1、PDF外文：http:/ 附录 A 外文翻译 -原文部分 Optimal Filtering based Shift Estimation for Fringe Pattern Profilometry by Generalized Analysis Model Abstract-This paper presents a new algorithm for fringe pattern parofilometry by utilizing generalized analysis model, called optimal filtering based shift
2、estimation (OFSE) method, which provides much lower complexity compared with traditional methods. Meanwhile, as OFSE is derived based on the generalized analysis model, the reconstruction results will not be influenced by the nonlinearity of fringe pattern projection and acquisition system. The effi
3、ciency of the proposed OFSE method is confirmed by simulation results, which show that the accuracy of three-dimensional reconstruction using digital fringe pattern profilemetry technique can be much improved and the computational complexity can be significantly reduced. 1.INTRODUCTION Fringe patter
4、n profilometry (FPP) is one of the most popular non-contact approaches to measuring three-dimensional object surfaces. With FPP, a Ronchi grating or sinusoidal grating is projected onto a three-dimensional diffuse surface, the height distribution of which deforms the projected fringe patterns and mo
5、dulates them in phase domain. Hence by retrieving the phase difference between the original and deformed fringe patterns, three-dimensional profilometry can be achieved. In order to obtain phase maps from original and deformed fringe patterns, research contributed various analysis methods, inc
6、luding Fourier Transform Profilometry (FTP)1,2, Phase Shifting Profilometry (PSP)7, Spatial Phase Detection(SPD)10, Phase Locked Loop(PLL)11 and other analysis methods12,13. In recent years, because of the simplicity and controllability, digital projectors have been widely used to yield fringe patte
7、rns for implementing FPP 14-17. However, when generating fringe patterns by using digital projectors, nonlinear distortions are unavoidably introduced and result in visible measurement errors 16,17, which has been theoretically analyzed by Hu et al.17.In order to eliminate the reconstruction errors
8、caused by nonlinear distortions, Guo et al.16 proposed a gamma correction based method to recover the distorted fringes. However, with this method, the precondition is that the projection system strictly matches the gamma distortion model. Moreover, as gamma coefficient varies with projection system
9、s, the correction coefficient has to be estimated for different systems or whenever the system condition changes.Hu et al. introduced a generalized analysis model, which revealed the essential relationship between the projected and the deformed fringe patterns. This model does not depend on the nonl
10、inear characteristics of projection systems17. Based on the mathematical model, Gradient-based Shift Estimation(GSE) algorithm17 and inverse function analysis(IFA) method18 have been presented to reconstruct accurate profiles from nonlinearly distorted fringe patterns. However, with IFA algorithm, t
11、he performance of IFA highly depends on the degree of the polynomial selected for curve fitting and in order to achieve high accuracy, higher degree polynomials has to be used, which accordingly leads to higher computational complexity. On the other hand, with GSE method, the height distribution of
12、the surface is calculated point-by-point, but not to simultaneously reconstruct the whole object profile. The height distribution of each point on the object surface has to be individually and independently measured, which results in substantial computation if the captured image has got relatively h
13、igh resolution. In addition, although GSE has a very strong ability to obtain precise surfaces without prior-knowledge of projection system, for some points of the surface, it needs very small learning rates and accordingly many times of iterations to achieve desired accuracy. In order to reduce com
14、putational complexity, in this paper ,we present a novel shift estimation approach to fringe pattern profilometry based on the design of optimal filters ,called optimal filtering based shift estimation(OFSE) algorithm. The proposed OFSE converts the original shift estimation problem into a new probl
15、em of estimating the parameters of a designed optimal filter. Thus, the parameters of a designed optimal filter. Thus, the parameter space can be significantly reduced and consequently economic computation can be achieved. Additionally, because of the use of generalized analysis model, same with GSE
16、 and IFA, proposed OFSE algorithm can be also utilized to accurately reconstruct object surfaces from nonlinearly distorted fringe patterns. This paper is organized as follows. In Section 2, we review the principles of FPP technique and generalized mathematical model for fringe pattern analysis. In
17、section 3, we design a optimal filtering scheme to accurately reconstruct height distribution of object surfaces. Meanwhile, based on generalized analysis model, an algorithm of estimating the parameters of the filter is derived. Simulation results are demonstrated in Section 4.Section 5 concludes t
18、his paper. 2.PRINCIPLES OF FPP AND GENERALIZED AND ANALYSIS MODELA A schematic diagram of a typical FPP system is shown in Fig.1. For simplicity , we consider a cross section of the object surface for a given y coordinate. Hence, the intensity of fringe patterns captured by CCD camera and the height
19、 distribution function can be expressed as a function with single variable x. Thus we use s(x) and d(x) to denote the intensity of the projected and deformed fringe pattern respectively and use h(x) to represent the height distribution of the object surface. Fig 1.Schematic diagram of fringe p
20、attern prafilometry (FPP) system Conventional fringe pattern analysis methods are based on Fourier expansion and fundamental component analysis, which is sensitive to nonlinear distortions and highly depends on the performance of filters and the characteristics of the profile. In order to solve this
21、 problem and accurately reconstruct three-dimensional surfaces from nonlinearly distorted fringe patterns, Hu et al. presented a generalized analysis model to explain the principle of FPP technique 17,18: ( ) ( ( )d x s x u x &nbs
22、p; (1) 00()() ()l u xhx d u x (2) Where ()ux represents the shift between the fringe patterns captured on the reference plane and the surface of the object, which
23、 varies with x coordinates .Eq.(1) reveals that the deformed signal ()dx is a shifted version of ()sx , and the shift function ()ux can be used to determine the object height distribution by Eq.(2). Therefore, the key to reconstructing three dimensional surface is to retrieve the s
24、hift function ()ux from ()sx and ()dx.In addition, because of its simplicity and generality, in following sections of this paper, the generalized model will be used for our algorithm derivation. 3、 OPTIMAL FILTERING BASED SHIFT ESTIMATION(OFSE) ALGORITHM As theoretically analyzed in 17,
25、nonlinear distortions of the projected and captured fringe pattern will unavoidably introduce measurement errors into the height distribution hp(x) calculated by FTP or PSP algorithm. On the other hand, although FTP and PSP can not make an accurate reconstruction when nonlinear distortion exists, ei
26、ther of them is still capable of roughly measuring a profile of an object. Therefore, ()hpx can be still employed as a pre- estimation of the theoretical value of the height distribution ()hx .Corresponding to ()hpx , ()pux denoting the pre-estimated shift signal can be calculated by simply invertin
27、g Eq.(2).obviously, compared to the theoretical shift distribution ()ux , ()pux is not accurate either. Thus, we can regard the errors as a sort of noise and further eliminate it by a well-designed digital filter. Hence, the shift estimation problem can be converted into a new problem of designing a
28、 filter to make its output be a precise estimation of the theoretical value of the shift distribution, Certainly, the design of the optimal filter should be based on generalized model given by Eq.(1) and (2).Additionally, as the number of the filter parameter is usually much fewer than number of sam
29、ple points of the shift distribution ()ux ,the parameter space will be significantly reduced, so that less computational complexity and faster convergence can be achieved. A.notation and representation Generally, after the pre-estimated signal ()pux has been filtered by a 2K+1 order spatial filter ()pk ,the output signal ()ux can be expressed as:
