Fast calculation of phase in spatial n-point phase-shifting techniques

Transcription

1 Fast calculation of phase in spatial n-point phase-shifting techniques Joanna Schmit and Katherine Creath Optical Sciences Center University of Arizona, Tucson, AZ ABSTRACT The n-point technique (also called the spatial carrier phase-shifting technique) is one of a few fringe pattern analysis techniques. We present modified algorithms for the n-point technique from which the phase modulo 2c can be calculated in a much shorter time versus conventional algorithms. The algorithms are based on spatial synchronous detection techniques and by analogy to those techniques the modified algorithms are named low-pass algorithms, as opposed to the conventional "high-pass" algorithms in the n-point technique. 1. INTRODUCTION Spatial phase measurement techniques' require only one fringe pattern to extract the phase encoded in fringes, making these techniques very attractive for analysis of dynamic events. Temporal phase measurement techniques' require at least three frames with fringes registered sequentially in time and are commonly used for analysis of static events. Fringe patterns are typically recorded by means of CCD camera. When we measure the phase using either of these two techniques, we ordinarily employ a phase-shifting technique for spatial measurements, the n-point technique, and for temporal measurements, the n-frame technique. These techniques use exactly the same algorithms, employing intensity information from at least three spatial or temporal points. For this reason the spatial phase shift from point to point must be the same as the temporal phase shift in points from frame to frame. The fringes in the n-frame technique can be of any shape. The large phase shift between spatial points requires the introduction of high carrier frequency fringes in n-point technique. For example, for a t/2 phase shift, four pixels per fringe are needed. In interferometric methods the carrier frequency of the fringes is obtained by tilting one of the wavefronts, which we call a carrier tilt. In this article we are concerned only with the spatial n-point phase-shifting technique. We call this technique conventional or "high-pass". Our modification of the conventional technique presented here allows us to calculate the phase modulo 2ir in a much shorter time and to automatically subtract the carrier tilt. Because the tilt is subtracted automatically, we call this technique "low pass". 102 / SPIE Vol /95/$6.OQ

2 2. CONVENTIONAL N-POINT TECHNIQUE The intensity distribution for a fringe pattern with carrier fringes is described by the equation: I(x,y) = a(x,y)+b(x,y)cos[f + (x,y)], (1) where a(x,y) is the DC intensity, b(x,y) is the fringe intensity modulation, f is the carrier frequency of fringes, and p(x,y) is the measured phase distribution. Generally, the phase in the n-point technique is calculated from the same algorithm for each set of n pixels in one direction. Each subsequent set of pixels is shifted by one pixel with respect to the previous set. The most common algorithms with t /2 phase shift between pixels are2: 3-point: = 2 arctan1 \I1 12) 3 (2) 4-point: = arctan (3) ( point: q' = arctanl \I ) 2, (4) where the subsequent intensities registered at the pixels are described by indices 1, 2, 3,..., in the same way for each new set of pixels. All the different algorithms for the n-point technique can be described by the general form: (N çü = arctanj-j, (5) where N is a numerator and D denominator. The general procedure described by Eq. (5) results in many modulo 2ic phase fringes because a large carrier tilt is introduced in phase; a schematicof n- point techniques and is represented in Fig. la. A phase unwrapping procedure could be employed to these phase fringes at this point but typically is not because very often there are less than four pixels per phase fringe, which in the presence of noise can lead to improper 2ir phase jump removal and the failure of the unwrapping procedure. For this reason the carrier phase tilt is first removed, Eq. (Ga), and the arctan function is calculated again to bring back the 2ic phase jumps, as shown in Eq. (Ga) and Fig. la. SPIE Vol /103

3 ,' (x,y) = ço(x,y) - 7t (Ga) sin[q' (x,y)] çø' '(x,y) = arctan cos[' (x,y)] (Gb) The x and y are integer numbers designating coordinates of the current pixel in the CCD camera. After such a procedure, only a few well-separated modulo 2t phase fringes without carrier tilt are obtained and the unwrapping procedure runs smoothly. The tilt-subtracting procedure and the additional arctan function require time, and according to this study are not necessary. We propose to calculate these few phase fringes without carrier tilt in one step (Fig. ib) by using the modified algorithm presented below. However, the advantage of the conventional technique is the ability to choose to subtract any tilt for optimal separation of modulo 2t phase fringes. HIGH-PASS LOW-PASS Intensity fringes with carrier frequenncy i'sriyim!uii x- Modulo 2ir phase with tilt p =arctan(n ID) 2i (p(x)=(p(x)-xjt/2 ç''=arctan(sin(p')/cos(q)) PModulo 2ir phase NO tilt 0 x 0 x Fig. la Fig. lb Fig. 1 Schematic of n-point techniques, a) high pass, and b) low-pass. 104 ISPIE Vol. 2544

4 3. HIGH- AND LOW-PASS SPATIAL SYNCHRONOUS DETECTION TECHNIQUE Phase-shifting techniques are based on synchronous detection techniques; they were introduced to optics from the telecommunications discipline by Bruning3. The n-point technique, with its conventional algorithms and the modified ones presented here, is really only a special case of the high- and low-pass spatial synchronous detection (SSD) techniques described by Womack4. He detailed the high-pass and low-pass SSD techniques, calling them quadrature sinusoidal and quadrature moire techniques respectively. The n-point technique, by analogy, we divide into highand low-pass n-point techniques. Our purpose is to present a series of low-pass n-point algorithms which reduce the processing time for modulo 2icphase calculation. First, we will briefly describe the SSD techniques, and then we will consider special cases of 3- and 4-point algorithms and all algorithms with ic/2 phase shift between spatial points - pixels on CCD camera. HIGH-PASS LOW-PASS L[J jj [SPECTRUM DOMAIN I MODULO 2ir PHASE 2i With Tilt No Tilt 0 x Fig. 2a Fig. 2b Fig. 2 Schematic of spatial synchronous detection techniques, a) high-pass and b) low-pass High-pass SSD technique The high-pass SSD technique involves a convolution of the measured quasi sinusoidal signal 1(x) (the spatial intensity distribution with carrier fringes, in our case along the x axis) with the filter function, which is a sine (and cosine) reference signal multiplied by the window function h(x). The reference signal has to have a frequency similar to that of the measured signal. The algorithm for the high-pass SSD technique and discrete sampling is given in Eq. (7). SPIE Vol / 105

5 tan[,(x)] = - i;1 Ii (x)sin[!! (x- xj)]h(x_ x) V Ii (x ) cos[ (x - x II. (7) According to this algorithm, each set of intensity samples is multiplied by exactly the same values of the filter function. As a result of this operation in the space domain, very closely spaced modulo 2ic phase fringes are obtained. In the frequency domain this operation corresponds to the filtering of the two sidelobes of the measured signal frequency spectrum- see Fig. 2a. A different spatial phase measuring technique, one which operates in the frequency domain, is the Fourier transform technique introduced by Takeda5. In this technique, so as to avoid a large number of modulo 2ic phase fringes, the first order of the measured signal is shifted towards the origin and the inverse Fourier transform is taken. This significant reduction in the number of modulo 2c phase fringes can be realized in the space domain using low-pass SSD, which we describe in the following section. 3.2 Low-pass SSD technique In the low-pass SSD technique the measured signal 1(x) is multiplied by the sine (and cosine) reference signal and then convolved with the window function h(x) as shown in Eq. (8). Ii (x)sin[ (x)j tan[ço" (x)} = i;1 2n 1 (x)cos[_ (x)] (8) However, for each set of samples new filter function values must be calculated, which requires additional numerical operations. The filter function values here are the sine (or cosine) values multiplied by the values of the window function h(x) for each set of pixels. This space domain operation corresponds to shifting the two sidelobes towards the origin in the frequency domain, and in the resulting phase we obtain nicely separated modulo 2c phase fringes - see Fig.2b. The synchronous detection technique generally compares the phase of a measured signal with the phase of a reference signal of similar frequencies. In the high-pass technique the phase for each set of measured signal samples is compared with the sinusoidal signal of the constant phase, while in the low-pass technique the phase for each consecutive set of samples is compared with the sinusoidal signal of the phase which changes linearly from set to set of samples. In these general SSD techniques a frequency for the reference signal should be chosen which is very close in frequency to the measured signal so as to avoid large phase errors. 106 /SPIE Vol. 2544

6 4. HIGH-PASS AND LOW-PASS N-POINT TECHNIQUES The n-point techniques are simply a special case of SSD techniques. In the n-point techniques the phase offset, the frequency and the sampling period of the reference signal are chosen so that the values of the filter functions for each set ofpixels are very simple point technique The 4-point (4-frame) technique is very often given as a simple case of high-pass SSD (temporal synchronous detection) technique. The phase offset of the reference signal is assumed to be zero, the sampling period (phase shift between samples) to be ic/2, and the frequency with four samples per period. The window function is assumed to be rectangular, always containing four samples of measured signal. In such case the filter functions for high-pass and low-pass techniques take only values 0, 1,-i and their graphic representation by black dots is given in Fig. (3). I Numerator] benominatoj II I4q(9).,q(3) II cos(x) -sin(x) 11,... 1 dl o o a 0! ' 01 1 : -I %._; -1 '..-'.%' /.%' ;' riioh II q(9b) II I I I I iiqm I I Ij II "W Ii II Fig. 3 Filter function values for 4-point, low-pass technique. The high-pass 4-point technique is described by Eq.(3). The low-pass 4-point technique is described by following Eqs (9a-d): (14 I2" = ço' '= = arctanf T T (9a) '1 13) = ' '=+i = ctan[ IJ' (9b) SPIE Vol / 107

7 (I -fl çz r = q" x=x+2 = ICtfl( T T i ' (9c) '\13 1) 3ir (i -L c, i; = ço' x=x+3 = arctant T TJ ' (9d) " 24 In the high-pass 4-point algorithm the same filter function values are used for each set of four pixels. In the low-pass technique the values change from set to set giving only four different sets of filter functions and are repeated for each four new sets of pixels. We clearly see that simple assumptions converted rather complicated algorithms (Eq. (7) and Eq. (8)) to simple ones (Eq. (3) and eqs. (9a d)). It should be noticed that the complexity of the low-pass algorithms is at least the same if not simpler than that of the high-pass algorithms (numerator and denominator in consequent algorithms are the same except for the signs). For the low-pass 4-point algorithm with ic/2 phase shift only these four equations need to be applied sequentially to each four sets of pixels. Further, we note that the additional tilt subtraction and arctan function are not needed and nicely separated phase fringes can be obtained. 4.3 N-point techniques with ic/2 phase shift In general, for any high-pass n-point technique a fairly simple low-pass technique exists. The most common n-point techniques, those with a ic/2 phase shift, are a family of techniques from which lowpass algorithms can be generated using a very simple rule. The family of n-frame techniques was presented by Schmit and Creath6 and can be read also as a family of n-point techniques. We notice that as in the 4-point low-pass algorithm only the numerators and denominators change places and signs according to the rule given in Eq. (10), resulting in four simple equations. Note that the position of "-" minus sign in Eq. (7) of high-pass algorithm is important. If this minus sign is incorporated into the numerator, then follow Eq. (10); if the minus sign is not included, then Eq. (lob) and Eq. (lod) switch places. Indices of ip" were this time omitted. co'1 = arctan(j, (loa) ço" = arctan I, (lob) co'1 = arctan( ) (loc) co" = arctan_j, (lod) 108/SPIE Vol. 2544

8 This rule follows from the sine and cosine functions character as they are shifted by t/2 with respect to each other. However, for n-point algorithms with a phase shift other than it/2, this rule does not apply and the number of equations in the low-pass algorithm is different point technique with 2ic13 phase shift If the assumed phase shift is different than ti2, then the number of the equations will change and each equation needs to be derived. The ideal carrier frequency of the fringes also changes. We show in Eq. (1 1) the algorithm for the low-pass 3-point technique with 2t/3 phase shift. This technique requires carrier frequency of fringes equivalent to 3 pixels per fringe. ( sj(i3 i2i l, = ço' XX arctanl2j _ 12 I3J ' (1la) ço 2ir,, = ço = arctan21 '2) (1ib) 4ui, (J(1-I3) co j = xxo+2 arctan21 i3j In this technique only three equations are needed for each three consequent sets of three pixels. (lic) 5. NUMBER OF EQUATIONS FOR LOW-PASS N-POINT TECHNIQUE The number of equations in a low-pass algorithm depends on the phase shift between pixels. This number can be calculated according to the rule: Number of equations 2t I phase shift. (12) The number of equations does not depend on the number of samples. For example, for any algorithm with ir/2 phase shift, the number of equations is four, while for algorithm with 2ic/3 phase shift, the number of equations is three. 6. REAL IMAGES We implemented the high- and low-pass 5-point algorithms in a grating (moire) interferometer. The fringes obtained in the grating interferometer were sinusoidal and showed no significant change in DC intensity or fringe modulation. The obtained phase maps for high- and low- pass techniques were identical, as shown in Fig. 4, indicating that no discernible difference exists in using either the highor low-pass technique. SPIE Vol / 109

9 !XPPiO - ZTO Date: ,26 D PLOT P-u 2042 a) b) IKPPiO - ZTO Date: PLOT IXPPIO Date: P-u : 2043 RNS : C) d) Fig. 4 Sinusoidal fringes registered in grating (moire) interferometer (a), retrieved phase by 5-point technique, (b) high-pass, (c) low-pass, and (d) their difference. Using the low-pass technique to calculate phase modulo 2ir allowed us to process the image 40% faster. The remainder of the procedure is exactly the same as for high-pass technique. The improvement in calculation time makes this technique very promising in real-time fringe pattern processing, especially for the measured phases of small slopes when an unwrapping procedure is unnecessary. 7. ERROR ANALYSIS OF HIGH AND LOW-PASS N-POINT TECHNIQUES The phase error analysis in the high-pass n-point techniques for sinusoidal fringes, which was presented by Creath and Schmit8' applies also to the low-pass n-point methods. The experimental verification was provided above. ilo/spievoi. 2544

10 8. CONCLUSIONS We described a low-pass n-point technique for single fringe pattern analysis. We provided complete algorithms for 4-point technique with r/2 phase shift and for 3-point with 3ir/2 phase shift; we further detailed a general procedure for deriving such low-pass algorithms, especially for a ic/2 phase shift. The advantage of the low-pass technique over a conventional (high-pass) n-point technique is a shorter calculation time of modulo 2t phase fringes and an automatically subtracted carrier tilt that allows for an easier phase unwrapping procedure while the simplicity of the algorithm is basically the same. 9. ACKNOWLEDGMENTS Thanks to Division of Optical Techniques at Warsaw University of Technology for use of their optical systems, especially to Maria Pirga for her assistance. 10. REFERENCES 1. M. Kujawinska, "Spatial Phase Measurement Methods," in Interferogram analysis: Digital Fringe Pattern Measurement Technique, ed. D. W. Robinson and G. T. Reid, (Institute of Physics Publishing, Bristol and Philadelphia, 1993), Chap.5, K. Creath, "Temporal phase measurement methods," in Interferogram analysis: Digital Fringe Pattern Measurement Technique, ed. D. W. Robinson and G. T. Reid, (Institute of Physics Publishing, Bristol and Philadelphia, 1993), Chap.4, J. H. Bruning, D. H. Herriot, J. E. Gallagher, D. P. Rosenfeld, A. D. White and D. J. Brangccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses", Appl. Opt., 13, (1974) 4. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng., 23(4), (1984) 5. M. Takeda, H. ma, and Kobayashi, "Fourier transform method of fringe pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, , (1982) 6. J. Schmit, K. Creath, "Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry," to be published in Appl. Opt. 7. K. Patorski, Handbook of Moire Fringe Technique, Elsevier Science Publisher, Amsterdam, Creath, J. Schmit, "Errors in spatial phase stepping techniques", proc. SPIE 2340, Interferometry'94, Warsaw, , (1994) SPIE Vol / 111