Application and Optimisation of the Spatial Phase Shifting ...

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164 Improvements on SPS Fig. 6.22: Comparison of sawtooth images from an out-of-plane tilt; left, ∆ϕ v (x,y) measured with one SOP, σ d =0.067 λ; right, ∆ϕ m (x,y) by merging of measurements from two SOPs, σ d =0.051 λ. An attempt to use the intensity-correcting formula (6.6) to derive a modulation criterion that includes speckle intensities remained unsuccessful. While both ∆ϕ v (x,y) and ∆ϕ h (x,y) were better than their noncorrected counterparts, ∆ϕ m (x,y) was slightly worse in terms of σ d . Apparently, the simple modulation analysis of (6.25) rejects unreliable pixels well enough, and the inclusion of speckle intensities tends to complicate the procedure. In the version of the system described here, the polariser is rotated manually. Of course, it could be replaced by an electro-optical device, so that the interferograms for both SOPs can be captured in subsequent video frames. Furthermore, with a polarising beamsplitter and two cameras, it would even be possible to record the two interferograms simultaneously. In that case, the phase compensating unit can be given up, provided SPS is used. 6.7 Extensions of SPS by temporal unwrapping While the reduced spatial resolution in ESPI does not seem to constitute practical limitations for SPS, the temporal resolution is increased in comparison with TPS by a factor of at least 3. This has been used for high-precision classical interferometry to obtain and average phase maps at a higher rate [Fre90b], and the single-frame measuring capability has enabled successful measurements of high-speed events [Kuj88, Sho90, Ped93]. 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 [Hun93a] and abbreviated by TPU. It has been used for profilometry [Tak94, Sal97, Joe98b] and shearography [vBru98] and was applied to ESPI deformation measurements in combination with TPS [vBru98, Hun99] and also with temporal FT evaluation [Joe98a]. A method utilising carrier fringes with TPU for a shearography ESPI system has recently been described in [Mar00]. The principle of TPU is shown in Fig. 6.23.
6.7 Extensions of SPS by temporal unwrapping 165 Dj ( x, y) mod 2p 6p F ( x, y) 4p 2p 2p n(x, y) n=1 n=2 n=3 x j O ( x,y, t) mod 2p 6 p F ( x, y, t) 4p 2p 2 pn(x, y, t) n=1 n=2 n=3 t Fig. 6.23: Principle sketch of spatial (top row) and temporal (bottom row) phase unwrapping; for details, see text. In spatial unwrapping, a sawtooth image (top, left), representing a temporal phase history ∆ϕ(x, y) mod 2π = (ϕ O (x, y, t f ) mod 2π–ϕ O (x, y, t i ) mod 2π) mod 2π, with i and f referring to undeformed and deformed object state, is converted to a continuous displacement phase Φ(x, y) (top, right) by appropriate additions of 2π, i.e. by finding the correct step function 2πn(x,y), n∈. This is done by a simple criterion: when the data satisfy the sampling theorem spatially, there will be no phase changes >π from pixel to pixel. If such a transition is detected nevertheless, it must then be a 02π jump that is wrapped back onto [0, π) by in- or decrementing n. This procedure along the x direction at an image row y is sketched in Fig. 6.23 in the centre of the upper row. Temporal unwrapping starts from an empty displacement map (bottom, left) and tracks the phase history of every pixel (x,y) in time by comparing it with an initial phase map ϕ O (x, y, t i ). The unwrapping criterion is applied temporally, as shown in the centre of the bottom row for some pixel (x, y); the temporal sampling rate must be high enough to keep differences of ϕ O (x, y, t) from frame to frame smaller than π on each pixel, this is, the sampling theorem must be fulfilled temporally. The phase differences are unwrapped by addition of 2πn(x, y, t) as required and used to continuously update Φ(x, y, t), which may conveniently be represented by grey levels as well, as on the right in the bottom row. The advantage of this method is that errors due to faulty – mostly badly 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 TPU can become a disadvantage: it accumulates data, including errors, and severely corrupted unwrapped phase maps Φ(x, y) cannot be restored a posteriori. It is therefore favourable to store both the temporally unwrapped data and several phase maps ϕ O (x, y, t). If the former are doubtful, the latter may yield conventional sawtooth images ∆ϕ (x, y) that can easily be unwrapped spatially when they contain few fringes.
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Application and Optimisation of the
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Table of Contents 1 1 INTRODUCTION
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3 1 Introduction The present era of
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Introduction 5 retrieval can help t
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7 2 Statistical Properties of Speck
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2.1 Experimental set-up 9 Gaussian
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2.2 First-order speckle statistics
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2.3 Second-order speckle statistics
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47 3 Electronic or Digital Speckle
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3.1 Subtraction-mode ESPI 49 some 4
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3.2 Phase-shifting ESPI 51 filled u
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3.2 Phase-shifting ESPI 53 known si
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3.2 Phase-shifting ESPI 55 α n =α
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3.2 Phase-shifting ESPI 57 But reme
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3.2 Phase-shifting ESPI 59 ϕ ϕ O,
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3.2 Phase-shifting ESPI 61 These fo
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3.2 Phase-shifting ESPI 63 ∞ ~ ~
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3.2 Phase-shifting ESPI 65 ∞ N
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3.2 Phase-shifting ESPI 67 The spec
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3.2 Phase-shifting ESPI 69 − 0. 2
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3.2 Phase-shifting ESPI 71 ν x 0 i
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3.2 Phase-shifting ESPI 73 to think
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3.3 Temporal phase shifting 75 Agai
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3.3 Temporal phase shifting 77 addi
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3.4 Spatial phase shifting 79 which
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3.4 Spatial phase shifting 81 3.4.1
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3.4 Spatial phase shifting 83 cross
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3.4 Spatial phase shifting 85 Fig.
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3.4 Spatial phase shifting 87 arran
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3.4 Spatial phase shifting 89 infor
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3.4 Spatial phase shifting 91 frequ
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3.4 Spatial phase shifting 93 altho
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3.4 Spatial phase shifting 95 error
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3.4 Spatial phase shifting 97 Since
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100 Quantification of displacement-
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110
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112 Comparison of noise in phase ma
Page 116 and 117: 114 Comparison of noise in phase ma
Page 136 and 137: 134 Improvements on SPS intensity o
Page 138 and 139: 136 Improvements on SPS 0.14 σ d /
Page 140 and 141: 138 Improvements on SPS we are conc
Page 142 and 143: 140 Improvements on SPS 6.2 Modifie
Page 146 and 147: 144 Improvements on SPS Fig. 6.7 te
Page 148 and 149: 146 Improvements on SPS Finally, bo
Page 150 and 151: 148 Improvements on SPS To use the
Page 152 and 153: 150 Improvements on SPS The improve
Page 154 and 155: 152 Improvements on SPS On the whol
Page 156 and 157: 154 Improvements on SPS In classica
Page 158 and 159: 156 Improvements on SPS generally n
Page 162 and 163: 160 Improvements on SPS uncertain,
Page 164 and 165: 162 Improvements on SPS To obtain p
Page 168 and 169: 166 Improvements on SPS This proced
Page 170 and 171: 168 Improvements on SPS 6.7.2 Long-
Page 172 and 173: 170 Improvements on SPS Fig. 6.28:
Page 174 and 175: 172 Improvements on SPS the heater
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Page 178 and 179: 176 Summary method to search for a
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Page 182 and 183: 180 References [Bot97] [Bra87] [Bro
Page 184 and 185: 182 References [Ett97] [Fac93] [Far
Page 186 and 187: 184 References [Har94] [Her96] [Het
Page 188 and 189: 186 References [Kuj89] [Kuj91a] [Ku
Page 190 and 191: 188 References [McLa86] [Mer83] J.
Page 192 and 193: 190 References [Rav99] I. Raveh, E.
Page 194 and 195: 192 References [Sur98b] [Sur98c] [S
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Page 198 and 199: 196 Appendix A: Counting events and
Page 200 and 201: 198 Appendix B: Real-time phase cal
Page 202 and 203: 200 Appendix C: Derivation of inten
Page 204 and 205: I 1 a r g ( S ~ ( ) ) 202 Appendix
Page 206 and 207: 204 Appendix D: Alternative error-c
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Page 212 and 213: 210 The author List of publications
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