensional geometric elements.

A modeling skeleton set S ⊂ IR3 can be represented as afield function g : IR3 7→ IR, defined as follows:

g(p) > 0, p ∈ skeleton S;g(p) = 0, otherwise;

(1)

where p is a point in IR3 Euclidean space. Let f : IR3 7→ IRbe a kernel function representing the field generated by asingle point in the skeleton, and x be a point in the skeletonS, then the total field contributed by the skeleton at a point pis the convolution of the functions f and g as follows:

F(p) = ( f ⊗g)(p) :=Z

Sg(x) f (p−x)dx (2)

A convolution surface with threshold T is then defined by

S = p|F(p)−T = 0,p ∈ IR3 (3)

Although Bloomenthal and Shoemake demonstrated theirresults using a cubic approximation to a Gaussian, we utilizethe Cauchy kernel and its closed-form solution as presentedby McCormack and Sherstyuk21, 28

f (p) =1

(1 + s2r2)2 (4)

where r = ‖p‖ and s is a parameter for controlling the kernelwidth. This allows us to develop an analytical polynomial-weighted model in the next section; the model can, however,be used with any kernel.

The model proposed by McCormack and Sherstyuk isof uniform distribution, i.e., the resulting surface has con-stant radius of influence along the skeletal element. To fa-cilitate the design of generalized cylindrical shapes withfewer skeletal elements, Jin et al.17, 18 proposed a polynomialweighted model and derived analytical solutions for line seg-ments, arcs, and quadratic curves.

For efficiency and robustness, we choose line segment asthe skeleton element in our system. To reduce the number



of segments required, which is critical in interactive design,we adopt the polynomial-weighted convolution model. Thismodel is briefly reviewed in the next subsection, and an en-hanced model is presented in Section 4.2 for more versatilecross-sectional design.

4.1. Polynomial-weighted convolution model

Let the skeleton S be an arc-length parameterized curvedsegment c(t) (0 ≤ t ≤ l). A convolution surface with a poly-nomial weight w(t) distribution can be defined as

F(p;c(·),w(·)) =Z l

0w(t) f (p− c(t))dt. (5)

For a line segment L(t) = b + t~a, 0 ≤ t ≤ l, the squareddistance from a point p to the line L(t) is given by r2(t) =‖b + t~a−p‖2 = d2 + t2 − 2ht, where ~d = p−b, d = ‖~d‖,and h =~d ·~a. Thus, for Cauchy kernel, equation (5) becomes

F(p;L,w) =

Z l

0

w(t)(1 +(d2 + t2 −2ht)s2)2 dt. (6)

In particular, we use linear weight distribution: varying fromw0 to w1 along the line segment, the field function is

F(p) = w0F1(p)+w1 −w0

lFt(p), (7)

where

F1(p) := F(p;L,1) = 12p2 [ h

s2h2+p2 + l−hs2(l−h)2+p2 ]

+ 12sp3 [tan−1( sh

p )+ tan−1(s(l−h)

p )],(8)

and

Ft(p) := F(p;L, t) =1

2s2 [ 1s2h2+p2 −

1s2(l−h)2+p2 ]+ hF1(p)

(9)

with p being a distance term p2 = 1 + s2(d2 −h2).

4.2. Enhanced convolution model

The polynomial-weighted model still produces only circularcross-section shapes. Since many objects in the real worldhave anisotropic shapes, it motivates us to enhance the orig-inal convolution model to achieve more versatile design.

We refer to the initial polynomial-weighted convolutionsurface with circular cross-section as the generic surface.The user designs a target surface by modifying the genericsurface. Thus the main issue is how to define a mappingfrom the field function of the generic surface to the fieldfunction of the target surface M : Fgeneric 7→ Ftarget . Forthe purpose of interactive design, the mapping procedureshould be intuitive and have low response time. Based onthe idea of spatial deformation, we assume that the mappingbetween the two field functions is effected by a transformT : Ωgeneric 7→ Ωtarget , where Ω∗ is the domain of function

F∗. That is, if Fgeneric and the corresponding T transformfunction of a line segment are given, then

Ftarget(·) = M(Fgeneric) = Fgeneric(T−1(·)). (10)

To evaluate a target surface, the implicit function evaluatorfirst transforms a point back to the generic surface space andthen evaluates the original convolution surface model to ob-tain a field value for the target surface.

To find a suitable representation for the transform T , wedefine a local coordinate system for each line segment l asshown in Figure 3. Let the y-axis be normal to the referenceplane supporting the silhouette curve, the z-axis be alongthe line segment direction, and the x-axis be in the directionl× n. Without losing generality, let the origin be the startpoint of the line segment. We can then decompose a spatialtransformation into three scalar mappings along these axisdirections

T = Tx · Ty · Tz, (11)

where Tx, Ty and Tz are transform functions defined on they-z plane, z-x plane, and x-y plane, respectively. In general,they can be curves defined in the B-spline form for achievingcomplex design; but for simplicity of interface, we adopt thefollowing decomposition:

T = Sx · Sy · Tz (12)

where the mapping Tz defines the cross-section shape per-pendicular to the silhouette supporting plane, and the map-pings Sx and Sy expand or contract the cross-section in the xand y directions, respectively. We define Tz in terms of polarcoordinate as follows:

Tz : (r,θ) 7→ (r′,θ′)

and define the scaling mappings as

Sx : x 7→ βxx Sy : y 7→ βyy.

Our system provides two ways for the user to specify Tz:sketch a closed profile curve or specify a parameter defininga super-quadric. The scaling factors βx and βy are usuallyequal to 1 in the first case, but are useful in the second case.

X

Y

Zn

Silhouette

SkeletonReference

Plane

Figure 3: Local coordinate system of a line segment

Class 1. Bezier curve model in polar coordinates. Weconsider the mapping image of the unit circle, i.e., R(θ) =Tz(1,θ). Then a class of transform functions Tz is given by

Tz : (r,θ) 7→ (r ·R(θ),θ), (13)



with the inverse of Tz easily found as follows:

T −1z : (r′,θ′) 7→ (r′/R(θ′),θ′). (14)

Our system allows the user to sketch R(θ) as the cross-section shape and automatically converts it to a Bezier curvein polar coordinates. The closed curve is constrained to be ofa star shape. For completeness, we include here some deriva-tions on polar-form Bezier curves. An n-degree polar-formBezier curve segment b(θ) with θs ≤ θ ≤ θe is defined as

b(θ) =n

∑i=0

riBn,i(θ−θs

θe −θs). (15)

where

Bn,i(t) =

(

ni

)

(1− t)n−it i, 0 ≤ t ≤ 1 (16)

are the Bernstein polynomials, ri are the control radii.Clearly, when r0 = · · · = rn = R0, the curve represents anarc of radius R0. A closed curve R(θ) (0 ≤ θ ≤ 2π) canbe constructed by stitching together several curve segments,with each segment bk(θ), θk ≤ θ≤ θk+1 (0≤ k < m) definedby the control radii rk,0, rk,1, . . ., rk,n. In our implementation,we let n = 3, m = 12, and θk = 2kπ/m. The system samplespoints on the sketched curve and fits the points with a closedpolar-form Bezier curve. By the endpoint interpolation prop-erty of Bezier curves, bk(θk) = rk,0 and bk(θk+1) = rk,n.To ensure that the segments connect up at their endpoints,rk,n = rk+1,0 (0 ≤ k < m) must be satisfied. The radii rk+1,0(0 ≤ k < m) are simply computed by averaging the nearesttwo sampling points at the angle θk, and the remaining rk, jare obtained by solving a least-squares problem (see Ap-pendix A). To achieve tangent continuity, the control radiican be modified according to the following sufficient condi-tion (see Appendix B for detail derivation)

rk,n = rk+1,0 =12[rk,n−1 + rk+1,1]. (17)

Class 2. Super-quadrics model. For simplicity of design-ing the commonly used elliptic and squarish cross-sectionshapes, our system provides an alternative way of specify-ing Tz in terms of super-quadrics. In parametric form, super-quadrics can be represented as

x = r sign(cosθ) |cosθ|1/α

y = r sign(sinθ) | sinθ|1/α (0 < θ < 2π)(18)

where sign(a) = 1 when a > 0, otherwise sign(a) =−1. Andα > 0 acts as a shape control parameter: when α = 1, we geta circle; when α < 1, the shape becomes star shaped withsharper corners; when α →∞, it converges to a square. Theadvantage of the super-quadrics model is that only one pa-rameter value has to be specified. The mapping Tz is thengiven by:

Tz : (r,θ) 7→ (r[|cosθ|α + | sinθ|α]−1/α,θ). (19)

Many objects have components that are of 2.5-dimension,

i.e., which can be defined by extruding a curve to someuniform thickness (Figure 4(b)). However, such componentsare not within the modeling domain of Equation 19 becausethe field function is not uniform along the line segment. Toachieve the extrusion effect along the y-axis direction, weintroduce the following extrusion mapping by modifying theprevious equation:

Tz : (√

(r′ cosθ′)2 + |r′ sinθ′|α,θ′) 7→ (r′,θ′). (20)

Here α ≥ 2 controls the stiffness of the shape just as in thecase of generic super-quadric. Since the term (r′ sinθ′)α actsas a soft threshold function at 1 along the y-axis, the result-ing shape has nearly uniform thickness of 2 for large valuesof α, say α = 50. Other extrusion thickness can be obtainedby combining the effect with the scaling mapping Sy.

Figure 4 shows an example of designing the tail of a car-toon style dinosaur. Sub-figures (b) and (c) use the super-quadrics model, and (d) illustrates the freeform cross-sectiondesign. In our experimental system, the shape parameters α,βx and βy are directly controlled through sliders.

5. Convolution Surface From Silhouette Curve

With our system ConvMo, the user designs a prototype shapeby incrementally adding new components via sketching sil-houette curves from different viewing directions. The systemgenerates a convolution surface from each silhouette, andthen allows the user to sculpt the surface into a more inter-esting shape by modifying its cross-section and performingother component-based editing.

To design an object, the user sketches the silhouette of acomponent as a simple curve with no self intersections inthe input canvas. The system then converts the curve into asimple polygon by sampling the input device movement. Toobtain stable input, filtering is required. Let W be the canvaswidth. We sample such that the length of each polygon edgeis at most µ0W , and at least µ1W if the edge forms an angle ofmore than 20 degree with the previous edge (we let µ0 = 0.05and µ1 = 0.01). If the polygon is self-intersecting, the useris requested to sketch another curve.

To generate a convolution surface from this simple poly-gon, two major issues have to be addressed:

• find a suitable skeleton with a small number of primitives;• determine the parameters of the field contribution of

skeletal primitives.

Briefly, our approach finds an approximate medial axis com-prising of line segments, and determines the weight at eachend of a line segment as a function of the segment length andthe distance to the silhouette polygon.

5.1. Skeleton extraction

There are many algorithms reported in the literature forfinding skeletons7, 23. A well known class of algorithms



(a) (b) (c) (d)

Figure 4: Designing the tail of dinosaur. (a) is the generic dinosaur shape. (b) uses Equation 20 for Tz with α = 100 and βy = 2,and (c) uses the standard super-quadric model for Tz with α = 2 and βy = 3. (d) uses the sketching method to obtain a Beziercurve for Tz.

finds the medial axis using Voronoi diagram or Delaunaytriangulation8. A simple method is to find a rough skeletonby applying the constrained Delaunay triangulation (CDT)29

and connecting the midpoints of the internal edges16, 25.Since this skeleton is not a medial axis (i.e., the locus of thecenters of maximally inscribed circles inside the 2D shape),additional computation is needed to find the proper radii ofinfluence to produce a good-fit convolution surface. In con-trast, we determine an approximate medial axis, comprisingof line segments, so that the parameters of the convolutionmodel can be more directly determined. Our algorithm in-volves two steps: (1) apply the CDT to the simple polygon,with the polygon edges constrained to be in the triangula-tion, and link up the circumcenters of all the triangles ac-cording to their connectivity (Figure 5(a)-(c)); (2) prune re-dundant circumcenters to obtain a simplified skeleton withfewer line-segment primitives (Figure 5(d)-(e)).

(a) initial stroke (b) CDT result (c) circumcenters

(d)influence circles (e) filtered skeleton (f) result surface

Figure 5: Generating convolution surface from silhouette

We refer to the initial polygon edges as the external edges,

and refer to the edges added by the Delaunay triangulationas the internal edges. A triangle in the CDT is then calleda junction triangle, sleeve triangle, or terminal triangle if ithas three, two or one internal edges, respectively25.

To prune the redundant circumcenters, we process the tri-angles in order, starting from a junction triangle until a ter-minal or another junction triangle is reached. Let p, q, r bethree most-recently-found circumcenters obtained in that or-der. The point r is pruned if its associated triangle is a sleevetriangle and it satisfies one of the following two criteria (Fig-ure 6).

Overlapping criterion The circumcircle of r lies mostlyinside the circumcircle of q. We test this criterion using

‖q− r‖+ τRr ≤ Rq,

where Rx denotes the radius at circumcenter x; τ = 0.6 ischosen so that a small but noticeable feature contributedby the point r would still be retained.

Ordering criterion The circumcircle of r lies on the half-plane defined by (q,n), where n is the normal to the com-mon edge of the triangles associated to q and r, directedtowards the triangle of r. We test it with

(r−q) ·n ≤ 0.

We remove the point q if the following criterion is satisfied.

Proportionality criterion The points p, q, and r are nearlycollinear, measured by

(q−p) · (r−q)

‖q−p‖‖r−q‖≥ cos(π/18),

and their circumradii vary proportionally to their distancesapart:

−0.1 <Rq −Rp

Rr −Rp−

‖q−p‖‖r−p‖

< 0.1



(a)Overlapping Criterion

(b)Ordering Criterion

Figure 6: Skeleton filtering criteria.

5.2. Constructing generic convolution surfaces

We determine the parameters that control the field contribu-tion of each skeletal line segment in two fitting steps: a localinferring step and a global fitting step.

In the local inferring step, for each line segment of length lwith endpoints p and q, we infer the weights w0 and w1 suchthat the field contribution produces a convolution surfacethat locally fits the silhouette curve well. Rewriting Equa-tion 7 in symmetric form, we have

F(p) = w0F(l−t)(p)+ w1Ft(p) (21)

where F(l−t)(p) = lF1(p)−Ft(p). We then set the field con-tribution at two selected points—the point sp at distance Rp

above p and the point sq at distance Rq above q (Figure 7)—to be equal to the desired implicit surface threshold T :

T = F(sp) = w0F(l−t)(sp)+ w1Ft(sp),

T = F(sq) = w0F(l−t)(sq)+ w1Ft(sq).(22)

Since Ft(sp) and F(l−t)(sq) are relatively small, we can as-sume that they are negligible, and approximately computethe two weights as follows:

w0 = T/F(l−t)(sp),

w1 = T/Ft(sq).(23)

This resulting isosurface, however, would bulge in the mid-dle beyond the silhouette boundary. Hence, to ensure that thesurface passes through smid , which is the point at a distance(Rp + Rq)/2 above (p + q)/2, we scale the weights w0 andw1 by a factor of κ:

κ =T

w0F(l−t)(smid)+ w1Ft(smid). (24)

For skeletal endpoints that have multiple connecting linesegments (i.e., points attributed to a sleeve or a junction tri-angle), we divide its weight by that number of line segmentsso as to cancel out the multiple field contributions.

Finally, we multiply each field contribution Fi(p) by a

l

p qpR qR

qSmidSpS

Figure 7: Convolution surface from a line segment.

Figure 8: The resulting implicit surfaces without (left) andwith (right) the global fitting step.

control scale λi and sum all the contributions from the skele-tal segments Li(i = 1,2, . . .,n):

F(p) =n

∑i=1

λiF(p;Li) (25)

Usually, λi is equal to 1. In the next section, we employ neg-ative λi for soft carving operation.

This local inferring step produces a convolution surfacethat, in general, does not interpolate the given silhouettecurve. Nevertheless in our experiments we have found thefitting to be adequate for the purpose of prototype design.Note that a convolution surface is the result of summingthe fields of all skeletal primitives, thus it is in fact a globalshape. We propose a global interpolation step for obtaininga tighter fitting (Figure 8).

Let the input stroke be consisting of points pi, . . .,pm.We solve the following constrained least-squares problemfor Λ = [λ1, . . .,λn]

>:

minΛ≥0

(FΛ−T)>(FΛ−T) (26)

where Fi = [F(p1;Li), . . .,F(pm;Li)]>, F = [F1, . . .,Fn],

and T = [T, . . .,T ]>. The condition Λ ≥ 0 here guaranteesthat the resulting surface has the correct topology. SinceF>F is semi-positive, this constrained quadratic convex pro-gramming problem can be solved by a simplified version ofHildreth-d’Esopo method9 . Nested within this algorithm is aGauss-Seidel iteration, and the problem size n is usually be-tween 10 to 100, thus the computation cost is small for thisfitting step.

Unfortunately, reflecting the general ‘over-fitting’ prob-lem of interpolation methods, the fairness of the global fit-ting surface could be worse than the non-interpolating onewhen the input silhouette has a zigzag shape. Therefore, thisglobal fitting step is kept as optional in our system.



Solid

SurfaceComponent

SurfaceComponent

. . .

Skel

Model Type

Model Params

0ω 1ω0P 1P

λ

Skel-Group

Material

Skel Indices

Silhouette

Project Plane

Skel List

Skel1Skel2

...

Polygonal Mesh

Skeli...

Skelj...

Skeln

Skel-Group List

Skel-Group1Skel-Group2

...Skel-Groupm

Figure 9: Data structure of our prototype design system.

6. Implementation Details

The design primitives are organized into hierarchical lev-els in our prototype system (Figure 9). In the top level, thesolid, or the design object, consists of several surface com-ponents. Each surface component is represented by a convo-lution surface defined by Equation 25. For example, the di-nosaur shape in Figure 4 has six surface components. Everyweighted component λiFi in Equation 25 corresponds to askeleton element or skel. Each skel has associated shape pa-rameters of the enhanced convolution surface model. Theseskels are naturally grouped into sets, named skel-groups, ac-cording to whether they belong to the same silhouette curve(or same set of silhouette curves consisting of a boundarycurve and one or more loop, see Section 6.3). Input informa-tion, such as projection plane, silhouette curve(s) and render-ing material, is stored in the skel-group structure. Currently,as a tradeoff between system flexibility and complexity, allthe operations are performed at the skel-group level. Basedon this structure, the system can provide a set of powerfuland flexible design operations. Since all the design opera-tions have compact representation, it is very easy to supportcopy, paste, ‘un-do’ and ‘re-do’ edit commands, which arevery common in practical user interfaces. All the represen-tation information can be saved to a simple script file. Thusthe whole design procedure can be recorded, and the usercan further modify the surface at the skel level if desired.

6.1. Merging new components

The user progressively builds a more complex object byadding new components in two ways. First, the user maydesign a separate surface component and simply attach it toan existing surface component. Second, the user can sketch anew silhouette curve and designate the resulting component(the associated skel-group) to be merged smoothly with anexisting surface component. To sketch a new silhouette, the

user first identifies the depth of the projection plane by draw-ing a small loop on an existing component, then rotates theobject to obtain a projection plane. This interface is similarto the extrusion operation of Teddy. The difference is that inour case the drawn loop serves only to indicate the depth ofthe projection plane, and not as the base loop for extrusion.

Similar to conventional modeling systems, we also pro-vide skel-group translate, rotate, clone and mirror opera-tions. The cross-section user interface is defined at this level.With these operations, the user can control the procedure ofdesigning new components more precisely.

6.2. Soft carving

We introduce a carving operation by defining a group ofcarving skels lying outside a surface boundary. Carving skelshave negative λi in Equation 25. The user first creates a dis-connected component as the carving tool by drawing a sil-houette, and then designates the resulting skels to be carv-ing skels by negating the λi values. By moving the carvingtool to the desired position and orientation overlapping withthe object to be carved, the desired shape is carved out bymerging the two components (Figure 10). This operation re-sembles the carving process familiar to artists: rough shapeis first created and details are created by carving out somevolume of mass.

Similar to the work of Wyvill et al.32, volume boolean op-erators can also be applied to the components. For instance,we can define the subtract operator as follows:

SA−B = x|min(FA(x)−T,T −FB(x)) = 0

to sharply cut a part from a shape. The output is similarto soft carving, however, since the resulting field functionis not continuous, the subtract operator sometimes causessampling problem in the polygonalization algorithm and re-sults in jaggy effect (see Figure 10(c)). Comparing with theBoolean operator, carving can produce smoother and softerresults. Wyvill et al. also proposed the super-elliptic blend-ing operator to obtain a soft blending effect. We consideredthis technique in our early experiments, but abandoned it dueto the high computation price.

6.3. Designing surfaces with holes

Surfaces with handles or holes can be easily created usingour system. One way to create a hole, such as the one ina torus, is by smoothly merging two components to meetat two locations. Another more direct way provided by oursystem is to let the user draw an inner silhouette loop withinan outer silhouette curve. Our skeleton extraction algorithmin Section 5.1 can handle such cases. For convenience, in-stead of drawing two nested curves, we also allow the userto directly specify a closed curve, approximated by line seg-ments, as the skeleton, and assign a radius to each skeletalpoint. The cylindrical part of the cup in Figure 12 is createdin this manner.



(a) (b) (c)

Figure 10: Carving vs. Boolean operation. (a) shows a whale shape and a carving tool. (b) and (c) are the results of soft carvingand hard subtraction operation, respectively. All the whales are polygonalized at the same resolution.

7. Design Examples

We demonstrate the capabilities of our system with examplesin Figure 11 to 13. In these figures, the grey curves are theinput silhouettes, the skels are drawn in green or blue lines,and the red lines indicate the βy scaling direction of the en-hanced convolution method. Table 1 lists some computationdetails for these objects. The chair shape contains some 2.5-dimension components, i.e., the rectangular seat and the fourlegs are designed using Equation 20. A variety of styles ofchairs can be easily obtained by changing the parameters ofthe transform mapping T . The cup shape is an example ofobjects with holes and handles. The cylindrical part of thecup is designed by sketching the top circular curve as theskeleton, assigning radius to each skeletal points, and set-ting βy to extrude the shape vertically. The rabbit and mousein Figure 13 and 1 are examples of attractive prototype ob-jects, which typically take a novice user about 10 minutes todesign using our system.

Model # Surface # Skel- # Skels MT time(sec)components groups (0.5/0.25)

Chair 1 6 32 1.7 / 6.1Cup 1 3 74 6.7 / 27.4Dinosaur 6 16 126 1.1 / 4.1Mouse 4 12 95 3.6 / 14.7Rabbit 8 13 178 7.9 / 31.6

Table 1: Model data. MT time refers to Marching Tetrahedrapolygonization time. The two resolutions used are 0.5 and0.25 cube-size. The timing results are obtained on a standard2.4GHz Pentium IV PC with 512MB memory.

8. Conclusions and Future Work

We have presented a system based on convolution surfacefor designing meaningful freeform shapes from silhouette

Figure 11: Chair shape. Skeleton structure (top left), genericsurface (bottom left), rigid-style produced by our enhancedconvolution model (top right), another style obtained by tun-ing the parameters of T (bottom right).

curves. The convolution surface model lends itself to intu-itive, robust, and versatile design operations.

Comparing with single-resolution polygonal meshes, theimplicit surface representation can be viewed as a continu-ous but compact representation. Meshes of arbitrary resolu-tion can be extracted by the marching tetrahedra algorithm.As the implicit function evaluation is still the main bottle-neck of our method, to achieve interactive response, we use



Figure 12: Cup shape. Skeleton structure and resulting sur-face.

a low sampling rate in the early stages of design, and onlyincrease the sampling rate for the final output. Most of theskels in an object can be adequately represented by the orig-inal polynomial-weighted convolution model; only a handfulof them requires the capability of the enhanced model. Weadd a condition switch to each skel so that the more expen-sive evaluation is only performed for the enhanced models.This strategy sped up the whole polygonization procedureby about 2-3 times in our experiments.

For simplicity and efficiency, we chose to use only linesegment as the skeleton primitive. A weakness of convolv-ing branching line segments is that bulges appear aroundjunctions. We handled this bulging problem by scaling thefield contributions at joints between line segments. Our sim-ple and practical solution produces satisfactory surfaces inmost of our experiments. The global fitting step also helpsto solve the bulging problem. An alternative solution maybe to include polygons in the skeleton and apply certainconstraints3, however, deriving a closed-form representa-tion of polynomial weighted convolution surface in this casewould be challenging and numerical method may be re-quired. From the system implementation point of view, thetradeoff between additional primitive and increased capabil-ities also needs careful consideration. Nevertheless, a com-parison study of these two branching methods is an interest-ing future work.

The spatial mappings introduced to deform the cross sec-tion of the convolution model anisotropically have subtleeffects on the resulting blend; for example, the continuityproperty is weaken, this problem, however, was observed tobe not serious in our experiments except for cross-sectionswith very sharp corners. A local evaluation technique thatignores the contribution of distant line segments may be asolution.

Due to the medial axis being sensitive to changes in in-put sample points, our skeleton extraction algorithm mayproduce unstable results for certain inputs. Future workmay include exploring other skeletonization algorithms1, 7 to

Figure 13: Rabbit shape. Input silhouettes and partial skele-ton structure (upper), and the rabbit rendered with cartoonshading (lower).

extract alternative skeletons, possibly containing parabolicarcs17 and polygons.

Acknowledgements

We wish to thank John Hughes for his constructive com-ments on an earlier version of the paper. We are also gratefulto the anonymous reviewers for careful reading and com-ments which helped improve the exposition of the paper.

References

1. O. Aichholzer and F. Aurenhammer. Straight skeletons forgeneral polygonal figures in the plane. In Proc. 2nd Annu. Int’l



Computing and Combinatorics Conf., pages 117–126, 1996.10

2. J. Bloomenthal. An implicit surface polygonizer. In P.S. Heck-bert, editor, Graphics Gems IV, pages 324–349. AcademicPress, Cambridge, 1994. 3

3. J. Bloomenthal. Bulge elimination in convolution surfaces.Computer Graphics Forum, 16(1):31–41, 1997. 10

4. J. Bloomenthal and K. Shoemake. Convolution surface. ACMSIGGRAPH 1991 Conference Proceedings, pages 251–256,1991. 3

5. D. Bourguignon, M.-P. Cani, and G. Drettakis. Drawing for il-lustration and annotation in 3D. In Computer Graphics Forum,volume 20, pages 114–122, 2001. 2

6. J.C. Carr, R.K. Beatson, J.B. Cherrie, T.J. Mitchell, W.R.Fright, B.C. McCallurm, and T.R. Evans. Reconstructionand representation of 3D objects with radial basis function.ACM SIGGRAPH 2001 Conference Proceedings, pages 67–76, 2001. 2

7. F. Chin, J. Snoeyink, and C. Wang. Finding the medial axisof a simple polygon in linear time. Discrete ComputationalGeometry, 21(3):405–420, 1999. 5, 10

8. M. deBerg, M. van Kreveld, M. Overmars, andO. Schwarzkopf. Computational Geometry—Algorithmsand Applications. Springer-Verlag, 2000. 6

9. D. A. D’Esopo. A convex programming procedure. NavalResearch Logistic Quarterly, 6:33–42, 1959. 7

10. H. Q. Dinh, G. Slabaugh, and G. Turk. Reconstructing sur-faces using anisotropic basis functions. In International Con-ference on Computer Vision 2001, pages 606–613, Vancouver,Canada, July 2001. 3

11. L. Eggli, C. Hsu, B. Bruderlin, and G. Elber. Inferring 3Dmodels from freehand sketches and constraints. Computer-Aided Design, 29(2):101–112, 1997. 1, 2

12. T. Galyean and J. F. Hughes. Sculpting: an interactive volu-metric modeling technique. ACM SIGGRAPH 1991 Confer-ence Proceedings, pages 267–274, 1991. 1

13. C. M. Grimm. Implicit generalized cylinders using profilecurves. In Implicit surface 1999, 1999. 2

14. I. J. Grimstead and R.R. Martin. Creating solid models fromsingle 2D sketches. Proc. of the Third Symposium on SolidModeling and Applications, pages 323–337, May 1995. 2

15. T. Igarashi and J. F. Hughes. Smooth meshes for sketch-basedfreeform modeling. In ACM Symposim on Interactive 3DGraphics, pages 139–142, Monterey, California, April 2003.1, 3

16. T. Igarashi, S. Matsuoka, and H. Tanaka. Teddy: A sketch-ing interface for 3D freeform design. ACM SIGGRAPH 1999Conference Proceedings, pages 409–416, 1999. 1, 2, 6

17. X. Jin and C. L. Tai. Analytical convolution surfaces forquadratic curve skeletons with polynomial weight distribu-tions. Visual Computer, 18(8):530–546, 2002. 3, 10

18. X. Jin, C. L. Tai, J. Feng, and Q. Peng. Convolution surfaces

for line skeletons with polynomial weight distributions. Jour-nal of Graphics Tools, 6(3):17–28, 2001. 3

19. O. Karpenko, J. F. Hughes, and R. Raskar. Free-form sketch-ing with variational implicit surfaces. Computer Graphics Fo-rum, 21(3), 2002. 1, 3

20. L. Markosian, J. M. Cohen, T. Crulli, and J. F. Hughes. Skin:A constructive approach to modeling free-form shapes. ACMSIGGRAPH 1999 Conference Proceedings, pages 393–400,1999. 1

21. J. McCormack and A. Sherstyuk. Creating and rendering con-volution surfaces. Computer Graphics Forum, 17(2):113–120,1998. 3

22. P. Michalik, D. Kim, and B. Bruderlin. Sketch- and constraint-based design of B-spline surfaces. In Solid Modeling 2002,pages 297–304, 2002. 1, 2

23. R. Ogniewicz and M. Ilg. Voronoi skeletons: theory and ap-plications. Proc. Computer Vision and Pattern Recognition,pages 63–69, June 1992. 5

24. S. Owada, F. Nielsen, K. Nakazawa, and T. Igarashi. Lec-ture Notes in Computer Science, volume 2733, chapter ASketching Interface for Modeling the Internal Structures of 3DShapes, pages 49–57. Springer-Verlag Heidelberg, 2003. 2

25. L. Prasad. Morphological analysis of shapes. CNLS Newslet-ter, 139:1–18, July 1997. 6

26. D. Pugh. Designing solid objects using interactive sketch in-terpretation. In ACM Symposium on Interactive 3D Graphics,pages 117–126, March 1992. 2

27. S.F. Qin, D. K. Wright, and I.N. Jordanov. From on-linesketching to 2D and 3D geometry: a system based on fuzzyknowledge. Computer-Aided Design, 32(14):851–866, 2000.1

28. A. Sherstyuk. Kernel functions in convolution surfaces: Acomparative analysis. The Visual Computer, 15(4):171–182,1999. 3

29. J.R. Shewchuk. Triangle: engineering a 2D quality mesh gen-erator and delaunay triangulator. First Workshop on AppliedComputational Geometry Proceedings, pages 124–133, 1996.6

30. O. Tolba, J. Dorsey, and L. McMillan. A projective draw-ing system. In ACM Symposium on Interactive 3D Graphics,pages 25–34, March 2001. 2

31. G. Turk and J.F. O’Brien. Modeling with implicit surfaces thatinterpolate. ACM Transactions on Graphics, 21(4):855–873,October 2002. 2

32. B. Wyvill, A. Guy, and E. Galin. Extending the CSG tree.warping, blending and boolean operations in an implicit sur-face modeling system. Computer Graphics Forum, 18(2):149–158, 1999. 8

33. R.C. Zeleznik, K.P. Herndon, and J.F. Hughes. Sketch: an in-terface for sketching 3D scenes. ACM SIGGRAPH 1996 Con-ference Proceedings, pages 163–170, 1996. 1



Appendix A: Approximating a sketched profile bypolar-form Bezier curves

Let PS = (R j,θ j)| j = 1,2, . . . , p be a point sequence sampledfrom the input profile. We want to fit an n-degree polar-form Beziercurve b(θ). Assume that the end control radii (r0,θb) and (rn,θe)are known. We minimize the following function

fPS(r1, . . . , rn−1) =p

∑j=1

(b(θ j)−R j)2 (27)

Let r = [r1, . . . , rn−1]> and Bi = [Bn,i(t1), . . . ,Bn,i(tp)]> (with ti =θi−θbθe−θb

) be the i-th column of the Bezier coefficient matrix B =

[B1, . . . ,Bn−1]. We rewrite Equation 27 into matrix form

minr

‖B · r− R‖22 (28)

where

R = [R1, . . . ,Rp]> − [B0 Bn]

[

r0

rn

]

.

Since it is a typical least-squares optimization problem, the solutionis as follows:

r = (B>B)−1B>R.

Appendix B: Derivative property of polar-form Beziercurves

Consider the following transformation of the polar-form Beziercurves

x(θ) = R(θ)cosθ,y(θ) = R(θ) sinθ.

(29)

By the chain rule, the first-order derivatives are

x′ = R′(θ)cosθ−R(θ) sinθ,y′ = R′(θ) sinθ + R(θ)cosθ.

(30)

From Equation 15

R′(θ) =n

θe −θs

n−1

∑i=0

4riBn,i(θ−θs

θe −θs), (31)

where 4ri = ri+1 − ri is the forward difference operator. Then atthe endpoints,

R′(θs) =n

θe −θs4r0, R′(θe) =

n

θe −θs4rn−1

To achieve tangent continuity for two consecutive j and j + 1 curvesegments, we must satisfy

(x,y)|θ j+1− = (x,y)|θ j+1+, (x′,y′)|θ j+1− = α(x′,y′)|θ j+1+,

where α > 0. By choosing α = 1, we obtain the following sufficientcondition:

4r j,n−1 = 4r j+1,0, (32)

which is equivalent to Equation 17.


prototype modeling from sketched silhouettestaicl/papers/convmo-final.pdf2 c.l. tai et al. /...

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