تجزیه و تحلیل حساسیت اعمال شده برای پروفیلومتر تبدیل فوریه بهبودیافته

Every estimation concerning model parameters has to deal with uncertainty, but its quantification is a complex task to solve, especially for the identification of the different uncertainty sources that affect the system. In this paper we provide an extensive uncertainty analysis on the calibration model of a Fourier-transform profilometry, studying the effect of the uncertainty parameters estimation on the final measuring result. The methods used for the classification are discrete derivatives and the global sensitivity analysis based on Monte Carlo simulations. The experimental results show that the uncertainty propagation of the system parameters to the output strongly depends on different system setups that may be chosen. This dependency is analysed and interpreted.

Fringe projection techniques are very popular thanks to their possible application in a lot of fields, such as industrial inspection, manufacturing, computer and robot vision, reverse engineering and medical diagnostics. The main qualities of these profilometry methods are non-contact and full field measurement, low cost and speed in obtaining the 3D information [1], [2], [3], [4], [5], [6] and [7]. One of the most used techniques is Fourier transform profilometry, which is based on the projection of a grid onto a surface and then viewed from another direction by a camera, which acquires the image [8]. The object topography deforms the fringe pattern: the corresponding image is acquired on the camera sensor plane and then processed to obtain depth information. The depth is extracted, through the Fourier-transform, from the phase difference between the grid projected on a reference plane and the same grid projected on the object surface. In comparison with other fringe techniques, which require more than one image for the 3D measurement, the advantages of the FTP are elaboration speed and need of only one deformed image [9], [10] and [11]. On the other hand, it needs to resolve the projected grid lines individually, and consequently has a strong requirement on the pixel spatial resolution of the recording device. Moreover, FTP requires frequency domain filtering whose consequence is fine detail reduction and resolution loss [12] and [13]. Phase-to-depth conversion is possible by means of a suitable formula that depends on the geometric model of the acquisition system, i.e. its geometric parameters and the carrier frequency of the grid. Obtaining the correct depth distribution is possible only if the geometric parameters are known. Theoretically the task is easy to be achieved, but in practice the estimation of these parameters is quite complex: for example the relative position between the projector and the camera cannot be fixed without a certain degree of uncertainty, but also the evaluation of the carrier frequency of the grid and the reference plane position is affected by errors. A calibration procedure is consequently necessary to overcome these limitations. Calibration methods proposed in literature till now may be distinguished in three categories: model-based, polynomial and neural networks. The model-based methods try to define the correct phase-to-depth conversion formula by means of the indirect determination of the system parameters [14], [15], [16], [17] and [18]. The polynomial methods otherwise use planes, placed at known positions, that are acquired and then processed to get phase distribution. The polynomial coefficients of the function that best fits the phase-depth data are estimated by a least squares algorithm to produce a calibration map [19], [20], [21], [22], [23], [24], [25] and [26]. Finally the last approaches apply several neural network methods to define the relationship between the inputs of the system (i.e. phase distribution) and the outputs of the system (i.e. depth distribution) with a black-box philosophy that is totally free from the geometric configuration [27]. In a previous paper we proposed a hybrid calibration method, between a model-based and a polynomial calibration process, based on an exhaustive geometric model, which describes the system with a generic relative position of the projector and the camera [28]. The parameters estimation is achieved by a minimization algorithm of the mean squared error between the nominal depth of some planes placed at well defined positions and the result of the conversion formula applied to the phase obtained from the same planes. This calibration method was chosen because it has many advantages: (a) The model is based on a well defined theory. (b) It is based on a complete geometric model that has a physical correspondence with the real measurement setup; the main consequence is the direct comparison between the parameters estimation and the geometric quantities to avoid macroscopic errors. (c) Both the geometric model and the pin-hole camera model include the main non-linearities, and then the errors due to the model simplifications are reduced. (d) The calibration method is easy to apply. A calibration method based on Mao’s geometric model had already been presented in a previous work [29]. The aim of this paper is to improve the problem analysis, studying the effect of the uncertainty parameters estimation on the final result. The question is how the uncertainty concerning the geometric parameters of the system (input parameters) can be extended to depth estimation (measurement output). The answer may only come out from a sensitivity analysis of the calibration model. It must be noted that the results will be applicable for all the calibration methods based on Mao’s geometric model (and its simplified version that corresponds to Takeda’s model). The aim is the definition of a guide to uncertainty analysis about all the most common system setups to use in the design stage for the optimization of the measure uncertainty.

In this paper we presented a sensitivity analysis of a Fourier-transform profilometry, performed both with the simple discrete derivative method and the more complex global sensitivity analysis, based on Monte Carlo simulations. Global sensitivity analysis is a powerful method that permits the evaluation of the uncertainty distribution from the input parameters to the output. The quantification of the single contribution of every input parameters uncertainty onto to the uncertainty distribution of the system output gives the opportunity to define a factor prioritization, i.e. which factor deserves further analysis or measurement, in order to improve the uncertainty of the estimated height distribution of the measured object. In this work we presented an extensive analysis to reasonable possible cases, in terms of system setup and measured height distribution. The result is a substantial description of the uncertainty propagation from which is possible to derive all the information related to a specific application of the system. The aim obtained is the definition of a guide useful to quantify the measure uncertainty according to the application, i.e. a different geometric disposition and the chosen calibration procedure.