1、相移技术五帧任意步距相移算法相移技术：五帧任意步距相移算法G. Stoilov & T. Dragostinov 光信息存储和处理的科学院中心实验室，保加利亚，索菲亚，邮政信箱1113，保加利亚9.5（收到1995.7.31修订版1996.8.2；出版1996.8.16）摘要The present paper offers a new formula for calculating the phase in interferometric measurements by the phase-stepping method. The proposed five-frame algorithm u
2、tilizes equal phase steps with an arbitrary value. Results are presented on computer simulation of the errors occurring at a phase-step different from 90and due to vibrations in the systeminterferometer-object. 1997 Elsevier Science Ltd.本文提供了一种新的相移法计算在干涉测量中的相位计算公式。该算法所提出的五步运算法则利用相移量任意的等步长相移。用计算机模拟该算
3、法得出的结果误差出现在相移步长不为90和系统中干涉对象的摆动。 1997 Elsevier科学有限公司1引言It is known that the phase-stepping method (PSM), used to calculate the phase in interferometric measurements, is based on the basic interferometric equation. When two plane waves interfere, the interferometer can be adjusted to produce an interf
4、erence pattern in the form of vertical fringes： （1） 已知的相移法（PSM）是用于计算干涉测量中的相位，是基于这个基本的干涉等式.当两个平面波的干涉，干涉结果可以被校正产生垂直条纹的干涉图案：where is the intensity of the nth picture, Z is the illuminating intensity, m is the modulation index, k is the spatial frequency, is the phase change, caused by the change in the
5、 measured quantity p, and is the angle of each phase step.其中是第n个图像的强度，Z表示背景强度，k是空间频率，表示相位这个相位的变化，由被测量量p的变化而变化，是每个相位的变化角。The phase is calculated by making at least three measurements of the intensity distribution at different phases. With the evolution of the phase-stepping method, several algorithms
6、 for phase calculation have become popular.计算相位时至少要在不同相位下做三次强度分布的测量。随着相移法的发展，几种相移法的计算已经变得很普遍。Three-frame method with a phase step 90：90度相移的三帧法： (2) Three-frame method with a phase step 120: 120度相移的三帧法： （3） Four-frame method with a phase step 90:90度相移的四帧法： C (4)Four-frame method with a phase step wit
7、h reduced error:减少误差90度相移四帧法： (5)Four-frame method with an arbitrary step:任意相移的四帧法： (6)Five-frame method with a phase step 90:90度相移的5帧法： (7) where and, are the measured intensities of each pixel at phase steps ,and, respectively. 和分别是在和相移下测的各像素的强度。According to the manner of motion of the phase-shift
8、ing mirror in the interferometer, four approaches are feasible: 根据对干涉仪中相移反映的运动方式，四种方法都是可行的：1. All phase-steps in all pixels of the image are equal ( = 60, 90or 120):1.图像中所有像素的全部相移都是相等的( = 60, 90or 120): where i = 0.N, j = 0.N; N is the number of phase-steps, and x,y and p,q are indices of a pair of
9、pixels in the picture. This is the ideal case that should give the best results. However, it is very difficult to achieve. 当i = 0.N, j = 0.N;N是相移的步数，x,y和p,q是图像中的一对像素指数。这是理想的情况下，能得到最好的结果。然而在现实中很难实现。2. The consecutive phase-steps are unequal, but the phase-steps with equal serial number are equal:2.连续
10、的相移步步宽是不同的，但相移步步骤序号是相同的。 This approach has less stringent requirements to the stability of the phase-setting device. In this case the phase steps are among the unknowns in the set of equations.这种方法的相位设定装置的稳定性要求不太严格。在这种情况下的相位的步骤是在方程组中的未知。3. All phase-steps at a specific pixel are equal, but the phase
11、-steps in different pixels are different:3.全部的相移步在特定的像素下是相等的，但在不同的像素中是不同的。 This is the case when the phase-shifting mirror does not move parallel to itself.这是当相移镜不动平行与自身的情况。4. All phase-steps in all pixels may be different. However, the proportionality between the changes in phase-steps of equal num
12、bers in different pixels is constant:4.在所有像素的所有相移步可能是不同的。然而，在不同的像素的序号相等的相移变化之间的比例是不变的： There is no data about this case in the literature; it may attract researchersinterest in the future.The majority of the existing phase-stepping methods are based on the first approach. It demands that the phase-s
13、tepping system be calibrated prior to each measurement in order to eliminate the possible drift due to changes in temperature and other ambient factors. Quite often the calibration procedure takes more time than the measurement itself. The change in the transfer function of the phase-shifting device
14、 leads to a change in sensitivity and, hence, to imprecision in setting the phase step. It is much easier to set steps equal to one another than to make them all equal to a certain value. This has caused the authors to examine the third approach more closely. The method proposed makes possible the u
15、se of an arbitrary phase step, maintaining the uniform error distribution in the whole phase range. The elimination of the need of frequent calibration of the system is very convenient for measurements under industrial conditions.在文献资料中没有这种情况的数据。它可能以后会吸引研究者的兴趣。已存在大部分的相移法都是基于第一种途径。它要求每次测量都要校准这个相移系统，以
16、消除相移可能导致温度变化和其它环境因素的带来的误差。经常的校准程序需要甚至比测量本身更多的时间。因此在设置相移步时，移相器的传递函数的变化导致的灵敏度和精度的变化。而且只是让一个相移与另一个同步比让他们都等于一个特定值容易多了。这更加急切的引发了作者研究的第三种方法。所提出的方法使得有可能使用一个任意相移步，在整个相位范围内保持均匀的误差分布。这消除了需要经常校准的系统，在工业条件下测量非常方便。 2 THEORETICAL ANALYSISFor the sake of convenience, eqn (1) for the five-frame method with an arbitr
17、ary step A can be rewritten as:为方便起见，等式（1）对于五帧任意步距法中A可改写为： (8) Similarly to the five-frame algorithm with a phase step of 90, the system is reduced to: 同样对于90度相移的五帧算法，系统可简化为： (9) Appropriate trigonometric transformations yield:利用三角变换得到： (10) Dividing the first equation by the second and solving with
18、 respect to the phase yields:用第二个等式讲第一个等式分离，解出相位。 (11)The correction term, dependent on A, can be calculated in the following way: 依靠A的校正项可以按以下方式计算： (12) (13) The trigonometric transformations yield:三角变换得： (14)Thus, the final formula becomes:这样最终算法变成： (15) 3 COMPUTER SIMULATION OF RESISTANCE TO PHAS
19、E-STEP VARIATIONS AND VIBRATIONS3对相移变化和振动性能的计算机模拟The different algorithms for phase calculations are compared by computer simulation of the measurements and assessment of the phase calculation error, , of the different formulae:对相位计算的不同算法通过计算机仿真比较不同公式的相位计算误差的测量和评估： (16)where f() is the phase intensi
20、ty dependence, is the inverse function depending on the calculation algorithm, is the initial phase in a certain point in space, is the phase-step error, is the error due to vibrations, and is the phase-step between the different frames.The plots in Fig. 1 illustrate the relationships between the ph
21、ase calculation error () and the initial phase of the pixel () in the three-, four-, and five-frame algorithms in the presence of vibrations and error in the setting of the phase-step.f（）是相位强度依赖性，是取决于计算算法的反函数，在空间某点的初始相位，是相移步误差，是振动误差， 是不同帧的相移步。图1中的坐标图说明相位计算误差与初始相位在三帧、四帧和五帧算法在设置相移步时存在振动和误差时的关系。Normall
22、y, when simulating the noise-generated error, Gaussian distribution of the random quantity is used. However, when this noise is caused by uniform rather than Gaussian vibrations, distribution of the error in measuring the intensity in each pixel is more suitable.通常，当模拟噪声产生的误差，使用高斯分布的随机量。然而，当这种噪声是由均匀
23、而非高斯振动，噪声等适合分布在在每个像素的强度测量误差。The computer simulation of phase unwrapping in the presence of vibrations shows that the error in the three-frame algorithm with a phase-step of 90changes 1.5 times depending on the initial phase of the measured pixel. This is explained by the fact that the probability of
24、 coincidence of the mean measured and real quantity, as proven by mathematical statistics, is higher for phase-steps 0and 180which are linearly dependent measurements (in the same direction). When phase-step of 120is used, the deviation due to vibrations becomes uniform, but the error from the inacc
25、urate step increases by 25%.在去掉振动的计算机仿真中表明改变1.5次时相移步为90度的三帧算法的误差依据测得的像素中的初始相位。这说明了概率的平均量和实际量一致性。通过数理统计证明，高阶相移步早0度和180度是线性相关的测量量（在相同的方向）。当使用相移步为120度时，振动的绝对误差变得均匀，但是误差从这有误的一步开始增加了25%。The use of the four-frame algorithm with a phase-step of 90diminishes the effect of vibrations by about 25% relative to
26、 the three-frame algorithm with a phase-step of 90and, in addition,increases the pulsation rate of the error because of the increase in the nember of measurements.The formula for the four-frame method with reduced error, suggested by Schwider, is less sensitive to the dispersion and slightly more se
27、nsitive to vibrations.相移步为90度的四帧算法比相移步为90度的三帧算法在振动的影响减少了约25%，并且由于测量次数的增加增加了错误的脉动率。Schwider提出的减少误差的四帧法对离散率更不敏感但对振动更敏感一点。Fig. 1. Error distribution as a function of the initial phase at the measured pixel for three-, four- and five-frame algorithms for different combinations of the phase-step error ()
28、 and vibrations (): a = 0.3, = 0; b = 0, = 0.1; c = 0.3, = 0.1. The curves are designated as follows: 3_90, three-frame algorithm with a phase-step 90; 3_120, three-frame algorithm with a phase-step 120; 4_90, four-frame algorithm with a phase-step 90; 4_L, four-frame algorithm with a phase-step 90w
29、ith reduced error; 4_n four-frame algorithm with an arbitrary phase-step; 4_a, four-frame algorithm with an arbitrary phase-step with averaging; 5_90, five-frame algorithm with a phase-step 90; 5_n, five-frame algorithm with an arbitrary phase-step.图一中，误差分布在三、四、五帧算法中是相移步误差（）和振动 ()不同组合的初始相位函数。a = 0.3
30、, = 0; b = 0, = 0.1; c = 0.3, = 0.1.曲线被命名为如下：3_90,是相移步为90度三帧算法。3_120,是相移步为120度三帧算法。4_90，是相移步为90度四帧算法。4_L,是减少误差的相移步为90度四帧算法。4_n 是任意相移步的四帧算法。4_a是任意平均相移步的四帧算法。5_90,是相移步为90度的五帧算法。5_n是任意相移步的五帧算法。The formula with compensation of the difference of the phase-step from 90, also suggested by Schwider, elimina
31、tes the systematic error completely. The introduction of more data (new steps) decreases the pulsations thanks to the greater number of samples, and distributes the error more uniformly with reference to the angle, since the angles in the trigonometric circumference are more uniformly positioned. However, this is not valid for the compensated four-frame algorithm (4_n) since there are zones in which the phase step is calculated very inaccurately.Although demanding a phase-step of 90, the five-frame algorithm seems insensitive to the phase-step er
