Superquadric reprentation of automotive parts applying part decomposition
Yan Zhang
Andreas Koschan
Mongi A.Abidi
University of Tenne
Department of Electrical and Computer Engineering
334Ferris Hall
Knoxville,Tenne37996-2100
E-mail:utk.edu
两位数乘法题500道
Abstract.Superquadrics are able to reprent a large variety of objects with only a few parameters and a single equation.We prent a superquadric reprentation strategy for automotive parts compod of3-D triangle meshes.Our strategy consists of two ma-jor steps of part decomposition an
d superquadricfitting.The origi-nalities of this approach include the following two features.First,our approach can reprent multipart objects with superquadrics suc-cessfully by applying part decomposition.Second,superquadrics re-covered from our approach have the highest confidence and accu-racy due to the3-D watertight surfaces utilized.A novel,generic3-D part decomposition algorithm bad on curvature analysis is also propod.Experimental results demonstrate that the propod part decomposition algorithm is able to gment multipart objects into meaningful single parts efficiently.The propod superquadric rep-rentation strategy can then reprent each individual part of the original objects with a superquadric model successfully.©2004 SPIE and IS&T.[DOI:10.1117/1.1762516]
1Introduction
Object reprentation denotes reprenting real-world ob-jects with known graphic or mathematical primitives that can be recognized by computers.This rearch has numer-ous applications in areas including computer vision,com-puter graphics,and rever engineering.An object can be reprented by three levels of primitives in terms of the dimensional complexity:volumetric primitives,surface el-ements,and contours.The primitive lected to describe the object depends on the complexity of the object and the tasks involved.As the highest level primitives,volumetric primitives can better repre
nt global features of an object with a significantly reduced amount of information com-pared with surface elements and contours.In addition, volumetric primitives have the ability to achieve the highest data compression ratio without losing the accuracy of the original data.The primarily ud volumetric primitives in-clude generalized cylinders,geons,and superquadrics.1Su-perquadrics are a generalization of basic quadric surfaces and they can reprent a large variety of shapes with only a few parameters and a single equation.An object initially reprented by thousands of triangle meshes can be repre-nted by only a small t of superquadrics.This compact reprentation can be applied to object recognition to aid, for example,automated depalletizing of industrial parts or robot-guided bin picking of mixed nuclear waste in a haz-ardous environment.The quality control of both tasks can be enhanced by employing superquadrics.Furthermore,the registration of multiview data is indispensable to measure the size of partially occluded objects or their distances from each other in veral image-bad quality control tasks.Su-perquadrics can be ud to efficiently register multiview range data of scenes with small overlap.2
Most early rearch on superquadric reprentation con-centrated on reprenting single-part objects from single-view intensity or range images by assuming that the objects have been appropriately gmented.3–13This category of rearch focud on the data-fitting process,including ob-jective function lection,fitting criteria measurements,and
Paper ORNL-007received Jul.30,2003;accepted for publication Feb.23,2004. 1017-9909/2004/$15.00©2004SPIE and
IS&T.Fig.1Real range image of a multipart object obtained from Ref.18. Journal of Electronic Imaging13(3),411–417(July2004).
Journal of Electronic Imaging/July2004/Vol.13(3)/411
convergence analysis.For complex,multipart objects or scenes,there are two major types of approaches in the re-arch literature.The first type of method incorporates an image gmentation step prior to the superquadric fitting.11–15The other type of method directly recovers su-perquadrics from a range image without
吴亦凡写真
pregmentation.16–19
Compared with superquadric repre-ntation of single-part object,the two types of methods can reprent more complex objects and have wider appli-cations in related tasks including robotic navigation,object recognition,and virtual reality.However,existing super-quadric reprentation methods have veral weakness.First,existing methods cannot handle arbitrary shapes or significant occlusions in the scene.Figure 1shows an ex-ample of the most complicated object that can be repre-nted by superquadrics appeared in the rearch literature.18
白月照流光
We obrve that the range image shown in Fig.1con-tains very few occlusions due to the simplicity of the ob-ject.In this ca,an optimal viewpoint can easily be found from which each part of the object is visible.When an automotive ,a complex,multipart object such as shown in Fig.2,is of interest,no existing methods can reprent this object correctly becau heavy occlusions are inevitable from any single viewpoint due to the complexity of the object.
The cond weakness of existing methods is that they utilize only single-view images.Again,for the automotive part shown in Fig.2͑a ͒,it is too difficult to find an optimal viewpoint from which all the parts are visible due to lf-occlusions and occlusions,as shown in Fig.2͑b ͒.In addi-tion,the confidence of recovered superquadrics is low due to incomplete single-view data utilized and the accuracy of the recovered models highly depends on the viewpoint ud to acquire the data.How complicated,multipart objects can
be reprented by superquadrics with high confidence and accuracy remains unknown from the literature.
In this paper,we propo an efficient strategy to repre-nt multipart objects with superquadrics.We also prent a novel 3-D part decomposition algorithm bad on curvature analysis to facilitate our s
uperquadric reprentation strat-egy.Experiments are shown for automotive parts compod of 3-D triangulated surfaces.
The remainder of this paper proceeds as follows.Section 2prents a superquadric reprentation approach for mul-tipart objects.Section 3propos the 3-D part decomposi-tion algorithm for triangle meshes.The experimental results are prented in Sec.4and Sec.5concludes the paper.
2
Superquadric Reprentation of Multipart Objects Utilizing Part Decomposition
A diagram for the propod superquadric reprentation al-gorithm is illustrated in Fig.3.Beginning with a multipart object compod of triangle meshes,we propo a part de-composition algorithm to gment the meshes into single parts.Next,each single part is fitted with a superquadric model.Utilizing part decomposition,the difficult repren-tation problem of complicated objects is solved.We u 3-D triangulated surfaces reconstructed from multiview range images as input so that the recovered superquadrics have significantly higher confidence than tho recovered from single-view images.In addition,our propod algo-rithms are generic and flexible in the n of triangle mesh handling ability since triangle meshes have been the stan-dard surface reprentation element
s in many computer-related areas.A triangulation step is required only if un-structured 3-D point clouds are
provided.
Fig.2Distributor cap:(a)photograph of the object,(b)rendering of 3-D triangulated surfaces scanned from view 1,and (c)rendering of 3-D triangulated surfaces scanned from view
2.
Fig.3Diagram of the propod superquadric reprentation strategy utilizing part decomposition.
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412/Journal of Electronic Imaging /July 2004/Vol.13(3)
2.1Introduction to Superquadrics
A t of superquadrics with various shape factors is shown in Fig.4.The implicit definition of superquadrics is ex-presd as 18F ͑x ,y ,z ͒ϭ
ͫͩx a 1
ͪ2/2
ϩͩy a 2
ͪ2/2ͬ
2/1
ϩͩz a 3
ͪ
2/1
ϭ1,
1,2͑0,2͒,
͑1͒
where (x ,y ,z )reprents a surface point of the superquad-ric,(a 1,a 2,a 3)reprent sizes in the (x ,y ,z )directions,and (1,2)reprent shape factors.To reprent a super-quadric model with global deformations in the world coor-dinate system,15parameters are needed.They are summa-rized as 18
∧ϭ͑a 1,a 2,a 3,1,2,,,,p x ,p y ,p z ,k x ,k y ,␣,k ͒,
͑2͒
where the first 11parameters define a regular superquadric.Parameters k x and k y define the tapering deformations and ␣and k define the bending deformations.Most approaches define an objective function and find the superquadric pa-rameters through minimizing this objective function.The objective function ud in this paper is 1
G ͑∧͒ϭa 1a 2a 3
͚i ϭ1
N
͓F 1
͑x c ,y c ,z c ͒Ϫ1͔2.͑3͒
华为平板怎么截图The Levenberg-Marquardt method 20was implemented to
minimize the objective function due to its stability and ef-ficiency.In addition,our superquadric fitting algorithm is able to recover superquadrics with global deformations from unstructured 3-D data points.3
Curvature-Bad 3-D Part Decomposition
Many tasks in computer vision,computer graphics,and re-ver engineering involve objects or models.The tasks become extremely difficult when the object of interest is ,it contains multiple parts.Part decompo-sition can simplify the original task performed on multipart objects into veral subtasks each dealing with their con-stituent single,much simpler parts.21,22While a significant amount of rearch for part decomposition of 2-D intensity or 2.5-D range images has been conducted over the last 2decades,23–25little effort has been made on part gmenta-tion of 3-D data.26,27Therefore,a novel 3-D part decompo-sition algorithm is propod in this paper.Figure 5illus-trates the difference between region gmentation and part decomposition.A scene consisting of a barrel on the floor is gmented into three surfaces by a region gmentation al-gorithm and two single-part objects by a part decomposi-tion algorithm.We can obrve that the scene can be rep-rented by two superquadrics,which is consistent with the part decomposition result.Therefore,we conclude that part decomposition is more appropriate for high-level tasks such as volumetric primitives-bad object reprentation and recognition.A diagram of the propod part decomposition algorithm is shown in Fig.6.
The propod part decomposition consists of four major steps:Gaussian curvature estimation,bound
ary detection,region growing,and postprocessing.Boundaries between two articulated parts are compod of points with highly negative curvature according to the transversality regularity.21,22The boundaries are therefore detected by thresholding estimated curvatures for each vertex.A component-labeling operation is then performed to grow nonboundary vertices into parts.Finally,a postprocessing step is performed to assign nonlabeled vertices,including boundary vertices,to one of the parts and merge parts con-taining fewer vertices than a prespecified threshold into their neighbor parts.This part decomposition algorithm is summarized as follows.
ˆAlgorithm 1…3-D part decomposition of triangle meshes …‰
ˆInput:‰Triangulated surfaces.
ˆStep 1.‰Compute Gaussian curvature for each vertex on the surface.
ˆStep 2.‰Label vertices of highly negative curvature
as
Fig.4Superquadrics with various shape
parameters.
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Fig.6Diagram of the propod 3-D part decomposition algorithm.关于活动的作文
Superquadric reprentation of automotive parts ...
Journal of Electronic Imaging /July 2004/Vol.13(3)/413
boundaries and the remaining vertices as eds.
ˆStep 3.‰Perform iterative region growing on each ed vertex.
ˆStep 4.‰Assign nonlabeled vertices to parts and merge parts having less than a prespecified number of vertices into their neighboring parts.
ˆOutput:‰Decompod single parts.
The major steps of this part decomposition algorithm are described respectively in the following ctions.
3.1Gaussian Curvature Estimation and Boundary
Detection
The Gaussian curvature for each vertex on a triangulated surface is estimated by K ͑p ͒ϭ
3͑2Ϫ͚i ϭ1N i ͒
i ϭ1N
A i
␦2͑p Ϫp i ͒,͑4͒
using the method propod in Ref.28.Variable p reprents
the point of interest,p i reprents a neighboring vertex of the point p ,and A i reprents the area of the triangle con-taining the point p .Variable i reprents the interior angle of the triangle at p ,and ␦is the Dirac delta function.The triangles sharing the vertex p are illustrated in Fig.7.After Gaussian curvature is obtained for each vertex on the surface,a prespecified threshold is applied to label ver-tices as boundary or ed.Vertices of highly negative cur-vature are labeled as boundaries between two parts accord-ing to the transversality regularity,21while the rest are labeled as eds.The threshold is critical and affects the performance of region growing.This threshold is deter-mined in a heuristic manner and depends on mesh resolu-tion.Two types of isolated vertices defined in this work according to their labels include:͑1͒a point that is labeled as boundary while all of its neighbors are labeled as eds and ͑2͒a point that is labeled as a ed while all of its neighbors are labeled as boundary.The isolated vertices are removed by changing their labels to be the same as tho of their neighbors.主板诊断卡
3.2Region Growing and Postprocessing
After the vertices are labeled,a region-growing step is per-formed on each vertex labeled as ed.Figure 8shows
triangle meshes around the point p .To illustrate the region growing process,a t of two-ring neighbor meshes around point p is shown in this figure.
Region growing is performed as follows.Starting from a ed vertex p ,the region number 1is first assigned to the vertex.Second,all the neighbors p i initially labeled as eds are then labeled with the same region number as the point p .The same labeling process is performed for each neighbor p i to label vertices p i j .This process terminates when the grown region is surrounded by boundary ,the neighbors of the edge vertices of the region are all labeled as boundaries (Ϫ1).This process is repeated for every other vertex labeled as ed ͑0͒,but not for a vertex that has been grown and labeled with one of the region numbers (1,2,...,N ).After all the ed vertices are assigned new labels,a postprocessing is performed for each bound-ary vertex.Given a ed point x ,all its neighbors x i are first sorted in an ascending order bad on their Euclidean distance to the point x .Next,a neighboring vertex x i ,which is the first point labeled with a region number ͑Ͼ0͒,is picked up.The boundary vertex x is then labeled the same as the vertex x i ,i.e.,the label of x is changed from Ϫ1to a region number (Ͼ0).Finally,with the exception of a few missing vertices,each vertex is labeled as 1,2,3,...,N ,the number of the parts.Missing vertices are usually located around boundaries between two articulated parts,and they are further assigned to parts during the post-processing step.
Finally,a postprocessing step is performed to assign the nonlabeled vertices to parts.For example,the vertex p is an unlabeled vertex and needs further postprocessing.Assum-ing that p i (i ϭ1,2,...,N )reprents a neighboring vertex of the point p ,the neighboring vertices are first lected if they have the same sign of curvature as that of the vertex p and belong to one of the gmented parts.Next,among tho neighbor vertices,the vertex that has the smallest Euclidean distance to the vertex p is lected as a target vertex.For example,the vertex p 1is assumed to be the target vertex of the vertex p .Vertex p is assigned the
same
Fig.7Curvature estimation for the vertex p utilizing triangle mesh
information.
Fig.8Region growing process for the vertex p .
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414/Journal of Electronic Imaging /July 2004/Vol.13(3)
Superquadric reprentation of
Fig.5Region and part gmentation of a synthetic scene:(a)rendering of a synthetic scene consist-
ing of a barrel on thefloor,(b)three gmented regions rendered in different colors,and(c)two
colors.
decompod parts rendered in different
view range images from the IVP Ranger system29and consists of37,171vertices and73,394tri-
angles.The part decomposition results consist of two parts:(a)photograph of the original object,(b)
rendering of the reconstructed mesh,(c)decompod parts rendered in different colors,and(d)two
colors.
recovered superquadrics rendered in different
multiview range images from the IVP Ranger system29and consists of58,975vertices and117,036
triangles.The part decomposition results consist of13parts:(a)photograph of the original object,(b)
rendering of the reconstructed mesh,(c)decompod parts rendered in different colors,and(d)
colors.
recovered superquadrics rendered in different
multiview range images from the IVP Ranger system29and consists of58,784vertices and117,564
triangles.The part decomposition results consist of nine parts:(a)photograph of the original object,(b)
rendering of the reconstructed mesh,(c)decompod parts labeled in different colors,and(d)recov-
ered superquadrics rendered in different colors.
祝尔慷Journal of Electronic Imaging/July2004/Vol.13(3)/415
label as vertex ,the same gmented part.Further-more,parts compod of fewer vertices than a specified threshold are merged with adjacent regions.
4Experimental Results
Experimental results on superquadric reprentation for multipart,automotive objects including a disk brake,a dis-tributor cap,and a water neck are shown in this ction. The meshes were reconstructed from multiview range im-ages scanned from the IVP Ranger System.29The recovered superquadrics were rendered in three dimensions using quad meshes.30Figure9shows the disk brake and its part decomposition and superquadric reprentation results.The reconstructed3-D triangulated surface shown in Fig.9͑b͒consists of37,171vertices and73,394triangles.Starting from this reconstructed mesh,our part decomposition algo-rithmfirst decompod the disk brake into two single parts, as shown in Fig.9͑c͒.Each decompod part was nextfit-ted to a superquadric model,as shown in Fig.9͑d͒.The decompod parts and recovered superquadrics are ren-dered in different colors.We obrve that our part decom-position algorithm successfully decompod the disk brake into its cons
tituent parts and the superquadric reprenta-tion strategy recovered correct superquadrics in terms of their size,shape,and orientation.Compared to the original triangle mesh reprentation consisting of37,171vertices and73,394triangles,the recovered superquadrics describe the disk brake with only22parameters͑11parameters for each superquadric without global deformations͒.This low reprentation cost of superquadric reprentation can sig-nificantly benefit tasks including virtual reality,object rec-ognition,and robotic navigation.However,the hole at the center of the disk brake failed to be reprented since su-perquadrics can only reprent objects with genus of zero.19 Figure10shows the distributor cap and its part decom-position and superquadric reprentation results.The recon-structed mesh shown in Fig.10͑b͒consists of58,975ver-tices and117,036triangles and was decompod into13 single parts,as shown in Fig.10͑c͒.We obrve that this decomposition result is consistent with human perception. The recovered superquadrics shown in Fig.10͑d͒correctly reprent the distributor cap.The recovered superquadric parameters and the ground truths for one of the small cyl-inders on top of the distributor cap are shown in Table1. We can obrve that the recovered superquadric parameters for this cylinder have the correct size and shape informa-tion when compared with the ground truth parameters of the object.In addition,superquadrics reprent the distribu-tor cap with only143floating numbers,while the original triangle mesh consists of58,975vertices and117,036tri-angles.
Figure11shows the water neck and its part decomposi-tion and superquadric reprentation results.The recon-structed mesh shown in Fig.11͑b͒consists of58,784ver-tices and117,564triangles and was decompod into nine single parts,as shown in Fig.11͑c͒.We obrve that the decompod parts are consistent with human perception. The recovered superquadrics shown in Fig.11͑d͒correctly reprent the water neck.The recovered superquadric pa-rameters and the ground truths for the handle,the ball,and the small cylinder next to the handle of the water neck are shown in Table2.From this table,we obrve that the recovered superquadric parameters have the correct size and shape information when compared with the ground truth parameters of the objects.Again,superquadrics repre-nt the water neck in a desirable accuracy with only99 parameters while the original triangle mesh consists of 58,784vertices and117,564triangles.
5Conclusions
This paper propod a superquadric reprentation ap-proach for multipart objects.Superquadrics can reprent objects in an acceptable accuracy with only a few param-eters,while other surface primitives and contours usually require thousands of reprentation elements.Such a com-pactness and low reprentation cost can significantly ben-efit tasks including virtual reality,object recognition,and robot ,it enables the tasks to run in a real-time manner.The advanta
ges of the propod super-quadric reprentation approach include:͑1͒it can success-fully reprent complicated,multipart objects byfirst de-composing them into single-part objects,and͑2͒the recovered superquadrics have the highest confidence and accuracy since the input we u are3-D triangulated sur-faces reconstructed from multiview range images.The in-completeness and ambiguities contained in single-view im-ages were eliminated during the multiview surface reconstruction process.We also propod a3-D part de-composition algorithm to decompo compound objects reprented by triangle meshes into their constituent single parts bad on curvature analysis.Considering the fact that the triangle mesh has been a standard surface reprentation element in computer vision and computer graphics,the pro-pod part decomposition algorithm is generic,flexible,and can facilitate computer vision tasks such as shape descrip-tion and object recognition.Furthermore,the part decom-position algorithm can gment a large number of triangle meshes͑over100,000͒in only conds on an SGI Octane workstation.
Table1Recovered superquadric parameters and ground truths for one of the small cylinders shown in Fig.10(d)where the unit is millimeters.
Parameters a
1
a2a312 Ground truths15.215.620.10.1 1.0 Superquadric parameters16.4515.6720.420.120.96Table2Recovered superquadric parameters and ground truths for the water neck shown in Fig.11(d)where the unit is millimeters. Object Parameters a
1
a2a312 Handle Ground truths39.739.417.60.1 1.0 Superquadric parameters40.2340.5866.830.130.98 Ball Ground Truths50.047.656.0 1.0 1.0 Superquadric parameters51.6247.5654.28 1.020.95 Cylinder Ground truths16.517.844.20.1 1.0 Superquadric parameters17.5617.9443.380.110.95
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