Accurate Reduction of Surface Noise
Henry W. Patton Center for Engineering Education and Practice Technical Project Report i

Accurate Reduction of Surface Noise
Henry W. Patton Center for Engineering Education and Practice Technical Project Report ii

Accurate Reduction of Surface Noise
Synopsis ........................................................................................................................................................... iii
1. Background ............................................................................................................................................... 1
2. Objectives .................................................................................................................................................. 1
3. Approach................................................................................................................................................... 1
4. Results ......................................................................................................................................................... 4
5. Conclusions .............................................................................................................................................. 13
6. Impact on Engineering Education ...................................................................................................... 13
7. Industrial Impact ..................................................................................................................................... 14
8. Acknowledgments ................................................................................................................................. 14
9. References .............................................................................................................................................. 15
Henry W. Patton Center for Engineering Education and Practice Technical Project Report iii

Accurate Reduction of Surface Noise
Transformation from physical objects to digital models is marked as a revolutionary step in human history toward the information age. Different sensors (contact or non-contact) can be used to facilitate such a transformation. With the wide use of non-contact optical sensors such as laser scanners in recent years, reconstruction of physical objects at fast acquisition rates has been practiced in industry, medicine, military, and other fields. However, there are several factors, including measurement noise, registration and calibration error, and occlusion, which limit the optical acquisition of 3D data. The best known and widely used non-contact optical sensors are based on triangulation [KLIN03]. The performance of this type of sensors is limited by coherent or speckle noise [DORS94]. Compared to tactile Coordinate
Measurement Machine (CMMs) or robotic arms, a low measurement accuracy is one of the most important factors that hinder the application of non-contact optical sensors in model reconstruction for high-fidelity industrial inspection, reverse engineering and analysis.
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Accurate Reduction of Surface Noise
Transformation from physical objects to digital models is marked as a revolutionary step in human history toward the information age. Different types of sensors (contact or non-contact) can be used to facilitate such a transformation. The main problem of contact sensors, including touch probes on CMMS or robotic arms, is their extremely low rate of data acquisition. With the proliferation of non-contact optical sensors such as laser scanners in recent years, fast reconstruction of physical objects has been practiced in industry, medicine, military, and many other fields. However, there are several factors, including measurement noise, registration and calibration error, and occlusion, which lead to a poor measurement accuracy and exclude these non-contact sensors from various tasks in manufacturing industry. For instance, the best known and widely used non-contact optical sensors are based on triangulation for close range (within 2m) measurements [KLIN03, BERN99b, VARA97] . The performance of these types of sensors is limited by coherent or speckle noise [DORS94]. With most laser scanners, the measurement uncertainty together with other factors results in a measurement accuracy equal to or worse than ±
0 .
005
inch/
127
m
(data based on 2003 3D Laser Scanner Survey by http://www.pobonline.com/) for metal surfaces at a standoff distance of 1m, while the detection of surface distortions on sheet metal panels normally requires an accuracy of ±
0 .
001
inch or
25
m
(mean value; personal communication with Mr. Maurice Lou, Ford Motor Company). The machining accuracy of a computer numerical control (CNC) machine nowadays can reach
5
m
±
.0
0002 inch or
(e.g., TRM by Fadal). This means that the measurement from these sensors can’t be used directly for high-precision manufacturing applications, including high-precision industrial inspection [SHENT00,
PERN02, FRAS99, PRIE99, LEEK00] and reverse engineering [VARA97, BERN99b, PAGE03, THOM99,
SARK91].
The main objective of the proposed research is to design a new surface denoising approach, which can accurately remove the measurement noises of non-contact optical sensors such that the measurement accuracy can be significantly and reliably improved to a level that is suited for many tasks in manufacturing applications such as high-precision industrial inspection and reverse engineering. The primary goal of a parallel education plan is to integrate the proposed research into the existing computer graphics curricula.
Although a considerable amount of advance has been made in the recent years in the area of surface denoising or surface smoothing, all existing algorithms still face two important problems that have not been well solved, as explained below. More importantly, these two problems directly influence the measurement accuracy of non-contact optical sensors, and therefore impede the application of these sensors in many tasks in manufacturing.
First, all existing iterative smoothing algorithms don’t have a robust convergence, i.e., they diverge from time to time, as illustrated in Figure 1. This divergence behavior makes the smoothing accuracy of existing algorithms rely greatly on a correct choice of termination threshold for an iterative smoothing
Henry W. Patton Center for Engineering Education and Practice Technical Project Report 1

Accurate Reduction of Surface Noise process. Under certain circumstances such as the process of an initially smooth model (Figure 1) or partially smooth model, such a termination threshold may not be successfully estimated, resulting in a failure of the algorithms. The smoothing accuracy is quantified by certain error metric between true geometry and denoised geometry, as explained in the second paragraph of Section 6.3. It indicates how well the surface noises are removed.
(a) original model
(b) convergence test of smoothing an initially smooth sphere
Figure 1. Smoothing of a perfectly smooth sphere.
Secondly, each of existing algorithms performs better than others only with a certain type of data models.
For instance, median filter produces the best smoothing accuracy at sharp edges, while Gaussian or mean filter performs better with noised data models that don’t contain sharp features. No single existing algorithm is available, which performs consistently best in terms of smoothing accuracy over a wide spectrum of different data models.
To overcome these two crucial problems of existing algorithms and to design a universal approach that is consistently better than existing algorithms in terms of convergence and smoothing accuracy, the following approach is proposed:
• Adopt a feature-preserving pre-smoothing (median filter) that does not require any threshold and implicitly retains the sharp features. Use G
1 geometric discontinuity and curvature threshold as an indicator for surface partitioning of feature and non-feature regions. Feature regions mean the areas in which either sharp edges or high curvatures exist, while the remaining parts are called the nonfeature regions.
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Accurate Reduction of Surface Noise
• Design a second-order predictor as an accurate indicator for guiding a surface smoothing process in non-feature regions. The main benefit of the proposed second-order predictor is a better accuracy and convergence with curved surfaces than the first-order predictors, mean-curvature flow, Gaussian and RBF predictors in existing algorithms.
• Adopt a median filter for feature regions. The main advantage over existing anisotropic diffusion algorithms is no feature over-enhancement due to inverse diffusion. Any over-enhancement is not suited for the manufacturing applications.
• Apply our second-order predictor in non-feature regions, while median filter is used in feature regions. This forms a hybrid approach that is expected to perform consistently better than existing algorithms with different types of noised data models in terms of convergence and smoothing accuracy.
The schematic of our proposed two-stage approach is shown in Figure 2, in which the left stage is executed first to generate surface partitions, and then in the right stage the noised data model is processed again with extra partitioning information obtained from the left stage. In this figure, the data models are considered to be in the format of polygonal meshes, which can be obtained from point-cloud data using either commercial software (e.g., Geomagic
BERN99, HOPP92, AMEN98]
TM ) or published algorithms such as [EDEL94,
. A denoised data model can be directly used for industrial inspection or can be further converted to NURBS surfaces for reverse engineering through segmentation and surface fitting [VARA97].
Noised Data
Model
Surface Partitioning by Geometric
Discontinuity
Feature
Regions
Noised Data
Model
Feature
Regions
Second-order
Predictor
Feature-Preserving
Pre-smoothing
Non-feature
Regions Non-feature
Regions
Median filter
Denoised
Data Model
Figure 2. Our proposed two-stage smoothing approach.
In addition, an interior-point method has been developed to overcome the shortcoming of median filter when sharp features are treated. It was proven that the median filter can’t guarantee removing local noises at the location of sharp features even though it can maintain all the global sharp features. The basic idea of the interior-point method is to utilize the median filter to maintain all the global sharp features in a pre-smoothing process, and then to conduct a partition on the basis of these global sharp features.
Finally, a c1 discontinuity is transformed to a number of c0 discontinuity problems constrained by the points at the sharp features .
Furthermore, a brand-new concept of sparing least-squares fitting was proposed in this study. The basic idea of the sparing least-squares fitting treatment is to avoid the least-squares fitting as much as possible because it is computationally time consuming. If the vertex density is very high in a local region, then the
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Accurate Reduction of Surface Noise local least-squares fitting of a quadratic or cubic curve segment in this region becomes unnecessary, because an approximation error becomes smaller even with a linear approximation. On the contrary, if the vertex density is very low in a local region, a local least-squares fitting becomes crucial to achieve a low approximation error, which in turns leads to a high denoising accuracy.
4.1 Hybrid Smoothing
The combination of the results from feature and non-feature noised regions as well as non-noised regions reflects the philosophy of our hybrid smoothing approach, in which we propose to choose a non-feature smoothing scheme as the best tool for non-feature noised regions, to use a feature smoothing scheme for feature noised regions, and to perform no action for non-noised regions. This provides much more flexibility than existing approaches such as anisotropic diffusion and bilateral filter. Figure 3 shows that our approach performs best among all existing approaches on objects with or without sharp features.
4.2 Boundary Smoothing
The boundary of a non-closed surface mesh may cause an extra problem in the convergence of smoothing algorithms. According to Figure 4(c), only our scheme shows a benign tendency of convergence. Since the target surface of mean curvature flow is a minimal surface or a surface with minimum surface area per volume, it would be very interesting to observe its smoothing behavior on a noised minimal surface.
Surprisingly, the mean curvature flow performs worst on a noised helicoid model among all testing algorithms, as shown in Figure 5(c). If you look back at Figure 3(c), the same thing happens on a noised sphere that is the case of minimum surface area per volume. Therefore, even though mean curvature flow is already a matured topic in mathematics and several theorems on regularity, global existence and convergence of the flow have been proven, it is not a good choice for surface denoising in terms of convergence and accuracy on the basis of our numerical experiments.
4.3 Smoothing of Complex Synthetic Objects
Figure 6(a) is a complex synthetic model (5,213 vertices and 10,342 triangles) that contains a jungle of quadric surfaces: several planes, one cone, one paraboloid, three half spheres, three cylinders and one torus. The noised model is shown in Figure 6(b), while Figures 6(c) through (f) are the smoothed model by using several testing algorithms, respectively. In terms of accuracy, our scheme and median filter perform best, while the Gaussian filter gives the worst result. The execution time of different algorithms is: MN – 0.34 sec/step; MD – 1.42 sec/step; GS – 1.08 sec/step; BL – 0.93 sec/step; MC – 0.7 sec/step; VL
– 0.14 sec/step; MQ – 3.6 sec/step.
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Accurate Reduction of Surface Noise
4.4 Smoothing of Real Objects
Figure 7(a) is a mountain model (16,384 vertices and 32,258 triangles) that was originally made by John
Moreland. After the addition of noises as in Figure 7(b), the smoothing results are illustrated in Figures 7
(c) through (f). In terms of accuracy, our scheme, median filter and volume-preserving Laplacian perform best, while the bilateral filter is the worst. In Figures 7(e) and (f), both mean curvature flow and bilateral smoothing give an over-smoothing result, which may be useful for surface simplification, but not for surface denoising. Our scheme retains a slightly more features than the Gaussian filter, as in Figures 7(c) and (d).
Figure 8(a) is a female model (51,726 vertices and 103,275 triangles) from Cyberware, Inc. After the addition of noises, we applied the noised model in Figure 8(b) to different smoothing algorithms. Median filter gives a similar result as our scheme except the side face is slightly rougher, as illustrated in Figure
8(d). Bilateral filter provides an over-smoothing result, while the mean curvature flow doesn’t produce a very smooth mesh with this highly-noised model in Figure 8(b). The running time for different smoothing algorithms is: MD – 15.76 sec/step; GS – 20.3 sec/step; BL – 130.8 sec/step; MC – 7.5 sec/step;
VL – 2.97 sec/step; MQ – 73.2 sec/step. Once again, bilateral filter and our scheme rank the first two in the aspect of the computation cost.
4.5 Interior-point Method
Figure 9 shows the denoising effect of two noised data models, an airplane and a submarine. As you can see that the noises are effectively removed while the sharp features are well maintained. The comparison among our interior-point scheme, our previous hybrid scheme, and the traditional anisotropic diffusion is given in Figure 10, in which the interior-point scheme is about 10% better than the hybrid scheme, and both are much better than the traditional anisotropic diffusion method. Figure 10 is the average result from four data models: an airplane, a fighter jet, a submarine and a dolphin. Two denoising error metrics, surface normal and vertex distance, are used, and both are scaled to a same level of magnitude. The smaller the scaled metrics, the higher the denoising accuracy becomes.
4.6 Sparing Least-squares Fitting
We define time saving as a ratio of the execution time of conventional least-squres fitting to the execution time of sparing least-squares treatment. Figure 11 shows the comparison between the traditional leastsquares fitting and our sparing least-squares fitting. The average time saving of our scheme for eight test cases is 2.93 times as good as the traditional least-squares fitting, while the average differences between the two approaches in distance and normal error metrics are 12.7% and 9.3%, respectively. It is reasonable to infer that the sparing least-squares treatment gains a significant reduction in computation time and loses slightly in denoising accuracy, compared to the conventional least-squares denoising.
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(a) noised
(b) smoothed
(c) convergence test
(d) noised
(e) smoothed
Accurate Reduction of Surface Noise
(f) convergence test
Items (b) and (e) are smoothing results of our scheme. igure 3.
F
Smoothing of a noised sphere and a noised box.
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Accurate Reduction of Surface Noise
(a) noised
(b) smoothed
(c) convergence test
Figure 4. Smoothing of a noised monkey saddle.
Item (b) is smoothing result of our scheme.
(a) noised
(b) smoothed (c) convergence test
Figure 5. Smoothing of a noised helicoid.
Item (b) is smoothing result of our scheme.
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Accurate Reduction of Surface Noise
(a) original
(b) noised
(c) our
(d) Gaussian
(e) volume-preserving
Laplacian
(g) surface normal error metric
(h) vertex distance error metric
(f) bilateral
Figure 6. Surface smoothing of a jungle of quadratic surfaces.
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Accurate Reduction of Surface Noise
(a) original model
(c) our scheme
(e) mean curvature
(b) noised surface
(d) Gaussian smoothing
(f) bilateral smoothing
(g) surface normal error metric (h) vertex distance error metric
Figure 7. Surface smoothing of a mountain model.
Data courtesy of John L. Moreland.
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Accurate Reduction of Surface Noise
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Accurate Reduction of Surface Noise
(a) original (b) noised (c) our
(d) median (e) bilateral (f) mean curvature
(g) surface normal error metric (h) vertex distance error metric
Figure 8. Surface smoothing of a female head. Data courtesy of Cyberware, Inc.
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Accurate Reduction of Surface Noise
Figure 9. Original noised models (left) and new models denoised by our interior-point method (right).
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Accurate Reduction of Surface Noise
Figure 10. Original noised models (left) and new models denoised by our interior-point method (right).
Figure 11. Performance indices of two smoothing schemes (LSF: traditional
least-squares fitting; SLSF: our sparing least-squares fitting).
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Accurate Reduction of Surface Noise
In the aspect of denoising accuracy, we have achieved a noticeable better performance than existing methods, including well-known median filter and anisotropic diffusion. Our schemes also overcome the limitation of some previous approaches, including the well-know surface intersection method in most point-based approaches, in handling objects that contain self intersections or singular points.
Furthermore, we achieved about twofold increase in the computational efficiency of least-squares fitting by introducing a brand-new concept, sparing least-squares fitting. These advances will play an essential role in broad applications of laser sensors in metrology and reverse engineering. The following publications were supported in part by this grant from the Henry W. Patton Center for Engineering
Education and Practice at the University of Michigan-Dearborn.
• Shen, J., Yoon, D., Shou, H., Zhao, D. 2006. A set of denoising algorithms for two-dimensional closed curves. Computer-Aided Design and Applications , Vol. 3, Nos. 1-4, pp. 1-10.
• H.Shou, Shen, J. and Yoon, J. 2006. “Robust plotting of polar algebraic curves, space algebraic curves and offsets of planar algebraic curves”, Reliable Computing , Vol. 12, No. 4, pp. 323-335.
• Shen, J., Maxim, B., Akingbehin, K. 2005. "Accurate Correction of Surface Noises of Polygonal
Meshes ." International Journal for Numerical Methods in Engineering Vol 64, No. 12, pp. 1678-1698.
• Yoon,D., Shen,J., et al. 2005. "Application of Sweeping Techniques to Reverse Engineering."
International Journal of Modeling and Simulation , Vol. 25, No. 4, pp. 278-284.
• Shen, J., Song, Y., Yoon, D. and Liu, S. 2006. “Hybrid Surface Partitioning of Finite Element Meshes”,
Proceedings of Computer Graphics and Virtual Reality 06 , pp. 182-188.
• Shen, J., Yoon, D., Song, Y. and Liu, S. 2006. “Repairing Polygonal Meshes for Volume Meshing”,
Proceedings of Computer Graphics and Virtual Reality 06 , pp. 189-195.
• Song, Y., Shen, J. and Yoon, D. 2006. “A Bipolar Model for Region Simplification”, Proceedings of
Computer Graphics and Virtual Reality 06 , pp. 147-160.
• Shou, H., Song, W., Shen, J., Martin, R. and Wang, G. 2006. “A Recursive Taylor Method for Ray
Casting Algebraic Surfaces”, Proceedings of Computer Graphics and Virtual Reality 06 , pp. 196-202.
• Shen, J., Yoon, D., Chi, L. and Murphey, Y. 2006. “Accurate Surface Denoising for Metrology and
Reverse Engineering”, Proceedings of 2006 NSF DMI Grantees and Research Conference.
Currently, there is a computer graphics course offered in the curriculum of both the Bachelor and Master
Degree Programs in the Department of Computer and Information Science at the University of Michigan-
Dearborn. This course covers topics related to different modeling techniques of curves and surfaces.
However, advanced application in automotive industry is not covered in the course due to the deficiency of related technical materials. The results of this research are being selected to complement the contents of reverse engineering and visualization.
Collaboration between the University of Michigan-Dearborn and the Dearborn Center for Mathematics,
Science, and Technology (DCMST) has been discussed among Dr. Jie Shen (University of Michigan-
Dearborn), Dr. Herman Boatin (Lead Teacher at DCMST) and Ms. Kim Shawver (Computer/Math
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Accurate Reduction of Surface Noise
Teacher at DCMST). A high-school student field trip from DCMST to the University of Michigan-
Dearborn campus to visit the virtual engineering laboratory was scheduled in the Fall of 2006.
The following impacts can be envisioned:
1.
A potential to bring the accuracy of optical sensors to a comparative level with touch sensors while taking advantage of high data acquisition rates and non-contact property, leading to a strong advantage to the industry. Accurate and robust non-contact sensor will have a major impact on highprecision inspection, process control and metrology.
2.
A first-time systematic study on denoising at geometric discontinuities (mainly C0 and C1), which will become a milestone in the area of surface denoising.
3.
A better understanding on the patterns of measurement noises, surface attributes, sampling resolutions and their effects on the denoising accuracy. This will enrich the knowledge across computational geometry and metrology.
4.
A new virtual simulation system that will facilitate the study on measurement uncertainty and scanning path of laser probing on CMMs, enriching the knowledge across computer graphics, sensor and simulation.
5.
A possible commercialization of the proposed approach, which will facilitate the technology transfer, leading to a broader impact on the industry.
Based upon the research results in this project, we have successfully obtained an NSF DMI grant (DMI-
0514900 ) of $114,116.00. This continuous research is in line with the PI’s long-term goal of fusing the technologies in computational geometry, computer graphics with finite element method, solid mechanics and optimization technology to establish new approaches for the next-generation of design and manufacturing, which will be useful in automotive and aerospace industry. In addition, a proposal to the
Michigan Economic Development Corporation has been through several reviewing stages. If it is awarded, it will help commercialize our research results.
This project is supported by a grant from the Henry W. Patton Center for Engineering Education and
Practice at the University of Michigan-Dearborn. Some polygonal mesh models were obtained from the computer graphics laboratory at Stanford University.
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Accurate Reduction of Surface Noise
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.
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