A novel technique for multi-scale smoothing of a free-form 3-D surface is presented. Complete triangulated models of 3-D objects are constructed automatically and using a local parametrization technique, are then smoothed using a 2-D Gaussian filter. Our method for local parametrization makes use of semigeodesic coordinates as a natural and efficient way of sampling the local surface shape. The smoothing eliminates the surface noise together with high curvature regions such as sharp edges, therefore, sharp corners become rounded as the object is smoothed iteratively. Our technique for free-form 3-D multi-scale surface smoothing is independent of the underlying triangulation. It is also argued that the proposed technique is preferrable to volumetric smoothing or level set methods since it is applicable to incomplete surface data which occurs during occlusion. Our technique was applied to closed as well as open 3-D surfaces and the results are presented here

Khalili, N

Mokhtarian, F

Yuen, P

English

A novel technique for multi-scale smoothing of a free-form 3-D surface is presented. Complete tri-angulated models of 3-D objects are constructed automatically and using a local parametrization technique, are then smoothed using a 2-D Gaus-sian filter. Our method for local parametrization makes use of semigeodesic coordinates as a natu-ral and efficient way of sampling the local surface shape. The smoothing eliminates the surface noise together with high curvature regions such as sharp edges, therefore, sharp corners become rounded as the object is smoothed iteratively. Our technique for free-form 3-D multi-scale surface smoothing is independent of the underlying triangulation. It is also argued that the proposed technique is pre-ferrable to volumetric smoothing or level set meth-ods since it is applicable to incomplete surface data which occurs during occlusion. Our technique was applied to closed as well as open 3-D surfaces and the results are presented here. 

Peter Yuen

Farzin Mokhtarian

Nasser Khalili

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Multi-scale 3-d surface description: open and closed surfaces

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Multi-Scale 3-D Surface Description: Open and Closed SurfacesPeter Yuen, Farzin Mokhtarian and Nasser KhaliliCentre for Vision, Speech, and Signal ProcessingDepartment of Electronic and Electrical EngineeringUniversity of Surrey, Guildford, England GU2 5XH, UKE-mail: P.Yuen@ee.surrey.ac.ukWeb: http://www.ee.surrey.ac.uk/Research/VSSP/demos/css3d/index.htmlAbstractA novel technique for multi-scale smoothing ofa free-form 3-D surface is presented. Complete tri-angulated models of 3-D objects are constructedautomatically and using a local parametrizationtechnique, are then smoothed using a 2-D Gaus-sian ﬁlter. Our method for local parametrizationmakes use of semigeodesic coordinates as a natu-ral and eﬃcient way of sampling the local surfaceshape. The smoothing eliminates the surface noisetogether with high curvature regions such as sharpedges, therefore, sharp corners become rounded asthe object is smoothed iteratively. Our techniquefor free-form 3-D multi-scale surface smoothing isindependent of the underlying triangulation. It isalso argued that the proposed technique is pre-ferrable to volumetric smoothing or level set meth-ods since it is applicable to incomplete surface datawhich occurs during occlusion. Our technique wasapplied to closed as well as open 3-D surfaces andthe results are presented here.1 IntroductionThis paper introduces a new technique for multi-scaleshape description of free-form 3-D surfaces representedby polygonal or triangular meshes. Complete 3-D mod-els of test objects have been used in our experiments.Such models can be constructed through automatic fu-sion of range images of the object obtained from diﬀer-ent viewpoints [3].Multi-scale descriptions have become very commonin computer vision since they oﬀer added robustnesswith respect to noise and object detail as well as providefor more eﬃcient processing. The multi-scale techniqueproposed here can be considered a generalization of ear-lier multi-scale representation theories proposed for 2-Dcontours [7, 8] and space curves [5]. However, the the-oretical issues are signiﬁcantly more challenging whenworking on free-form 3D surfaces.In our approach, diﬀusion of the surface is achievedthrough convolutions of local parametrizations of thesurface with a 2-D Gaussian ﬁlter. Semigeodesic coordi-nates [2] are utilized as a natural and eﬃcient way of lo-cally parametrizing surface shape. The most importantadvantage of our method is that unlike other diﬀusiontechniques such as volumetric diﬀusion [4] or level setmethods [11], it has local support and is therefore appli-cable to partial data corresponding to surface-segments.This property makes it suitable for object recognitionapplications in presence of occlusions.The organization of this paper is as follows. Sec-tion 2 gives a brief overview of previous work on 3-Dobject representations including the disadvantage(s) ofeach method. Section 3 describes the relevant theoryfrom diﬀerential geometry and explains how a multi-scale shape description can be computed for a free-form3-D surface. Section 4 covers implementation issues en-countered when adapting semigeodesic coordinates co-ordinates to 3-D triangular meshes. Section 5 presentsdiﬀusion results and discussion. Section 6 contains theconcluding remarks.2 Literature SurveyPolyhedral approximations [1] ﬁt a polyhedral objectwith vertices and relatively large ﬂat faces to a 3-D ob-ject. Their disadvantage is that the choice of verticescan be quite arbitrary which renders the representationnot robust. Smooth 3-D splines [13] can also be ﬁtted to3-D objects. Their shortcomings are that the choice ofknot points is again arbitrary and that the spline param-eters are not invariant. Generalized cones or cylinders[12] as well as geons [9] approximate a 3-D object usingglobally parametrized mathematical models, but theyare not applicable to detailed free-form objects. Multi-view representations [10] are based on a large numberof views of a 3-D object obtained from diﬀerent view-points, but diﬃculties can arise when a non-standardview is encountered. In volumetric diﬀusion [4] or levelset methods [11], an object is treated as a ﬁlled area orvolume. The object is then blurred by subjecting it tothe diﬀusion equation. The boundary of each blurredobject can then be deﬁned by applying the Laplacianoperator to the smoothed area or volume. The majorshortcoming of these approaches is lack of local support.In other words, the entire object data must be available.This problem makes them unsuitable for object recog-nition in presence of occlusion. A form of 3-D surfacesmoothing has been carried out in [14] but this methodhas drawbacks since it is based on weighted averagingusing neighboring vertices and is therefore dependent onthe underlying triangulation.3 Semigeodesic Parametrizationon a 3-D SurfaceA crucial property of 2-D contours and space curves (or3-D contours) is that they can be parametrized globallyusing the arclength parameter. However, free-form 3-Dsurfaces are more complex. As a result, no global co-ordinate system exists on a free-form 3-D surface whichcould yield a natural parametrization of that surface.Indeed, studies of local properties of 3-D surfaces arecarried out in diﬀerential geometry using local coordi-nate systems called curvilinear coordinates or Gaussiancoordinates [2]. Each system of curvilinear coordinatesis introduced on a patch of a regular surface referred toas a simple sheet. A simple sheet of a surface is obtainedfrom a rectangle by stretching, squeezing, and bendingbut without tearing or gluing together. Given a para-metric representation r = r(u,v) on a local patch, thevalues of the parameters u and v determine the positionof each point on that patch.3.1 Geodesic LinesBefore the semigeodesic coordinates can be described,it is necessary to deﬁne geodesic lines on a regular 3-Dsurface. The following deﬁnitions are useful [2]:Deﬁnition. A geodesic line or a geodesic of a surfaceis a curve whose geodesic curvature is zero at every point.Geodesic curvature is the magnitude of the vector ofgeodesic curvature.Deﬁnition. The vector of geodesic curvature of acurve C lying on a surface S at a point P on C is ob-tained by projecting the curvature vector of C at P onthe tangent plane to S at P.Deﬁnition. The curvature vector of a curve C atpoint P is of the same direction as the principal normalvector at P and of length equal to the curvature of thecurve at P.Deﬁnition. The principal normal vector of a curveC at point P is perpendicular to C at P and lies inthe osculating plane at P. The plane with the highestpossible order of contact with the curve C at point P iscalled the osculating plane at P.The following crucial property of geodesic lines is ac-tually utilized to construct geodesics on 3-D triangularmeshes:Minimal property of geodesics: An arc of ageodesic line C passing through a point P and lyingentirely in a suﬃciently small neighborhood of a pointP of a surface S of class C2 is the shortest join of Pwith any other point of C by a curve lying in the neigh-borhood.3.2 Semigeodesic CoordinatesSemigeodesic coordinates can be constructed in the fol-lowing way at a point P on a surface S of class C2:• Choose a geodesic line C through point P in anarbitrary direction.• Denote by v the arclength parameter on C, suchthat P corresponds to the value v = 0.• Take further through every point of C the geodesicline L perpendicular to C at the correspondingpoint.• Denote by u the arclength parameter on L.The two parameters u and v determine the positionof each point in the domain swept out by these geodesiclines. It can be shown that in a suﬃciently small neigh-borhood of the point P, semigeodesic coordinates canalways serve as curvilinear coordinates in a regular para-metric representation of S [2]. The orthogonal carte-sian coordinates in the plane are a special case of semi-geodesic coordinates on a ﬂat surface.3.3 Gaussian Smoothing of a 3-D sur-faceThe procedures outlined above can be followed to con-struct semigeodesic coordinates at every point of a 3-Dsurface S. In case of semigeodesic coordinates, localparametrization yields at each point P:r(u,v) = (x(u,v), y(u,v), z(u,v)).The new location of point P is given by:R(u,v,σ) = (X(u,v,σ), Y(u,v,σ), Z(u,v,σ))whereX(u,v,σ) = x(u,v) ⊗ G(u,v,σ)Y(u,v,σ) = y(u,v) ⊗ G(u,v,σ)Z(u,v,σ) = z(u,v) ⊗ G(u,v,σ)G(u,v,σ) =12πσ2 e−(u2+v2)2σ2and ⊗ denotes convolution.This process is repeated at each point of S and thenew point positions after ﬁltering deﬁne the smoothedsurface. Since the coordinates constructed are valid lo-cally, the Gaussian ﬁlters always have σ = 1.3.4 Multi-Scale Description of a 3-Dsurface via diﬀusionIn order to achieve multi-scale descriptions of a 3-D sur-face S, it is smoothed according to the process describedin section 3.3. The smoothed surface is then consideredas the input to the next stage of smoothing. This pro-cedure is then iterated many times to obtain multi-scaledescriptions of S. This process is equivalent to diﬀusionsmoothing∂S∂t= H nsince the Gaussian function satisﬁes the heat equation.In the equation above, t is time, H is mean curvature,and n is the surface normal vector. t can be regardedas the number of iterations.4 Implementation on a 3-D Tri-angular MeshThe theory explained in the previous section must beadapted to a 3-D triangular mesh. Semigeodesic coordi-nates involve construction of geodesic lines. Clearly thesegment of a geodesic that lies on any given triangle isa straight line. Two situations must be considered:• Extension of a geodesic when it intersects an edge• Extension of a geodesic when it intersects a vertexFigure 1 illustrates the ﬁrst case and ﬁgure 2 illus-trates the second case.Theorem 1 addresses the ﬁrst situation.Theorem 1. Suppose a geodesic intersects an edgee shared by triangles T1 and T2. The extension of thisgeodesic beyond e is obtained by rotating T2 about eso that it becomes co-planar with T1, extending thegeodesic in a straight line on T2, and rotating T2 aboute back to its original position.Theorem 2 addresses the second situation.Theorem 2. Suppose a geodesic arrives at a vertexV of the mesh. Deﬁne the normal vector n at V asthe average of the surface normals of all the trianglesincident on V weighted by the incident angle. Let Qbe the plane formed by the geodesic incident on V andn. The extension of this geodesic beyond V is found byintersecting Q with the mesh.Proofs of these theorems are given in [6].RotationAxis for rotationDG’GA segment of thegeodesic LineThe triangle GDC rotatesΨ1the geodesic LineA segment ofΨIat the common edge CDto produce a flatsurface G’DECB’BCφRotated segment(2) On the flat surface G’DEC, the straight Line between Point A and Point B’ is a geodesic line.(1) Angle    is constant before and after the rotation.Figure 1: A geodesic intersects a triangle edgeSecond neighbourWclockwiseAnti-αin iLN(B)FCDPrPSS B 001 1IFigure 2: A geodesic intersects a triangle vertex4.1 Implementation of semigeodesic co-ordinatesSemigeodesic coordinates are constructed at each ver-tex of the mesh which becomes the local origin. Thefollowing procedure is employed:1. Construct a geodesic from the origin in an arbitrarydirection such as the direction of one of the incidentedges.2. Construct the other half of that geodesic by ex-tending it through the origin in the reverse direc-tion using the procedure outlined in theorem 2.3. Parametrize that geodesic by the arclength param-eter at regular intervals to obtain a sequence ofsample points.4. At each sample point on the ﬁrst geodesic, con-(a) A 3-D triangular meshArbitrary Geodesic LineSecond family of LinesBWhere "o"s are the semigeodesic coordinates and "B" is the current vertex.(b) Semigeodesic coordinatesFigure 3: A triangular mesh and semigeodesic coordi-nates for an area at the back of the head at about eyelevelstruct a perpendicular geodesic and extend it inboth directions.5. Parametrize each of the geodesics constructed inthe previous step by the arclength parameter atregular intervals.Figure 3 shows a triangular mesh and an example ofsemigeodesic coordinates.4.2 Semigeodesic coordinates on opensurfacesQuite often, due to occlusion or complex object shape, itis not possible to construct complete and closed surfaces.As a result, the algorithm described above should bemodiﬁed to make it also applicable to open surfaces.The algorithm for smoothing an open surface is deﬁnedin the following way:1. Grid construction and smoothing at internal ver-tices is carried out as on closed surfaces. Anygeodesic line that reaches the boundary will stop.The last sample point at or near the boundary willbe duplicated until the grid is ﬁlled. Likewise, ifsome geodesic lines can not be constructed, the lastgeodesic line near the boundary will be duplicateduntil the grid is ﬁlled.2. If the vertex V of triangle T resides on the bound-ary, measure the angle α between the two edges ofT that are incident on V . Choose the ﬁrst geodesicline as the bisector of α. Only half of the ﬁrstgeodesic line is constructed because the other halffalls outside the surface boundary.3. At the same vertex, construct another geodesic lineperpendicular to the ﬁrst one.4. One of those geodesic lines might soon intersect theboundary, so compare the lengths of those lines andchoose the longer one. This allows the maximumsize grid to be constructed.5. Construct the second family of geodesic lines asperpendicular to the longer geodesic line deter-mined above.6. As before, any geodesic line that reaches theboundary will stop, and the last sample point ator near the boundary will be duplicated until thegrid is ﬁlled.5 Results and DiscussionThe smoothing routines were implemented entirely inC++. Complete triangulated models of 3-D objectsused for our experiments are constructed at our cen-ter [3]. Each iteration of smoothing on a surface with1000 vertices took about 0.5 second of CPU time onan UltraSparc 170E. In order to experiment with ourtechniques, both simple and complex 3-D objects withdiﬀerent numbers of triangles were used. The ﬁrst testobject was a simple cube with 98 vertices. The smooth-ing results using semigeodesic coordinates are shown inFigure 4. The original cube becomes rounded iterativelyand ﬁnally changes to a sphere after ﬁve iterations.The second test object was a foot with 500 vertices.The smoothing results are shown in ﬁgure 5. The footalso becomes rounded iteratively and evolves into an el-lipsoidal shape after 100 iterations. We then examinedour technique with more complex 3-D objects. Figure6 shows the third test object which was a telephonehandset with 1000 vertices. Notice that the surfacenoise is eliminated iteratively with the object becomingsmoother gradually and the handset almost disappears(a) Original(b) 1 iteration(c) 5 iterationsFigure 4: Smoothing of the Cubeafter 15 iterations. The fourth test object was a chairwith 1000 vertices as shown in ﬁgure 7 where again thelegs of the chair disappear as was seen for the telephonehandset. Therefore, from these results one can concludethat the proposed technique is eﬀective in eliminatingthe surface noise as well as removing surface detail andit is also independent of the underlying triangulation.The result is gradual simpliﬁcation of object shape. An-imation of surface diﬀusion can be observed at:http://www.ee.surrey.ac.uk/Research/VSSP/demos/css3d/Our smoothing technique was also applied to a num-ber of open/incomplete surfaces. Figure 8 shows thesmoothing results obtained on a part of the foot object(a) Original(b) 10 iterations(c) 30 iterations(d) 100 iterationsFigure 5: Diﬀusion of the Footshown in ﬁgure 5. Figure 9 shows the results obtainedon a part of the telephone handset shown in ﬁgure 6.This object also has a triangle removed in order to gen-erate an internal hole. Figure 10 shows the smoothingresults obtained on the lower part of the chair objectshown in ﬁgure 7. Figure 11 shows smoothing resultsobtained on a partial rabbit.6 ConclusionsA novel technique for multi-scale smoothing of a free-form triangulated 3-D surface was presented. This wasachieved by convolving local parametrizations of the sur-face with 2-D Gaussian ﬁlters iteratively. Our methodfor local parametrization made use of semigeodesic coor-dinates as natural and eﬃcient ways of sampling the lo-cal surface shape. The smoothing eliminated the surface(a) Original(b) 2 iterations(c) 6 iterations(d) 12 iterations(e) 15 iterationsFigure 6: Diﬀusion of the telephone handset(a) Original(b) 2 iterations(c) 3 iterations(d) 4 iterationsFigure 7: Smoothing of the Chair(a) Original(b) 2 iterations(c) 5 iterationsFigure 8: Diﬀusion of the partial Footnoise and small surface detail gradually, and resulted ingradual simpliﬁcation of object shape. The method wasindependent of the underlying triangulation. Our ap-proach is preferrable to volumetric smoothing or levelset methods since it is applicable to incomplete surfacedata which occurs during occlusion.References[1] O D Faugeras and M Hebert. The representation, recog-nition, and locating of 3-d objects. International Jour-nal of Robotics Research, 5(3):27–52, 1986.[2] A Goetz. Introduction to Diﬀerential Geometry.Addison-Wesley, 1970.[3] A Hilton, A J Stoddart, J Illingworth, and T Windeatt.Implicit surface-based geometric fusion. Computer Vi-sion and Image Understanding, 69(3):273–291, 1998.[4] J J Koenderink. Solid Shape. MIT Press, Cambridge,MA, 1990.[5] F Mokhtarian. A theory of multi-scale, torsion-basedshape representation for space curves. Computer Visionand Image Understanding, 68(1):1–17, 1997.[6] F Mokhtarian, N Khalili, and P Yuen. Multi-scale 3-dfree-form surface smoothing. In Proc British MachineVision Conference, pages 730–739, 1998.(a) Original(b) 3 iterations(c) 5 iterationsFigure 9: Diﬀusion of the partial telephone handset[7] F Mokhtarian and A K Mackworth. Scale-based de-scription and recognition of planar curves and two-dimensional shapes. IEEE Trans Pattern Analysis andMachine Intelligence, 8(1):34–43, 1986.[8] F Mokhtarian and A K Mackworth. A theory of multi-scale, curvature-based shape representation for planarcurves. IEEE Trans PAMI, 14(8):789–805, 1992.[9] M Pilu and R Fisher. Recognition of geons by paramet-ric deformable contour models. In Proc ECCV, pages71–82, Cambridge, UK, 1996.[10] M Seibert and A M Waxman. Adaptive 3-d objectrecognition from multiple views. IEEE Trans PatternAnalysis and Machine Intelligence, 14:107–124, 1992.[11] J A Sethian. Level Set Methods. Cambridge UniversityPress, 1996.[12] B I Soroka and R K Bajcsy. Generalized cylinders fromserial sections. In Proc IJCPR, 1976.[13] A J Stoddart and M Baker. Reconstruction of smoothsurfaces with arbitrary topology adaptive splines. InProc ECCV, 1998.[14] G Taubin. Curve and surface smoothing without shrink-age. In Proc International Conference on Computer Vi-sion, pages 852–857, 1995.(a) Original(b) 2 iterations(c) 3 iterations(d) 4 iterationsFigure 10: Smoothing of the partial Chair(a) Original(b) 2 iterations(c) 5 iterations(d) 9 iterationsFigure 11: Smoothing of the rabbit head

Multi-scale 3-D Surface Description: Open and Closed Surfaces

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