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Y. Zhang, A. Koschan, and M. Abidi, "Superquadrics Based 3D Object Representation of Automotive Parts Utilizing Part Decomposition," Proc.
of SPIE 6th International Conference on Quality Control by Artificial Vision, TN, Vol. 5132, pp. 241-251, Gatlinburg, TN, May 2003.
Superquadrics based 3D object representation of automotive parts
utilizing part decomposition
Yan Zhang∗ , Andreas Koschan, and Mongi Abidi
334 Ferris Hall, Dept. Electrical and Computer Engineering
University of Tennessee, Knoxville, TN 37996-2100
ABSTRACT
We present a new superquadrics based object representation strategy for automotive parts in this paper. Starting from a
3D watertight surface model, a part decomposition step is first performed to segment the original multi-part objects into
their constituent single parts. Each single part is then represented by a superquadric. The originalities of this approach
include first, our approach can represent complicated shapes, e.g., multi-part objects, by utilizing part decomposition as a
preprocessing step. Second, superquadrics recovered using our approach have the highest confidence and accuracy due
to the 3D watertight surfaces utilized. A novel, generic 3D part decomposition algorithm based on curvature analysis is
also proposed in this paper. The proposed part decomposition algorithm is generic and flexible due to the popularity of
triangle meshes in the 3D computer community. The proposed algorithms were tested on a large set of 3D data and
experimental results are presented. The experimental results demonstrate that our proposed part decomposition algorithm
can segment complicated shapes, in our case automotive parts , efficiently into meaningful single parts. And our
proposed superquadric representation strategy can then represent each part (if possible) of the complicated objects
successfully.
Keywords: Superquadrics, 3D object representation, automotive parts, 3D part decomposition
1. INTRODUCTION
Object representation denotes representing real world objects with known graphic or mathematic primitives that can be
recognized by computers. This research has numerous applications for object-related tasks in areas including computer
vision, computer graphics, reverse engineering, etc. The shape of 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, they have the ability to achieve the highest data
compression ratio without losing the accuracy of the raw data. For these reasons volumetric primitives are the most
efficient features to represent objects in 3D. The primarily used volumetric primitives include generalized cylinders,
geons, superquadrics, etc 1 . As a subclass of generalized cylinders, superquadrics are a family of geometric solids, which
can be interpreted as a generalization of basic quadric surfaces and solids. With only a few parameters, they can
represent a large variety of standard geometric solids as well as smooth shapes. This makes superquadrics very
convenient for object representation. Superquadrics are also very efficient for representing three-dimensional surface
data. Opposed to a mesh representation of an object with thousands of triangles, the same object may be represented by a
single superquadric or a small set of superquadrics which are uniquely defined by 11 parameters each. This compact
object representation can be efficiently used for 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 mentioned above can be enhanced by employing superquadrics. Furthermore, the registration of multi-view
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 range data of multi-object scenes
with small overlap 2 .
∗
[email protected]; phone: (865) 974-9213; fax: (865) 974-5459
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Most early research on superquadric representation concentrated on representing single-part objects from single-view
intensity or range images by assuming that the image has been pre-segmented into single-part simple objects 3-13 . This
category of research focused on the data fitting processes including objective functions, fitting criteria measurements,
convergence speed, etc. For complex, multi-part objects or scenes, there are two major types of approaches. The first
type of methods incorporates an image segmentation step priori to the superquadric representation 11-15 . The other type of
methods directly recovers superquadrics from a range image without pre-segmentation 16-19 . Compared with the early
work on single-part object representation, these types of methods can represent more complex objects and have wider
applications in related tasks including robotic navigation, object recognition, etc. However, there are several weaknesses
for existing superquadric representation methods for complex, multi-part objects. First, exis ting methods cannot handle
arbitrary shapes or occlusions in the scene. Fig.1 shows an example of the most complicated object that can be
represented by superquadrics using existing methods 18 .
Fig. 1. A real range image of a multi-part object obtained from 18.
It can be observed that the range image shown in Fig.1 contains very few occlusions between different parts (objects)
because the viewpoint from which the image was acquired has been very well selected. When an automotive part, a
complex, multi-part object as shown in Fig.2, is of interest no existing methods can represent this part correctly because
heavy occlusions cannot be avoided from any single viewpoint.
(a)
(b)
Fig. 2. Images of a distributor cap. (a) A color picture and (b) a set of 3D calibrated range data from a single view.
The second weakness of existing methods is they only utilize single-view images. Again, for the automotive part shown
in Fig. 2 (a), it is too difficult to choose an optimal viewpoint from which all the parts are visible due to self-occlusions
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and occlusions as shown in Fig. 2 (b). 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, multi-part 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 multi-part objects with superquadrics utilizing part
decomposition as a preprocessing step. We also present a novel 3D part decomposition algorithm based on curvature
analysis. Experiments were conducted on a large number of 3D triangulated surfaces.
The remainder of this paper proceeds as follows. In Section 2, a superquadric representation method is proposed for
multi-part objects. In Section 3, the 3D part decomposition algorithm is proposed. The experimental results are presented
in Section 4. Finally, conclusions are presented in Section 5.
2. SUPERQUADRIC REPRESENTATION OF MULTI-PART OBJECTS UTILIZING PART
DECOMPOSITION
The diagram for this algorithm is illustrated in Fig. 3.
Part 1
A multi-part object
3D part
decomposition
Single SQ recovery
SQ 1
Single SQ recovery
SQ 2
……
Part N
Single SQ recovery
SQ N
Fig. 3. Diagram of the proposed superquadric representation strategy utilizing part decomposition.
Beginning with a complete 3D watertight surface composed of triangle meshes, we propose a part decomposition
algorithm to segment the model into single parts. Next, each single part is fitted with a superquadric model. The reasons
that we use complete, watertight 3D triangulated models as input data include first, the complete 3D model can provide
sufficient information for superquadric fitting and recovered superquadrics, therefore, have higher confidence when
compared with single-view images. Second, both the proposed representation algorithm and the proposed part
decomposition algorithm are generic and very flexible due to the triangle meshes used since triangle meshes have been
the standard surface representation elements in many computer-related areas. A triangulation step is needed if only
unstructured 3D point clouds are provided.
2.1 Introduction to superquadrics
Shapes superquadrics can represent are shown in Fig. 4. The implicit definition of superquadrics is expressed as
20
ε1
2
2
2 ε

 2
ε2
ε2
ε1






x
y
z
F ( x , y , z) =   +    +   = 1, ε 1 , ε 2 ∈ (0,2).
a
 a2  
 a3 
 1 

(1)
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Fig. 4. Shapes represented by superquadrics with various shape parameters.
Where ( x, y, z) represents a surface point of the superquadric, ( a1, a2 , a3 ) represent sizes in the ( x, y, z) directions, and
(ε1 , ε 2 ) represent shape factors. To represent a superquadric model in the world coordinate system, 11 parameters are
needed. They are summarized as
∧ = ( a1 , a2 , a3 , ε1 , ε 2 , φ ,θ , ϕ , p x , p y , p z )
(2)
Most approaches define an objective function and find the parameters of superquadrics through minimizing this
objective function. The mostly commonly used objective function is 1
N
G ( ∧) = a1a2 a3 ∑ ( F ε1 ( xc , y c , z c ) − 1) 2 .
i =1
The Levenberg-Marquardt method
efficiency.
20
(3)
has been primarily used to minimize the objective function due to its stability and
3. CURVATURE BASED 3D PART DECOMPOSITION
Many tasks in computer vision, computer graphics, and reverse engineering are performed based on objects or models.
Those object-based tasks become extremely difficult when the object is complicated, e.g., it contains multiple parts. Part
decomposition can simplify the original task performed on multi-part 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 2D
intensity or 2.5D range images has been conducted over the last two decades 23-25 , little effort has been made on part
segmentation of 3D data 26, 27 . Therefore, a novel 3D part decomposition algorithm is proposed in this paper. Fig. 5
illustrates the difference between region segmentation and part decomposition. The cylinder is seen as segmented into
three surfaces by a region segmentation algorithm but remains one part for a part decomposition algorithm. Part
decomposition is more appropriate for high-level tasks such as object recognition, especially for 3D models. The
diagram of the proposed part decomposition algorithm is shown in Fig. 6.
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Fig. 5. The difference between region segmentation and part decomposition.
3D triangulated
surface
Curvature
estimation
Boundary
detection
Region
growing
Postprocessing
Fig. 6. Diagram of the proposed 3D part decomposition algorithm.
The proposed algorithm 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 based on the transversality regularity 21, 22 . These boundaries are therefore detected by thresholding estimated
Gaussian curvatures for each vertex. A component labeling operation is then performed to grow non-boundary vertices
into parts. A postprocessing step is performed to assign non-labeled vertices to one of the parts according to the sign of
curvature and the smallest distance to neighbor vertices. Additionally, a threshold, that is the minimum number of
vertices of a part, is used to eliminate parts composed of too few vertices. This part decomposition algorithm is
summarized as follows.
{Algorithm 1 (3D Part decomposition)}
{Input:} Triangulated 3D surface.
{Step 1.} Compute Gaussian curvature for each vertex on the surface.
{Step 2.} Label vertices of highly negative curvature as boundaries using a pre-specified threshold. Label the remaining
vertices as seeds.
{Step 3.} Eliminate isolated vertices.
{Step 4.} Perform iterative region growing on each seed vertex.
{Step 5.} Assign non-labeled vertices to parts, and eliminate parts having less than a pre-specified number of vertices.
{Output:} Point clouds (3D unstructured data points) of various decomposed single parts written to separate 3D files.
The major steps of this part decomposition algorithm are described respectively in the following sections.
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3.1 Gaussian curvature estimation and boundary detection
The mesh-based curvature estimation method proposed in 28 is implemented in this work to estimate Gaussian curvature
for each vertex of a triangle mesh. Since only the sign of surface curvature is used in our proposed decomposition, this
simple curvature estimation is sufficient for our application. As shown in Fig. 7, Gaussian curvature of the vertex p is
computed as 28
N
K ( p) =
3( 2π − ∑ θ i )
i =1
N
δ 2 ( p − pi ) ,
(4)
∑ Ai
i =1
where p represents the point of interest, pi represents one of the mesh neighbors of the point p, and Ai represents the
area of the corresponding triangle. θ i represents the interior angle of the triangle at p, and δ is the Dirac delta function.
Fig. 7. Curvature calculation for the vertex p utilizing triangle mesh information.
After Gaussian curvature is obtained for each vertex on the surface, a specified threshold is applied to label vertices as
boundary or seed. Vertices of highly negative curvature are labeled as boundaries between two parts while the rest are
labeled as seeds belonging to potential object parts according to the transversality regularity 21 . The threshold is critical
and affects the performance of later region growing. This threshold is determined in a heuristic way depending on object
and mesh resolution. According to the label of each vertex, isolated vertices are then removed. As noted previously, two
types of isolated vertices defined in this work include: (i) a point which is labeled as boundary while all of its neighbors
are labeled as seeds and (ii) a point which is labeled as a seed while all of its neighbors are labeled as boundary. The
labels of isolated vertices are changed to be the same as their neighbors.
3.2 Region growing and postprocessing
After the vertices are labeled, a region-growing operation is performed on each vertex labeled as seed. Fig. 8 shows
triangle meshes around the point p. To illustrate the region growing process, a two-ring neighborhood of the mesh
around point p is shown in this figure. Region growing is performed as follows. Starting from a seed vertex p, the unique
region label is first assigned to the vertex. Second, all the neighbors pi labeled as seeds initially are then labeled with
the same region number as the point p. The same labeling process is performed for each neighbor pi to label vertices
pij . 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. This process is repeated for each seed vertex, but not for a vertex
which has been grown and already labeled uniquely. After the seed vertices are assigned new labels, a post-processing
step is performed for each boundary vertex. Given a seed point x, its mesh neighbors xi are first sorted in ascending
order based on their Euclidean distance to the point x. Next, the vertex x k is selected from the ordered neighbors where
x k is the first vertex in the list with a region label and not a boundary label. The boundary vertex x is then labeled the
same as the vertex x k , i.e., the label of x is changed from boundary to the unique region label of x k . Finally, with the
exception of a few missing vertices, each vertex should now have a region label and thus assigned to different object
parts.
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Fig. 8. Region growing process for the vertex p.
Finally, a post-processing step is performed to assign the remaining vertices that have not been labeled. For example, the
vertex p is an unlabeled vertex and needs further post-processing. Assuming that pi (i = 1, 2, ..., N) represents a
neighbor vertex of the point p, the neighbor 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 which has the
smallest Euclidean distance to the vertex p is selected as a target vertex. For example, the vertex p1 is assumed to be the
target vertex of the vertex p. Finally, the vertex p is assigned the same label of the vertex p1 , i.e., the same segmented
part. Furthermore, parts that are composed of fewer vertices than a specified threshold are merged with adjacent regions.
4. EXPERIMENTAL RESULTS
Experimental results on both part decomposition and subsequent superquadric representation are shown in this section.
Fig. 9 shows a 3D triangulated teapot model, the model with labeled negative Gaussian curvatures, and the decomposed
parts of the teapot. Boundary vertices with negative Gaussian curvature are labeled blue as shown in Fig. 9 (b). Fig. 9 (c)
labels decomposed parts in different colors and it can be observed that all the parts including the cap, the handle, the
body, the spout, and the small block at the bottom are correctly decomposed from the teapot.
(a)
(b)
(c)
Fig. 9. Part decomposition of a teapot. (a) The 3D triangulated surface of a teapot, (b) the model with curvature labeled as blue, and
(c) decomposed parts.
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Fig. 10 shows results of both part decomposition and subsequent superquadric representation for the distributor cap. The
3D triangulated surface for the object was reconstructed from multi-view range images using the RANGER scanner 29 .
Fig. 10 (c) shows that the five small cylinders, the base, two small screws of the distributor cap are all correctly
segmented from the original distributor cap model. The only part that is not decomposed from the distributor cap is the
long small cylinder linked to the base. Instead, this small cylinder is assigned to the base part after the decomposition.
This is because the mesh smoothing step during the surface reconstruction step also smoothes out the boundary between
the small cylinder and the base of the distributor cap. Therefore, the small cylinder is decomposed into the same part as
the base instead of a single part of its own. Fig. 10 (d) shows five recovered and rendered superquadrics for the small
cylindrical parts on the top of the distributor cap. The recovered superquadric parameters and the ground truth values of
one of the small cylinders are shown in Table 1 as an example. It can be observed that the superquadrics have correct
size and shape information when compared with the ground truth parameters of the object.
(a)
(b)
(c)
(d)
Fig. 10. Part decomposition and superquadric representation of an automotive part . (a) The 3D triangulated surface of a distributor
cap, (b) the model with curvature labeled as blue, (c) decomposed parts, and (d) recovered and rendered superquadrics.
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Table 1. Recovered superquadric parameters and ground truth values for one of the small cylinders shown in Fig. 10 (d). GT: ground
truth values. RP: recovered parameters. Units: mm.
a1
a2
a3
ε1
ε2
GT
15.2
15.6
20.1
0.1
1.0
RP
16.45
15.67
20.42
0.12
0.96
Fig. 11 shows a reconstructed 3D model of a water neck, the model with labeled negative Gaussian curvatures, the
decomposed parts, and superquadric representation. Fig. 11 (c) shows that the cylindrical handle, the ball, the base, the
small cylinder next to the handle, and several small screws of the water neck are all correctly segmented from the
original model although the boundaries are detected redundantly. This demonstrates that the proposed part
decomposition algorithm can handle 3D triangulated models with various smoothness and resolutions very well. And it
is insensitive to detected boundaries. Fig. 11 (d) shows three recovered and rendered superquadrics for the handle, the
ball, and the small cylinder. The recovered superquadric parameters and the ground truth values for the handle of the
water neck are shown in Table 2. Again, we observe correct size and orientation information for the recovered
superquadrics.
(a)
(c)
(b)
(d)
Fig. 11. Part decomposition and superquadric representation of an automotive part. (a) The 3D triangulated surface of a waterneck, (b)
the model with curvature labeled as blue, (c) decomposed parts, and (d) recovered and rendered superquadrics.
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Table 2. Recovered superquadric parameters and ground truth values for handle shown in Fig. 11 (d). GT: ground truth values. RP:
recovered parameters. Units: mm.
a1
a2
a3
ε1
ε2
GT
39.7
39.4
17.6
0.1
1.0
RP
40.23
40.58
66.83
0.13
0.98
5. CONCLUSIONS
This paper proposed an effective approach to represent multi-part objects with superquadrics utilizing part
decomposition as a preprocessing step. The advantages of our method include: (i) the proposed representation approach
can represent very complicated, multi-part objects by decomposing them into their constituent single parts, and (ii)
superquadrics recovered using the proposed representation method have the highest confidence and accuracy due to the
reconstructed 3D triangulated surface utilized. The ambiguities contained in single-view data have been eliminated
during the surface reconstruction step by utilizing multi-view data sets . A novel 3D part decomposition algorithm is also
proposed to decompose compound objects represented by triangle meshes into their constituent parts based on curvature
analysis. Considering the fact that the surface polygonal mesh has been a standard representation form in computer
vision and computer graphics, the proposed part decomposition algorithm is generic, flexible, and has wide application
in high level computer vision tasks such as shape description, object recognition, and 3D reconstruction. Furthermore,
the algorithm can handle a large number of triangle meshes (over 100,000) in only seconds when implemented on a SGI
Octane workstation. Even though not every decomposed part of the original object can be represented by a superquadric
due to its representation limitation, the superquadric parameters recovered for some parts of the object will significantly
benefit the object recognition task.
6. ACKNOWLEDEGMENTS
This work was supported by the University Research Program in Robotics under grant DOE-DE-FG02-86NE37968, by
the DOD/TACOM/NAC/ARC Program, R01-1344-18, and by FAA/NSSA Program, R01-1344-48/49.
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