Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities

doi:10.1364/OE.20.010896

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Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities

Jing-Feng Weng and Yu-Lung Lo »View Author Affiliations

Optics Express, Vol. 20, Issue 10, pp. 10896-10920 (2012)
http://dx.doi.org/10.1364/OE.20.010896

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Abstract

For 3D objects with height discontinuities, the image reconstruction performance of interferometric systems is adversely affected by the presence of noise in the wrapped phase map. Various schemes have been proposed for detecting residual noise, speckle noise and noise at the lateral surfaces of the discontinuities. However, in most schemes, some noisy pixels are missed and noise detection errors occur. Accordingly, this paper proposes two robust filters (designated as Filters A and B, respectively) for improving the performance of the phase unwrapping process for objects with height discontinuities. Filter A comprises a noise and phase jump detection scheme and an adaptive median filter, while Filter B replaces the detected noise with the median phase value of an N × N mask centered on the noisy pixel. Filter A enables most of the noise and detection errors in the wrapped phase map to be removed. Filter B then detects and corrects any remaining noise or detection errors during the phase unwrapping process. Three reconstruction paths are proposed, Path I, Path II and Path III. Path I combines the path-dependent MACY algorithm with Filters A and B, while Paths II and III combine the path-independent cellular automata (CA) algorithm with Filters A and B. In Path II, the CA algorithm operates on the whole wrapped phase map, while in Path III, the CA algorithm operates on multiple sub-maps of the wrapped phase map. The simulation and experimental results confirm that the three reconstruction paths provide a robust and precise reconstruction performance given appropriate values of the parameters used in the detection scheme and filters, respectively. However, the CA algorithm used in Paths II and III is relatively inefficient in identifying the most suitable unwrapping paths. Thus, of the three paths, Path I yields the lowest runtime.

1. Introduction

The performance of interferometric microscopy systems is constrained by depth of field and diffraction limitations. As a result, the interferograms generated by interferometric microscopy systems invariably contain noise. For example, environmental effects or contamination of the interferometric system result in so-called residual noise [1

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45(10), 3240–3251 (2007). [CrossRef]

, 2

J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express 19(4), 3086–3105 (2011). [CrossRef] [PubMed]

]. Residual noise causes the phase values inconsistency in the wrapped phase map, and thus is most commonly avoided by the various methods which can estimate the correct phase values from the inconsistency [1

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45(10), 3240–3251 (2007). [CrossRef]

–4

A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens. 40(3), 699–709 (2002). [CrossRef]

]. The interferogram may also contain speckle noise; namely a random noise mixed with the interference pattern within the map. Speckle noise is commonly removed (or reduced) using an experimental setup [5

B. F. Pouet and S. Krishnaswamy, “Technique for the removal of speckle phase in electronic speckle interferometry,” Opt. Lett. 20(3), 318–320 (1995). [CrossRef] [PubMed]

, 6

I. Moon and B. Javidi, “Three-dimensional speckle-noise reduction by using coherent integral imaging,” Opt. Lett. 34(8), 1246–1248 (2009). [CrossRef] [PubMed]

] or a numerical-filtering algorithm [7

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer, 1984).

–11

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008). [CrossRef]

]. Finally, for objects containing height discontinuities, noise is commonly produced at the lateral surfaces of the discontinuity due to the depth of field and diffraction limitations of the interferometric system. In practice, the presence of the filtering noise algorithms used in the interferogram results in a smearing of the 2π phase jumps when the filtered interferograms are converted into a wrapped phase map. As a result, robust phase unwrapping algorithms and filtering algorithms are required in order to improve the quality of the reconstructed image.

Existing phase unwrapping algorithms can be broadly categorized as either temporal [12

H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997). [CrossRef] [PubMed]

–14

D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. 44(35), 7515–7521 (2005). [CrossRef] [PubMed]

], spatial [15

W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef] [PubMed]

–27

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602–063608 (2011). [CrossRef]

], or period-coding [28

K. Liu, Y. C. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18(5), 5229–5244 (2010). [CrossRef] [PubMed]

]. Spatial phase unwrapping algorithms can be further classified as path-dependent (e.g., the MACY algorithm [15

W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef] [PubMed]

]), path-independent (e.g., the cellular automata (CA) algorithm [16

D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4(1), 267–280 (1987). [CrossRef]

–18

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998). [CrossRef]

]), or miscellaneous [19

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). [CrossRef]

–27

]. First, in path-dependent algorithms, the unwrapping path starts from a fixed pixel and simply follows a fixed line. Thus, a short runtime is achieved. However, the unwrapping error accumulates line (path)-by-line (path) if the corresponding lines in the wrapped phase map contain noise. As a result, such an accumulation error causes MACY to provide a poor unwrapping performance when applied to such objects containing the noise at height discontinuities. Second, Path-independent phase unwrapping algorithms search for all possible unwrapping paths between two pixels. Thus, the accumulated unwrapping error is improved. However, the unwrapping speed is far slower than that of path-dependent schemes. For example, CA is time-consuming and may fail to converge if the huge number of noise at the lateral surface of the height discontinuity causes the algorithm to miss the most correct and non-noisy unwrapping paths. Finally, the miscellaneous algorithms are continually presented to resolve one of three types of noise (i.e. speckle noise, residual noise, and noise at the height discontinuities) [19

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). [CrossRef]

–27

]. The branch cut is the common method for the phase unwrapping algorithm [19

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). [CrossRef]

, 20

T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14(10), 2692–2701 (1997). [CrossRef]

]. Importantly, the inconsistent pixels marked by the branch cut are avoided by the corresponding phase unwrapping algorithms. However, the branch cut is easily invalid when encountering the situation with the high speckle or high residual noise [21

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012). [CrossRef] [PubMed]

]. Furthermore, the branch cut easily fails with the noise at the height discontinuities [27

]. Accordingly, the problem of the high speckle noise is studied by simultaneously performing the phase unwrapping and linear-filtering algorithms [21

–24

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 5(2), 147–151 (2008). [CrossRef]

]. Also, the spinning algorithm solves the problem of the high residual noise by avoiding the bad pixels [25

S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007). [CrossRef] [PubMed]

]. Similarly, the noise at height discontinuities is discussed by the methods of digital point array [26

L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. 50(4), 043605–043612 (2011). [CrossRef]

] and Laplace phase derivative variance [27

From above, the present miscellaneous algorithms discuss only one of three types of noise. However, in this study, the three paths (Path I, Path II and Path III) are proposed for removing not only the detection errors but also the three types of noise. Unwrapping the phase maps of 3D objects with height discontinuities is also studied. The three paths are based on two filters, designated as Filter A and Filter B, respectively. The filters are based in turn on the detection scheme proposed in [2

]. The detection scheme is not only capable of detecting the three types of noise described above, but also preserves the phase jumps in the wrapped phase map. However, the scheme may generate detection errors. That is, the noise and phase jump maps may include pixels which are not in fact noisy pixels or phase jump pixels, respectively. These detection errors may cause the image reconstruction process to fail. Accordingly, in Filters A and B, the phase values of the noise and detection error pixels are replaced by suitable median values. Thus, Filters A and B not only remove the noisy pixels without smearing the 2π edges of the phase jumps, but also remove the detection error pixels. Note that the practice of replacing the phase values of the noise and detection error pixels with median values differs from the approach taken in the temporal and spatial methods proposed in [12

H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997). [CrossRef] [PubMed]

, 13

J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993). [CrossRef] [PubMed]

, 16

D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4(1), 267–280 (1987). [CrossRef]

, 19

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). [CrossRef]

, 20

T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14(10), 2692–2701 (1997). [CrossRef]

, 25

S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007). [CrossRef] [PubMed]

], in which the reconstruction process simply avoids the “bad” pixels within the wrapped phase map. In the three image reconstruction paths proposed in this study, Filter A is used before phase unwrapping to remove the majority of the noise and detection errors from the wrapped phase map. Filter B is then used during the phase unwrapping process to remove any noise and detection errors missed by Filter A. Path I combines Filters A and B with the path-dependent MACY algorithm, while Paths II and III combine the two filters with the path-independent CA algorithm. In Path II, the CA algorithm is applied to the whole wrapped phase map, while in Path III, the CA algorithm is applied to multiple sub-maps of the wrapped phase map. The CA algorithm is generally implemented on an array processor in order to reduce the runtime [17

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991). [CrossRef]

]. However, the use of an array processor results in errors at the stitching line. Accordingly, in Path III, stitching errors are reduced by artificially extending the area of each wrapped phase sub-map prior to the unwrapping process and then cropping the redundant region containing the stitching error pixels during the subsequent sub-map meshing process.

The remainder of this paper is organized as follows. Section 2 introduces the theoretical background to the detection scheme and filtering operations used in the present study. Section 3 describes the integration of the detection scheme and the two filters to form three alternative reconstruction paths (i.e., Paths I, II and III). Section 4 presents a series of simulation results which demonstrate the effectiveness of the three reconstruction paths in unwrapping the phase maps of 3D objects with height discontinuities given appropriate values of the detection scheme and filter parameters. Section 5 presents the results of an experimental investigation into the precision, robustness and sensitivity of the three reconstruction paths when applied to typical 3D samples with height discontinuities and surface roughness. Finally, Section 6 presents some brief concluding remarks.

2. Underlying principles of detection scheme and filtering operations

2.1 Noise and phase jump detection scheme

The noise and phase jump detection scheme presented in [2

] is based on four comparative phase parameters (S1-S4), i.e.,

\begin{matrix} \begin{array}{r} S 1 (i, j) = [\frac{ϕ (i + 1, j) - ϕ (i, j) - σ_{Α, Β}}{2 π}] + [\frac{ϕ (i + 1, j + 1) - ϕ (i + 1, j) + σ_{Α, Β}}{2 π}] \\ + [\frac{ϕ (i, j + 1) - ϕ (i + 1, j + 1) - σ_{Α, Β}}{2 π}] + [\frac{ϕ (i, j) - ϕ (i, j + 1) + σ_{Α, Β}}{2 π}] \\ S 2 (i, j) = [\frac{ϕ (i + 1, j) - ϕ (i, j) + σ_{Α, Β}}{2 π}] + [\frac{ϕ (i + 1, j + 1) - ϕ (i + 1, j) - σ_{Α, Β}}{2 π}] \\ + [\frac{ϕ (i, j + 1) - ϕ (i + 1, j + 1) + σ_{Α, Β}}{2 π}] + [\frac{ϕ (i, j) - ϕ (i, j + 1) - σ_{Α, Β}}{2 π}] \\ S 3 (i, j) = [\frac{ϕ (i + 1, j) - ϕ (i, j) + σ_{Α, Β}}{2 π}] + [\frac{ϕ (i + 1, j + 1) - ϕ (i + 1, j) + σ_{Α, Β}}{2 π}] \\ + [\frac{ϕ (i, j + 1) - ϕ (i + 1, j + 1) - σ_{Α, Β}}{2 π}] + [\frac{ϕ (i, j) - ϕ (i, j + 1) - σ_{Α, Β}}{2 π}] \\ S 4 (i, j) = [\frac{ϕ (i + 1, j) - ϕ (i, j) - σ_{Α, Β}}{2 π}] + [\frac{ϕ (i + 1, j + 1) - ϕ (i + 1, j) - σ_{Α, Β}}{2 π}] \\ + [\frac{ϕ (i, j + 1) - ϕ (i + 1, j + 1) + σ_{Α, Β}}{2 π}] + [\frac{ϕ (i, j) - ϕ (i, j + 1) + σ_{Α, Β}}{2 π}] \end{array} \end{matrix}

(1)

where (i, j) is the current pixel position,

ϕ

is the corresponding phase value, and [ ] indicates a rounding operation. In addition,

σ_{Α, Β}

is a threshold parameter with a value in the range of

0 < σ_{Α, Β} < π

. Note that in the present study,

σ_{Α}

denotes the threshold value used in Filter A, while

σ_{Β}

denotes the threshold value used in Filter B. (Note also that

σ_{Α, Β} = σ_{Α} = σ_{Β}

In computing parameters S1-S4, the four neighboring pixels are processed in the counter-clockwise direction, i.e.,

ϕ (i + 1, j) - ϕ (i, j),

ϕ (i + 1, j + 1) - ϕ (i + 1, j),

ϕ (i, j + 1) - ϕ (i + 1, j + 1),

and

ϕ (i, j) - ϕ (i, j + 1)

. The computation process generates a phase range which describes the absolute-maximum phase difference in the whole phase map, i.e.,

π - σ_{Α, Β}

. If the absolute phase difference between any two neighboring pixels,

| P D |

, is less than

π - σ_{Α, Β}

, the detection scheme yields a result of

S 1 (i, j) = S 2 (i, j) = S 3 (i, j) = S 4 (i, j) = 0

, and the current pixel (i.e., (i,j)) is designated as a “good” pixel. Conversely, if the absolute phase difference between any two neighboring pixels is greater than

π - σ_{Α, Β}

, one or more of parameters S1, S2, S3 or S4 has a value other than zero, and the current pixel is designated as a “bad” pixel. The detection scheme may yield an outcome in which one or more of S1-S4 is not equal to zero, but the sum of S1-S4 is equal to zero. In such a case, the current pixel is defined as a phase jump pixel and is categorized as a “good” pixel. The results obtained using the detection scheme are expressed in the form of two maps, namely a noise map and a phase jump map, which mark the positions of the bad (i.e., noisy) pixels and phase jump pixels, respectively. Note that in practice, a suitable value of the threshold parameter

σ_{Α, Β}

in Eq. (1) is obtained by adjusting the value of

σ_{Α, Β}

and observing the corresponding changes in the noise and phase jump maps. The optimal value of

σ_{Α, Β}

is then taken when the value of

σ_{Α, Β}

not only results in the maximum number of noise in the noise map but also the maximum number of the phase jump pixels in the phase jump map.

2.2 Filter A – detection scheme combined with adaptive median filter

In the image reconstruction paths proposed in this study, the majority of the noise in the raw wrapped phase map is removed using Filter A, which comprises the detection scheme given in Eq. (1) and the adaptive median filter proposed in [29

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997). [CrossRef]

]. The adaptive median filter utilizes a 5 × 5 mask corresponding to five different positions relative to a phase jump, namely far from a phase jump, close to a phase jump to a higher phase region, close to a phase jump to a lower phase region, straddling a phase jump to a higher phase region, and straddling a phase jump to a lower phase region. In applying the filter, the wrapped phase data interval [0, 2π] is divided into three equal sub-intervals. Based on the number of mask pixels within each of the three subintervals, the good pixels detected by detection scheme will decide one of the five different positions. In the decision position, the median phase value of the mask-center position is then calculated based on the phase values of the good pixels in the five positions and a set of weighting parameters. Finally, the phase values of the good pixels within the mask retain their original phase values, while the phase values of the bad pixels are replaced by the median phase value.

In [29

], the weighting parameters were assigned values of β = 1 and γ = 0.7, which could influence the classification of good and bad pixels indirectly. In this study, besides β = 1 and γ = 0.7, two weighting parameters, expressed as

N_{good pixels} \leq G

, were re-denoted clearly since they were important and directly influenced the filtering results. Where

N_{good pixels}

is the number of good pixels in the 5 × 5 mask and

G

is a positive integer. And thus, the phase value of the central pixel in the 5 × 5 mask was replaced by the corresponding median phase value if the number of good pixels

N_{good pixels}

was fewer than the integer of . However, from an inspection of [29

], the re-denoted parameter of was approximate to 6 in this study. Although the parameter = 6 was not the optimal parameter in this study, this value of the parameter was also assigned to demonstrate the non-optimal image reconstructions. Accordingly, the suitable parameter of = 20 was found and was assigned to demonstrate the feasible filtering results and image reconstructions. The difference between the phase value of a good pixel and its median phase value was defined by the parameter

σ_{g}

, which was assigned a value of 0.05. If the difference between the actual phase value of a supposedly good pixel and its expected phase value was greater than this threshold setting, the phase value of the good pixel was simply replaced with the corresponding expected phase value.

2.3 Filter B – detection scheme combined with noisy pixel replacement mechanism

In the reconstruction paths proposed in this study, Filter B is applied during the phase unwrapping process to remove any noise missed by Filter A. As shown in Fig. 1 , in implementing the filter, the phase map is processed pixel-by-pixel using a sliding N × N mask centered at the pixel of interest, i.e., pixel (i_c, j_c). The phase unwrapping procedure commences by marking the pixels within the phase map as either “good” or “bad” using the detection scheme given in Eq. (1). In the subsequent filtering operation, the phase of any pixel identified as a good pixel and having a deviation of less than

σ_{g}

from its expected phase value is left unchanged. Conversely, for any pixel erroneously identified as a good pixel (i.e., classed by the detection scheme as a good pixel but actually having a phase deviation of more than

σ_{g}

from the expected phase value), the phase value of the pixel is replaced by the expected phase value. Finally, the phase values of all the pixels identified by the detection scheme as bad pixels are replaced by the median value of the good pixels within the mask.

Fig. 1 N × N mask used in Filter B during phase unwrapping process.

2.4 Implementation of CA algorithm using array processor and additional sub-map area (using in Path III)

The runtime performance of the CA unwrapping algorithm can be improved by implementing the algorithm an on array processor [17

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991). [CrossRef]

]. However, stitching errors are invariably produced when meshing the unwrapped maps generated by the individual processors. In the present study, this problem is resolved by partitioning the wrapped phase map into multiple sub-maps and then extending each map in the row and column directions. For example, in Fig. 2 , each sub-map is extended by an additional 3 pixels in the row and column directions, respectively. The extended sub-maps are unwrapped using the CA algorithm in the usual manner. The additional areas of the unwrapped phase sub-maps are then cropped and the remaining regions of the sub-maps are stitched together in order to construct the complete unwrapped phase map.

Fig. 2 Extension of wrapped phase sub-maps by 3 pixels in row and column directions.

3. Integration of filtering and phase unwrapping algorithms for image reconstruction

As shown in Fig. 3 , the present study proposes three image reconstruction paths based on the noise and phase jump detection scheme described in Section 2.1, Filters A and B described in Sections 2.2 and 2.3, and the MACY or CA phase unwrapping algorithms presented in [15

W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef] [PubMed]

] and [16

D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4(1), 267–280 (1987). [CrossRef]

], respectively. In all three paths, the noisy wrapped phase map is filtered initially by Filter A. Note that Filter A is applied repeatedly to the wrapped phase map; with the number of repetitions being determined via an inspection of the noise map. In Path I, the noise-reduced phase map is unwrapped initially in the row direction using the MACY algorithm. During the unwrapping process, Filter B₁ is applied to remove any noise missed by Filter A and any row-unwrapping errors. The MACY algorithm is then re-applied to unwrap the phase map in the column direction. Filter B₂ is then applied once again to remove any noise missed by Filter A and any column-unwrapping errors. Finally, the unwrapped phase map (i.e., the image reconstruction result) is obtained. In Path II, the wrapped phase map obtained from Filter A is unwrapped by the CA algorithm. As shown in Fig. 3, the CA algorithm acts on the uncut wrapped phase map and involves a cycle of local and global iterations separated by Filter B₁. Once the local / global iteration procedure terminates, Filter B₂ is applied and the unwrapped phase map is obtained. In Path III, the noise-reduced phase map obtained from Filter A is partitioned into several sub-maps. As described in Section 2.4, an additional border area is added to each sub-map, and each map is then unwrapped using the CA algorithm. As in Path III, the unwrapping procedure involves a cycle of local and global iterations integrated with Filter B₁. On completion of the iterative cycle, the border areas are cropped from the unwrapped sub-maps, and the maps are then meshed in order to construct the full unwrapped phase map. Finally, Filter B₂ is applied to obtain the final image reconstruction results.

Fig. 3 Flowchart of three image reconstruction paths.

4. Simulation results

The performance of the three reconstruction paths was evaluated by conducting a series of MATLAB simulations on a PC equipped with an AMD Athlon 64 × 2 4400 + 2.31GHz dual-core processor and 2 GB of RAM. In performing the simulations, five interferograms of a sample with height discontinuities were produced using MATLAB software and were then merged into a single raw wrapped phase map with dimensions of 294 × 246 pixels (rows × columns). In generating the wrapped phase map, three types of noise were introduced. Speckle noise (Noise A) was generated using the “imnoise” function in MATLAB with an intensity parameter setting of 0.08 (i.e., “imnoise (each of five interferograms, 'speckle', 0.08)”). Residual noise (Noise B) was produced using the “imnoise” salt and pepper noise function with an intensity parameter setting of 0.35 (i.e., “imnoise (each of five interferograms, 'salt & pepper', 0.35)”). Finally, noise at the lateral surface of the height discontinuities (Noise C) was produced using a self-written program based on the phase signal corresponding to the low and high positions of the discontinuity and the “imnoise” salt and pepper noise function with an intensity parameter setting of 0.01.

Figures 4(a) and 4(b) show the noisy wrapped phase map of the simulated object and the phase difference values of the pixels within pixel column 125, respectively. As shown in Fig. 4(a), the phase map contains Noise A throughout the whole map, Noise B within the square and circular regions, and Noise C along the solid red line. In addition, the phase map includes a phase jump within the region indicated by the ellipse. Observing the results obtained for the noise map and phase jump map, respectively, given different values of , the optimal value of

σ_{Α}

was determined to be 2.4 (see Figs. 5(a) and 5(b)). An inspection of Figs. 4(a) and 5(a) shows that the positions of Noises A, B and C in the noisy wrapped phase map correspond to the pixel positions marked in the noise map. Similarly, comparing Figs. 4(a) and 5(b), it is seen that the phase jump region indicated by the ellipse in Fig. 4(a) corresponds to the pixel positions marked in the phase jump map. However, a few detection errors are also observed in the phase jump map within the areas corresponding to the square and circular regions in Fig. 4(a) (i.e., the Noise B regions).

Fig. 4 (a) Noisy wrapped phase map. (b) Phase difference values of pixels in pixel column 125 in noisy wrapped phase map.

Fig. 5 Detection results obtained from Filter A for noisy wrapped phase map in Fig. 4(a) given threshold parameter of

σ_{Α}

= 2.4: (a) noise map and (b) phase jump map.

In performing the simulations, the threshold parameters in Filters A and B were set to

σ_{Α, Β} = σ_{Α} = σ_{Β} = 2.4

; parameters β and γ in Filter A were assigned values of β = 1 and γ = 0.7, respectively;

σ_{g}

was specified as

σ_{g}

= 0.05; the critical number of good pixels was assigned a default value of G = 20; the size of the sliding window in Filter A was specified as 5 × 5 pixels; and the size of the sliding window in Filter B was set as 9 × 9 pixels. In the Path III reconstruction process, the wrapped sub-maps were extended by an additional 3 pixels in both the row and the column directions, respectively. Finally, Filter A was applied twice in order to remove most of the noise and detection errors from the wrapped phase map prior to unwrapping.

4.1 Application of Filter A to noisy wrapped phase map

Figure 6(a) shows the noise-reduced wrapped phase map obtained by filtering the phase map shown in Fig. 4(a) twice using Filter A. Figure 6(b) shows the phase difference values of the pixels within pixel column 125 of the noise-reduced phase map. Finally, Figs. 7(a) and 7(b) show the noise map and phase jump map obtained when assigning the threshold parameter in the detection scheme a value of

σ_{A}

= 2.4. Comparing the results presented in Fig. 7(a) with those presented in Fig. 5(a), it is seen that the majority of the pixels corresponding to Noise A and Noise B are removed following the repeated application of Filter A. However, it is also noted that the noise map still contains noisy pixels at the lateral surfaces of the discontinuity (i.e., Noise C). Therefore, further processing by Filter B is required.

Fig. 6 (a) Noise-reduced wrapped phase map after repeated filtering by Filter A with detection threshold setting of = 2.4. (b) Phase difference values of pixels in pixel column 125 of noise-reduced wrapped phase map in Fig. 6(a).

Fig. 7 Detection results obtained when applying Filter A twice: (a) noise map and (b) phase jump map.

To demonstrate the effectiveness of Filter B in removing noise at the lateral surfaces, Subsections 4.2.1 and 4.2.2 present the results obtained from the Path I and Path II reconstruction paths, respectively, when unwrapping the noise-reduced wrapped phase map shown in Fig. 6(a) with and without Filter B. Similarly, in Subsections 4.2.1 and 4.2.2, to demonstrate that the value of G influenced the filtering results of Filter A, two values of G = 20 and G = 6 were used in Path I and Path II reconstruction routes.

4.2 Application of Filter B to noise-reduced wrapped phase map

4.2.1 Path I

Figure 8(a) shows the unwrapping results obtained when applying the MACY algorithm and Filter B to the noise-reduced wrapped phase map shown in Fig. 6(a). Note that the unwrapping process commences with the pixels located in pixel row 90. The upper and lower plots in Fig. 8(b) show the phase difference values of the pixels in pixel column 125 and pixel row 203, respectively. Note that the two cross-sections shown in Fig. 8(b) pass through the center of the circular (Noise B) region in Fig. 4(a). And in the lower plot in Fig. 8(b), the phase difference of the height discontinuity is found to be around 2.5 rads. Finally, an inspection of Figs. 8(a) and 8(b) shows that the Path I reconstruction process successfully reconstructs the simulated 3D object with height discontinuities.

Fig. 8 (a) Unwrapping results obtained using Path I (G = 20). (b) Phase difference values of pixels in pixel column 125 (upper) and pixel row 203 (lower).

Figure 9(a) presents the unwrapping results obtained from the Path I reconstruction process when Filter B is not applied. (Note that the parameter settings in Filter A and the MACY algorithm are identical to those used in Fig. 8.) It can be seen that the presence of Noise C missed by Filter A in the noise map (see Fig. 7(a)) causes the unwrapping process to fail. Figure 9(b) shows the unwrapped phase map obtained from Path I (with Filter B) when the value of G in Filter A (i.e., the critical number of good pixels within the mask) is reduced from G = 20 (Fig. 9(a)) to G = 6. A comparison of Figs. 9(b) and 8(a) shows that a lower value of G results in an unwrapping error in the circled region of the discontinuity (see Fig. 9(b)). In other words, a value of G = 20 in Filter A results in a better unwrapping performance than a value of G = 6.

Fig. 9 (a) Unwrapping results obtained using Path I without Filter B (G = 20). (b) Unwrapping results obtained using Path I with Filter B (G = 6).

In general, the three cases considered in this Subsection (i.e., (1) with Filter B, G = 20; (2) without Filter B, G = 20; and (3) with Filter B, G = 6) confirm the importance of Filter B in removing Noise C from the unwrapped phase map and improving the reconstruction performance of the MACY algorithm as a result. In addition, the results indicate that a higher value of G in Filter A leads to an improved reconstruction performance. The runtimes of the Path I reconstruction process were found to be 15 s without Filter B and 32 s with Filter B, respectively, which were measured from the noisy-wrapped phase map to the unwrapped phase map in the flowchart. In other words, Path I not only yields a good reconstruction performance (given appropriate parameter settings), but also has a short runtime.

4.2.2 Path II

Figure 10(a) presents the unwrapping results obtained using the Path II reconstruction method based on the CA algorithm and Filters A and B. The phase difference values of the pixels in pixel column 125 and pixel row 203 are shown in Fig. 10(b). In the lower plot in Fig. 10(b), the phase difference of the height discontinuity is found to be approximately 2.5 rads. Therefore, the results presented in Fig. 10 confirm that Filter B successfully removes the noise at the lateral surface of the height discontinuities and therefore enables the CA algorithm to converge. An inspection of the runtime data revealed that the CA unwrapping procedure involved 375 local iterations and 3 global iterations. Moreover, the total runtime measured from the noisy-wrapped phase map to the unwrapped phase map was found to be 1376 s.

Fig. 10 (a) Unwrapping results obtained using Path II (G = 20). (b) Phase difference values of pixels in pixel column 125 (upper) and pixel row 203 (lower).

Figure 11(a) shows the reconstruction results obtained when using Path II without Filter B. (Note that the parameter settings in Filter A and the CA algorithm are identical to those considered in Fig. 10.) The results reveal that the failure of Filter A (G = 20) to remove all of Noise C from the wrapped phase map prevents the CA algorithm from converging when Filter B is not applied. The runtime of Path II (G = 20) without Filter B was found to be 3247 seconds. Figure 11(b) shows the reconstruction results obtained using Path II given a lower value of G = 6 in Filter A. In this case, a convergent solution is obtained. In other words, the noise missed by Filter A does not prevent the CA algorithm from determining the most suitable unwrapping paths. An inspection of the runtime data revealed that the CA unwrapping procedure involved 375 local iterations and 3 global iterations. Moreover, the total runtime was found to be 1299 s. In other words, given the application of Filter B, a successful unwrapping result is obtained for both G = 20 and G = 6.

Fig. 11 (a) Unwrapping results obtained using Path II without Filter B and with G = 20. (b) Unwrapping results obtained using Path II with Filter B and G = 6.

Overall, the results presented in Figs. 10 and 11 for the CA algorithm confirm the importance of Filter B in compensating for the inability of Filter A to remove all of the noise from the lateral surface of the discontinuities. Moreover, the unwrapping results with both G = 20 and G = 6 are successful in Path II. However, in Path I, the parameter G = 20 provides a better reconstruction performance. Therefore, for consisting with the value of G in Path I and Path II, the parameter of G = 20 was assigned for the following simulations and experiments.

4.2.3 Path III

In implementing the Path III reconstruction procedure, the noise-reduced wrapped phase map produced by Filter A was cut into 6 parts in the row direction and 6 parts in the column direction. And thus, each sub-map had dimensions of 49 × 41 (rows × columns) pixels. Figure 12 shows the sub-map corresponding to pixel rows 99~147 and pixel columns 83~123 in the noise-reduced wrapped phase map. Each sub-map was extended by an additional 3 pixels in the row and column directions; giving a total sub-map area of 52 × 44 pixels. The simulations focused specifically on the noise and phase jump data of the pixels near the stitching line (marked by the green line in Fig. 12.)

Fig. 12 Noise-reduced wrapped phase sub-map extending from pixel rows 99~147 and pixel columns 83~123. Note that the green line indicates the stitching line.

The noise-reduced wrapped phase map shown in Fig. 6(a) was unwrapped using the Path III reconstruction procedure both with and without an additional area added to each sub-map, respectively. The standard deviation of the phase values of the pixels along the stitching line was then calculated for both phase maps. The standard deviation of the phase values of the corresponding pixels in the wrapped phase sub-map was also calculated for reference purposes. The CA phase unwrapping process removes only the discontinuous 2π jumps in the wrapped phase map. Thus, the phase unwrapping process has no effect on the standard deviation of the phase values of the pixels along the stitching line. Table 1 compares the standard deviation results for the three stitching lines. It is seen that the standard deviation for the stitching line in the sub-map with an additional area is closer to the reference value of 0.12 than that for the stitching line in the sub-map with no additional area. In other words, the additional area reduces the stitching error when the unwrapped sub-maps are meshed to reconstruct the complete unwrapped phase map.

Table 1 Standard Deviation of Phase Values of Pixels along Stitching Lines in Wrapped and Unwrapped Phase Sub-Maps

	Wrapped phase sub-map (Reference value)	Unwrapped phase sub-map with additional area	Unwrapped phase sub-map without additional area
Standard Deviation	0.12	0.07	0.28

Figure 13(a) shows the image reconstruction results obtained using Path III with an additional area added to each sub-map. Figure 13(b) shows the phase difference values of the pixels in pixel column 125 (upper) and pixel row 203 (lower). The phase difference of the height discontinuity is found to be about 2.5 rads. The results presented in Fig. 13 confirm that the Path III reconstruction procedure successfully removes the noise missed by Filter A and therefore enables the CA algorithm to converge. The runtime for the Path III reconstruction process was found to be 92 s, i.e., much faster than the 1376 s runtime of Path II. Therefore, Path III not only improves the runtime performance compared to Path II, but also reduces the stitching error; thereby enhancing the quality of the reconstruction results. Again, in Fig. 13(a), the pixels, located at pixel row 1 and pixel columns 1~246, are not evaluated by an additional sub-map area (see Fig. 2 in Subsection 2.4), and thus these pixels contain a few of errors.

Fig. 13 (a) Unwrapping results obtained using Path III (G = 20). (b) Phase difference values of pixels in pixel column 125 (upper) and pixel row 203 (lower).

4.2.4 Summary of simulation results for Paths I, II and III

Table 2 summarizes the simulation results obtained for Paths I, II and III given values of G = 20 or G = 6 in Filter A and with and without Filter B, respectively. The results show that Paths I, II and III all enable the successful reconstruction of 3D objects with height discontinuities. In other words, Filters A and B collectively remove the noise in the wrapped phase map and correct the detection errors produced by the detection scheme proposed in [2

] irrespective of the unwrapping algorithm used (i.e., MACY or CA). For Path I with G = 20, the noise propagation problem in the MACY algorithm is resolved and the runtime is only 32 s. For Path I with G = 6, the propagating-unwrapping error has only a minor effect on the unwrapping performance, and thus the unwrapping results are still acceptable. For Path II, the CA algorithm achieves a convergent solution for both G = 20 and G = 6. However, the runtime is significantly longer than that for Path I, i.e., 1376 s and 1299 s, respectively. For Path III, a convergent solution is once again obtained for G = 20. Moreover, the runtime is reduced from 1376 s to 92 s. In other words, compared to Path II, the Path III reconstruction route not only improves the quality of the reconstruction results, but also reduces the runtime. For all three paths, the phase difference of the height discontinuity in the unwrapped phase map is around 2.5 rads. Accordingly, Filters A and B remove the noise at the height discontinuities and retain the phase values neighboring at the height discontinuity. For the parameter of G = 20 or G = 6, it was seen that a lot of the bad pixels (i.e. the speckle noise and residual noise) could reduce the good pixels to be less than 20 pixels (or 6 pixels) in the 5 × 5 filtering mask in Filter A. And thus, a parameter setting of G = 20 (or G = 6) in Filter A filtered the speckle noise and residual noise effectively. However, the bad pixels (i.e. the detection errors and the noise at the height discontinuities), fewer than above two types of noise, still could reduce the good pixels to be less than 20 pixels (G = 20), but not be less than 6 pixels (G = 6). Accordingly, for removing the detection errors and the noise at the height discontinuities, a parameter setting of G = 20 was more suitable than G = 6. Overall, for three types of noise and detection errors, a value of G = 20 in Filter A results in a better reconstruction performance than G = 6, which have been demonstrated in Paths I and II routes. The two right-most columns in Table 2 show that the Path I and Path II reconstruction routes both fail when Filter B is not applied. In other words, the importance of Filter B in removing the noise missed by Filter A (particularly the noise at the lateral surfaces of the discontinuity) is confirmed.

Table 2 Summary of Simulation Results for Paths I, II and III

	Path I		Path II		Path III	Path I without Filter B (G = 20)	Path II without Filter B (G = 20)
	G = 20	G = 6	G = 20	G = 6	G = 20	Path I without Filter B (G = 20)	Path II without Filter B (G = 20)
Simulation Outcome	Success	Accept	Success	Success	Success	Failure	Failure
Runtime	32 seconds	32 seconds	1376 seconds	1299 seconds	92 seconds	15 seconds	3274 seconds
Phase Difference of Height Discontinuities	About 2.5 rads		About 2.5 rads		About 2.5 rads	×	×

4.2.5 Effect of Filter B on Path III reconstruction performance

In order to evaluate the effect of Filter B on the Path III reconstruction performance, the wrapped sub-map shown in Fig. 12 was unwrapped by the CA algorithm with and without an additional area added to each sub-map, respectively, and without applying Filter B. The standard deviations of the phase values of the pixels in the stitching lines in the two unwrapped phase maps are shown in Table 3 . It is seen that the standard deviation value for the sub-map with an additional area is identical to the reference value (0.12). However, that for the sub-map with no additional area is slightly higher than the reference value. Thus, the results show that even in the absence of Filter B, the use of an extended sub-map area is beneficial in improving the image reconstruction results.

Table 3 Standard Deviation of Phase Values of Pixels in Stitching Lines in Wrapped and Unwrapped Phase Sub-maps (without Filter B)

	Wrapped phase sub-map (Reference value)	Unwrapped phase sub-map with additional area	Unwrapped phase sub-map without additional area
Standard Deviation	0.12	0.12	0.13

In Table 3 (without Filter B), the standard deviation value obtained when using the additional area is identical to the reference standard deviation value (0.12). In Table 1 (with Filter B), the difference between the two standard deviation values is just 0.05 (i.e., 0.07 and 0.12, respectively). In other words, it is inferred that Filter B introduces only a marginal error in the unwrapped phase results.

5. Experimental setup and results

The performance of the three reconstruction paths was evaluated experimentally using a Mirau interferometric system comprising a microscope (OLYMPUS BH2-UMA) fitted with a 20XDL objective lens (NA 0.4, WD 4.7, Nikon JAPAN), a CCD camera (JAI CV-A11, monochrome progressive scan 1/3 inch), and a lens (6-60 mm, F:1.6, Kenko, JAPAN). In performing the experiments, two different illumination sources were used, namely a white-light LED (LP-1201H-3-IO, EXLITE) and a 632.8 nm He-Ne laser (MODEL 1135P, JDS Uniphase). The He-Ne laser source produced the speckle noise and caused the quality of the interferograms poor. Meanwhile, the white-light source did not produce the speckle noise and resulted in the good quality of interferograms. Accordingly, the poor and good qualities of interferograms were used to verify the ability of filtering and phase unwrapping algorithms. For both light sources, the spatial resolution of the interferometer was around 0.3 µm between two neighboring pixels.

The experiments were performed using three different samples. The precision of the different reconstruction paths was evaluated using a standard step height sample with a discontinuity height of 80 nm (see Subsection 5.1). Meanwhile, the robustness of the three reconstruction paths was investigated using a TaSiN sample with a rough surface (see Subsection 5.2). Finally, the sensitivity of the reconstruction paths was examined using a Si sample containing two different height discontinuities (see Subsection 5.3).

In performing the experiments, the value of parameter

σ_{Α, Β}

in Filters A and B was set as = 2.75. In observing the step height sample, Filter B was implemented using a 5 x 5 mask. However, in observing the TaSiN and Si samples, the mask size was increased to 11 x 11. pixels. All the other reconstruction parameters were assigned the values given in the third paragraph in Section 4.

5.1 Precision evaluation of three reconstruction paths using sample with perpendicular phase jump lines

Figures 14(a) and 14(b) show the raw wrapped phase maps obtained for the standard step height sample using the white-light source and the He-Ne laser source, respectively. Note that in both cases, the phase maps have dimensions of 140 × 250 pixels (row × column). Figures 15(a) and 15(b) show the noise map and phase jump map obtained for the raw wrapped phase map shown in Fig. 14(a) using a threshold parameter setting of = 2.75 in the detection scheme. It is seen in Fig. 15(a) that the noise at the lateral surfaces of the step discontinuity is detected precisely and that only a few detection errors exist. Similarly, in the phase jump map shown in Fig. 15(b), the phase jumps are accurately detected and are orientated perpendicularly to the lateral surfaces. Figures 16(a) and 16(b) show the detection results for the wrapped phase map obtained using the He-Ne laser source. The noise map (Fig. 16(a)) shows that the speckle noise and the noise at the lateral surfaces are successfully detected. Meanwhile, the phase jump map (Fig. 16(b)) shows that the phase jumps are also successfully detected. However, some detection errors occur (see the circled region, for example).

Fig. 14 Raw wrapped phase map of standard step height sample obtained using: (a) white-light source and (b) He-Ne laser.

Fig. 15 Detection results obtained for standard step height sample given white-light source and

σ_{Α, Β}

= 2.75. (a) Noise map and (b) phase jump map.

Fig. 16 Detection results obtained for standard step height sample given laser source and = 2.75. (a) Noise map and (b) phase jump map.

The wrapped phase map obtained using the white-light source (Fig. 14(a)) was unwrapped using the Path I, Path II and Path III reconstruction routes, respectively. The image reconstruction results are presented in Figs. 17(a) , 18(a) and 19(a) , respectively. The corresponding distributions of the sample height in pixel column 125 and pixel row 70 are shown in Figs. 17(b), 18(b) and 19(b). The results presented in Figs. 17(a)–19(a) show that all three paths yield the acceptable and successful reconstruction results.

Fig. 17 (a) Path I reconstruction results for standard step height sample given white-light source. (b) Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower).

Fig. 18 (a) Path II reconstruction results for standard step height sample given white-light source. (b) Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower).

Fig. 19 (a) Path III reconstruction results for standard step height sample given white-light source. (b) Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower).

The precision of the three reconstruction routes was evaluated by determining the height discontinuity between two pixels located at row and column pixel coordinates of (70, 35) and (70, 120), respectively. In the unwrapping results obtained using the Path I, Path II and Path III reconstruction routes, the height discontinuities were found to be 80.94 nm, 80.94 nm, and 79.84 nm, respectively. All three results are close to the actual value of the step height, i.e., 80 nm. Accordingly, in three reconstruction routes, Filters A and B not only remove the noise but also retain the precision of the height discontinuity. For the Path I reconstruction route, the runtime was found to be 23 s. For the Path II and III reconstruction routes, the runtimes were found to be 518 s and 293 s, respectively. In other words, Path I results in the fastest reconstruction performance, while Path II results in the slowest. It is noted that the experimental results obtained for the relative runtime performance of the three schemes is consistent with the simulation results presented in Section 4.

The wrapped phase map obtained using the He-Ne laser source (Fig. 14(b)) was also unwrapped using the Path I, Path II and Path III reconstruction routes, respectively. The image reconstruction results obtained using Path I are presented in Fig. 20(a) . Figure 20(b) shows the variations of the sample height in pixel column 125 and pixel row 70, respectively. The pixel height distributions obtained in the Path II and Path III reconstruction routes are illustrated in Figs. 21(a) and 21(b), respectively. (Note that the image reconstruction results obtained using Paths II and III are not presented here since they are both very similar to those shown in Fig. 20(a).) Figs. 20(b), 21(a) and 21(b) show that the speckle noise produced by the laser source is successfully removed in all three reconstruction routes. Moreover, the height discontinuity between pixels (70, 35) and (70, 120) is found to be 80.63 nm for the Path 1 reconstruction process, 80.63 nm for Path II and 79.91nm for Path III. In every case, the height discontinuity is close to the actual value of 80 nm. The runtimes of the three reconstruction paths were found to be 26 s, 479 s and 185 s, respectively. In other words, the relative runtime performance of the three reconstruction routes when using a laser light source is identical to that when using a white-light source.

Fig. 20 (a) Path I reconstruction results for standard step height sample given laser source. (b) Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower).

Fig. 21 Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower) given laser source and: (a) Path II reconstruction route and (b) Path III reconstruction route

Table 4 summarizes the experimental reconstruction results obtained for the standard step height sample. It is seen that for both light sources, the Path I runtime is shorter than that of either Path II or Path III, while the Path III runtime is shorter than that of Path II. In addition, it is observed that the height discontinuity values obtained in the three reconstruction routes are all close to the actual step height value (i.e., 80 nm). In other words, all three paths yield the acceptable and successful reconstruction results.

Table 4 Summary of Experimental Results Obtained for Standard Step Height Sample

	Path I with white-light source	Path I with He-Ne laser	Path II with white-light source	Path II with He-Ne laser	Path III with white-light source	Path III with He-Ne laser
Runtime (seconds)	23	26	518	479	293	185
Experiment Outcome	Success	Success	Success	Success	Success	Success
80nm Height Discontinuities (nm)	80.94	80.63	80.94	80.63	79.84	79.91

5.2 Robustness evaluation of three reconstruction paths using sample with non-straight phase jump lines

The robustness of the Path I and Path III reconstruction routes was evaluated using a TaSiN sample with a rough surface. The sample roughness and contamination within the interferometric system result in significant residual noise and detection errors. Moreover, the phase jump lines are curved rather than straight, which represents the sample with the more complex surface than that introduced in Subsection 5.1. Figures 22(a) and 22(b) present the detection results obtained for the TaSiN sample using the white-light source and a threshold parameter setting of = 2.75. It is seen that the noise map (Fig. 22(a)) contains residual noise, noise at the lateral surfaces, and detection errors. Detection errors are also observed in the phase jump map (Fig. 22(b)). It is seen that these detection errors are mainly coincident with the regions of residual noise in Fig. 22(a). In addition, it is observed that the phase jump lines in the central region of the phase jump map are curved rather than straight. As a result, the phase jump lines and the noise lines at the lateral surfaces form two different angles, namely an oblique angle and a perpendicular angle. Figure 23(a) shows the noisy wrapped phase map of the rough TaSiN sample, while Fig. 23(b) shows the noise-reduced wrapped phase map following filtering (twice) by Filter A. A comparison of the two figures shows that Filter A successfully removes the noise at the lateral surfaces of the sample (see the regions marked by white arrows in the two figures). Figure 24(a) shows the Path I reconstruction results. The phase difference values of the pixels in pixel column 120 and pixel row 170 are shown in Fig. 24(b). (Note that two red lines of pixel column 120 and pixel row 170 intersect at the center position of the residual noise (see Fig. 22(a)). Figure 24 shows that the missed noise is marked, but does not adversely affect the Path I unwrapping procedure. In other words, the MACY algorithm achieves a reliable solution. The corresponding runtime was found to be 76 s.

Fig. 22 Detection results obtained for rough TaSiN sample given white-light source and = 2.75. (a) Noise map and (b) phase jump map.

Fig. 23 (a) Raw wrapped phase map of TaSiN sample obtained using white-light source. (b) Noise-reduced wrapped phase map of TaSiN sample. Note that the arrows indicate the position of noise at the lateral surface.

Fig. 24 (a) Path I reconstruction results obtained for TaSiN sample given white-light source. (b) Phase difference values of pixels in pixel column 120 (upper) and pixel row 170 (lower).

Figures 25(a) –25(c) show the detection results and reconstruction results obtained using Path I given the He-Ne laser source and a parameter setting of = 2.75. Figure 25(a) shows that the noise map contains all three types of noise. Moreover, the phase jump map shown in Fig. 25(b) contains detection errors. However, as shown in Fig. 25(c), a successful reconstruction result is still obtained. The corresponding runtime was found to be 90 s.

Fig. 25 Path I detection results and reconstruction results given laser source and threshold parameter setting of

σ_{Α, Β}

= 2.75 in detection scheme. (a) Noise map, (b) phase jump map, and (c) reconstructed image.

The Path III route was used to reconstruct the region of the sample characterized by an oblique angle between the phase jump lines and the horizontal noise line (see Fig. 26(b) ). Note that the residual noise crossed one phase jump line in Figs. 26(a) and 26(b), rather than between two phase jump lines (see Figs. 22(a) and 22(b)). Accordingly, the detection and reconstruction procedures were applied to a limited region of this wrapped phase map, i.e., pixel coordinates 250 x 300 (row x column) rather than pixel coordinates 300 x 400 (row x column). In implementing the Path III reconstruction process, the sample was illuminated using the white-light source and the wrapped phase map was cut evenly into 2 parts in the row direction and 3 parts in the column direction. The corresponding detection results and reconstruction results are presented in Figs. 26(a)~26(c). Figure 26(a) shows that the noise map contains residual noise, noise at the lateral surfaces and detection errors. As shown in Fig. 26(b), the detection errors produced by the residual noise straddle and contaminate one phase jump. By contrast, the detection errors are located between the two phase jump lines in Fig. 23(b). The contaminated phase jump line in Fig. 26(b) causes the evaluation of the reconstruction to be more difficult than that of the normal phase jump line in Fig. 22(b). Moreover, Fig. 26(b) shows that the marked phase jump line is irregular and therefore causes the evaluation even more difficult. Nonetheless, the results presented in Fig. 26(c) confirm that the CA algorithm achieves a convergent solution. The corresponding runtime was found to be 646 s.

Fig. 26 Path III detection results and reconstruction results given white-light source and threshold parameter setting of = 2.75 in detection scheme (a) Noise map, (b) phase jump map, and (c) reconstructed image.

In general, the results presented in this Subsection have shown that in the case of samples characterized by non-straight phase jump lines (e.g., samples with surface roughness), the detection errors may straddle the phase jump lines or lie in between them. However, irrespective of the location of the detection errors, the errors are successfully removed in the Path I and Path III reconstruction routes. The two paths also ensure the effective removal of all three types of noise from the wrapped phase map. In other words, both paths are robust toward noise and detection errors and achieve a reliable reconstruction performance as a result. However, the Path II route fails to converge since a lot of noise causes the algorithm to miss the most correct and non-noisy unwrapping paths. Compared to the Path I result of Fig. 24(a), the corresponding Path II result, as shown in Fig. 27(a) , is not convergent. Again, compared to the Path I result of Fig. 25(c), the corresponding and non-convergent Path II result is shown in Fig. 27(b). However, as shown in Fig. 27(c), the Path II result is convergent, which is similar to the Path III result of Fig. 26(c). The runtimes of Figs. 27(a) and 27(b) were found to be 10631 s and 10674 s, which both involved 1500 local and global iterations. And thus, the Path II route fails to converge. An inspection of the runtime data of 2269 s in Fig. 27(c) revealed that the CA unwrapping procedure involved 468 local iterations and 4 global iterations. Therefore, compared to the runtime of 646 s in the Path III route, the Path II route spent more runtime (i.e. 2269 s) to converge. In other words, the Path II route is more difficult to converge than the Path III route.

Fig. 27 Comparisons in the Path II reconstruction results. (a) Corresponding Path I result of Fig. 24(a), (b) corresponding Path I result of Fig. 25(c), and (c) corresponding Path III result of Fig. 26(c).

5.3 Sensitivity evaluation of Path I and Path III reconstruction paths using sample with two different height discontinuities

The sensitivity of the Path I and Path III reconstruction paths was evaluated using a sample containing two discontinuities with different heights. Path II reconstruction route was not demonstrated since the path-independent CA algorithm caused it to fail (or be difficult) to converge, as described in Subsection 5.2. Moreover, Path III also used the CA algorithm and the runtime of Path III was shorter than that of Path II. Therefore, this subsection demonstrated the representative reconstruction paths, Path I for the path-dependent MACY algorithm and Path III for the path-independent CA algorithm. Figures 28(a) and 29(a) present the reconstruction results obtained using Path I and Path III, respectively, with a white-light source. Note that in implementing the Path III reconstruction route, the wrapped phase map was cut evenly into 3 parts in the row direction and 4 parts in the column direction. Figure 28(b) shows the phase difference values of the pixels in pixel column 190 and pixel row 120 of the reconstructed phase map shown in Fig. 28(a). Similarly, Fig. 29(b) shows the phase difference values of the pixels in pixel column 190 and pixel row 120 of the reconstructed phase map shown in Fig. 29(a). It is seen that Paths I and III both yield a reliable solution for the reconstructed image. The Path I and Path III runtimes were found to be 60 s and 521 s, respectively. Figure 30(a) shows the Path I reconstruction results obtained using the laser illumination source. Figure 30(b) shows the phase difference values of the pixels in pixel column 190 and pixel row 120. Again, it is seen that the Path I reconstruction route achieves a reliable solution. The runtime was found to be 66 s.

Fig. 28 (a) Path I reconstruction results for sample with two different height discontinuities given white-light source. (b) Phase difference values of pixels in pixel column 190 (upper) and pixel row 120 (lower).

Fig. 29 (a) Path III reconstruction results for sample with two different height discontinuities given white-light source. (b) Phase difference values of pixels in pixel column 190 (upper) and pixel row 120 (lower).

Fig. 30 (a) Path I reconstruction results for sample with two different height discontinuities given laser source. (b) Phase difference values of pixels in pixel column 190 (upper) and pixel row 120 (lower).

In Figs. 28(b), 29(b) and 30(b), the height discontinuity to the right side of the sample (i.e., between the pixels located at (120, 350) and (120, 200) (row, column), respectively) is equal to 3.84, 3.59, 5.80 rads, respectively. Meanwhile, the height discontinuity to the left side of the sample (i.e., between the pixels located at (120, 30) and (120, 120) (row, column), respectively) is equal to 2.76, 2.51, 2.23 rads, respectively. For the case of the right-hand discontinuity, the discontinuity value obtained using a laser source (5.80 rads) is greater than that obtained using a white-light source (3.84 or 3.59 rads). The higher value of the height discontinuity is the result of the laser source which produces the speckle noise and the serious noise at the discontinuities. However, there is no significant difference in the phase difference values of the height discontinuity to the left side of the sample. And thus, from all of the reliable results of the right-higher and left-lower discontinuites in Figs. 28, 29, and 30, Filters A and B correctly remove the noise at the different height discontinuities without seriously contaminating the phase values which are neighboring with the noise at height discontiuties. Therefore, Paths I and III reconstruction routes are sufficiently sensitive to detect and unwrap the discontinuities with different heights irrespective of the illumination source.

6. Conclusions

This study has proposed three reconstruction paths (Paths I, II and III) for unwrapping the phase maps of 3D objects with height discontinuities. Path I is based on the MACY algorithm [15

W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef] [PubMed]

], while Paths II and III are based on the CA algorithm [16

D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4(1), 267–280 (1987). [CrossRef]

]. In Path II, the CA algorithm operates on the complete wrapped phase map. However, in Path III, the wrapped phase map is partitioned into multiple sub-maps. An additional border area is added to each map, and the CA algorithm then operates on each sub-map individually. Finally, the individual unwrapping results are meshed in order to reconstruct the full unwrapped phase map. All three paths utilize two filters (Filters A and B) to remove the noise in the wrapped phase map and to correct detection errors. Filter A comprises the noise and phase jump detection scheme proposed in [2

] and the adaptive median filter proposed in [29

]. Meanwhile, Filter B replaces the detected noise with the median phase value of the pixels within an N × N mask centered on the noisy pixel. In all three reconstruction routes, Filter A is applied repeatedly prior to unwrapping in order to remove the majority of the noise within the wrapped phase map and to correct the detection errors generated by the detection scheme [2

]. Filter B is then applied to eliminate any remaining noise or detection errors during the subsequent unwrapping procedure. Collectively, the two filters enable virtually all of the noise to be removed from the wrapped phase map (even that at the lateral surfaces of the height discontinuities). Thus, the unwrapping path can uniquely cross the height discontinuities. Accordingly, given appropriate values of the filter parameters, all three reconstruction paths enable the wrapped phase map to be successfully unwrapped.

The simulation results have confirmed the ability of Filters A and B to remove residual noise, speckle noise and noise at the lateral surfaces of the height discontinuity from the wrapped phase map. In addition, it has been shown that a higher value of G (the critical number of good pixels in the mask used in Filter A) results in an improved reconstruction performance. The results have also revealed that Filter B introduces only a minor error into the reconstruction results. Additionally, it has been shown that Path III, in which the wrapped phase map is partitioned into multiple sub-maps, yields a better reconstruction result than Path II. Finally, the results have shown that Path I yields the shortest runtime, while Path II yields the longest.

The precision, robustness and sensitivity of the three reconstruction paths have been evaluated experimentally. The results have shown that for a standard step height sample, all three paths yield the reliable solutions when using a white-light source or a He-Ne laser source. In addition, it has been shown that for a sample with surface roughness (characterized by curved rather than straight phase jump lines), Paths I and III both enable a successful reconstruction of the sample image. However, Path II fails (or is difficult) to converge. Finally, the results have revealed that for a sample containing two discontinuities with different heights, Paths I and III yield a reliable solution irrespective of the illumination source applied. However, Path II once again fails to converge. As in the simulation studies, the experimental results have shown that Path I results in the shortest runtime, while Path II results in the longest. Notably, for all three samples, the reconstruction process is accomplished using the same set of detection scheme and filter parameters (other than the mask size in Filter B). In other words, the three reconstruction paths provide a robust means of unwrapping the phase maps of 3D samples with different geometry and surface roughness characteristics.

In this study, there are many parameters. However, for different cases, the parameters are always set with fixed values except the threshold parameter and the mask size of Filter B. In one simulation case, is equal to 2.4 and the mask size of Filter B is 9 × 9. In three experimental cases, is equal to 2.75 and the mask size of Filter B is 5 × 5 or 11 × 11. Therefore, the suggestion is to keep the fixed values of the parameters and only search the parameter of and the mask size of Filter B. First, the parameter of is decided by comparing the noise map and phase jump map. Second, the mask size of Filter B is decided by corresponding to the size of residual noise. The reason is that all of mask size (i.e. 5 × 5, 9 × 9, 11 × 11) can filter the detection errors, speckle noise, the noise at height discontinuities except the residual noise. As a result, the mask size of Filter B needs to be larger (i.e. from 5 × 5 to 9 × 9, or 11 × 11) when the size of the residual noise is lager.

Overall, the simulation and experimental results presented in this study show that given appropriate values of the parameters used in the detection scheme and Filters A and B, respectively, Path I and Path III reconstruction routes enable the noisy wrapped phase maps of typical experimental samples to be unwrapped in a quick and efficient manner. In the future, the application of three reconstruction paths is toward the samples with the more complex-geometrical shapes or the other types of the image reconstruction containing the noise problems, such as digital holographic microscopy (DHM) [30

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38(34), 6994–7001 (1999). [CrossRef] [PubMed]

] and digital image correlation (DIC) [31

T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985). [CrossRef]

Acknowledgments

The authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under grant NSC 98-2221-E-006-053-MY3.

References and links

1.	R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45(10), 3240–3251 (2007). [CrossRef]
2.	J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express 19(4), 3086–3105 (2011). [CrossRef] [PubMed]
3.	R. Smits and B. Yegnanarayana, “Determination of instants of significant excitation in speech using group delay function,” IEEE Trans. Speech Audio Process. 3(5), 325–333 (1995). [CrossRef]
4.	A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens. 40(3), 699–709 (2002). [CrossRef]
5.	B. F. Pouet and S. Krishnaswamy, “Technique for the removal of speckle phase in electronic speckle interferometry,” Opt. Lett. 20(3), 318–320 (1995). [CrossRef] [PubMed]
6.	I. Moon and B. Javidi, “Three-dimensional speckle-noise reduction by using coherent integral imaging,” Opt. Lett. 34(8), 1246–1248 (2009). [CrossRef] [PubMed]
7.	J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer, 1984).
8.	R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge Univ. Press, 1989).
9.	H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999). [CrossRef]
10.	I. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Springer, 1990).
11.	M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008). [CrossRef]
12.	H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997). [CrossRef] [PubMed]
13.	J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993). [CrossRef] [PubMed]
14.	D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. 44(35), 7515–7521 (2005). [CrossRef] [PubMed]
15.	W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef] [PubMed]
16.	D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4(1), 267–280 (1987). [CrossRef]
17.	A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991). [CrossRef]
18.	H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998). [CrossRef]
19.	R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). [CrossRef]
20.	T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14(10), 2692–2701 (1997). [CrossRef]
21.	M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012). [CrossRef] [PubMed]
22.	J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012).
23.	X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5(3), 296–304 (2011). [CrossRef]
24.	J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 5(2), 147–151 (2008). [CrossRef]
25.	S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007). [CrossRef] [PubMed]
26.	L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. 50(4), 043605–043612 (2011). [CrossRef]
27.	H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602–063608 (2011). [CrossRef]
28.	K. Liu, Y. C. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18(5), 5229–5244 (2010). [CrossRef] [PubMed]
29.	A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997). [CrossRef]
30.	E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38(34), 6994–7001 (1999). [CrossRef] [PubMed]
31.	T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985). [CrossRef]

OCIS Codes
(100.2000) Image processing : Digital image processing
(100.6890) Image processing : Three-dimensional image processing
(100.5088) Image processing : Phase unwrapping

ToC Category:
Image Processing

History
Original Manuscript: January 10, 2012
Revised Manuscript: April 5, 2012
Manuscript Accepted: April 23, 2012
Published: April 26, 2012

Virtual Issues
Vol. 7, Iss. 7 Virtual Journal for Biomedical Optics

Citation
Jing-Feng Weng and Yu-Lung Lo, "Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities," Opt. Express 20, 10896-10920 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-10-10896

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References

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens.45(10), 3240–3251 (2007). [CrossRef]
J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express19(4), 3086–3105 (2011). [CrossRef] [PubMed]
R. Smits and B. Yegnanarayana, “Determination of instants of significant excitation in speech using group delay function,” IEEE Trans. Speech Audio Process.3(5), 325–333 (1995). [CrossRef]
A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens.40(3), 699–709 (2002). [CrossRef]
B. F. Pouet and S. Krishnaswamy, “Technique for the removal of speckle phase in electronic speckle interferometry,” Opt. Lett.20(3), 318–320 (1995). [CrossRef] [PubMed]
I. Moon and B. Javidi, “Three-dimensional speckle-noise reduction by using coherent integral imaging,” Opt. Lett.34(8), 1246–1248 (2009). [CrossRef] [PubMed]
J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer, 1984).
R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge Univ. Press, 1989).
H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun.162(4-6), 205–210 (1999). [CrossRef]
I. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Springer, 1990).
M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain44(3), 239–247 (2008). [CrossRef]
H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt.36(13), 2770–2775 (1997). [CrossRef] [PubMed]
J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt.32(17), 3047–3052 (1993). [CrossRef] [PubMed]
D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt.44(35), 7515–7521 (2005). [CrossRef] [PubMed]
W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt.22(23), 3898–3901 (1983). [CrossRef] [PubMed]
D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A4(1), 267–280 (1987). [CrossRef]
A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng.14(1), 25–37 (1991). [CrossRef]
H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng.30(6), 487–502 (1998). [CrossRef]
R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988). [CrossRef]
T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A14(10), 2692–2701 (1997). [CrossRef]
M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express20(3), 2556–2561 (2012). [CrossRef] [PubMed]
J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012).
X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig.5(3), 296–304 (2011). [CrossRef]
J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens.5(2), 147–151 (2008). [CrossRef]
S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express15(13), 8059–8064 (2007). [CrossRef] [PubMed]
L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng.50(4), 043605–043612 (2011). [CrossRef]
H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng.50(6), 063602–063608 (2011). [CrossRef]
K. Liu, Y. C. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express18(5), 5229–5244 (2010). [CrossRef] [PubMed]
A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng.36(9), 2466–2472 (1997). [CrossRef]
E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt.38(34), 6994–7001 (1999). [CrossRef] [PubMed]
T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech.25(3), 232–244 (1985). [CrossRef]

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