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Matched data storage in ESPI by combination of spatial phase shifting with temporal phase unwrapping
Optics & Laser Technology 32 (2000) 235–240
www.elsevier.com/locate/optlastec
Matched data storage in ESPI by combination of spatial phase shifting with temporal phase unwrapping Jan Burke, Heinz Helmers ∗ Carl von Ossietzky Universitat Oldenburg, FB Physik Postfach 2503, D-26111 Oldenburg, Germany Received 3 April 2000; accepted 25 May 2000
Abstract We combine the spatial phase-shifting technique with the real-time fringe counting capability of temporal phase unwrapping to provide simple solutions for some practical tasks in ESPI. First, we develop a method for automatically matched data storage intervals and apply this technique to a long-term observation of a biological object with strongly varying deformation rate. Second, we easily c 2000 obtain on-line displacement and deformation data during the observation of a complexly structured discontinuous object. Elsevier Science Ltd. All rights reserved. Keywords: Electronic speckle pattern interferometry (ESPI); Spatial phase shifting; Temporal phase unwrapping
1. Introduction In interferometry, spatial phase shifting (SPS) has proven its value especially in the measurement of highspeed events [1–3] due to its single-frame phase-measuring capability. Also, its potential for increased temporal resolution compared to temporal phase shifting (TPS) has been utilised, for instance to obtain phase maps at a higher rate [4]. But not only can the phase front be monitored at video real time, it can additionally be tracked and unwrapped pixelwise in time, which immediately yields displacement and deformation data and possibly eliminates the need for a-posteriori data processing. This approach is known as temporal phase unwrapping [5]. It has been used for pro lometry [6] and was recently applied to ESPI in combination with TPS [7,8] and also with temporal Fourier transform evaluation [9]. In this paper, we employ temporal phase unwrapping with SPS [10,11]. We use this combination rstly to automatically generate matched data storage intervals for the long-term observation of objects with strongly varying deformation rate. Secondly, we demonstrate that temporal phase unwrapping can be very valuable in the deformation and displacement analysis of discontinuous objects. ∗ Corresponding author. Tel.: +49-441-798-3512; fax: +49-441798-3576. E-mail address: [email protected] (H. Helmers).
Let us introduce the following notation: every object state is represented by a two-dimensional phase map ’(x; y) mod 2; the sux “mod 2” will be omitted for clarity in the following discussion. A certain object deformation is described by a phase di erence ’d (x; y) = ’f (x; y) − ’i (x; y) between an initial (index i) and a nal (index f) object state. In spatial phase unwrapping ’d (x; y) has to be converted to a continuous phase map, d (x; y); by nding an appropriate step function S(x; y) = 2n(x; y); n ∈ Z. Temporal phase unwrapping does this along the time axis for each pixel separately by nding a step function S(x; y; t) = 2n(x; y; t). This is usually done by assuming that the phase maps ’(x; y) are recorded according to the sampling theorem, i.e. phase di erences greater than ± between same pixels in consecutive frames do not occur. If such a transition is detected in ’(x; y) nevertheless, it must be a 0 ↔ 2 wrap that is accounted for by in- or decrementing S(x; y; t). The advantage of this method is that errors due to faulty — mostly under-modulated — pixels will not spread across the image as this may be the case for spatial unwrapping. In longer monitoring sequences however, the advantage of temporal unwrapping can become a disadvantage: it accumulates data, including errors, and severely corrupted phase maps d (x; y) cannot be restored a-posteriori. It would therefore be favourable if both, the temporally unwrapped data and several phase maps ’(x; y) were stored for future processing. From these phase maps conventional
c 2000 Elsevier Science Ltd. All rights reserved. 0030-3992/00/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 4 9 - 9
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J. Burke, H. Helmers / Optics & Laser Technology 32 (2000) 235–240
Fig. 1. Grey-scale phase map for an object tilt around the y-axis with “slow” pixel clusters.
sawtooth images ’d (x; y) can be calculated that can be spatially unwrapped a-posteriori. This procedure requires a storage interval t for the phase maps ’(x; y) which is matched to the possibly varying rate of object deformation and displacement. We will show, that the implicit fringe counting capability of temporal phase unwrapping can be used to drive such a matched course of t automatically. As far as we know, this issue has only once been dealt with before on a basis of speckle decorrelation analysis [12]; however we intend to limit the number of fringes in ’d (x; y) instead of the speckle decorrelation. Moreover, in ESPI one often deals with objects consisting of several independent parts that may undergo di erent displacements and deformations. However, sawtooth fringes do not allow to determine rigid-body movements or the sign of the deformation itself, unless the fringe orders are tracked by additional devices like, for instance, a phase stabilisation unit [13]. We will demonstrate how temporal phase unwrapping delivers these data for each object part automatically.
2. Temporal phase unwrapping of speckle phase ÿelds The use of temporal phase unwrapping is not entirely straightforward in speckle interferometry; we will therefore brie y consider the cumulative impact of speckle noise on displacement data. Not surprisingly, pixels with low modulation caused by low object speckle intensity lead to problems also in this application. The statistical uctuations of the calculated phase for those pixels should yield a displacement of zero when monitored over a sucient number of frames. It was however observed that even for longer observation sequences with hundreds of frames, some of these pixels seemed to change their phase constantly in one direction; both signs of changes were present. In a fringe counting procedure, this means that those pixels would trigger data storage even when no actual displacement has occurred. Therefore such outliers have to be suppressed, and as usual in speckle interferometry, a low-pass lter or pixel rejection criterion can serve to do so. There are sophisticated and well-founded ltering schemes [14] that give excellent rejection of noise in the displacement map over long, albeit not in nite, times of observation [15]. For reasons of processing speed, the
ltering scheme used here is simpler. The accumulated phase (x; y; t) of a pixel at time t is considered bad when it di ers by more than from the accumulated phase of at least one out of its nearest neighbours. In that case, (x; y; t) := ((x; y − 1; t) + (x; y + 1; t) + (x − 1; y; t) +(x + 1; y; t))=4
(1)
and the outlier is eliminated. Another problem occurs in the observation of real displacements. While noisy pixels are not necessarily detected as such in every frame, their calculated (x; y; t) will not follow the true course. If such a (x; y; t) is included in the ltering operation (Eq. (1)), its error will propagate into the surrounding pixels. In the long run, this will lead to pixel clusters whose (x; y; t) is dragged behind, i.e. will be between zero — from where all observations start — and the true value. An example is given in Fig. 1, where an out-of-plane tilt about the y-axis has been tracked. The zero displacement is marked by the white line and the displacement range is +5 at the left and −5 at the right side. The background intensity tends to deceive the eye; indeed, the “slow” pixel clusters on the left are brighter than those on the right, which means that the sign of motion is correctly determined for all of them, but the measured displacement is underestimated. When spatial averaging is applied to every frame, this behaviour can be greatly reduced by pixel weighting [15]. In this work, the bad pixel clusters are removed by ltering a-posteriori. 3. Experimental set-up For the experiments, we used a quasi-out-of-plane speckle interferometer with SPS as shown in Fig. 2. The light from a 50-mW HeNe laser ( ≈ 633 nm) is split by BS1. The object light is expanded by MO1 and collimated by a lens L1, which serves to illuminate the object with a plane wave. M3 directs the light onto the object at an ◦ angle of ≈ 11:5 to the surface normal. The light spot on the object has a diameter of about 10 cm, of which only (28:5 mm)2 are imaged onto the CCD sensor by the lens L2 with a magni cation of M =0:26. The polarisation lter PF attenuates the reference light to the extent required. By MO2 the reference wave is coupled into a bre. The bre is held in place by a bent syringe tube and guides the
J. Burke, H. Helmers / Optics & Laser Technology 32 (2000) 235–240
Fig. 2. Quasi-out-of-plane speckle interferometer with SPS.
reference light onto the sensor. The distance x of the bre end relative to the centre of the aperture A deter◦ mines the spatial phase shift x . It is set to x = 120 per column of the CCD sensor and calibrated by the Fourier method [11]. The speckle interferograms were recorded with an analogue CCD camera (Adimec MX12P) and digitised to 8 bits by a frame grabber (Data Translation DT3852). The phase calculation and unwrapping loop executed on a double-i860 50 MHz processor board (Alacron FT200), allowing for a frame rate of about 0.5 Hz for an image size of 800 × 600 pixels.
4. Matched data storage in long-term observations Temporal phase unwrapping is well suited to utilise the fringe order count n(x; y; t) to generate matched data storage intervals t. From the continuously updated values (x; y; t), the extreme values max and min are extracted in every run of the temporal phase unwrapping loop. When the di erence max − min exceeds a certain threshold T , it is assumed that the corresponding sawtooth phase map ’d (x; y) = ’f (x; y) − ’i (x; y) between the present phase distribution ’f (x; y) and the stored initial one, ’i (x; y), would show m fringes with m = T =2. In that case ’f (x; y) is stored and re-labelled
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’i (x; y); (x; y; t) is cleared and the procedure begins anew. By this technique, we obtain a sequence of few-fringe sawtooth images that constitute no problem for spatial unwrapping. Of course, the most convenient data evaluation would be to accumulate (x; y; t) throughout the whole observation, whereby it may even become obsolete to save phase maps ’(x; y) regularly. But with the type of lter used here (Eq. (1)), it is safer to eliminate accumulated noise or accidental errors (e.g. by abrupt stress relaxation in the interferometer) by clearing (x; y; t) when a phase map is stored. Thereby the continuous tracking of phases (x; y; t) is given up, but the propagation of errors is being limited to one value of (x; y; tl ) corresponding to one storage interval t = tl − tl−1 . Nevertheless, the phase and, if usable, added up maps (x; y; tl ) may be stored P later on to yield (x; y; ttotal ) = (x; y; tl ). To test this approach of dynamic data storage, we examined a biological test object whose likely deformation is not known in advance. The white spot on a fresh chestnut was found to be quite co-operative for interferometry: its surface is reasonably re ective and maintains speckle correlation over sucient time intervals. We expected the displacements to proceed most rapidly at the beginning of the experiment because the object will relax in its holder. Also, the loss of water from the surface should result in a constant shrinking, relatively fast initially and then levelling o . The surface changes of the chestnut were monitored over some days from shortly after its fastening in the interferometer until the deformation had settled somewhat. Besides the matched storage of phase maps ’(x; y) whenever the threshold of m = 5 fringes was reached, additional ones were stored at the steady rate of 1 frame per 10 min to study possible performance di erences between the methods. Fig. 3 provides an overview of the deformation dynamics. The black curves (left vertical scale) show the courses of the matched and static storage intervals t versus time after the beginning of the observation. The white curves (right vertical scale) show the corresponding courses of the hard disk space required for storing the phase maps.
Fig. 3. Courses of the matched and static storage intervals t (black curves, left vertical scale) during several days; for static storage t is xed to 10 min. The white curves indicate the hard disk space required in Mbytes=day (right vertical scale).
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Fig. 4. Comparison of matched vs. static data acquisition. (a) sequence of sawtooth images calculated from 5 automatically saved phase maps ’f (x; y) (matched t), leading to the resulting grey-scale height map (b) when spatially unwrapped, converted to heights and added. (c) sawtooth image for almost the same displacement calculated from only one phase map ’f (x; y) (static t) with resulting height map (d).
Fig. 5. Sawtooth image for matched data storage (a) and corresponding sequence of sawtooth images from static storage interval (b).
The matched data storage went through several phases: in the rst 3 h, the chestnut appeared to settle in its springloaded holder and short storage intervals t were necessary. After ≈ 15 h, the deformation slowed down and the matched storage intervals t were accidentally similar to the static ones in the time period between ≈ 25 h and ≈ 60 h. After ≈ 60 h, a distinct slowing down of the shrinkage took place, and the matched storage interval t remained around 20 –25 min for the rest of the observation. Hence, temporal unwrapping could help to avoid undersampling initially and to save disk space later on. To illustrate the value of this approach, we shall consider images from the two situations. In Fig. 4, a comparison of a 10-min deformation measurement at t ≈ 7 34 h is shown. This deformation is decomposed into ve parts (Fig. 4a) by the matched phase map acquisition, and the corresponding sawtooth images indeed show m ≈ 5 fringes each. The incremental sawtooth images [16] can all be spatially unwrapped with no problems, and the corresponding height data can be added to yield a awless deformation map (Fig. 4b). Depending on the individual phase gradients, the sum of these incremental phase
Fig. 6. White-light image of historical brick.
maps may contain well below 25 fringes, but not more. The single sawtooth image (Fig. 4c) from the static data storage indeed contains only ≈ 19 fringes. Their strongly
uctuating density causes problems in spatial unwrapping, so that some height assignments are faulty in the result (Fig. 4d).
J. Burke, H. Helmers / Optics & Laser Technology 32 (2000) 235–240
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Fig. 7. Sawtooth images (left) and corresponding grey-scale height maps achieved by temporal phase unwrapping for a heating (top) and a cooling period (bottom). Numerical values denote maximum and minimum displacements.
After t ≈ 65 h, the situation is reversed: the deformation is oversampled by the static acquisition, which generates a large amount of super uous data. Fig. 5 gives an example from t ≈ 79 h. At that stage of the experiment, the automatic storage interval had expanded to t ≈ 32 min. Consequently, the fringe density in the images from the xed-rate series is unnecessarily low, disk space is wasted and the data evaluation gets more laborious. In Fig. 5, we also nd a hint that our cautious decision to regularly reset (x; y; tl ) after each storage is justi ed. As the object deformation grows slower, t becomes larger, more noise is accumulated in the temporally unwrapped data, reduces the accuracy and also triggers storage too early: in Fig. 5a, we nd less than 4 fringes instead of m ≈ 5. 5. Measurement of absolute displacements and deformations of discontinuous objects Especially in the investigation of historical material, one frequently encounters cracks in the surface under inspection [17] and it is important to know the relative motion of neighbouring sub-areas of the object. As a realistic specimen of an aged material, a slice of a historical brick (≈ 2 cm thick) was observed under temperature changes. The heat source was an infrared radiator positioned some 30 cm behind the object. Fig. 6 shows a white-light image of the measuring eld.
When this sample is subjected to cycles of alternately 15 min of heating from the backside and 15 min of cooling, the resulting deformations reveal 9 separately moving sub-areas with rather di erent fringe densities and complicated boundaries, as Fig. 7 demonstrates. The shown displacements each have evolved in time intervals of ≈ 10 min. For the heating period, ’i (x; y) was stored at an ambient temperature T1 when the heater was switched on, while for the cooling period ’i (x; y) was stored at an ambient temperature T2 when the heater was switched o . While it would be very laborious to de ne and spatially unwrap all the regions separately, it is even impossible to deduce their relative heights from the sawtooth images (Figs. 7a and c). When such displacements are monitored with temporal phase unwrapping, the problems are overcome. Without the need to t data from different sub-areas together, a complete pro le of the surface changes is obtained. One can, and should, test its reliability by checking the obtained tilts for consistency with those following from the number of sawtooth fringes. On removing the above-mentioned “slow” pixel clusters, the data from temporal phase unwrapping did not deviate by more than 0:1 from the spatial unwrapping of the corresponding sawtooth images. According to the height maps from temporal phase unwrapping (Figs. 7b and d), the deformations that have developed in the heating period are almost reversed during cooling, apart from some remaining displacements and deformations that are clearly emphasised by an addition of
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Fig. 8. Overall displacement and deformation after heating and cooling. Residual tilts are visible in the sawtooth image (left); rigid-body displacements are revealed in the grey-scaled height map (right). Arrows mark locations of possible misinterpretations of the sawtooth image.
the height data from the two states. Most of the remaining displacement is presumably caused by an ambient temperature at the end of the cooling period that di ered from T1 . On the other hand, the sum of the sawtooth images lacks important information, as demonstrated in Fig. 8. At the black arrow, a substantial piston-type displacement is not discernible from the sawtooth image (Fig. 8a), while at the white arrow, the nearly matching fringe positions almost conceal the step of ∼ = 0:3 m (1 fringe) that has actually remained. On the contrary, the results based on the height maps from the temporally unwrapped data are unambiguous (Fig. 8b). They need no interpretation and thus allow an easier assessment of object changes. 6. Summary We have shown that the combination of temporal phase unwrapping and spatial phase shifting can be used to simplify some tasks in speckle interferometry. To make this method of on-line deformation and displacement observation reliable, care has to be taken to eliminate outliers from the current data for which we used a nearestneighbour ltering. In a long-term observation of a biological object, temporal phase unwrapping was employed to save one interferogram per ve fringes. Thereby the data storage rate was adapted to the actual object deformation rate. Indeed, the reliability of the data increased in the fast-deformation part, and data redundancy could be avoided when the deformation calmed down. Finally, we demonstrated that temporal phase unwrapping can deliver unambiguous displacement and deformation data of complexly structured objects; the phase gradients thus obtained were consistent with those from the corresponding sawtooth images to within ±0:1. These two additions on ESPI techniques should prove quite useful in practice. Acknowledgements We are pleased to thank the Deutsche Forschungsgemeinschaft, DFG, for nancial support of this work.
References [1] Kujawinska M, Robinson D. Multichannel phase-stepped holographic interferometry. Appl Opt 1998;27:312–20. [2] Shough D, Kwon O, Leary D. High-speed interferometric measurements of aerodynamic phenomena. Proc SPIE 1990;1221:394–403. [3] Pedrini G, P ster B, Tiziani H. Double-pulse electronic speckle interferometry. J Mod Opt 1993;40:89–96. [4] Freischlad K, Kuchel M, Wiedmann W, Kaiser W, Mayer M. High precision interferometric testing of spherical mirrors with long radius of curvature. Proc SPIE 1990;1332:8–17. [5] Huntley J, Saldner H. Temporal phase-unwrapping algorithm for automated interferogram analysis. Appl Opt 1993;32:3047–52. [6] Saldner H, Huntley J. Temporal phase unwrapping: application to surface pro ling of discontinuous objects. Appl Opt 1997;36: 2770–5. [7] van Brug H. Temporal phase unwrapping and its application in shearography systems. Appl Opt 1998;37:6701–6. [8] Huntley J, Kaufmann G, Kerr D. Phase-shifted dynamic speckle pattern interferometry at 1 kHz. Appl Opt 1999;38:6556–63. [9] Joenathan C, Franze B, Haible P, Tiziani H. Speckle interferometry with temporal phase evaluation for measuring large object deformation. Appl Opt 1998;37:2608–14. [10] Williams DC, Nassar NS, Banyard JE, Virdee MS. Digital phase-step interferometry: a simpli ed approach. Opt Laser Technol 1991;23:147–50. [11] Bothe T, Burke J, Helmers H. Spatial phase shifting in ESPI: minimization of phase reconstruction errors. Appl Opt 1997;36:5310–6. [12] Gulker G, Helmers H, Hinsch KD, Holscher C. Automatic timing of ESPI fringe storage in long-term measurements based on speckle correlation formalism. Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, FRINGE ’93, Bremen. Berlin: Akademie Verlag, 1993. p. 411–16. [13] Brozeit A, Burke J, Helmers H. Active phase stabilisation in ESPI without additional optical components. Opt Commun 2000;173: 95–100. [14] Huntley J. Random phase measurement errors in DSPI. Opt Lasers Eng 1997;26:131–50. [15] Coggrave C, Huntley J. Real-time speckle interferometry fringe analysis system. Proc SPIE 1999;3744:464–73. [16] Floureux T. Improvement of electronic speckle fringes by addition of incremental images. Opt Laser Technol 1993;25:255–8. [17] Gulker G, Helmers H, Hinsch KD, Meinlschmidt P, Wol K. Deformation mapping and surface inspection of historical monuments. Opt Lasers Eng 1996;24:183–213.
www.elsevier.com/locate/optlastec
Matched data storage in ESPI by combination of spatial phase shifting with temporal phase unwrapping Jan Burke, Heinz Helmers ∗ Carl von Ossietzky Universitat Oldenburg, FB Physik Postfach 2503, D-26111 Oldenburg, Germany Received 3 April 2000; accepted 25 May 2000
Abstract We combine the spatial phase-shifting technique with the real-time fringe counting capability of temporal phase unwrapping to provide simple solutions for some practical tasks in ESPI. First, we develop a method for automatically matched data storage intervals and apply this technique to a long-term observation of a biological object with strongly varying deformation rate. Second, we easily c 2000 obtain on-line displacement and deformation data during the observation of a complexly structured discontinuous object. Elsevier Science Ltd. All rights reserved. Keywords: Electronic speckle pattern interferometry (ESPI); Spatial phase shifting; Temporal phase unwrapping
1. Introduction In interferometry, spatial phase shifting (SPS) has proven its value especially in the measurement of highspeed events [1–3] due to its single-frame phase-measuring capability. Also, its potential for increased temporal resolution compared to temporal phase shifting (TPS) has been utilised, for instance to obtain phase maps at a higher rate [4]. But not only can the phase front be monitored at video real time, it can additionally be tracked and unwrapped pixelwise in time, which immediately yields displacement and deformation data and possibly eliminates the need for a-posteriori data processing. This approach is known as temporal phase unwrapping [5]. It has been used for pro lometry [6] and was recently applied to ESPI in combination with TPS [7,8] and also with temporal Fourier transform evaluation [9]. In this paper, we employ temporal phase unwrapping with SPS [10,11]. We use this combination rstly to automatically generate matched data storage intervals for the long-term observation of objects with strongly varying deformation rate. Secondly, we demonstrate that temporal phase unwrapping can be very valuable in the deformation and displacement analysis of discontinuous objects. ∗ Corresponding author. Tel.: +49-441-798-3512; fax: +49-441798-3576. E-mail address: [email protected] (H. Helmers).
Let us introduce the following notation: every object state is represented by a two-dimensional phase map ’(x; y) mod 2; the sux “mod 2” will be omitted for clarity in the following discussion. A certain object deformation is described by a phase di erence ’d (x; y) = ’f (x; y) − ’i (x; y) between an initial (index i) and a nal (index f) object state. In spatial phase unwrapping ’d (x; y) has to be converted to a continuous phase map, d (x; y); by nding an appropriate step function S(x; y) = 2n(x; y); n ∈ Z. Temporal phase unwrapping does this along the time axis for each pixel separately by nding a step function S(x; y; t) = 2n(x; y; t). This is usually done by assuming that the phase maps ’(x; y) are recorded according to the sampling theorem, i.e. phase di erences greater than ± between same pixels in consecutive frames do not occur. If such a transition is detected in ’(x; y) nevertheless, it must be a 0 ↔ 2 wrap that is accounted for by in- or decrementing S(x; y; t). The advantage of this method is that errors due to faulty — mostly under-modulated — pixels will not spread across the image as this may be the case for spatial unwrapping. In longer monitoring sequences however, the advantage of temporal unwrapping can become a disadvantage: it accumulates data, including errors, and severely corrupted phase maps d (x; y) cannot be restored a-posteriori. It would therefore be favourable if both, the temporally unwrapped data and several phase maps ’(x; y) were stored for future processing. From these phase maps conventional
c 2000 Elsevier Science Ltd. All rights reserved. 0030-3992/00/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 4 9 - 9
236
J. Burke, H. Helmers / Optics & Laser Technology 32 (2000) 235–240
Fig. 1. Grey-scale phase map for an object tilt around the y-axis with “slow” pixel clusters.
sawtooth images ’d (x; y) can be calculated that can be spatially unwrapped a-posteriori. This procedure requires a storage interval t for the phase maps ’(x; y) which is matched to the possibly varying rate of object deformation and displacement. We will show, that the implicit fringe counting capability of temporal phase unwrapping can be used to drive such a matched course of t automatically. As far as we know, this issue has only once been dealt with before on a basis of speckle decorrelation analysis [12]; however we intend to limit the number of fringes in ’d (x; y) instead of the speckle decorrelation. Moreover, in ESPI one often deals with objects consisting of several independent parts that may undergo di erent displacements and deformations. However, sawtooth fringes do not allow to determine rigid-body movements or the sign of the deformation itself, unless the fringe orders are tracked by additional devices like, for instance, a phase stabilisation unit [13]. We will demonstrate how temporal phase unwrapping delivers these data for each object part automatically.
2. Temporal phase unwrapping of speckle phase ÿelds The use of temporal phase unwrapping is not entirely straightforward in speckle interferometry; we will therefore brie y consider the cumulative impact of speckle noise on displacement data. Not surprisingly, pixels with low modulation caused by low object speckle intensity lead to problems also in this application. The statistical uctuations of the calculated phase for those pixels should yield a displacement of zero when monitored over a sucient number of frames. It was however observed that even for longer observation sequences with hundreds of frames, some of these pixels seemed to change their phase constantly in one direction; both signs of changes were present. In a fringe counting procedure, this means that those pixels would trigger data storage even when no actual displacement has occurred. Therefore such outliers have to be suppressed, and as usual in speckle interferometry, a low-pass lter or pixel rejection criterion can serve to do so. There are sophisticated and well-founded ltering schemes [14] that give excellent rejection of noise in the displacement map over long, albeit not in nite, times of observation [15]. For reasons of processing speed, the
ltering scheme used here is simpler. The accumulated phase (x; y; t) of a pixel at time t is considered bad when it di ers by more than from the accumulated phase of at least one out of its nearest neighbours. In that case, (x; y; t) := ((x; y − 1; t) + (x; y + 1; t) + (x − 1; y; t) +(x + 1; y; t))=4
(1)
and the outlier is eliminated. Another problem occurs in the observation of real displacements. While noisy pixels are not necessarily detected as such in every frame, their calculated (x; y; t) will not follow the true course. If such a (x; y; t) is included in the ltering operation (Eq. (1)), its error will propagate into the surrounding pixels. In the long run, this will lead to pixel clusters whose (x; y; t) is dragged behind, i.e. will be between zero — from where all observations start — and the true value. An example is given in Fig. 1, where an out-of-plane tilt about the y-axis has been tracked. The zero displacement is marked by the white line and the displacement range is +5 at the left and −5 at the right side. The background intensity tends to deceive the eye; indeed, the “slow” pixel clusters on the left are brighter than those on the right, which means that the sign of motion is correctly determined for all of them, but the measured displacement is underestimated. When spatial averaging is applied to every frame, this behaviour can be greatly reduced by pixel weighting [15]. In this work, the bad pixel clusters are removed by ltering a-posteriori. 3. Experimental set-up For the experiments, we used a quasi-out-of-plane speckle interferometer with SPS as shown in Fig. 2. The light from a 50-mW HeNe laser ( ≈ 633 nm) is split by BS1. The object light is expanded by MO1 and collimated by a lens L1, which serves to illuminate the object with a plane wave. M3 directs the light onto the object at an ◦ angle of ≈ 11:5 to the surface normal. The light spot on the object has a diameter of about 10 cm, of which only (28:5 mm)2 are imaged onto the CCD sensor by the lens L2 with a magni cation of M =0:26. The polarisation lter PF attenuates the reference light to the extent required. By MO2 the reference wave is coupled into a bre. The bre is held in place by a bent syringe tube and guides the
J. Burke, H. Helmers / Optics & Laser Technology 32 (2000) 235–240
Fig. 2. Quasi-out-of-plane speckle interferometer with SPS.
reference light onto the sensor. The distance x of the bre end relative to the centre of the aperture A deter◦ mines the spatial phase shift x . It is set to x = 120 per column of the CCD sensor and calibrated by the Fourier method [11]. The speckle interferograms were recorded with an analogue CCD camera (Adimec MX12P) and digitised to 8 bits by a frame grabber (Data Translation DT3852). The phase calculation and unwrapping loop executed on a double-i860 50 MHz processor board (Alacron FT200), allowing for a frame rate of about 0.5 Hz for an image size of 800 × 600 pixels.
4. Matched data storage in long-term observations Temporal phase unwrapping is well suited to utilise the fringe order count n(x; y; t) to generate matched data storage intervals t. From the continuously updated values (x; y; t), the extreme values max and min are extracted in every run of the temporal phase unwrapping loop. When the di erence max − min exceeds a certain threshold T , it is assumed that the corresponding sawtooth phase map ’d (x; y) = ’f (x; y) − ’i (x; y) between the present phase distribution ’f (x; y) and the stored initial one, ’i (x; y), would show m fringes with m = T =2. In that case ’f (x; y) is stored and re-labelled
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’i (x; y); (x; y; t) is cleared and the procedure begins anew. By this technique, we obtain a sequence of few-fringe sawtooth images that constitute no problem for spatial unwrapping. Of course, the most convenient data evaluation would be to accumulate (x; y; t) throughout the whole observation, whereby it may even become obsolete to save phase maps ’(x; y) regularly. But with the type of lter used here (Eq. (1)), it is safer to eliminate accumulated noise or accidental errors (e.g. by abrupt stress relaxation in the interferometer) by clearing (x; y; t) when a phase map is stored. Thereby the continuous tracking of phases (x; y; t) is given up, but the propagation of errors is being limited to one value of (x; y; tl ) corresponding to one storage interval t = tl − tl−1 . Nevertheless, the phase and, if usable, added up maps (x; y; tl ) may be stored P later on to yield (x; y; ttotal ) = (x; y; tl ). To test this approach of dynamic data storage, we examined a biological test object whose likely deformation is not known in advance. The white spot on a fresh chestnut was found to be quite co-operative for interferometry: its surface is reasonably re ective and maintains speckle correlation over sucient time intervals. We expected the displacements to proceed most rapidly at the beginning of the experiment because the object will relax in its holder. Also, the loss of water from the surface should result in a constant shrinking, relatively fast initially and then levelling o . The surface changes of the chestnut were monitored over some days from shortly after its fastening in the interferometer until the deformation had settled somewhat. Besides the matched storage of phase maps ’(x; y) whenever the threshold of m = 5 fringes was reached, additional ones were stored at the steady rate of 1 frame per 10 min to study possible performance di erences between the methods. Fig. 3 provides an overview of the deformation dynamics. The black curves (left vertical scale) show the courses of the matched and static storage intervals t versus time after the beginning of the observation. The white curves (right vertical scale) show the corresponding courses of the hard disk space required for storing the phase maps.
Fig. 3. Courses of the matched and static storage intervals t (black curves, left vertical scale) during several days; for static storage t is xed to 10 min. The white curves indicate the hard disk space required in Mbytes=day (right vertical scale).
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Fig. 4. Comparison of matched vs. static data acquisition. (a) sequence of sawtooth images calculated from 5 automatically saved phase maps ’f (x; y) (matched t), leading to the resulting grey-scale height map (b) when spatially unwrapped, converted to heights and added. (c) sawtooth image for almost the same displacement calculated from only one phase map ’f (x; y) (static t) with resulting height map (d).
Fig. 5. Sawtooth image for matched data storage (a) and corresponding sequence of sawtooth images from static storage interval (b).
The matched data storage went through several phases: in the rst 3 h, the chestnut appeared to settle in its springloaded holder and short storage intervals t were necessary. After ≈ 15 h, the deformation slowed down and the matched storage intervals t were accidentally similar to the static ones in the time period between ≈ 25 h and ≈ 60 h. After ≈ 60 h, a distinct slowing down of the shrinkage took place, and the matched storage interval t remained around 20 –25 min for the rest of the observation. Hence, temporal unwrapping could help to avoid undersampling initially and to save disk space later on. To illustrate the value of this approach, we shall consider images from the two situations. In Fig. 4, a comparison of a 10-min deformation measurement at t ≈ 7 34 h is shown. This deformation is decomposed into ve parts (Fig. 4a) by the matched phase map acquisition, and the corresponding sawtooth images indeed show m ≈ 5 fringes each. The incremental sawtooth images [16] can all be spatially unwrapped with no problems, and the corresponding height data can be added to yield a awless deformation map (Fig. 4b). Depending on the individual phase gradients, the sum of these incremental phase
Fig. 6. White-light image of historical brick.
maps may contain well below 25 fringes, but not more. The single sawtooth image (Fig. 4c) from the static data storage indeed contains only ≈ 19 fringes. Their strongly
uctuating density causes problems in spatial unwrapping, so that some height assignments are faulty in the result (Fig. 4d).
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Fig. 7. Sawtooth images (left) and corresponding grey-scale height maps achieved by temporal phase unwrapping for a heating (top) and a cooling period (bottom). Numerical values denote maximum and minimum displacements.
After t ≈ 65 h, the situation is reversed: the deformation is oversampled by the static acquisition, which generates a large amount of super uous data. Fig. 5 gives an example from t ≈ 79 h. At that stage of the experiment, the automatic storage interval had expanded to t ≈ 32 min. Consequently, the fringe density in the images from the xed-rate series is unnecessarily low, disk space is wasted and the data evaluation gets more laborious. In Fig. 5, we also nd a hint that our cautious decision to regularly reset (x; y; tl ) after each storage is justi ed. As the object deformation grows slower, t becomes larger, more noise is accumulated in the temporally unwrapped data, reduces the accuracy and also triggers storage too early: in Fig. 5a, we nd less than 4 fringes instead of m ≈ 5. 5. Measurement of absolute displacements and deformations of discontinuous objects Especially in the investigation of historical material, one frequently encounters cracks in the surface under inspection [17] and it is important to know the relative motion of neighbouring sub-areas of the object. As a realistic specimen of an aged material, a slice of a historical brick (≈ 2 cm thick) was observed under temperature changes. The heat source was an infrared radiator positioned some 30 cm behind the object. Fig. 6 shows a white-light image of the measuring eld.
When this sample is subjected to cycles of alternately 15 min of heating from the backside and 15 min of cooling, the resulting deformations reveal 9 separately moving sub-areas with rather di erent fringe densities and complicated boundaries, as Fig. 7 demonstrates. The shown displacements each have evolved in time intervals of ≈ 10 min. For the heating period, ’i (x; y) was stored at an ambient temperature T1 when the heater was switched on, while for the cooling period ’i (x; y) was stored at an ambient temperature T2 when the heater was switched o . While it would be very laborious to de ne and spatially unwrap all the regions separately, it is even impossible to deduce their relative heights from the sawtooth images (Figs. 7a and c). When such displacements are monitored with temporal phase unwrapping, the problems are overcome. Without the need to t data from different sub-areas together, a complete pro le of the surface changes is obtained. One can, and should, test its reliability by checking the obtained tilts for consistency with those following from the number of sawtooth fringes. On removing the above-mentioned “slow” pixel clusters, the data from temporal phase unwrapping did not deviate by more than 0:1 from the spatial unwrapping of the corresponding sawtooth images. According to the height maps from temporal phase unwrapping (Figs. 7b and d), the deformations that have developed in the heating period are almost reversed during cooling, apart from some remaining displacements and deformations that are clearly emphasised by an addition of
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Fig. 8. Overall displacement and deformation after heating and cooling. Residual tilts are visible in the sawtooth image (left); rigid-body displacements are revealed in the grey-scaled height map (right). Arrows mark locations of possible misinterpretations of the sawtooth image.
the height data from the two states. Most of the remaining displacement is presumably caused by an ambient temperature at the end of the cooling period that di ered from T1 . On the other hand, the sum of the sawtooth images lacks important information, as demonstrated in Fig. 8. At the black arrow, a substantial piston-type displacement is not discernible from the sawtooth image (Fig. 8a), while at the white arrow, the nearly matching fringe positions almost conceal the step of ∼ = 0:3 m (1 fringe) that has actually remained. On the contrary, the results based on the height maps from the temporally unwrapped data are unambiguous (Fig. 8b). They need no interpretation and thus allow an easier assessment of object changes. 6. Summary We have shown that the combination of temporal phase unwrapping and spatial phase shifting can be used to simplify some tasks in speckle interferometry. To make this method of on-line deformation and displacement observation reliable, care has to be taken to eliminate outliers from the current data for which we used a nearestneighbour ltering. In a long-term observation of a biological object, temporal phase unwrapping was employed to save one interferogram per ve fringes. Thereby the data storage rate was adapted to the actual object deformation rate. Indeed, the reliability of the data increased in the fast-deformation part, and data redundancy could be avoided when the deformation calmed down. Finally, we demonstrated that temporal phase unwrapping can deliver unambiguous displacement and deformation data of complexly structured objects; the phase gradients thus obtained were consistent with those from the corresponding sawtooth images to within ±0:1. These two additions on ESPI techniques should prove quite useful in practice. Acknowledgements We are pleased to thank the Deutsche Forschungsgemeinschaft, DFG, for nancial support of this work.
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