AN ENGINEERING SHAPE BENCHMARK FOR 3D MODELS
Proceedings of IDETC/CIE 2005ASME 2005 International Design Engineering Technical Conferences &Computers and Information in Engineering ConferenceSeptember 24-28, 2005, Long Beach, California, USADETC2005/CIE-85612AN ENGINEERING SHAPE BENCHMARK FOR 3D MODELSNatraj IyerSubramaniam JayantiKarthik RamaniPurdue Research and Education Center for Information Systems in Engineering (PRECISE)585 Purdue MallSchool of Mechanical EngineeringPurdue University, West Lafayette INABSTRACTThree dimensional shape searching is a problem of currentinterest in several different fields, especially in the mechanicalengineering domain. There has been a large body of work indeveloping representations for 3D shapes. However, there hasbeen limited work done in developing domain dependentbenchmark databases for 3D shape searching. In this paper, wepropose a benchmark database for evaluating shape basedsearch methods relevant to the mechanical engineering domain.Twelve feature vector based representations are comparedusing the benchmark database. The main contributions of thispaper are development of an engineering shape benchmark andan understanding of the effectiveness of different shaperepresentations for classes of engineering parts.1. INTRODUCTIONShape-based retrieval of 3D data has applications invarious disciplines such as computer vision [1], artifactsearching [2], molecular biology [3], and chemistry [4]. The 3Dshape searching area has so far been dominated by research incomputer vision and computer graphics, where researchershave extensively studied the “shape matching” problem.Extensive reviews of these methods are available in [5-7].Recently, there has been a lot of interest in shape-based searchmethods for the mechanical engineering domain [8-16]. In thispaper, we will focus mainly on a benchmark for searchingoverall shape of engineering objects - the 3D EngineeringShape Benchmark (ESB). Based on a classification of 801engineering components, we have evaluated 12 different shapedescriptors that are described in Section 2 of this paper.One of the few publicly available benchmarks for 3Dmodels includes the Princeton Shape Benchmark (PSB) [15].However, models in the PSB represent mostly multimediaobjects for computer graphics and vision applications. Ourfocus in this paper is on developing a benchmark specificallytailored towards the mechanical engineering CAD domain. Ourhypothesis in the development of a benchmark database forengineering is that shapes in the engineering domain havedifferent characteristics than multimedia shapes such as trees,humans and animals. Two key differentiating characteristics ofengineering shapes as compared to multimedia shapes are: (a)engineering shapes have high genus as compared to multimediashapes, and (b) distances between multimedia shapes are moreapparent as compared to engineering shapes. Engineeringshapes often have high genus and contain important features ofvarious types, including holes, tunnels, cavities, ribs, andhelixes. Their numbers as well as relative positioning are animportant factor in the resemblance of two shapes [33], unlikemultimedia where the overall shape is more important. As aresult, representations developed for multimedia objects maynot perform well for engineering objects. For example,structure-preserving representations in multimedia such asshock graphs [34] and Reeb graphs [17] are widely used andwork well for multimedia shapes of Genus-0. However, issuesrelated to topological sensitivity of Reeb graphs leads tosignificant number of false positives in engineering databases[17].We hypothesize that the performance of various shaperepresentations will be different than the results presented usingPSB. The National Design Repository developed by Regli et al.[11] provides a large database of engineering parts in variousCAD formats. However, currently the repository contains noclassification scheme analogous to the PSB database. Anobvious difficulty with creating a benchmark database is that nobenchmark can encompass the whole range of naturallyoccurring or man-made shapes. This is especially true in theengineering domain since the very nature of engineeringdesign, and the increasing complexity of product design haveforced engineers to design and manufacture increasinglycomplex and creative shapes. In addition, it is difficult tocategorize engineering shapes into semantic classes. Forexample, it is easy to see that dinosaurs and humans belong to1Copyright 2005 by ASME
different categories of shapes, but it is more difficult tocharacterize the shapes in Figure 1 into different categories.FlangeGimble RingLock NutFigure 1: Examples of parts that have similar shapebut different functions.Figure 2: Parts from the “Beds” category from thePrinceton Shape Benchmark ([18]).Another limitation with using PSB for the engineeringdomain is that PSB classifies models primarily on the basis offunction, and secondarily based on shape. Most objects createdin the multi-media domain can be classified into a categorysuch as “bed”, “tree” or “airplane” purely based on theirfunction; however, in the engineering domain the existence ofmany varieties of semi-standard and one-of-a-type componentsmakes it impossible to give names to objects or to classify theminto different functional categories. For example, in Figure 2beds with different geometric shapes are in the same category.However, for the engineering domain a primarily functionbased classification does not seem logical because parts withdifferent functions may have similar shape as seen in Figure 1.For example, in Figure 1, gimble rings are used in lightingfixtures for fastening; lock nuts are used primarily for locking,while flanges are used for connecting two components. Onecannot always explicitly state the function of an engineeringcomponent purely based on its shape. As a result, we believethat a function-based classification for the engineering domainis a daunting proposition.Another major difference between a multimedia databaseand an engineering database is the motivation behind the searchprocess. In the multimedia domain, the search is performedprimarily for reusing the models for different scene creation.On the other hand, in the engineering domain, search mayallow designers to not only reuse the 3D CAD model but alsoassociated information (such as manufacturing and analyses)thereby reducing product development time. As a result, thesearch system must be capable not only of distinguishingbetween overall shapes, but also consider manufacturing andlocal shape features in its similarity metric.The rest of this paper is organized as follows: Section 2presents a brief overview of shape representations that arecompared using our database; Section 3 describes theconstruction and evaluation of our ESB database along withevaluation methodologies; Section 4 presents a detailedanalysis of results along with discussion, and finally Section 5presents the conclusions.2. SHAPE REPRESENTATIONSA comprehensive review of various shape representationsand search techniques for 3D shapes is available in [5-7]. Basedon these papers existing shape representations can be classifiedinto two classes: feature vector based and topology based.Topology based representations extract information from thetopology of a part such as eigenvalues [8, 12], skeletal graphs[9, 12] and Reeb graphs [16] for comparison. In this paper wehave only evaluated representations based on feature vectors.Our future work will include benchmarking of topology basedmethods for 3D shape search. We describe below the twelvefeature vectors that we have used for benchmarking against theESB.2.1. Moment Invariants (MI)The three second-order moment invariants for the modelare stored as a feature vector. Moment invariants are by natureindependent of orientation. For every voxel in the modeltranslation, rotational and scale invariant second order momentsare calculated as described below.κ lmn are calculated as described in Eq.(1) as: x ylκ lmn mz n ρ ( x, y, z ) dx dy dz l m n 253µ000(1)where µlmn are central moments after translation, given byEq. (2) as:µlmn ( x xˆ ) ( y yˆ )lm( z zˆ ) n ρ ( x, y, z ) dxdydz l , m, n 1, 2,3,.(2)The integrals above are approximated by summation of thecontribution of every voxel to the moment. Since thecharacteristic function is invariant under rotation, thecharacteristic function of the matrix M (Eq. (3)) of translationand scale invariant second order moments is RST (RotationScale-Translation) invariant.κ110κ101 κ 200 Λ (3)Μ κ110κ 020 Λκ 011 κ101κ 011κ 002 Λ After evaluating the characteristic function for this matrix,the three moment invariants that are calculated are described inEq. (4).I1 κ 200 κ 020 κ 0022I 2 κ 002κ 200 κ 002κ 020 κ 200κ 002 κ10122 κ 011 κ110(4)I 3 κ 200κ 020κ 002 2κ110κ 011κ101 κ κ2101 02022κ 200 κ110κ 002 κ 0112.2. Principal Moments (PM)The principal moments for a 3D model are the threeeigenvalues µ xx µ yy µ zz of the moment matrix M asshown below in Eq. (5).2Copyright 2005 by ASME
µ 200M µ110 µ101µ110µ020µ011µ101 µ xxµ011 0µ002 00µ yy00 0 µ zz (5)2.3. Spherical Harmonics (SH)Spherical Harmonics are a decomposition of a sphericalfunction by finding the Fourier transform on a sphere [16]. Thetheory of spherical harmonics says that any spherical functionf (θ , φ ) can be decomposed as the sum of its harmonics as seenin Eq. (6):f (θ , φ ) almYl m (θ , φ )l 0 m l(6)0 θ π , 0 φ 2π2.8. 2.5D Harmonics (2.5D)2.5D Harmonics is a new feature vector proposed in [18]for representing 2D engineering drawings and 3D models. Inthis work, 3D model is first converted into a set of 2D viewsthrough a robust pose estimation algorithm described in [1819]. The intuition for transforming the problem to 2D space isthat many engineering shapes are created using orthographicprojections and hence are amenable to orthographic projections.Each 2D view is represented as a spherical function bytransforming it from 2D space into 3D space. Then a fastspherical harmonics transformation is employed to get arotation invariant descriptor.where alm are the Fourier coefficients and Yl m (θ , φ ) are thesolutions to the normalized Laplace’s equation in sphericalcoordinates. The spherical harmonic coefficients can be used toreconstruct an approximation of the underlying object atdifferent levels. Similar to moments, a partial yet accuratedescription of the part can be obtained by using a limited subsetof Fourier coefficients. Intuitively, we expect this method toperform especially well for objects with radial symmetry,because of the spherical decomposition.2.4. Surface Area and Volume (SAV)In a general shape-based search system, shaperepresentations are required to be independent of translation,rotation and size. However, in the mechanical engineeringdomain, the surface area and volume of a component haveserious implications on the manufacturability of an object. Forthe same volume, thin-walled components such as manifoldsand tubular parts often have higher surface area for the samevolume compared to prismatic components. Due to theirrelevance to design and manufacturing we include theserepresentations in our tests.2.5. Surface Area to Volume Ratio (SVR)In addition to SAV, we tested a separate feature vectorsurface area to volume ratio (SVR). Our hypothesis was thatthis feature will distinguish between thin walled and prismaticcomponents and can be used to prune the database using amulti-step search approach.2D viewBounding Box Bounding SphereRay Casting3D representationof 2D drawingFigure 3: Procedure of converting a 2D view into the2.5D Harmonics representation.Thus, the shape searching problem is reduced to severalsimple steps, such as sampling, normalization, and distancecomputation between the descriptors. d (7)ϕi arctan i r For example, in our tests we used a bandwidth of 64 for the2.5D spherical harmonics method, i.e., the descriptor of adrawing contains 64 signatures.2.9. 3D Shape Distribution (3DS)Shape distributions represent the shape signature as aprobability distribution sampled from a shape functionmeasuring the geometric properties of a 3D model [20].Selection of the shape function is the primary step in thistechnique.2.6. Geometric Ratios (GR)We also included the two aspect ratios for a 3D model inour tests due to the simplicity of computation and its relevanceto classification. Again the assumption here is that the aspectratios will serve as good search filters.2.7. Crinkliness and Compactness (CC)Crinkliness and Compactness are two feature vectors usedin [17]. Compactness is defined as the non-dimensional ratio ofthe volume squared over the cube of the surface area.Crinkliness is defined as the surface area of the model dividedby the surface area of a sphere having the same volume as themodel.Figure 4: Shape Functions used in the calculation ofshape distributions.Figure 3 illustrates typical geometry based shape functionsas explained below:A3: Measures the angle among three random points on thesurface of a 3D modelD1: Measures the distance between a fixed point and onerandom point on the surfaceD2: Measures the distance between two random points on thesurface3Copyright 2005 by ASME
D3: Measures the square root of the area of the triangle amongthree random points on the surfaceD4: Measures the cube root of the volume of the tetrahedronamong four random points on the surface2.10. Orthogonal Main Views (OMV)Jiantao and Ramani [18-19] recently proposed a newmethod to obtain shape signatures of 3D models afterautomatically obtaining their three orthogonal main views.Subsequently, a statistics based approach represents the shapeof the 2D views as a distance distribution between pairs ofrandomly sampled points. The 2D shape distributions thusobtained are used to compare 3D objects.This kind of space partitioning is especially suitable forvoxelized data, as cells and voxels are of the same shape, i.e.cells can be regarded as coarse voxels. Each of these partitionsis assigned to one or several bins in a histogram based ondifferent models. We tested a solid angle based similaritymodel in this paper. Below, we give more details about thesolid angle based similarity model.Figure 7: Computing the solid angle histogram for a3D model.Sample equidistant points onouter contour2D shape distributionFigure 5: Procedure of converting a 2D view into theOrthogonal Main Views representation.The methods proposed in sections 2.8 and 2.10 have manyvaluable properties: transform invariance, efficiency, androbustness. Experiments show that they can not only be appliedto vector drawings, but can also be applied to scanneddrawings. The insensitivity to noise allows for the user’s causalinput, thus supporting a freehand sketch-based retrieval userinterface.2.11. Convex Hull Histogram (CHH)In this new method we compute the 3D convex hull for agiven model using the Quickhull [25] algorithm. Then we builda histogram of the pairwise distances based on the pointsobtained from the convex hull [26].Let K c , r be a set of voxels that describes a 3D voxelizedsphere with central voxel c and radius r. For each surface-voxelv of an object o the Solid-Angle value is computed as follows.The voxels of o which are inside K v , r are counted and dividedby the size of K v , r , i.e. the number of voxels of K v , r . Theresulting measure is called the Solid-Angle value SA(v, r ) andcan be computed as follows:Kv,r V oSA(v, r ) (8)Kv,rThe solid angle value of each cell is transferred into threebins - surface voxels, inside voxels and no voxels. Due to itsuse of the discretization in rectangular coordinates we expectthis method to represent prismatic and flat-thin walled shapeswell.10.93. A 3D ENGINEERING SHAPE BENCHMARK (ESB)This section describes the processes of acquiring 3Dmodels for the benchmark database, classification of 3D modelsand evaluation of shape representations described in section 2.Normalized Distance0.80.70.60.50.40.30.20.103D Model3D Convex HullConvex Hull HistogramFigure 6: Procedure of converting a 3D model into theConvex Hull Histogram representation.The number of histogram bins is set based on the accuracyneeded for similarity searching. This histogram is alsonormalized and stored in the database for comparison. Modelsare retrieved based on the L1 norm for similarity searching[27].2.12. Solid Angle Histogram (SAH)The Solid-Angle method measures the concavity and theconvexity of geometric surfaces. It is described in more detailin [24]. Histograms are usually based on a completepartitioning of the data space into disjoint cells whichcorrespond to the bins of the histograms. The three dimensionaldata space is divided into axis parallel, equisized partitions.3.1 Model AcquisitionSince we are benchmarking shape-based search systems,we classified 3D models primarily based on shape. It isimportant to note that we have eliminated duplicates from ourdatabase, indicating that no two models in the database are thesame. This removes bias from the database, since most shaperepresentations satisfy the condition of self-similarity. The 3Dmodels in the database were acquired from a variety of sourcesincluding the National Design Repository [25], websites on theinternet [27, 28] and industry. One of the major difficulties inbuilding benchmark databases for engineering arises due to theproprietary nature of many engineering designs. While allstandard components have CAD models made freely availableto the researchers, most of the semi-standard and non-standarddesigns are not available for a public benchmark database. As aresult, most engineering shape search systems have been testedon freely available CAD models. However, it is difficult toestimate the scalability of shape search algorithms methods to4Copyright 2005 by ASME
real-world, non-standard components used in industry. In orderto overcome this difficulty, we acquired proprietary designsfrom a heavy machinery manufacturer. Of the 801 models inthe ESB, 76 models are proprietary and have a high degree ofshape complexity. We have therefore provided the remaining725 models for public use and encourage other researchers andinstitutes to use ESB for testing new methods and algorithmsfor shape based search. We will continue to add models to ourESB to encompass a wider variety of shapes. Our work is thefirst engineering benchmark for search.Each 3D model in the ESB has CAD files in two differentformats (STL and OBJ) and an associated thumbnail image(JPG). Models from the ESB can be downloaded along with ing.purdue.edu/precise/esb.html. Most models inthe ESB are of non-trivial complexity in terms of design andmanufacturing.3.2. Model ClassificationWe believe that users of a shape based search system arelikely to search a database of previous parts with some intent inmind. For the purposes of a benchmark database, shapes can beclassified into the most granular level (e.g. Gears, Handlesetc.). In order to keep our benchmark database as general aspossible, we used the classification methodology developed bySwift and Booker for the purposes of cost estimation andprocess planning [28]. The models for the benchmark databasewere classified by six individuals unrelated to the research,with varying degrees of training in Mechanical Engineering.Similar to the classification methodology of the PrincetonShape Benchmark, we provided individuals with thumbnailimages of 3D models for classification. In case of doubts, therespective 3D models were also provided to users.A total of 1,391 3D models were partitioned into threesuper-classes, namely: Solids of Revolution: Part envelope is largely a solid ofrevolution Prismatic: Part envelope is largely a rectangular or cubicprism, and Flat-Thin Wall: Parts with flat-thin walled sections andshell-like componentsWithin each super-class, models were classified intoclusters of similar shapes. A model was included in a particularcategory only when the six individuals agreed upon it. Thisclassification process continued iteratively until all the 1,391models were exhausted. Trivial models, as well as categorieswith less than 4 models were excluded from the ESB. The finalclassification consisted of 801 models classified into 42categories of similar parts such as “Discs”, “T-shaped parts”and “Bracket-like parts.” A list of super-classes along with theirrespective sub-classes is seen in Table 1.3.3. Evaluation MethodologyWe used standard evaluation procedures from informationretrieval, namely precision-recall curves and average precision.We also retrieved models randomly to ensure that every shaperepresentation performed better than random retrievals (RDM).Definitions of precision and recall are illustrated in Figure 8.Precision-recall (PR) curves describe the relationship betweenprecision and recall for an information retrieval method. Weused standard techniques of constructing PR curves from theNIST TREC standards [29].Table 1: Classification of Models in ESB.Flat Thin Wall ComponentsBracketlike parts18Clips4Contact Switches8Curved Housings9Doors7Rectangular Housings14Slender Thin Plates12Thin Plates23Total95Prismatic PartsBearing BlocksContoured SurfacesHandlesL BlocksLong Machine ElementsMachined BlocksMachined PlatesMotor BodiesPrismatic StockRocker Arms (*)Slender LinksSmall Machined BlocksT shaped partsThick PlatesThick Slotted platesU shaped partsTotal751871594973610131215122025260Solids of Revolution90 degree bends (*)Bearing like partsBolt like partsContainer like partsCylindrical PartsDiscsFlange like partsGearlike partsLong pinsMore than two openings (*)Non 90 bends (*)NutsOil pans (*)PostsPulley like partsRound, Change at endShelled TubesSpoked WheelsTotal412053104351153658981981112211615446A perfect retrieval retrieves all relevant modelsconsistently at each recall level, producing a horizontal line atprecision 1.0. However in practice, precision decreases withincreasing recall. The closer a PR curve tends to the horizontalline at precision 1.0, the better the information retrievalmethod. If the PR curves for two information retrieval methodscross each other or are very close to each other, it is difficult tomake judgements about the relative effectiveness of eachmethod. In this paper, we also attempt to quantify theperformance of various representation methods with respect toa base method (in this case, 3D shape distributions) as anAverage of Differences (AOD). Although this is not a standardpractice in information retrieval, we find that it gives relevantresults in the context of this paper. We calculated the averageof differences between the precision values of 3D shapedistributions and the method under investigation. This averageperformance is expressed as a percentage of the performance ofthe base cision X/YRelevantDocuments(Z)Recall X/ZFigure 8: Precision and Recall Calculations.4. RESULTSWe evaluated the precision at various levels of recall foreach of the 12 shape representation methods to generate PRcurves. We found that all shape representation methodsperforms better than the random retrieval method as seen inFigure 9.It was found that on average, the two methods based on 2Dviews (2.5D Harmonics and Orthogonal Main View)outperform other methods consistently. This is similar toconclusions drawn from [15], where the Lightfield Descriptorsbased on 2D projections work better than other 3D methods. Itis interesting to note that traditional engineering drawings alsouse 2D projections to represent 3D models. Sphericalharmonics and two histogram based methods - Solid Angle5Copyright 2005 by ASME
Histogram and Convex Hull Histogram also perform better than3D shape 0.50.60.70.80.91RecallFigure 9: Precision-Recall Calculations for 12 shaperepresentations.10.9with 64 harmonic coefficients, thereby capturing more shapecontent than other feature vectors.On an average, the base method for AOD, 3D shapedistributions, performs 5.57 times better than random retrieval.Table 2 shows the AOD of various methods as a percentagevalue. From this point forward, we only consider the top sixshape representations based on the AOD for comparison ofdifferent methods. The results for each super-class arepresented below.4.1. Flat-Thin Walled ComponentsFor the Flat-Thin walled components class, the methodsbased on 2D drawings outperformed other methods.Surprisingly, 3D shape distributions and Surface Area andVolume performed better than the rest of the three methodsbased on more complex feature vectors, viz., SH, CHH, andSAH. Although simple, the SAV performs better for this superclass because thin walled components have higher surface areasand lower volume, and these features are more explicitlycaptured in the SAV compared to other point-based methods.The PR curves for these methods for the Flat Thin Walledcomponents super class is shown in Figure 11.2.5D HarmonicsOrthogonal Main ViewSpherical HarmonicsHull HistogramSolid Angle Histogram3D Shape Distribution0.80.710.9Orthogonal Main View0.6Precision2.5D Harmonics0.83D Shape DistributionSolid Angle Histogram0.70.5Surface Area and VolumeGeometric RatiosSurface Area Volume .30.40.50.60.70.80.910.1RecallFigure 10: Precision-Recall curves with confidenceintervals for the top 6 shape representations.Table 2: AOD for 12 shape representations with 3Dshape distributions as a reference.MethodAOD2.5D Harmonics81.15%Orthogonal Main View72.50%Spherical Harmonics56.04%Hull Histogram33.87%Solid Angle Histogram27.00%3D Shape Distributions0.00%Surface Area and Volume-13.17%Geometric Ratios-14.10%Crinkliness and Compactness-14.38%Surface Area Volume Ratio-19.14%Principal Moments-35.24%Moment Invariants-41.66%Clearly, histogram based methods outperform featurevector based methods such as Moment Invariants and PrincipalMoments. This is because histogram based methods capturemore of the shape content than feature vectors. The onlyexception is spherical harmonics, which approximates a shape000.10.20.30.40.50.60.70.80.91RecallFigure 11: Precision-Recall curves for Flat-ThinWalled Components.4.2. Prismatic PartsFor this super class, there were four methods thatperformed consistently better than 3D shape distributions,namely the two 2D view based methods (2.5D and OMV),Spherical Harmonics and Solid Angle Histograms. However,the 2D view based methods did not show a markedimprovement in performance compared to other methods forthis super class. While the 2D view based methods performedconsistently well, two other methods viz. spherical harmonics(SH) and Solid Angle histograms (SAH) performed comparableat higher recall levels (after 0.3 and 0.5 recall respectively) aswe had expected. Results can be seen in Figure 12.4.3. Solids of RevolutionBoth methods based on 2D drawings (2.5D and OMV)performed significantly better than other methods on average.Spherical Harmonics and the Convex Hull Histogramperformed better than 3D shape distributions. The PR curvesfor these methods for the Solids of Revolution super class isshown in Figure 13.6Copyright 2005 by ASME
As with a text based search engine such as Google, userswould like the most relevant 3D models to appear early on inthe search results. This is equivalent to saying that we needhigher precision at low levels of recall. We ranked methodsbased on their AODs until a recall value at 0.25. The results areshown in Table 3. Clearly, the shape representations that holdmore shape content are better at retrieving more relevantmodels as compared to the retrieval size. Performing a similaranalysis for each of the three super-classes, we found that SAVgives better precision than 3D shape distributions for flat thinwalled parts. Not surprisingly, for all three super-classes, bothmethods based on 2D views outperformed other methods at lowrecall levels.12.5D Harmonics0.9Orthogonal Main ViewSolid Angle Histogram0.8Spherical Harmonics3D Shape Distribution0.7Hull HistogramSurface Area Volume RatioPrincipal MomentsCrinkliness and CompactnessMoment Invariants-29.02%-29.09%-31.12%-31.27%5. DISCUSSIONWe believe that for the engineering domain, it is importantto analyze which shape representations perform well for aparticular part category, which may seem contrary to the viewsof researchers in computer vision and graphics. This problem ispartly also because engineering shapes have higher levels ofshape complexity as compared to other shapes. Part repositoriesin a single engineering division often hold similar kinds ofparts. For example, a division of a car company that designsconnecting rods will hold similar connecting rods, whileanother division that designs steering columns will hold a lot ofsimilar steering .50.60.70.80.91RecallFigure 12: Precision-Recall curves for PrismaticParts.12.5D HarmonicsOrthogonal Main ViewSpherical HarmonicsSolid Angle HistogramHull Histogram3D Shape DistributionCrinkliness and .10.20.30.40.50.60.70.80.91RecallFigure 13: Precision-Recall curves for Solids ofRevolution.Table 3: AOD for 12 shape representations until 25%recall with 3D shape distributions as a reference.MethodAOD2.5D Harmonics49.04%Orthogonal Main View41.96%Spherical Harmonics25.19%Hull Histogram23.29%Solid Angle Histogram13.28%3D Shape Distribution0.00%Surface Area and Volume-16.35%Geometric Ratios-18.92%6. CONCLUSIONSIn summary, we have developed a publicly availableengineering shape benchmark (ESB) for comparing variousshape based search algorithms. ESB includes a set of 725models in two formats (STL and OBJ) along with associatedJPG images and a classification schema. All this
Shape Benchmark (ESB). Based on a classification of 801 engineering components, we have evaluated 12 different shape descriptors that are described in Section 2 of this paper. One of the few publicly available benchmarks for 3D models includes the Princeton Shape Benchmark (PSB) [15
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