shape differentiation of freeform surfaces using a similarity measure based on an integral of...

Computer-Aided Design 40 (2008) 311–323www.elsevier.com/locate/cad

Shape differentiation of freeform surfaces using a similarity measure basedon an integral of Gaussian curvature

Jing Fua,∗, Sanjay B. Joshia, Timothy W. Simpsona,b

a Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, PA, USAb Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA, USA

Received 3 October 2005; accepted 4 November 2007

Abstract

Freeform surfaces are popularly used to design and model complex 3D objects. These 3D models are stored as computerized models indatabases. To facilitate data retrieval and shape matching, a major challenge lies in defining and computing the level of similarity between two ormore freeform surfaces. In order to explore the useful 3D information associated with the surfaces, an integrated approach based on the integralof Gaussian curvature is proposed to develop the measures of similarity of freeform surfaces. Specifically, the integral of Gaussian curvatureis mapped into the 2D space, and a shape-based measure is developed using statistical methods to compute the level of similarity. For smoothsurfaces, a fast approximation algorithm is developed to calculate the curvature of individual subregions. In cases where the target surface has acomplex topological structure or a smooth surface is not available, the integral of Gaussian curvature for the discrete surface is first calculatedat each vertex, followed by mapping onto a 2D spherical coordinate. The distance measure focuses on the local geometry, which is critical toinvestigate models with a certain level of resemblance such as products in a family. This proposed approach can be applied to surfaces undervarious transformations, as well as 3D data from various sources.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Freeform surfaces; Similarity measure; Data mining

1. Introduction

The advancement of mathematical foundations and compu-tational algorithms in Computer Aided Design (CAD) has al-lowed users to model or design surfaces of unprecedented com-plexity. The shape of these surfaces has critical impact beyondgeometry. For example, the surface shape of kinetics-relatedproducts is often optimized by engineering mechanics, andmost handheld tools are shaped based on extensive ergonomicexperiments. Systematic comparison and analysis of the modelshape is needed to enhance the design phase as well as theoverall product lifecycle management [1]. Moreover, fitted orreconstructed surfaces based on advanced 3D data acquisitionmethods, such as Coordinate Measuring Machine (CMM) withmechanical probe or laser scanning, allow users to not only ob-tain the dimensional data but also to explore the relationship

∗ Corresponding author. Tel.: +1 814 865 9859.E-mail address: [email protected] (J. Fu).

0010-4485/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2007.11.006

between geometry and functions at different scales. The ac-quired data is often in discrete format and lacks clearly definedtopologic structures. In routine Reverse Engineering (RE) ap-proaches, the surfaces obtained are in the form of polygonalmeshes, which are reconstructed from point clouds. By scan-ning a damaged part and comparing it with the models in CADdatabase, users can promptly locate the appropriate part for re-placement [2].

The ability to automatically compare 3D surfaces becomescritical for the purpose of data classification and exploration.For surfaces whose normal is constant or can be represented bya limited number of parameters (e.g., spherical or cylindricalsurface), the direct approach to classification is based on alimited number of parameters (e.g., the radius of sphericalsurface). However, freeform surface models (surface normalchanging continuously) are often constructed and representeddifferently. A considerable amount of noise can also beintroduced if the data comes from reconstruction of digitizeddata. Although several approaches have been proposed forshape-based geometric comparison and search, most of

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312 J. Fu et al. / Computer-Aided Design 40 (2008) 311–323

(a) Model A: Power flash. (b) Model B: High quality.

(c) Model C: Water & sports.

Fig. 1. Three products in single-use camera family with the scanned models and D2 shape histograms.

them cannot be directly applied to analyze similar surfaces.Differences in local shape are more important since the globalshape signature is insensitive for products with commontopology or contours. In Fig. 1, the designs and scanned datafrom three single-use cameras are presented. The 2D photos arefrom the product website (www.kodak.com), and the associatedmeshes are obtained by scanning the cover surface of the realproducts. Based on human perception, the geometries of thefirst two models (Model A: Power Flash and Model B: HighQuality) are significantly different from the third one (Model C:Water & Sports) with regard to the shapes’ impacts on humanperceptions. However, the associated D2 shape histograms [3]of all the three models are similar to each other, and hencewould be considered as similar models due to the lack of a moredetailed comparison.

In this paper, we propose a novel approach to differentiatethe shape of surface patches based on the distribution ofGaussian curvature at local geometry. The challenges andrequirements for such a methodology are as follows.

1.1. Local shape/features

Most existing approaches based on signature or distributionsonly focus on the overall geometric properties of the input

models, which provide a computational advantage to searchlarge databases for similar designs. These tools often involveextraction of properties at an individual point followed byanalyzing a large sample of points from the model. A significantamount of variation in shape and/or topology is allowed in theseapproaches in order to achieve high performance. However,for products with overlapping functional requirements (such asproducts in a family), it is not uncommon to share a generalcontour or surface with limited modification. The results basedon global signature can be obscure, since dramatic geometric ortopologic change is rare among these designs.

If a significant portion of the model is planar, then thevalues of surface curvature or approximated curvature may bedominated by extremely small values approaching zero. In thiscase, the spatial distribution of the non-planar features will havea larger impact on the shape comparison. In-depth interrogationon shape is required rather than only comparing the globaldistributions/signatures.

1.2. Geometric representation

The information obtained through advanced data acquisitionapproaches, e.g., 3D reconstructed surface, is often in simplegeometric forms such as a point cloud or polygonal mesh.

http://www.kodak.com

J. Fu et al. / Computer-Aided Design 40 (2008) 311–323 313

The proliferation of online 3D data, such as Virtual Reality(VR) and Augmented Reality (AR), is mainly driven bypolygonal meshes. These low-level formats cause much fewercompatibility problems compared to more sophisticated ones,such as Boundary Representation (B-Rep) and ConstructiveSolid Geometry (CSG), especially when performing cross-platform operations. However, parametric surfaces (B-Spline,NURBS, etc.) are still popularly applied in design andmodeling. The surface curvature is well defined in parametricsurfaces, which cannot be computed analytically in a mesh.Although computing surface curvature is not critical foranalyzing prismatic parts, the prevalence of complex surfacesused in today’s designs demands new analysis tools to considerfreeform geometries.

In order to provide a complete tool for the differentiationof surfaces, we assume that the target surface patches to beanalyzed are in the form of either a parametric surface S ora triangular mesh M . The triangular mesh can be easily appliedto reconstructed data, since triangulation methods, such asDelaunay method, have been well developed for the originalpoint cloud. Although the design features in a particular systemmay not be directly shown in this case, it is not difficult toconvert a parametric surface to a polygonal mesh-based formatwith existing algorithms. Proposed methods and algorithmsbased on these two representations are flexible enough to beapplied in each stage of product design as well as on differentplatforms.

1.3. Consistency/robustness

Compared to a surface matching approach in whichfinding the proper transformation is critical, quantifying thelevel of resemblance is the most demanding task in thiswork. The calculated distance value should be consistentwith human perception. Moreover, there are a number ofoperations frequently used in the design phase, such asrigid transformation, scaling and mirroring. These operationshave limited impact on human perception, and the shape isoften considered as unchanged. However, obtaining accuratesimilarity measures can be a challenge after applying thesetransformations, and an ideal differentiation approach shouldbe invariant or insensitive to such transformations. Due tothe nature of the data acquisition process, it is expected thata significant amount of noise will be introduced, and partialor excessive data is also likely to be obtained. The proposedmethodology should be insensitive to small perturbations aswell as noise. For example, if a small portion of the surfaceis modified, or a small change has been applied to the shape,then the percentage of change in similarity measure should alsobe limited.

In this paper, we propose a novel methodology todifferentiate surface patches based on the distribution of theintegral of Gaussian curvature. The data can also be obtainedfrom various sources including surface reconstruction orretrieval of CAD models from design databases. For a full patchparametric surface, a fast approximation algorithm is proposedto calculate the integral of Gaussian curvature in constructed

subregions. For a smooth surface with irregular boundaries or adiscrete mesh, the integral of Gaussian curvature is computedat discrete vertices followed by mapping to a 2D sphericalcoordinate. This 2D distribution is used to represent the originalshape, which is less susceptible to noise and frequently usedtransformations. Compared to prior research in which globalsignatures are used for comparison, the proposed approachfocuses on local shape change of the designs, which is criticalfor comparing models having a certain level of resemblancebetween each other. Moreover, the computation is also efficientand only dependent on the tessellation of the original mesh, andthis allows the proposed methodology to be widely adoptedacross different platforms. Previous work and challenges arediscussed in Section 2, and the proposed methodology isintroduced in Sections 3 and 4. Implementations and examplesare shown in Section 5, and Section 6 provide the discussionsas well as future research directions.

2. Related research

2.1. Shape similarity measure

In order to quantify surface differentiation, the level ofdifference or similarity is defined as a similarity measure, whichprovides a value bearing some correlation with the perceptualdifference of the two geometric objects [4]. However, theconcept of similarity of geometrical models is ambiguous.A similarity measure based on a distance function d(A, B)

is often used to quantify the similarity level between twogeometric features A and B, as long as the function has metricproperty and matches human notions of shape resemblance [5].For practical computation, the geometric objects are oftentransformed to other forms, such as histograms and graphs tofacilitate the application of the distance function.

Nevertheless, the similarity measure can be more looselydefined but still satisfies the following properties:

(1) Non-negativity, d(A, B) > 0(2) Symmetry, d(A, B) = d(B, A)

(3) The more similar the shape of two geometric objects A andB, the smaller d(A, B).

d(A, B)is often scaled to [0, 1), where 0 represents a perfectmatch and extreme variations cause the value to approach 1.The third property is related to human perception, and it isoften subjective depending on the target group. One approachto justify the proposed similarity measures is to apply it to aset of examples followed by comparison with empirical studies.The alternative is to focus on matching geometric objects asan optimization problem involving similarity measures as theobjective function. For example, given geometric objects A andB for matching, a transformation T can be found, such thatsimilarity measure d(T (A), B) of T (A) and B is minimized.Surface matching and finding similarity measure are closelyrelated topics and often share the same techniques. In caseswhere the exact transformation T is the main aim, a surfacematching approach is often applied.


2.2. Literature review

Numerous applications of measuring shape similarity canbe found in computer vision, computer inspection, etc., andreviews on this topic have been presented [1,6]. To focus onour problem, we review the techniques that are closely relatedto differentiation of surface shape and are presented in threecategories: surface matching, topology-based and shape-based.

2.2.1. Surface matchingSurface matching problems focus on the transformation

for practical use, rather than the distance value itself. Forexample, finding the rigid transformation which minimizesthe difference between two surfaces is useful for modelregistration for scanned data. In Huang’s research [7], givenan original freeform surface, a rigid transformation is appliedto the input surface or discrete points, in order to find a bestmatch for the original base one. Since six degrees of freedom(three translations and three rotations) are required for rigidmotion in 3D space, a minimization problem for mean squaredistance can be constructed considering the six parameters asvariables. Ko et al. [8] propose a similar approach to findthe best transformation, but mean and Gaussian curvature ofsample points are used as variables. Surazhsky and Elber [9]develop a matching algorithm to be applied in shape morphingprocess. Unlike rigid transformation introduced previously, unitnormal fields of the two surfaces are computed and usedfor comparison. With the parameterization of the surfaces asvariables, the similarity measure of the unit normal fields canbe computed accordingly and serves as the objective function.Avagyan et al. [2] propose a novel approach to search formatched parts based on scanned data. The actual model to beused is assumed to be either identical to or a superset of thescanned model. After prescreening the models, searching isconducted by changing the orientations (rigid transformation).This approach is useful to find parts for replacement basedon damaged ones if the contained geometric information issufficient.

However, to obtain the similarity measure by surfacematching often implies that the discrepancy of the surfaces tobe matched is only due to the rigid transformation and ignoresscaling or reflections/mirroring. In other words, the point-to-point comparison of the surfaces to be tested in Euclideanspace should be identical (or close enough to each other)after proper translation and rotation. The transformations, suchas uniform scaling and mirroring are commonly used in theproduct design process, and conclusion of a low level ofsimilarity will be drawn if the previous matching algorithmsare applied. Furthermore, it is also computationally intensive tofind the scaling factor or symmetrical axis/plane.

2.2.2. Topology-basedTopology-based approaches are widely used in feature

recognition and process planning, and the main idea is togenerate and apply different forms of a graph that cansubstitute the geometric and topological structure of the objects,including the relationship between subfeatures. Using boundary

representation (B-Rep), which is based on the graph structureof the objects, recognition or comparison of geometric objectscan be reduced to the operations on the graph structure, such asretrieval and comparison of subgraphs [10–13]. One challengefor graph-based approaches is that the graph representationmay not be unique due to the complexity of the geometry andtopology of the target model, thus the conclusion of similaritylevel between two geometric objects can be erroneous if thegraph is not constructed properly.

A variant approach is to construct a unique skeleton from aninput model. One popular skeleton structure is based on MedialAxis Transformation (MAT) [14–16], which is defined as thelocus of an inscribed sphere of maximal diameter as it rollsinside the solid model. Another type of skeleton is based on aReeb graph where a continuous scalar function is defined on theobject to determine the skeleton [17–20], and the topologicalinformation can be preserved by using an appropriate function.However, the main focus of these skeleton-based approaches isto investigate topological information rather than geometricalshape, and in general the complete solid models are comparedwithout considering individual surfaces or local features.Although MAT can also be constructed on a surface based onthe geodesic distance [21], limitations of speed and stabilityprevent the application to freeform surfaces/meshes, and littlepractical use of the resulting skeleton is found regarding thelevel of similarity.

2.2.3. Shape-basedThe main focus of shape-based approaches is the geometric

shape of a feature, such as the spatial distribution of curvesand surfaces, rather than the structure of multiple features. Formodels in polyhedral meshes, signatures based on curvature canbe obtained by evaluating a set of these vertices, followed bycomputing the distance function between these signatures [22–24]. Distributions of the distance between random points areused to represent the shape of the surface, and a similaritymeasure can be computed by comparing the associatedshape histograms [3,25]. One or multiple histograms can beconstructed by using different distance functions, which arebased on the Euclidean metrics on a set of vertices. Thecomputation is straightforward, and this approach can beapplied for search and retrieval in a large database.

More sophisticated approaches are proposed to process largemesh data. Harmonic-based methods are applied to substitutefor the Euclidean metrics. The approaches use Fourier functionsto represent the shape of the geometric models, which can beeither curves or surfaces [26,27]. Large sample of curvatureat discrete point on the curve/surface is extracted followed byanalysis, and the original geometric objects are transformedinto digital signals for classification and comparison.

Compared to the methodologies in the previous twocategories, shape-based approaches show some promise for theaim to differentiate surface patches. The geometric comparisonis reduced to 1 or 2D histograms, and the histogram orsignatures can be constructed and compared based on statisticalapproaches, which are less susceptible to noise or perturbations.The limitation is that high similarity levels of two histograms


may not be sufficient to make conclusions for similar shapesas in the example in Fig. 1. Especially for family productdesign with similar contours or common features, the spatialdistribution of the local features is more important than aglobal signature. Also Euclidean metrics and surface curvatures(Gaussian and mean curvature) are not dimensionless, andthe corresponding results are subject to scaling operations. Inthe next section, we present the mathematical foundations formodeling freeform surfaces, followed by introduction of theproposed methodology in Section 4.

3. Mathematical background of surface

3.1. Differential geometry of smooth surface

Given a regular parametric surface r(u, v) = [x(u, v),

y(u, v), z(u, v)]T, parameter u and v are restricted to certainintervals. Without loss of generality, the interval can be [0, 1];so u ∈ [0, 1] and v ∈ [0, 1]. Suppose that a curve C on thesurface is defined by C = r(u(t), v(t)), the arc length s of curveC can be obtained by the first fundamental form:

(ds)2= I = dr · dr = Edu2

+ 2Fdudv + Gdv2

E = ru · ru F = ru · rv G = rv · rv.(1)

Subscripts u, v denote the partial derivatives. Suppose that Pis a point on curve C , t and n are the unit tangent vector and unitnormal vector at P . The curvature vector k can be decomposedto two orthogonal vectors:

k =dtds

= kn + kg = knN + kg(N × t), (2)

where N is the surface unit normal defined by N =ru×rv

|ru×rv |, and

the second fundamental form can be defined by:

II = −dr · dN = Ldu2+ 2Mdudv + Ndv2

L = N · ruu M = N · ruv N = N · rvv.(3)

The normal curvature can be represented by:

kn =III

=L + 2Mλ + Nλ2

E + 2Fλ + Gλ2 , where λ =dv

du. (4)

Two roots, k1 and k2, representing extreme values of kn canbe obtained by treating λ as the variable:

k1k2 = K =L N − M2

EG − F2 . (5)

K is called Gaussian curvature [28] which is an intrinsicproperty of the surface at point P , since k1 and k2 are theextrema of a function of λ. This also means that K is onlyrelated to the surface and invariant of the curves passing throughP . Gaussian curvature is widely used to represent properties ata certain point of the surface, and a large sample of Gaussiancurvature collected from the target surface can serve as adiscrete version of the surface shape, and used for comparisonor other applications.

3.2. Integral of Gaussian curvature

Theorem 1 (Gauss–Bonnet Theorem [28]). S is an oriented,smooth surface and R be a simple region of S. α : I → Ssuch that ∂ R = α(I ). Assume that α is positively oriented,parameterized by arc length s, and let α(s0), . . . , α(sk) andθ0, . . . , θk be, respectively, the vertices and the external anglesof α. Then

k∑i=0

∫ Si+1

Si

kg(s)ds +

∫∫R

K dσ +

k∑i=0

θi = 2π, (6)

where kg(s) is the geodesic curvature of the regular arcs of α

and K is the Gaussian curvature of R. Instead of evaluatingcurvature at a single point, total curvature

∫∫R K dσ shows the

aggregate Gaussian curvature over a certain closed region R andmeasures the developability of R. Suppose that the region Rhas uniform Gaussian curvature K R , then

∫∫R K dσ is reduced

to K R A where A is the area of R and equals∫∫

R dσ . If regionR represents a portion of a sphere of radius r , K R A =

1r2 A,

then the total curvature is proportional to the area and rangesfrom 0 to 4π . For simplicity of representation in later sections,we denote:

K T=

∫∫R

K dσ. (7)

3.2.1. Invariance propertiesGaussian curvature K is an intrinsic property of the

surface and well known for its invariance property to rigidtransformation; however, K is not invariant to uniform scalingtransformation.

Theorem 2. Suppose that a uniform scaling transformationcentered at the origin is applied to surface r(u, v), such asr = Sr = λr, where S is a non-singular diagonal matrix with λ

as each entry s, then K =Kλ2 .

Proof. The first fundamental form coefficients of the scaledsurface are functions of u, v, s :

E = ru · ru = λ2ru · ru = λ2 E,

correspondingly F = λ2 F, G = λ2G.

N is the scaled surface unit normal and unchanged since

N =ru × rv∣∣ru × rv

∣∣ =ru × rv

|ru × rv|= N.

Then the second fundamental form coefficients of the scaledsurface is:

L = N · ruu = λN · ruu = λL ,

correspondingly M = λM, N = λN .

So we obtain

K =L N − M2

E G − F2=

λ2

λ4

L N − M2

EG − F2 and

K =1

λ2 K . � (8)


Fig. 2. (a) A vertex and related variables and (b) a degenerate vertex due to afeature (hole).

Theorem 2 shows that Gaussian curvature K on a smoothsurface is dependent on the scale factor, and this implies that Kis not a good candidate for a similarity measure if target surfaceat different scales is to be compared.

Theorem 3. Integral of Gaussian curvature K T is invariant ofuniform scaling and mirroring.

Proof (Uniform Scaling). As for surface area dσ =√EG − F2dudv, according to Eqs. (5) and (7):

K T=

∫∫R

L N − M2√

EG − F2dudv. (9)

Suppose uniform scaling transformation is applied to surfacer(u, v), such as r = Sr = λr.

K T=

∫∫R

L N − M2√E G − F2

dudv =

∫∫R

L N − M2√

EG − F2dudv

= K T. � (10)

Proof (Mirroring). Suppose that mirroring transformation isapplied to surface r(u, v). For an arbitrary point r = r −

2Dn, where n is the normal vector of the symmetrical planeand D is the distance from r to the symmetrical plane. Forany point r, D and n are constant; therefore, ru = ru andrv = rv . Consequently, there is no change in the first andsecond fundamental forms, and K T

= K T based on Eq. (5).Theorem 3 provides a theoretical base to support application ofK T to represent the surface shape, which is invariant to uniformscaling and mirroring transformations. Also Theorem 1 showsthat computation of K T can be reduced to only studying theboundary of the closed region. �

3.3. Gaussian curvature on a discrete surface

In cases where a parametric surface has irregularboundaries/topologies or a smooth surface is not available, asimilar study can be conducted on discrete surface which alsobears the same shape information. In this paper, we assume thetarget surface, if discrete, is a triangular mesh Mconsisting ofa set of vertices V = {vi } ⊂ R3, a set of edges connectingthe vertices E = {e j = v j1v j2}, and a set of triangles T =

{tk = ∆vk1vk2vk3} (see Fig. 2(a)). The angles of incident edgesat vi are defined as {αi

1, αi2, . . . α

idi

} while di is the degree of

vertex vi . Although the mesh M is not C2 differentiable as with

smooth surfaces, the integral of Gaussian curvature with respectto vertex i can be approximated by [29,30]:

K Ti =

∫ K ds

S≈ 2π −

j=di∑j=1

αij . (11)

Computing the values of K Ti by Eq. (11) is straightforward

and only dependent on the tessellation of the mesh. Thedistribution of K T

i , if the sample is sufficiently large, is alsoa promising candidate to represent the shape comparable toK T for a smooth surface. The summation of K T

i values ateach vertex in region R approximates K T obtained for smoothsurface region R.

For discrete surfaces, the computed integral of Gaussiancurvature (K T

i ) is only an approximated value, and numericalerrors do exist if it is converted from smooth surfaces. Further-more, the proposed computation may provide misleading in-formation due to (1) the noise in the data acquisition process,e.g., digital scanning; (2) the geometric and topologic featuresembedded in the mesh. For example, the boundary of the coversurface and the associated topological features (e.g., holes,boundaries) often result in edges not shared with two triangles(see Fig. 2(b)). Surface curvature for smooth surfaces is not welldefined in this case. For the vertices in this case, the K T

i valuescomputed by Eq. (11) are directly linked to the curvature of theboundary curves instead of surface curvature only. In this paper,those vertices, whose computed K T

i value cannot represent thesurface shape, are defined as degenerate vertices. When con-ducting the similarity comparison, it is necessary that the de-generate vertices and the computed K T

i value be identified andtreated differently.

4. Methodology

Due to its close relationship with human cognition,distribution of curvature is widely used for comparing shapesimilarity as discussed in the previous sections. Differentdistance functions are applied to the curvature distributionafterwards to quantify the similarity level. In this paper, weuse the distribution of K T instead of K due to its invariantproperties and ease of computation/approximation.

4.1. Similarity of parametric surface

4.1.1. Computation of integral of Gaussian curvatureThe core idea in the proposed methodology is to compute

the integral of Gaussian curvature value (K T) for each regionof interest instead of computing the Gaussian curvature. Forsmooth surfaces, the Gauss–Bonnet Theorem (Theorem 1)provides the theoretical foundation to obtain K T values byonly focusing on the boundary curves, and a fast approximationalgorithm can be developed. A common parametric surface hasfour boundary curves by setting u or v to 0, and accordingto Theorem 1, the following equation can be derived:

K T=

∫∫R

Kdσ = 2π −

4∑i=1

θi −

4∑i=1

∫ Si

Si−1

κg(s)ds, (12)


Fig. 3. Four-sided patch bounded by iso-parametric curves.

Fig. 4. Discrete points sampled on a surface curve.

where the θi are the turning angles at the four corners(see Fig. 3), which can be calculated using vector arithmetic.The integral of κg can be calculated by numerical methods.Curve C is converted into a set of n P connected line segments.Suppose that points Pi (i = 0, 1, . . . n P ) are those consecutivepoints on C (see Fig. 4). ϕ(Pi) = 6 P∗

i−1PiP∗

i+1, where P∗

i−1,P∗

i+1 are the respective projections of Pi−1, Pi+1 on the tangentplane of S at Pi (see Fig. 4). Thus∫ Sb

Sa

kg(s)ds =

b−1∑i=a+1

(π − ϕ(Pi)). (13)

4.1.2. Subregion constructionIn order to compute multiple K T values at different locations

on the surface, a dividing process is applied based on theparameterization of the surfaces. Constructing iso-parametriccurves with u,v values uniformly distributed in the parameterdomain, namely, S(ui , v)(ui = i/m, i = 0, 1 . . . m) and

S(u, v j ) (vi = j/n, j = 0, 1 . . . n) (see Fig. 5). Without loss ofgenerality, we can assume that m = n.

By applying the boundary representation structure, loopsmodeled as a substructure of the surface can be used fortrimming. This operation can be useful since a parametricsurface normally has a square domain, which leads to inevitablesingular points at corners. One challenge is to consider thecase where the discrepancy is mainly caused by the portion(s)trimmed off, which is no longer a valid part in the model.Minimum Enclosing Region (MER) is defined as a rectangularregion (see Fig. 5(b)) [uL , uU ], [vL , vU ] that can enclose all ofthe valid portions of the surface. It can be simply obtained bythe extreme values of the valid portions in parametric space.

4.1.3. Surface similarity measureAfter the previous steps, n×n four-sided patches/subregions

constructed from the surface are obtained, and the K T value foreach patch can be computed by the aforementioned approach.The complex 3D surface now is mapped to a 2D matrix whoseentry Ki j represents the K T value in a certain portion of thesurface, and it is also possible to construct an n × n greyscale image based on the matrix to provide a visual tool forstudying curvature. Suppose that two matrices S = G(A) andT = G(B) of equal size n × n, with each entry of K A

i j and

K Bi j , are calculated from surfaces A and B respectively. If K A

and K B are average values for K Ai j and K B

i j , then the correlationcoefficient R(S, T) can be defined as:

Rn(S, T) =

n∑i=1

n∑j=1

(K Ai j − K A)(K B

i j − K B)√n∑

i=1

n∑j=1

(K Ai j − K A)

2√

n∑i=1

n∑j=1

(K Bi j − K B)

2.(14)

It is necessary to assume that the parameter domain istransformed, such as r(u, v) = r(u, 1−v), but with each cornermatching another corner. For the 2 × 2 example in the Fig. 9,the input matrix compared with original one can have 8 setupsafter transformations (see Fig. 6). The corresponding diagonalcorners are always fixed, and the same situation applies tohigher dimensional matrix, which results in 8 different setups.Suppose that matrix T has the orientations as T(i)(i =

Fig. 5. (a) Generation of 10x10 subregions for full patch surface and (b) Minimum Enclosing Region (MER) of partial surface.


Fig. 6. Possible setups for finding maximum similarity level.

Fig. 7. Mapping of the original mesh to a 2D spherical coordinate.

1, 2, . . . 8), then the correlation coefficients are calculated as:

R(i)n (S, T) = Rn(S, T(i)), (i = 1, 2, . . . 8). (15)

The similarity measure for surfaces is then defined as:

d Sn (A, B) = 1 − maxin

{∣∣∣R(i)n (G(A), G(B))

∣∣∣} ,

(i = 1, 2, . . . 8). (16)

4.2. Similarity of discrete surface

4.2.1. PreprocessingThe discrete surfaces to be processed are assumed to be

in a triangular mesh, and the steps based on smooth surfacescannot be directly applied. The construction of the subregionsrequires a different approach since the target surfaces often haveirregular boundaries, and no u, v parameters can assist in thedividing process.

4.2.1.1. Define orientation. If the orientations of the model arenot well defined, then domain knowledge in the data acquisitionphase is useful to align the target surfaces, or an exhaustivesearch is required based on similarity measures computed ateach orientations [2]. In this paper, orientation is assumed tobe predetermined for all discrete surfaces, e.g., for a familyproducts, the orientations of the models are often fixed duringthe design phase.

4.2.1.2. Remove degenerate vertices. For a common vertexvi , the integral Gaussian curvature K T

i based on Eq. (11) canbe misleading due to numerical errors. These values do notrepresent the surface shape and will have significant effects onthe shape comparison. We define a threshold value Ke whileall of the vertices satisfying K T

i > Ke are considered asdegenerate. In this paper, Ke = 2π/3, and any vertices withK T

i > 2π/3 are not to be considered when computing surfaceshape similarity.

4.2.2. Construction of 2D spherical mappingAfter excluding the degenerate points, the vertices are

mapped to a 2D plane (x–y) (see Fig. 7). The result is a 2Dpoint cloud, and each vertex has coordinates (xi , yi ) with theassociated K T

i . It is straightforward to transform the verticesto spherical coordinates (ri , θi ) where the origin is definedas the geometric center of the point cloud (x, y) and ri =√

(xi − x)2 + (yi − y)2. The distribution of K Ti with reference

to the center of the point cloud is utilized to represent the shapeof the original mesh.

4.2.3. Similarity measure of a discrete surfaceThe vertices in spherical coordinates can be sorted by ri and

grouped based on the percentiles. After properly constructinga data structure of the discrete surface, the calculationinvolves O(n) computations to construct the 2D mapping andO(n log(n)) computations to construct the percentile contours.To divide the vertices into n concentric regions, the j th regioncontains the vertices with ri ∈ [r j , r j+1), while r j is the(

100( j−1)n + 1

)th percentile of distribution {ri }. Thus we obtain

n groups of vertices as {vi }i∈I1 , {vi }i∈I2 , . . . , {vi }i∈In whilei ∈ I j indicates vertex vi belongs to the j th group. The numberof vertices in each set I j , j = 1, 2, . . . , n is identical. Thisapproach guarantees that a certain proportion of the verticeswill be grouped together based on their distance to the center,without considering the scaling factor. For each region j ,the integral of Gaussian curvature K T

j is computed as the

summation of all the K Ti :

K Tj =

∑i∈I j

K Ti . (17)

A vector K = [K T1 , K T

2 , . . . K Tn ] is obtained, with each entry

representing the integral of curvature in each concentric region.Differentiation of the original meshes is reduced to comparingthe K values of a pair of meshes, which is computationallyequivalent to comparing two vectors. This proposed signaturefocuses on the local change of Gaussian curvature, which isoften ignored in previous shape distribution approaches. In thispaper, we define a distance function as:

d(K1, K2) = 1 − r(K1, K2), (18)

where r(K1, K2) is the Pearson correlation coefficient and d ∈

[0, 1). Similarly, a d value close to zero indicates that the twocompared meshes have a similar shape, and an extreme lowlevel of similarity is implied if d approaches to 1. It should be


Fig. 8. Car door example for shape differentiation.

Fig. 9. 2D images and 3D scanned data of single-use camera family.

noted that there can be numerous ways to compare 2D sphericalmappings. We only propose one distance function that simplymeasures the correlation between two vectors representing K T

jvalues in each concentric region. Empirically, we find that theresults are more consistent with human perception, when thenumber of concentric regions n, is chosen between 5 and 10.

5. Implementation

The proposed methodology is applied to a number ofexamples in different domains to test its effectiveness. Thesmooth surface in the CAD models is represented usingB-Splines, if no further description is given. The discretesurfaces are modeled as triangular meshes such as a STL fileformat. Some of the discrete surfaces are obtained by digitalscanning (3D Digital EScan system), and triangular meshes areconstructed based on the scanned point cloud. The proposedalgorithm is programmed in C++ and visualized using OpenGL.

5.1. Car door example

Fig. 8 shows a CAD model of a door from a toy cardesign (see Fig. 8(a)), which is modeled using B-Splines. Forcomparison, the model is transformed by uniform scaling (1:2)and mirroring (see Fig. 8(b)). Furthermore, a bulge is createdon the original model in the valid region (see Fig. 8(c)) or inthe trimmed-off region (see Fig. 8(d)). The minimum enclosingregion is u ∈ [0, 0.8], v ∈ [0, 1].

The result (see Table 1) shows that models in Figs. 8(a) and(b) have high d values, implying that the proposed methodologyis invariant of scaling and mirroring. For models in Figs. 8(a)

Table 1Result of car door example

d S(i, j) Whole surface comparison Partial surfacecomparison (MER)

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

(b) 0.01 – – 0.01 – –(c) 0.82 0.82 – 0.78 0.78 –(d) 0.67 0.67 0.93 0.07 0.07 0.88

Table 2The numerical result of the similarity matrix of single-use cameras

PD Zoom HQ PF Water Average d

PD 0.00 0.28 0.06 0.12 0.69 0.24Zoom 0.28 0.00 0.45 0.67 0.65 0.48HQ 0.06 0.45 0.00 0.11 0.99 0.33PF 0.12 0.67 0.11 0.00 0.80 0.38Water 0.69 0.65 0.99 0.80 0.00 0.80

and (d), d values become relatively low when comparingthe entire patch. The proposed methodology provides a moresubstantial result (0.07) without whole surface comparison(by the defined MER), which indicates the higher level ofresemblance of these two models in Fig. 8(a) and (d).

5.2. Differentiation of scanned models

The cover surfaces of five Kodak single-use cameras werescanned followed by construction of the triangular surface(see Fig. 9). Table 2 and Fig. 10 show the results based onthe proposed methodology. Two models, Zoom and Water


Fig. 10. Similarity matrix for single-use cameras. Lightness indicates thedissimilarity between models.

& Sports, can be easily differentiated by the proposedmethodology, since their geometries are unique compared tothe other three. It should be noted that the Zoom has a stretchedlens due to its functional requirements, but its 2D photoimage is not sufficient to explicitly show that. To study thedifferentiation potential of one model within the whole family,the average distance measure against all the other models canbe used as an index for the level of commonality within thefamily of products, and the two models mentioned above havesubstantially higher values (0.48 and 0.80).

From these results, the remaining three camera models (PlusDigital, HQ and Power Flash) have small distance measure(d < 0.2) between each other, which is consistent withhuman perception. This finding implies that it is not trivial forcustomers to differentiate any pair of these three models withregard to their cover surface shapes. The average values of these

three models are also low (0.24, 0.33 and 0.38) compared to theother two models (0.48 and 0.80).

5.3. Mining a CAD database

In order to investigate further integration with a CADdatabase, such as an online product catalog, the proposedmethodology is applied to evaluate a number of productdesigns from different sources to show its performance ondesign differentiation. The methodology can provide valuableinformation about the designs, from not only designed userowned models but also counterparts from competitors.

A computer pointing device is scanned (see Fig. 11), andthe associated mesh is used as the base model. Four pointingdevices from a CAD design database (see Fig. 12), which arefrom the same category as the scanned product, are applied forcomparison. Based on the previous shape histogram approach,the scanned model is significantly varied from those of the CADmodels (see Fig. 13), and no conclusive results can be obtained.One possible cause is that the data from the database representsthe whole model compared to the partial surface obtained fromscanning. Furthermore, local geometry, such as the dramaticcurvature change in Model D, is not considered in shapehistogram approach. This limits the potential to differentiatethese models in which detailing is important.

Fig. 14 shows the result based on the proposed similaritycomparison methodology. The spherical mapping of theoriginal meshes and the percentile contours are plotted. ModelC is considered as the design most similar in shape (d = 0.06)

compared to the base (scanned) model. The other extreme caseis model D (d = 0.98), and it is considered to be easilydifferentiated from the original scanned model. Assume thatthe four CAD models represent the existing product family

Fig. 11. (a) Scanning a pointing device and (b) the constructed mesh and (c) the 2D spherical mapping.

Fig. 12. CAD models of pointing devices in database for shape differentiation.


Fig. 13. Designs of pointing devices in CAD database and the associated D2 shape histograms.

Fig. 14. Results of the proposed methodology applied to the CAD database.

or products in a category; the result implies that introducinga design such as the scanned model may result in visualconfusion especially compared with Model C. Modification onshape design or additional enhancement on other differentiationcriteria, such as adding new functions, is recommended.

5.4. Robustness

5.4.1. Number of trianglesTo show the robustness of the proposed methodology,

one single-use model (HQ) is scanned twice with different


Fig. 15. Comparison of the results by changing the mesh size.

scanner settings. Another mesh consisting of 30 150 trianglesis obtained compared to the original one of 17 995 triangles.Despite the mesh size increase, the distance values againstothers have similar patterns as shown in Fig. 15. Limitedvariations can be found, which can be due to the differencesin the manual setup of the scanner. The only exception is theresult when compared with model Water. A significant changeof d value can be found when the size of HQ is changed. Thisimplies that the d value becomes more sensitive to size changewhen the two models to compare are quite different in shape. Assuch, the meshes to be compared are suggested to be modeledin triangles of similar size, and a logarithmic transformation canbe introduced on value d to minimize this effect.

5.4.2. Additional factorsThe proposed methodology is invariant to uniform scaling

and rigid transformation, due to the properties of Gaussiancurvature. To be eligible for practical use, robustness of theproposed methodology should also be considered. Three pairsof models are randomly chosen from the single-use cameramodels. Noise is added to the original data by perturbingthe individual K T values by 10%, or randomly deleting 10%of the vertices, while all the other conditions remain thesame. Factorial experiment design of 3 factors is conductedby calculating corresponding d values with settings listedin Table 3.

Table 4 shows the results from the experiment, wherePair is the only main factor identified as significant (p =

0.000). This finding implies that the proposed methodology isable to distinguish different pairs of input surfaces, which isalso supported in the previous sections. Moreover, the othertwo main factors, Noise and Parameter n, are considered asinsignificant (p = 0.581 and p = 0.851). It can be concludedthat the proposed methodology is insensitive to added noiseor varying the parameters. The effect of Noise coupled withParameter n is also considered as significant (p = 0.048). Byreviewing the original data, it is found that increasing n usuallyresults in some increase in the distance value d , since moredetails of the curvature values are compared by constructingmore subregions. However, this trend does not exist in the dataafter 10% reduction (Level 3 of Noise factor). This findingimplies that reduction of the original data may affect thedifferentiation process due to missing details, although theeffect is limited (for data after 10% reduction).

Table 3Settings of the experiment design

Factor Level Description

Pairs 1, 2, 3 Three random pairs of modelsNoise 1, 2, 3 No Noise, 10% Perturbation, 10% ReductionParameter 1, 2 n = 5 or n = 10

Table 4Estimated effects and coefficients

Term DF SS F p

Pair 2 2.383 3545.585 0.000*Noise 2 0.000 0.624 0.581Parameter n 1 0.000 0.040 0.851Pair*Noise 4 0.001 0.758 0.603Pair*Parameter n 2 0.005 7.724 0.042*Noise*Parameter n 2 0.005 7.148 0.048*Error 4 0.001

Total 17 2.394

6. Closing remarks

In this paper, a novel methodology is proposed to comparefreeform surfaces. Unlike some previous approaches thatmainly consider the global difference between models, theproposed methodology focuses on the curvature change in localgeometry, which is critical to differentiate among models withcommon contours. Furthermore, the proposed methodologyfocuses on comparisons by investigating the models in eithersmooth or discrete format. This feature allows the proposedmethodology to be widely applied to different systems, in orderto support different activities in product design and 3D dataanalysis. Digitally obtained models can be combined with CADdesigns, so that users can conduct benchmarking with existingmodels in the database. With the explosion of 3D data, theproposed methodology can be applied as an agent to comparethe novelty of designs or enhance the differentiation process ofthe acquired 3D data.

The proposed methodology is also invariant to rigidtransformation and scaling, two process that are routinelyused when designing consumer products. With regard toits performance, the result is also consistent with humanperception as shown in the examples, and the robustness testdemonstrated that the proposed methodology is insensitive tonoise in section. Computation of the statistical index is trivial,and all of the computations in the examples are completedwithin seconds.

The target surfaces/meshes to be compared are assumedto be constructed properly in order to obtain curvaturevalues which are fairly uniformly sampled. Extremely irregularsurfaces can be a challenge and may require smoothing oroptimization in advance. Although it is often suggested thatless curvature change is aesthetically preferred for a designedsurface (i.e., lower K values), the aim in this study is toprovide a discriminate analysis based on curvature instead ofquantifying the level of beauty. As a matter of fact, it is not


sufficient to conclude that a design of lower K values is morepreferred. The number of subregions or concentric regions nranges from 5 to 10 in this paper, which is considered to matchbest with human perception empirically. This implies that onlya limited amount of curvature change can be recognized ina design, while a high frequency of curved features may notsupport design differentiation much. More domain knowledgeis required to decide the best size of n for appropriate shapedifferentiation.

Finally, it should be noted that the proposed methodologyis for freeform surfaces, wherein the distribution of Gaussiancurvature (non-zero) can be interpreted as the embedded shape.As such, it is not applicable for studying prismatic modelsor models that have complicated topology with plain localgeometry. For discrete surfaces, the Gaussian curvature that iscomputed is only an approximation, which is highly dependenton the quality of the triangulated surface. Large variationor patterns in the dimensions of these triangles may lead toincorrect conclusions.

Acknowledgements

Professor Simpson would like to acknowledge the supportby NSF under grant# IIS 0325402. Any opinions, findings andconclusions or recommendations presented in this paper arethose of the authors and do not necessarily reflect the views ofthe National Science Foundation. The authors thank Dr. FabriceAlizon for providing the scanned models.

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