Performance analysis of 3-D shape measurement algorithm with a short baseline projector-camera system
© Liu and Li; licensee Springer. 2014
Received: 9 December 2013
Accepted: 24 March 2014
Published: 28 August 2014
A number of works for 3-D shape measurement based on structured light have been well-studied in the last decades. A common way to model the system is to use the binocular stereovision-like model. In this model, the projector is treated as a camera, thus making a projector-camera-based system unified with a well-established traditional binocular stereovision system. After calibrating the projector and camera, a 3-D shape information is obtained by conventional triangulation. However, in such a stereovision-like system, the short baseline problem exists and limits the measurement accuracy. Hence, in this work, we present a new projecting-imaging model based on fringe projection profilometry (FPP). In this model, we first derive a rigorous mathematical relationship that exists between the height of an object’s surface, the phase difference distribution map, and the parameters of the setup. Based on this model, we then study the problem of how the uncertainty of relevant parameters, particularly the baseline’s length, affects the 3-D shape measurement accuracy using our proposed model. We provide an extensive uncertainty analysis on the proposed model through partial derivative analysis, relative error analysis, and sensitivity analysis. Moreover, the Monte Carlo simulation experiment is also conducted which shows that the measurement performance of the projector-camera system has a short baseline.
Keywords3-D shape measurement Projector-camera Short baseline Binocular stereovision FPP
Noncontact optical measurement methodology has been widely used in many industrial applications, such as industry inspection and 3-D printing manufacturing. Among these mature optical 3-D measurement techniques, the structured light technique has been widely used in recent years due to its good characteristics of high precision, flexibility, and robustness to texture-less object surface reconstruction. Numbers of works have been presented in this issue –. According to a different model, the methods for obtaining 3-D shape information with a structured light system can be simply divided into two main categories: one common way is using conventional binocular stereo vision or named ‘CSV’ model and the other strategy is adopting fringe projection profilometry (FPP) technique. In the first category, the projector is always treated like a camera. In this model, projector and camera in the system are always required to be pre-calibrated before 3-D shape measurement task. Many camera calibration methods are available to be utilized directly ,. However, for projector calibration, even with the latest accurate calibration methods using active target and phase-shifting technique ,, the accuracy of projector calibration can hardly reach as the same level as the camera. One of the intuitive reasons is that the parameters of a projector cannot be calibrated individually without the help of camera. Thus, the error propagation from camera calibration process is unavoidable and the overall system calibration accuracy is limited. The biased calibrated parameters of camera and projector will decrease the measurement accuracy. Particularly, in a short baseline arrangement system, the bias from feature point localization on the image will also be magnified with biased parameters. Therefore, how to accurately calibrate a short baseline arrangement system is more critical than a general configuration system which usually has a much larger baseline. In the second category, the projector is commonly regarded as grating optical device. The height information is obtained from the phase-to-height mapping relationship between phase distribution and geometric parameters of the system. Hence, the projector is not needed to be pre-calibrated anymore. In some presented phase-to-height model, even the camera is also not required to be pre-calibrated ,.
In this paper, our method falls into the second category. We propose a generic FPP-based projecting-imaging model and explore the relationship between the phase distribution, height information, and geometric parameters of the system. Based on the proposed model, we then study the problem of how the uncertainty of relevant parameters, the length of baseline in particular, affects the 3-D shape measurement accuracy. In other words, we focus on the performance analysis of the 3-D shape measurement according to our proposed model particularly when the system has a short baseline.
Background and literature review
Conventional stereovision model
Equation 6 shows that an accurate measurement result is depending on the four factors: (1) accurate determination of the pixel point m cu (u c , v c ) T on camera’s image plane, (2) unbiased camera’s parameters, (3) unbiased projector’s parameters, and (4) accurate determination of the pixel point which corresponds to the pixel point on the projector’s image plane. The first two conditions are easy to be satisfied with available camera calibration algorithm , and well-developed image processing technique ,. However, the latter two are much more difficult to be achieved. Furthermore, since the projector cannot ‘capture’ like a camera, how to determine the corresponding pixel point on the projector’s image plane is a challenge. If the pixel point on the projector’s image plane is biased, as Figure 2 shows, the measurement error will be enlarged in a short baseline arranged Pro-Cam system.
FPP-based phase-to-height mapping model
Research design and methodology
In the following, a phase-to-height mapping model is presented for parameters’ influence analysis. In this model, a rigorous mathematical relationship that exists between the height of an object’s surface, the phase difference distribution map, and the parameters of the setup is firstly derived. Based on this model, we then study the problem of how the uncertainty of relevant parameters, particularly the baseline’s length, affects the 3-D shape measurement accuracy. The uncertainty analysis on the proposed model including partial derivative analysis, relative error analysis, and sensitivity analysis are performed. Moreover, the Monte Carlo simulation experiment is also conducted.
Our proposed projecting and imaging model
Influence of the length of baseline b
It is important to note that the parameters in the following analysis are also falling into this range. The results shown in Figure 5 indicate that in the two cases (the baseline b = 30 mm and the baseline b = 120 mm), the measurement error becomes larger as the height of target object increases. However, in the red curve which indicates the shorter baseline case (b = 30 mm), the measurement error is smaller (less than 0.1); meanwhile, the relationship between ∂h(x, y)/∂b and h is almost linear. On the other hand, when the baseline changes to a larger value (b = 120 mm), the relationship between the measurement error ∂h(x, y)/∂b and h is nonlinear and as the height of the object increases the measurement error increases faster than the shorter baseline (b = 30 mm). In particular, when the height of target object equals to 100 mm, the maximum value of the measurement error is larger than 0.5.
Relative measurement error analysis
Equation 25 indicates that the relative measurement error δh/h is a function of the length of baseline b, the parameters K 1, K 2, K 3, and their variation δK 1, δK 2, δK 3.
Results and discussion
We present a global sensitivity analysis that permits the evaluation of the uncertainty distribution from other input parameters (i.e., L p , L c , θ 1, θ 2, α) to the output (the height information) with respect to the different baseline lengths. In this method, we use four objects with different heights for the experiment. After that, in order to obtain the distribution map of height value, we change the length of baseline and its relevant parameters (i.e., θ 1, θ 2, α), but keep the other parameters unchanged. It is worth noticing that the unchanged parameters in the evaluating process are randomly selected from the given range.
In this paper, 3-D shape measurement error analysis is performed based on a short baseline Pro-Cam system. The first model is based on conventional stereovision technique. Through analysis, we obtain that as the baseline becomes shorter, the two main factors, which are the inherent biased parameters of the projector and unavoidable biased pixel point localization on the projector’s image plane, have more uncertainty. Therefore, the measurement accuracy is further destroyed. In the second one, we propose a FPP technique-based projecting-imaging model. After deriving a new phase-to-height mapping relationship, measurement error which mainly refers to the height error is analyzed with respect to the length of baseline through partial derivative analysis, relative measurement error analysis, and sensitivity analysis. From the analysis result, we conclude that the smaller the baseline, the more sensitive the system is and the relative measurement error is smaller when if the same biases are introduced in the calibrated parameters. The Monte Carlo simulation experimental results also demonstrate the same measurement result using the proposed model under the short baseline configuration.
JL received B.S. and M.S. degrees from the Department of Applied Physics, Sichuan University, Chengdu, China, in 2008 and 2011, respectively. Currently, he is a Ph.D. student in the Department of Mechanical and Biomedical Engineering at City University of Hong Kong, Hong Kong. His research interests include Photoelectric Information processing, 3-D measurement, 3-D reconstruction, and robot vision. YFL received the Ph.D. degree in robotics from the Department of Engineering Science, University of Oxford, Oxford, U.K., in 1993. From 1993 to 1995, he was a Postdoctoral Research Associate in the Department of Computer Science, University of Wales, Aberystwyth, U.K. He joined City University of Hong Kong, Hong Kong, in 1995 where he is currently a Professor in the Department of Mechanical and Biomedical Engineering. His research interests include robot vision, sensing, and sensor-based control for robotics.
This work was supported by City University of Hong Kong (Project No. 7002829) and the National Natural Science Foundation of China (No. 61273286).
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