Development of spatial and temporal phase evaluation techniques in digital holography 5

CHAPTER FIVE RESULTS AND DISCUSSION 5.1 Spatial phase evaluation analysis 5.1.1 Wrapped phase map extraction As mentioned in Section 3.1.1, the wrapped phase map can be determined by t

CHAPTER FIVE RESULTS AND DISCUSSION

5.1 Spatial phase evaluation analysis

5.1.1 Wrapped phase map extraction

As mentioned in Section 3.1.1, the wrapped phase map can be determined by the conventional digital phase subtraction method, and can also be obtained by using the proposed complex phasor method With the proposed method, direct phase manipulation is effectively avoided, so better results for the extracted phase map can

be expected as shown in the following descriptions

Figure 5.1(a) shows an unfiltered wrapped phase map between the initial and deformed states Figure 5.1(b) shows the central row (along the horizontal direction)

of Fig 5.1(a) As can be seen in Fig 5.1(b), the wrapped phase map is noisy Figure 5.2(a) shows a wrapped phase map obtained by the proposed complex phasor method with an average filter Figure 5.2(b) shows the central row of Fig 5.2(a) Figure 5.3(a) shows a wrapped phase map obtained by the complex phasor method with a median filter Figure 5.3(b) shows the central row of Fig 5.3(a) It can be seen from Figs 5.1-5.3 that the phase map extracted by complex phasor method is much smoother, and complex phasor method with an average filter performs better than that with a median filter in the suppression of speckle noise Only the central part of the phase difference map calculated by complex phasor method is investigated Phase maps [as shown in Figs 5.1(a), 5.2(a) and 5.3(a)] correspond to the deformation of the plate, and a

comparison between conventional methods and the proposed method in a determination of higher-order displacement derivatives is presented in Section 5.1.2

5.1.2 Determination of displacement derivative

5.1.2.1 First-order displacement derivative

As described in Section 3.1.2.1, after the complex amplitude corresponding to the deformation distribution is determined, the first-order displacement derivative can be further obtained by using the proposed method Figure 5.4(a) shows a wrapped phase map corresponding to ∂ ∂w ξ' without any filtering Figure 5.4(b) shows a wrapped phase map corresponding to ∂ ∂w ξ' by filtering Fig 5.4(a) using a direct phase manipulation with an average filter It can be seen from Fig 5.4(b) that the blur increases Figures 5.4(c) and 5.4(d) show two wrapped phase maps corresponding to

'

∂ ∂ filtered by sine/cosine transformation with average and median filters, respectively A 3×3 filtering window in the algorithms is employed In the filtering methods, the iterative cycle is 3 The results shown in Figs 5.4(c) and 5.4(d) are obtained by processing the wrapped phase map in Fig 5.4(a) with the sine/cosine transformation method It is seen from Figs 5.4(c) and 5.4(d) that in this study, conventional sine/cosine transformation method does not work well

Figure 5.5(a) shows a wrapped phase map corresponding to ∂ ∂w ξ' obtained

by complex phasor method with a median filter It can be seen from Fig 5.5(a) that the results are not satisfactory Figure 5.5(b) shows a wrapped phase map corresponding to ∂ ∂w ξ' obtained by complex phasor method with an average filter

A shearing value of 56 pixels is used in Figs 5.4, 5.5(a) and 5.5(b) It can be seen that the phase map obtained by the proposed method with an average filter shows a higher quality than previously proposed methods (Schnars and Jüptner, 1994b; Liu, 2003).In the conventional sine/cosine transformation, each pixel is given an equal weight with fixed amplitude without considering the reliability of the pixel In the complex phasor

method with an average filter, direct phase manipulation is avoided, so better results are clearly observed [see Fig 5.5(b)] In the filtering methods, the iterative cycle is also 3 If the iterative cycle increases slightly, the results can be improved using the sine/cosine transformation method but are still not as good as those using the proposed complex phasor method with an average filter However, the iterative cycle

is kept relatively small to ensure that dense fringes do not become blurred

Main advantages of the proposed method in the determination of displacement derivatives are: (1) Sign ambiguity can be overcome but without the need to discriminate the cases as in the conventional digital phase subtraction method (Kreis, 2005); (2) real and imaginary parts of complex values Γ( , , 2)m n Γ*( , ,1)m n

and A'(m,n)A*(m,n) are respectively processed, and direct phase manipulation is avoided; (3) a filter (such as an average filter) can be defined easily; (4) a larger and more controllable range of the sensitivity is achieved

In the proposed complex phasor method, an averaged point is dominated by pixels with relatively large amplitudes A pixel with the largest amplitude will have the highest influence on the results However, a complex phasor can not be lined up in

a definite way, and low-pass filters such as a median filter are not suitable as real amplitudes vary within the filtering window (Ströbel, 1996; Ghiglia and Pritt, 1998) The appropriate selection of a window size in the average filter is also important, and the proposed method is more suitable for the wrapped phase maps with low-density fringes When the phase maps are dense, iterative cycles should be relatively small in order not to smear out dense fringes

As the two-dimensional short-time Fourier transform is further applied, a wrapped phase map corresponding to ∂ ∂w ξ' is obtained as shown in Fig 5.5(c)

Since noise usually has small coefficients, coefficients lower than the preset threshold can be fully eliminated In the proposed two-dimensional short-time Fourier transform method, the preset threshold is 3, and the values of σξ' and ση' are set at 10 After the filtered phase map is obtained, an unwrapping algorithm can be carried out in order to correct the 2π jumps The unwrapped phase map obtained can correspond to the first-order displacement derivative [according to Eq (3.18)] Figure 5.5(d) shows

an unwrapped phase distribution for Fig 5.5(c) with a quality-guided (phase derivative variance) unwrapping algorithm (Ghiglia and Pritt, 1998) It can be seen that two eyes in the conventional shearography is obtained in digital holography, and the first-order displacement derivative can be calculated by Eq (3.18)

Figure 5.4 Wrapped phase maps corresponding to∂ ∂w ξ' (a) obtained by complex phasor method without any filtering; (b) after directly filtering with an average filter; (c) after sine/cosine transformation with an average filter; (d) after sine/cosine transformation with a median filter

Figure 5.5 (a) A wrapped phase map corresponding to∂ ∂w ξ'obtained by complex phasor method with a median filter; (b) a wrapped phase map

corresponding to∂ ∂w ξ'obtained by complex phasor method with an

average filter; (c) a wrapped phase map obtained by processing the

phase map in Fig.5.5(b) with short-time Fourier transform algorithm; (d)

a continuous phase distribution

The effectiveness of the proposed two-dimensional short-time Fourier transform is due to the following reasons (Mallat, 1999): (1) The coherence between the kernel and the wrapped phase map results in relatively large coefficients; (2) the redundancy of short-time Fourier transform algorithm can make it more robust to the thresholding operation

To show the advantages of the proposed method, the central rows of wrapped phase maps are extracted Figures 5.6(a)-5.6(d) show the central rows of the wrapped phase maps in Figs 5.4(a), 5.4(c), 5.5(b) and 5.5(c), respectively It can be seen from Figs 5.6(a)-5.6(d) that better results are obtained by using the proposed method

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5.1.2.2 Second-order displacement derivative

As mentioned in Section 3.1.2.2, after the complex amplitude corresponding to order displacement derivative is obtained, the second-order displacement derivatives can be further determined by using the proposed method Figure 5.7(a) shows an extracted wrapped phase map obtained by the proposed complex phasor method An average filter with 3×3 pixels window is used A 3D plot of the continuous phase map that corresponds to the displacement of the test plate is shown in Fig 5.7(b)

first-(b)

(a)

It can be seen from Fig 5.8(a) that the wrapped phase map is noisy Since the phase map contains 2 jumps, a phase unwrapping algorithm is required to determine a π

continuous phase map However, when the wrapped phase map is noisy, the unwrapping operation is usually unsuccessful Hence, two strategies are usually adopted to solve this problem One is to improve the filtering algorithm, while another

is to develop a more powerful unwrapping algorithm In this case study, the first strategy is considered, and a filtering algorithm is developed to remove the noise Figure 5.8(b) shows a wrapped phase map corresponding to the curvature ∂2w ∂ξ'2using the proposed complex phasor method with an average filter After the phase map [Fig 5.8(b)] is obtained, conventional sine/cosine transformation is used to further remove the noise Figure 5.8(c) shows a filtered wrapped phase map corresponding to the curvature along the ξ' direction using sine/cosine transformation with a 3×3 average filter Iterative cycle is set as 19 It can be seen

from Fig 5.8(c) that a high-quality phase map is obtained However, it is obvious that some noise still exists at the center The 3D plot of an unwrapped phase map is shown

The proposed method can also be used to determine the curvature along the '

η direction Figure 5.9(a) shows a phase map corresponding to the curvature

'

∂ ∂ without any filtering Figure 5.9(b) shows a wrapped phase map obtained

by the complex phasor method Figure 5.9(c) shows a filtered wrapped phase map corresponding to the curvature along the η' direction using sine/cosine transformation An average filter with a 3×3 window is also used, and the iterative

cycle is set as 15 A 3D plot of an unwrapped phase map for the wrapped phase map

in Fig 5.9(c) is shown in Fig 5.9(d) In this case, each shifting value is 30 pixels

Figure 5.10(a) shows an extracted wrapped phase map corresponding to the twist ∂2w ∂ ∂ξ η' ' without using any filter Figure 5.10(b) shows a wrapped phase map after filtering The wrapped phase map shown in Fig 5.10(b) is obtained by directly filtering the phase map in Fig 5.10(a) using the conventional sine/cosine transformation approach with a 3×3 average filter The iterative cycle in the filtering algorithm is set as 25 Figure 5.10(c) shows a wrapped phase map corresponding to the twist 2

(c) a phase map obtained by using complex phasor method; (d) a

filtered phase map obtained by using sine/cosine transformation

filtering on the phase map in Fig 5.10(c); (e) a continuous phase map;

(f) a 3D plot of the continuous phase map

Figure 5.10(d) shows a filtered wrapped phase map which is obtained by

filtering the phase map in Fig 5.10(c) using sine/cosine transformation approach with

a 3×3 average filter The iterative cycle is also 25 It can be seen that a higher-quality phase map is obtained by using the complex phasor method The small void part in the bottom left corner of Fig 5.10(d) might be solved with a more appropriate cropping of the experimental data Figure 5.10(e) shows a continuous phase map, and Fig 5.10(f) shows a 3D plot of the continuous phase map

Two-dimensional short-time Fourier transform is applied to further process the wrapped phase map so as to suppress speckle noise The ranges of ξ and η are estimated by Fourier transform of a phase map These ranges are adjusted based on the window size such that the most of energy is employed In this study, both the values of σξ' and ση' are set as 10 Figure 5.11(a) shows a wrapped phase map corresponding to the curvature along the 'ξ direction filtered by short-time Fourier transform with a threshold value of 6 It is seen that the central dashed line in Fig 5.8(c) disappears, which indicates that noise is suppressed Figure 5.11(b) shows a wrapped phase map corresponding to the curvature along the 'η direction filtered by short-time Fourier transform with a threshold value of 6 The effect of speckle suppression is obvious in Fig 5.11(b) Figure 5.11(c) shows a wrapped phase map corresponding to the twist ∂2w ∂ ∂ξ η' ' filtered by short-time Fourier transform with

a threshold value of 4 The wrapped phase maps in Figs 5.11(a)-5.11(c) are obtained using short-time Fourier transform on the wrapped phase maps in Figs 5.8(b), 5.9(b) and 5.10(c) The results illustrate that the proposed method can effectively suppress the speckle noise, which are better than the previously proposed method (Liu, 2003) After wrapped phase maps are extracted, continuous phase maps can be obtained using an unwrapping algorithm (Ghiglia and Pritt, 1998) Note that the determination

of higher-order displacement derivatives is proposed based on a numerical concept

Figure 5.11 (a) A phase map corresponding to curvature ∂2w ∂ξ'2filtered by

short-time Fourier transform; (b) a phase map corresponding to

curvature∂2w ∂η'2 filtered by short-time Fourier transform; (c) a

phase map corresponding to the twist ∂2w ∂ ∂ξ η' ' filtered by short- time Fourier transform

5.1.3 Fringe density estimation

As mentioned in Section 3.1.3, a new method based on short-time Fourier transform is proposed to estimate the fringe density of a wrapped phase map before phase unwrapping in digital holographic interferometry In this study, both simulation and experimental results are presented to show the feasibility and effectiveness of the proposed method Figure 5.12(a) shows a simulated circular wrapped phase map without noise, and Fig 5.12(b) shows a corresponding continuous phase distribution Figure 5.12(c) shows the central row (along the horizontal direction) of Fig 5.12(a)

(c)

As can be seen in Figs 5.12(a) and 5.12(c), a fringe-density distribution of the wrapped phase map is symmetric, and the values of the wrapped phase map are in the range of 0 to 2π Using the proposed method, a fringe-density distribution is estimated as shown in Fig 5.12(d) It can be seen from Fig 5.12(d) that the fringe-density distribution obtained is symmetric and smooth Compared with the previous work (Quan et al., 2005), the proposed method has higher noise-immunity due to the localization characteristic of the proposed short-time Fourier transform It is noteworthy that the wrapped phase map is converted into complex exponential signals before implementing short-time Fourier transform

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To further verify the proposed method, another simulated vertical wrapped phase map is shown in Fig 5.13(a) Figure 5.13(b) shows the corresponding continuous phase distribution Figure 5.13(c) shows the central row (along the horizontal direction) of Fig 5.13(a) As can be seen in Figs 5.13(a) and 5.13(c), the fringes on the left side is less dense than those on the right side Using the proposed method, a fringe-density distribution along the central row of Fig 5.13(a) is shown in Fig 5.13(d) As can be seen in Fig 5.13(d), the proposed method can effectively estimate the fringe-density distribution The negative values in the estimated fringe-density distribution have been converted to positive values for clarity

(a) (b)

To further demonstrate the feasibility and effectiveness of the proposed method, wrapped phase maps with noise are also analyzed Figure 5.14(a) shows a simulated circular wrapped phase map with noise Figure 5.14(b) shows a fringe-density distribution along the central row of Fig 5.14(a) It is shown in Fig 5.14(b) that the fringe-density distribution is smooth As the mean filter (K=9) is used, a filtered

fringe-density distribution is shown in Fig 5.14(c) In the mean filter, to avoid large errors at the boundary, the phase values can be extended at its left- and right-edges In practice, symmetrical or zero-padding extension can be employed, or a linear predictive extrapolation can be selected In this study, the values around the boundary are preserved and are not processed

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(a)

Figure 5.15(a) shows a simulated vertical phase map with noise Figure 5.15(b) shows a fringe-density distribution along the central row of Fig 5.15(a) Figure 5.15(c) shows a filtered fringe-density distribution with the mean filter (K=7) It can be seen

in Figs 5.15(b) and 5.15(c) that the proposed method can effectively estimate the fringe-density distribution Note that the image sizes of Figs 5.12(a), 5.13(a), 5.14(a) and 5.15(a) are 256×256pixels

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Average relative errors for the results in Figs 5.14(b) and 5.15(b) are 3.83% and 1.27%, respectively For the circular wrapped phase map, a point in the ideal

fringe-density map whose value is zero is ignored in the error calculation It is illustrated that the proposed method is accurate in the fringe density estimation The errors existing in the estimated fringe-density distributions may be due to the following factors: (1) The estimation of fringe density is not in the direction of fringes; (2) errors may occur in the local areas where the dominant frequency has drastic changes In addition, as can be seen in Figs 5.14(c) and 5.15(c), the mean filter does not make an obvious improvement on the fringe-density distribution In some cases, if there is multiplicative noise, the noise may remove the original fringe density information Hence, a pre-processing to recover the fringe density information becomes necessary (Quan et al., 2005)

The proposed method is also investigated using experimental data Figures 5.16(a) and 5.16(b) show two digital holograms at initial and deformed states captured

by a CCD, respectively The CCD has an array of 1024×1024 pixels, and the size of each pixel is 4.65µm In the experiment (shown in Fig 4.1), an aluminum plate which is clamped around all the edges is chosen as a specimen, and a central point load is applied The distance between the plate and the CCD is 92.5 cm, and the laser wavelength is 632.8 nm The holograms are processed by using a numerical reconstruction algorithm with pure Fourier transform A phase difference map between the initial and deformed states is determined by digital phase subtraction method, and the phase difference map obtained contains 2π jumps The wrapped phase map extracted is shown in Fig 5.16(c) The size of the wrapped phase map is 218

227× pixels The central row of Fig 5.16(c) is shown in Fig 5.16(d) It is seen from Fig 5.16(d) that the wrapped phase map is noisy Figure 5.16(e) shows a fringe-density distribution along the central row of Fig 5.16(c) The fringe-density distribution obtained is smooth, and no further post-processing is required

(a) (b)

(c)

An experiment with a deformed cantilever beam is also used to verify the proposed method Two holograms are captured at two states with a CCD (512×480pixels, and pixel size of 7.4µm) The wrapped phase map obtained by using digital phase subtraction method is shown in Fig 5.17(a) The size of the wrapped phase map

is 304×71 pixels Figure 5.17(b) shows the central row (along the horizontal direction) of Fig 5.17(a) It can be seen in Fig 5.17(b) that the wrapped phase map is noisy Figure 5.17(c) shows a fringe density distribution along the central row of Fig 5.17(a) The fringe density distribution is smooth and no any post-processing is required It should be emphasized that negative values in the estimated fringe-density distribution are also converted to positive values In addition, the proposed method may also be used to detect discontinuous points of a wrapped phase map

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In some cases, fringe density distributions in both x and y directions need to be measured by using the proposed method, so an overall fringe-density distribution is estimated by the detected fringe densities in both the x and y directions In addition, the magnitudes of short-time Fourier transform can also be used as a weight as those

in continuous wavelet transform (Quan et al., 2005) Finally, the measurement of the overall fringe density may be more accurate in the fringe direction

5.2 Temporal phase evaluation analysis

5.2.1 Continuous deformation analysis

As mentioned in Section 3.2.1, an improved short-time Fourier transform is proposed,

so a window size is adaptively determined for each sequence To study continuous deformation, a series of holograms is recorded by a high-speed CCD Figure 5.18(a) shows a typical reconstructed image of a hologram using pure Fourier transform reconstruction algorithm Since the illumination area is limited, an area of interest containing 265×64 pixels [see the rectangular box in Fig 5.18(a)] is properly cropped Figure 5.18(b) shows phase variation on points A, B, C, and D [see Fig 5.18(a)] along the time axis based on complex phasor method and short-time Fourier transform Figure 5.18(c) shows phase variations along section E-E [see Fig 5.18(a)]

at instants 2.8 s and 3.04 s using the complex phasor method and short-time Fourier transform As can be seen in Figs 5.18(b) and 5.18(c), the phase variation for the pixels can be effectively determined using the combination of complex phasor method and short-time Fourier transform Figure 5.18(d) shows window size variations along the 20th and 40th rows in the area of interest using the proposed method Note that a window size is given for each pixel along the time axis (called a sequence)

Figures 5.19(a) and 5.19(b) show 3D plots of continuous phase maps at instant 1.2 s based on conventional temporal phase unwrapping and short-time Fourier transform, respectively Figures 5.20(a) and 5.20(b) show 3D plots of continuous phase maps at instant 2.8 s based on conventional temporal phase unwrapping and short-time Fourier transform, respectively The integration method in the proposed short-time Fourier transform is applied to determine the continuous phase distribution Before continuous phase maps are obtained in Figs 5.19 and 5.20, wrapped phase maps are obtained by using complex phasor method with an average filter As can be seen in Figs 5.19 and 5.20, the proposed method using complex phasor approach and short-time Fourier transform is valid and has higher noise-immunity compared with the conventional temporal processing method When the conventional temporal phase unwrapping is employed, the errors still accumulate along the time axis and can not

be effectively eliminated or suppressed In addition, this comparison between the conventional and proposed methods also confirms that the proposed method is accurate No spatial filtering algorithm is used after the temporal operations in Figs 5.19 and 5.20, and the area of interest is cropped before the determination of a complex phasor

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Figure 5.18 (a) A typical reconstructed image of a hologram; (b) phase variation

on points A, B, C, and D; (c) phase variation along section E-E; (d) window size variations

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Figure 5.19 3D plots of continuous phase maps at instant 1.2 s based on (a)

complex phasor method and temporal phase unwrapping; (b) complex phasor method and short-time Fourier transform

Figure 5.20 3D plots of continuous phase maps at instant 2.8 s based on (a)

complex phasor method and temporal phase unwrapping; (b) complex phasor method and short-time Fourier transform

In practice, a fixed window size may also be used to determine relatively satisfactory results, and a relatively large window size is more robust against noise However, it is not easy to find a fixed window size to process the signals The fixed window size is usually determined based on a trade-off between the accuracy of phase

(a)

(b) (a)

(b)

approximation and noise-immunity In this study, the main procedure in complex phasor method is to filter the imaginary and real parts of a complex amplitude before the temporal operations As the complex phasor method is combined with short-time Fourier transfrom algorithm for extracting instantaneous frequency, if the amplitude variation of the complex phasor is not small, a normalization operation should be applied

Among the advanced analysis methods, continuous wavelet transform and Wigner-Ville distribution (Mallat, 1999) may also be effective for processing the fringe patterns However, continuous wavelet transform does not perform well for a noisy fringe pattern (Qian, 2007) Wigner-Ville distribution suffers from cross-terms (Mallat, 1999), so the actual support of a signal in the time-frequency domain is disturbed In contrast, short-time Fourier transform possesses the characteristics of free cross-terms and computational simplicity, thus it has received wide-spread use in practice (Mallat, 1999; Qian, 2007)

If the data are relatively noisy, the phase distribution obtained by using an integration of the extracted instantaneous frequency might contain some accumulated errors Hence, the filtering algorithmsmay be applied In this study, the average filter

is used to filter the imaginary and real parts of the complex phasor before temporal operations If the filtering algorithm is not selected properly, the phase distributions will contain large errors In addition, the effects of the spatial filter used before and after temporal operations may be further investigated Compared with the conventional temporal phase unwrapping method, the proposed method using complex phasor method and short-time Fourier transform is a time-consuming process However, with a rapid development of the computer technology, this disadvantage would be inconspicuous

5.2.2 Surface profiling with multi-illumination method

5.2.2.1 Profiling using short-time Fourier transform

The proposed method using an improved short-time Fourier transform (mentioned in Section 3.2.1) is also applied to study surface profiling of an object with height steps The top view of the specimen and an area of interest for analysis have been presented

in Fig 4.10 The digital holographic technique also has shading problem similar to all triangulation techniques As shown in Fig 4.7 (Chapter 4), maximum sensitivity is obtained if the angle between the illumination direction and the observation direction

is close to 90 degree However, the flat illumination can cause shadows due to surface variations The optimum illumination direction is usually determined by a trade-off between the maximum sensitivity and the minimum shadows in the reconstructed image (Schnars and Jueptner, 2005) In this study, only an area of interest with the minimum shading is cropped (indicated in Fig 4.10), and one of the twin images is selected before a complex phasor is determined Using a high-speed CCD camera,

546 consecutive digital holograms are recorded, and 401 consecutive holograms are selected for the analysis In the results, the 100th frame is shown

Figure 5.21(a) shows a wrapped phase map using complex phasor method with

an average filter Figure 5.21(b) shows a wrapped phase map using digital phase subtraction and sine/cosine transformation method with an average filter Figure 5.21(c) shows a wrapped phase map by directly filtering the phase extracted by digital phase subtraction method In Fig 5.21(c), a 3×3 window for the average filter is employed As can be seen in Figs 5.21(a)-5.21(c), the wrapped phase map extracted

by complex phasor method is clearer It should be emphasized that the phase itself is not a signal but rather a property of the signal (Ghiglia and Pritt, 1998) If the filtering algorithm is directly applied to the phase, some unexpected results [see Fig 5.21(c)]

would be obtained In the complex phasor method, a filtering algorithm is applied to the imaginary and real parts of a signal instead of the phase itself, thus a direct phase manipulation is avoided Figures 5.22(a) and 5.22(b) show magnified continuous phase maps using temporal phase unwrapping approach and short-time Fourier transform, respectively The complex phasor method with an average filter is used in Figs 5.22(a) and 5.22(b) As can be seen in Figs 5.22(a) and 5.22(b), the gray-scale maps are able to characterize the height differences of the specimen, and the results obtained by complex phasor and short-time Fourier transform are better than that using complex phasor and temporal phase unwrapping in the speckle suppression

(a) (b)

Figure 5.22 Continuous phase maps obtained by (a) complex phasor method and temporal phase unwrapping; (b) complex phasor method and short- time Fourier transform

Figure 5.23(a) shows a 3D plot of an unwrapped phase map using digital phase subtraction and temporal phase unwrapping approach Conventional sine/cosine transformation with an average filter is used to filter the extracted phase maps Figure 5.23(b) shows a 3D plot of an unwrapped phase map by complex phasor and temporal phase unwrapping approach An average filter is used to filter the imaginary and real parts of complex value Γ(m,n,t)Γ*(m,n,0) Figure 5.23(c) shows a 3D plot of a continuous phase map by complex phasor and short-time Fourier transform As can be seen in Figs 5.23(a)-5.23(c), the techniques using temporal phase unwrapping or short-time Fourier transform are able to detect the height steps of the specimen However, the result obtained by complex phasor and temporal phase unwrapping is not as good as that using complex phasor and short-time Fourier transform In this comparison, the noise can be effectively reduced by using the short-time Fourier

A

A

transform due to its localization characteristic Note that after all the temporal operations, no spatial filtering method is applied

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Figure 5.23 3D plots of continuous phase maps by using (a) digital phase

subtraction and temporal phase unwrapping; (b) complex phasor method and temporal phase unwrapping; (c) complex phasor method and short-time Fourier transform

It is found that the selected filtering method can also affect the quality of the phase distribution Figure 5.24(a) shows a 3D plot of an unwrapped phase map using

a median filter based on digital phase subtraction and temporal phase unwrapping