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Journal of Electronic Imaging 13(3), 411 – 417 (July 2004).

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Y. Zhang, A. Koschan, and M. A. Abidi, "Superquadric Representation of Automotive Parts Applying Part Decomposition," Journal of Electronic Imaging, Special Issue on Quality Control by Artificial Vision, Vol. 13, No. 3, pp. 411-417, July 2004.

Superquadric representation of automotive parts applying part decomposition Yan Zhang Andreas Koschan Mongi A. Abidi University of Tennessee Department of Electrical and Computer Engineering 334 Ferris Hall Knoxville, Tennessee 37996-2100 E-mail: [email protected]

Abstract. Superquadrics are able to represent a large variety of objects with only a few parameters and a single equation. We present a superquadric representation strategy for automotive parts composed of 3-D triangle meshes. Our strategy consists of two major steps of part decomposition and superquadric fitting. The originalities of this approach include the following two features. First, our approach can represent multipart objects with superquadrics successfully by applying part decomposition. Second, superquadrics recovered from our approach have the highest confidence and accuracy due to the 3-D watertight surfaces utilized. A novel, generic 3-D part decomposition algorithm based on curvature analysis is also proposed. Experimental results demonstrate that the proposed part decomposition algorithm is able to segment multipart objects into meaningful single parts efficiently. The proposed superquadric representation strategy can then represent each individual part of the original objects with a superquadric model successfully. © 2004 SPIE and IS&T. [DOI: 10.1117/1.1762516]

1 Introduction Object representation denotes representing real-world objects with known graphic or mathematical primitives that can be recognized by computers. This research has numerous applications in areas including computer vision, computer graphics, and reverse engineering. An object can be represented by three levels of primitives in terms of the dimensional complexity: volumetric primitives, surface elements, and contours. The primitive selected to describe the object depends on the complexity of the object and the tasks involved. As the highest level primitives, volumetric primitives can better represent global features of an object with a significantly reduced amount of information compared 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 used volumetric primitives include generalized cylinders, geons, and superquadrics.1 Superquadrics are a generalization of basic quadric surfaces and they can represent a large variety of shapes with only a

Paper ORNL-007 received Jul. 30, 2003; accepted for publication Feb. 23, 2004. 1017-9909/2004/$15.00 © 2004 SPIE and IS&T.

few parameters and a single equation. An object initially represented by thousands of triangle meshes can be represented by only a small set of superquadrics. This compact representation 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 hazardous 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 several image-based quality control tasks. Superquadrics can be used to efficiently register multiview range data of scenes with small overlap.2 Most early research on superquadric representation concentrated on representing single-part objects from singleview intensity or range images by assuming that the objects have been appropriately segmented.3–13 This category of research focused on the data-fitting process, including objective function selection, fitting criteria measurements, and

Fig. 1 Real range image of a multipart object obtained from Ref. 18.

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Zhang, Koschan, and Abidi

Fig. 2 Distributor 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.

convergence analysis. For complex, multipart objects or scenes, there are two major types of approaches in the research literature. The first type of method incorporates an image segmentation step prior to the superquadric fitting.11–15 The other type of method directly recovers superquadrics from a range image without presegmentation.16 –19 Compared with superquadric representation of single-part object, these two types of methods can represent more complex objects and have wider applications in related tasks including robotic navigation, object recognition, and virtual reality. However, existing superquadric representation methods have several weaknesses. First, existing methods cannot handle arbitrary shapes or significant occlusions in the scene. Figure 1 shows an example of the most complicated object that can be represented by superquadrics appeared in the research literature.18 We observe that the range image shown in Fig. 1 contains very few occlusions due to the simplicity of the object. In this case, an optimal viewpoint can easily be found from which each part of the object is visible. When an automotive part, i.e., a complex, multipart object such as shown in Fig. 2, is of interest, no existing methods can represent this object correctly because heavy occlusions are inevitable from any single viewpoint due to the complexity of the object. The second weakness of existing methods is that they utilize only single-view images. Again, for the automotive part shown in Fig. 2共a兲, it is too difficult to find an optimal viewpoint from which all the parts are visible due to selfocclusions and occlusions, as shown in Fig. 2共b兲. In addition, 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 used to acquire the data. How complicated, multipart objects can

be represented by superquadrics with high confidence and accuracy remains unknown from the literature. In this paper, we propose an efficient strategy to represent multipart objects with superquadrics. We also present a novel 3-D part decomposition algorithm based on curvature analysis to facilitate our superquadric representation strategy. Experiments are shown for automotive parts composed of 3-D triangulated surfaces. The remainder of this paper proceeds as follows. Section 2 presents a superquadric representation approach for multipart objects. Section 3 proposes the 3-D part decomposition algorithm for triangle meshes. The experimental results are presented in Sec. 4 and Sec. 5 concludes the paper.

2

Superquadric Representation of Multipart Objects Utilizing Part Decomposition

A diagram for the proposed superquadric representation algorithm is illustrated in Fig. 3. Beginning with a multipart object composed of triangle meshes, we propose a part decomposition algorithm to segment the meshes into single parts. Next, each single part is fitted with a superquadric model. Utilizing part decomposition, the difficult representation problem of complicated objects is solved. We use 3-D triangulated surfaces reconstructed from multiview range images as input so that the recovered superquadrics have significantly higher confidence than those recovered from single-view images. In addition, our proposed algorithms are generic and flexible in the sense of triangle mesh handling ability since triangle meshes have been the standard surface representation elements in many computerrelated areas. A triangulation step is required only if unstructured 3-D point clouds are provided.

Fig. 3 Diagram of the proposed superquadric representation strategy utilizing part decomposition.

412 / Journal of Electronic Imaging / July 2004 / Vol. 13(3)

Superquadric representation of automotive parts . . . N

G 共 ∧ 兲 ⫽a 1 a 2 a 3

兺 关 F ␧ 共 x c ,y c ,z c 兲 ⫺1 兴 2 . 1

i⫽1

共3兲

The Levenberg-Marquardt method20 was implemented to minimize the objective function due to its stability and efficiency. In addition, our superquadric fitting algorithm is able to recover superquadrics with global deformations from unstructured 3-D data points.

共2兲

3 Curvature-Based 3-D Part Decomposition Many tasks in computer vision, computer graphics, and reverse engineering involve objects or models. These tasks become extremely difficult when the object of interest is complicated, e.g., it contains multiple parts. Part decomposition can simplify the original task performed on multipart objects into several subtasks each dealing with their constituent single, much simpler parts.21,22 While a significant amount of research for part decomposition of 2-D intensity or 2.5-D range images has been conducted over the last 2 decades,23–25 little effort has been made on part segmentation of 3-D data.26,27 Therefore, a novel 3-D part decomposition algorithm is proposed in this paper. Figure 5 illustrates the difference between region segmentation and part decomposition. A scene consisting of a barrel on the floor is segmented into three surfaces by a region segmentation algorithm and two single-part objects by a part decomposition algorithm. We can observe that the scene can be represented 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-based object representation and recognition. A diagram of the proposed part decomposition algorithm is shown in Fig. 6. The proposed part decomposition consists of four major steps: Gaussian curvature estimation, boundary detection, region growing, and postprocessing. Boundaries between two articulated parts are composed of points with highly negative curvature according to the transversality regularity.21,22 These 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 containing fewer vertices than a prespecified threshold into their neighbor parts. This part decomposition algorithm is summarized as follows.

where the first 11 parameters 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 parameters through minimizing this objective function. The objective function used in this paper is1

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. 4 Superquadrics with various shape parameters.

2.1 Introduction to Superquadrics A set of superquadrics with various shape factors is shown in Fig. 4. The implicit definition of superquadrics is expressed as18 F 共 x,y,z 兲 ⫽

冋冉 冊 冉 冊 册 x a1

2/␧ 2



y a2

2/␧ 2 ␧ 2 /␧ 1



冉 冊 z a3

2/␧ 1

⫽1,

␧ 1 , ␧ 2 苸 共 0,2兲 ,

共1兲

where (x,y,z) represents a surface point of the superquadric, (a 1 ,a 2 ,a 3 ) represent sizes in the (x,y,z) directions, and (␧ 1 ,␧ 2 ) represent shape factors. To represent a superquadric model with global deformations in the world coordinate system, 15 parameters are needed. They are summarized as18 ∧⫽ 共 a 1 ,a 2 ,a 3 ,␧ 1 ,␧ 2 , ␾ , ␪ , ␸ , p x , p y , p z ,k x ,k y , ␣ ,k 兲 ,

Fig. 6 Diagram of the proposed 3-D part decomposition algorithm.

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Zhang, Koschan, and Abidi

Fig. 7 Curvature estimation for the vertex p utilizing triangle mesh information.

boundaries and the remaining vertices as seeds. ˆStep 3.‰ Perform iterative region growing on each seed vertex. ˆStep 4.‰ Assign nonlabeled vertices to parts and merge parts having less than a prespecified number of vertices into their neighboring parts. ˆOutput:‰ Decomposed single parts. The major steps of this part decomposition algorithm are described respectively in the following sections. 3.1 Gaussian Curvature Estimation and Boundary Detection The Gaussian curvature for each vertex on a triangulated surface is estimated by K共 p 兲⫽

N ␪i兲 3 共 2 ␲ ⫺ 兺 i⫽1 N 兺 i⫽1 Ai

␦ 2 共 p⫺ p i 兲 ,

共4兲

using the method proposed in Ref. 28. Variable p represents the point of interest, p i represents a neighboring vertex of the point p, and A i represents the area of the triangle containing the point p. Variable ␪ i represents 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 vertices as boundary or seed. Vertices of highly negative curvature are labeled as boundaries between two parts according to the transversality regularity,21 while the rest are labeled as seeds. The threshold is critical and affects the performance of region growing. This threshold is determined in a heuristic manner and depends on mesh resolution. Two types of isolated vertices defined in this work according to their labels include: 共1兲 a point that is labeled as boundary while all of its neighbors are labeled as seeds and 共2兲 a point that is labeled as a seed while all of its neighbors are labeled as boundary. The isolated vertices are removed by changing their labels to be the same as those of their neighbors. 3.2 Region Growing and Postprocessing After the vertices are labeled, a region-growing step is performed on each vertex labeled as seed. Figure 8 shows 414 / Journal of Electronic Imaging / July 2004 / Vol. 13(3)

Fig. 8 Region growing process for the vertex p .

triangle meshes around the point p. To illustrate the region growing process, a set of two-ring neighbor meshes around point p is shown in this figure. Region growing is performed as follows. Starting from a seed vertex p, the region number 1 is first assigned to the vertex. Second, all the neighbors p i initially labeled as seeds 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 vertices, i.e., the neighbors of the edge vertices of the region are all labeled as boundaries (⫺1). This process is repeated for every other vertex labeled as seed 共0兲, but not for a vertex that has been grown and labeled with one of the region numbers (1,2,...,N). After all the seed vertices are assigned new labels, a postprocessing is performed for each boundary vertex. Given a seed point x, all its neighbors x i are first sorted in an ascending order based on their Euclidean distance to the point x. Next, a neighboring vertex x i , which is the first point labeled with a region number 共⬎0兲, 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 ⫺1 to a region number (⬎0). 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 postprocessing 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. Assuming that p i (i⫽1,2,...,N) represents a neighboring vertex of the point p, the neighboring vertices are first selected if they have the same sign of curvature as that of the vertex p and belong to one of the segmented parts. Next, among those neighbor vertices, the vertex that has the smallest Euclidean distance to the vertex p is selected as a target vertex. For example, the vertex p 1 is assumed to be the target vertex of the vertex p. Vertex p is assigned the same

Superquadric representation of automotive parts . . .

Fig. 5 Region and part segmentation of a synthetic scene: (a) rendering of a synthetic scene consisting of a barrel on the floor, (b) three segmented regions rendered in different colors, and (c) two decomposed parts rendered in different colors.

Fig. 9 Superquadric representation for a disk brake. The triangle mesh is a reconstruction from multiview range images from the IVP Ranger system29 and consists of 37,171 vertices and 73,394 triangles. The part decomposition results consist of two parts: (a) photograph of the original object, (b) rendering of the reconstructed mesh, (c) decomposed parts rendered in different colors, and (d) two recovered superquadrics rendered in different colors.

Fig. 10 Superquadric representation for a distributor cap. The triangle mesh is a reconstruction from multiview range images from the IVP Ranger system29 and consists of 58,975 vertices and 117,036 triangles. The part decomposition results consist of 13 parts: (a) photograph of the original object, (b) rendering of the reconstructed mesh, (c) decomposed parts rendered in different colors, and (d) recovered superquadrics rendered in different colors.

Fig. 11 Superquadric representation for a water neck. The triangle mesh is a reconstruction from multiview range images from the IVP Ranger system29 and consists of 58,784 vertices and 117,564 triangles. The part decomposition results consist of nine parts: (a) photograph of the original object, (b) rendering of the reconstructed mesh, (c) decomposed parts labeled in different colors, and (d) recovered superquadrics rendered in different colors.

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Zhang, Koschan, and Abidi Table 1 Recovered superquadric parameters and ground truths for one of the small cylinders shown in Fig. 10(d) where the unit is millimeters.

Table 2 Recovered superquadric parameters and ground truths for the water neck shown in Fig. 11(d) where the unit is millimeters. Object

Parameters Ground truths Superquadric parameters

a1

a2

a3

␧1

␧2

15.2 16.45

15.6 15.67

20.1 20.42

0.1 0.12

1.0 0.96

Parameters

a1

a2

a3

␧1

␧2

Handle

Ground truths 39.7 39.4 17.6 0.1 1.0 Superquadric parameters 40.23 40.58 66.83 0.13 0.98

Ball

Ground Truths 50.0 47.6 56.0 1.0 1.0 Superquadric parameters 51.62 47.56 54.28 1.02 0.95

Cylinder

Ground truths 16.5 17.8 44.2 0.1 1.0 Superquadric parameters 17.56 17.94 43.38 0.11 0.95

label as vertex p 1 , i.e., the same segmented part. Furthermore, parts composed of fewer vertices than a specified threshold are merged with adjacent regions. 4

Experimental Results

Experimental results on superquadric representation for multipart, automotive objects including a disk brake, a distributor cap, and a water neck are shown in this section. The meshes were reconstructed from multiview range images scanned from the IVP Ranger System.29 The recovered superquadrics were rendered in three dimensions using quad meshes.30 Figure 9 shows the disk brake and its part decomposition and superquadric representation results. The reconstructed 3-D triangulated surface shown in Fig. 9共b兲 consists of 37,171 vertices and 73,394 triangles. Starting from this reconstructed mesh, our part decomposition algorithm first decomposed the disk brake into two single parts, as shown in Fig. 9共c兲. Each decomposed part was next fitted to a superquadric model, as shown in Fig. 9共d兲. The decomposed parts and recovered superquadrics are rendered in different colors. We observe that our part decomposition algorithm successfully decomposed the disk brake into its constituent parts and the superquadric representation strategy recovered correct superquadrics in terms of their size, shape, and orientation. Compared to the original triangle mesh representation consisting of 37,171 vertices and 73,394 triangles, the recovered superquadrics describe the disk brake with only 22 parameters 共11 parameters for each superquadric without global deformations兲. This low representation cost of superquadric representation can significantly benefit tasks including virtual reality, object recognition, and robotic navigation. However, the hole at the center of the disk brake failed to be represented since superquadrics can only represent objects with genus of zero.19 Figure 10 shows the distributor cap and its part decomposition and superquadric representation results. The reconstructed mesh shown in Fig. 10共b兲 consists of 58,975 vertices and 117,036 triangles and was decomposed into 13 single parts, as shown in Fig. 10共c兲. We observe that this decomposition result is consistent with human perception. The recovered superquadrics shown in Fig. 10共d兲 correctly represent the distributor cap. The recovered superquadric parameters and the ground truths for one of the small cylinders on top of the distributor cap are shown in Table 1. We can observe that the recovered superquadric parameters for this cylinder have the correct size and shape information when compared with the ground truth parameters of the object. In addition, superquadrics represent the distributor cap with only 143 floating numbers, while the original triangle mesh consists of 58,975 vertices and 117,036 triangles. 416 / Journal of Electronic Imaging / July 2004 / Vol. 13(3)

Figure 11 shows the water neck and its part decomposition and superquadric representation results. The reconstructed mesh shown in Fig. 11共b兲 consists of 58,784 vertices and 117,564 triangles and was decomposed into nine single parts, as shown in Fig. 11共c兲. We observe that the decomposed parts are consistent with human perception. The recovered superquadrics shown in Fig. 11共d兲 correctly represent the water neck. The recovered superquadric parameters and the ground truths for the handle, the ball, and the small cylinder next to the handle of the water neck are shown in Table 2. From this table, we observe that the recovered superquadric parameters have the correct size and shape information when compared with the ground truth parameters of the objects. Again, superquadrics represent the water neck in a desirable accuracy with only 99 parameters while the original triangle mesh consists of 58,784 vertices and 117,564 triangles. 5 Conclusions This paper proposed a superquadric representation approach for multipart objects. Superquadrics can represent objects in an acceptable accuracy with only a few parameters, while other surface primitives and contours usually require thousands of representation elements. Such a compactness and low representation cost can significantly benefit tasks including virtual reality, object recognition, and robot navigation, e.g., it enables these tasks to run in a real-time manner. The advantages of the proposed superquadric representation approach include: 共1兲 it can successfully represent complicated, multipart objects by first decomposing them into single-part objects, and 共2兲 the recovered superquadrics have the highest confidence and accuracy since the input we use are 3-D triangulated surfaces reconstructed from multiview range images. The incompleteness and ambiguities contained in single-view images were eliminated during the multiview surface reconstruction process. We also proposed a 3-D part decomposition algorithm to decompose compound objects represented by triangle meshes into their constituent single parts based on curvature analysis. Considering the fact that the triangle mesh has been a standard surface representation element in computer vision and computer graphics, the proposed part decomposition algorithm is generic, flexible, and can facilitate computer vision tasks such as shape description and object recognition. Furthermore, the part decomposition algorithm can segment a large number of triangle meshes 共over 100,000兲 in only seconds on an SGI Octane workstation.

Superquadric representation of automotive parts . . .

Acknowledgments This work was supported by the University Research Program in Robotics under Grant No. DOE-DE-FG0286NE37968, by the Department of Defense/U.S. Army Tank-automotive and Armaments Command/National Automotive Center/Automotive Research Center Program R01-1344-18, and by the Federal Aviation Administration National Safe Skies Alliance Program R01-1344-48/49.

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23. H. Rom and G. Medioni, ‘‘Part decomposition and description of 3D shapes,’’ in Proc. Int. Conf. Pattern Recognition, pp. 629– 632 共1994兲. 24. M. Bennamoun, ‘‘A contour-based part segmentation algorithm,’’ in Proc. Int. Conf. Acoustics, Speech, and Signal Processing, Vol. 5, pp. 41– 44 共1994兲. 25. K. Koara, A. Nishikawa, and O. Miyazaki, ‘‘Hierarchical part decomposition method of articulated body contour,’’ in Proc. Int. Conf. Intelligent Robots and Systems, Vol. 3, pp. 2055–2060 共2000兲. 26. K. Wu and M. D. Levine, ‘‘3D part segmentation using simulated electrical charge distributions,’’ IEEE Trans. Pattern Anal. Mach. Intell. 19, 1223–1235 共1997兲. 27. A. Mangan and R. Whitaker, ‘‘Partitioning 3D surface meshes using watershed segmentation,’’ IEEE Trans. Vis. Comput. Graph. 5共4兲, 308 –321 共1999兲. 28. C. Lin and M. Perry, ‘‘Shape description using surface triangulation,’’ in Proc. Int. Conf. Computer Vision: Representation and Control, pp. 38 – 43 共1982兲. 29. User Documentation: MAPP2500 Ranger PCI System, Version 1.6, Integrated Vision Products, Sweden 共2000兲. 30. J. Wernecke, The Inventor Mentor: Programming Object-oriented 3D Graphics with Open Inventor, Addison-Wesley, Reading, MA 共1994兲. Yan Zhang received her BS and MS degrees in electrical engineering from Huazhong University of Science and Technology, China, in 1994 and 1997, respectively, and her PhD degree in electrical engineering from the University of Tennessee, Knoxville, in 2003. Her research interests include 3-D image processing, computer vision, and pattern recognition.

Andreas Koschan received his MS degree in computer science and his PhD in computer engineering from the Technical University Berlin, Germany, in 1985 and 1991, respectively. He is currently a research associate professor with the Department of Electrical and Computer Engineering, the University of Tennessee, Knoxville. His work has primarily focused on color image processing and 3-D computer vision including stereo vision and laser range finding techniques. He is a coauthor of two textbooks on 3-D image processing and a member of IS&T and IEEE. Mongi A. Abidi is a W. Fulton Professor with the Department of Electrical and Computer Engineering, the University of Tennessee, Knoxville, which he joined in 1986. Dr. Abidi received his MS and PhD degrees in electrical engineering in 1985 and 1987, both from the University of Tennessee, Knoxville. His interests include image processing, multisensor processing, 3-D imaging, and robotics. He has published over 120 papers in these areas and coedited the book Data Fusion in Robotics and Machine Intelligence (Academic Press, 1992). He is the recipient of the 1994 to 1995 Chancellor’s Award for Excellence in Research and Creative Achievement and the 2001 Brooks Distinguished Professor Award. He is a member of the IEEE, the Computer Society, the Pattern Recognition Society, SPIE and the Tau Beta Pi, Phi Kappa Phi, and Eta Kappa Nu honor societies. He also received the First Presidential Principal Engineer Award prior to joining the University of Tennessee.

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