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

Jing-Feng Weng; Yu-Lung Lo

doi:10.1364/OE.20.010896

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, 2]. 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 [1–4]. 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, 6] or a numerical-filtering algorithm [7–11]. 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 [12–14], spatial [15–27], or period-coding [28]. Spatial phase unwrapping algorithms can be further classified as path-dependent (e.g., the MACY algorithm [15]), path-independent (e.g., the cellular automata (CA) algorithm [16–18]), or miscellaneous [19–27]. 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) [19–27]. The branch cut is the common method for the phase unwrapping algorithm [19, 20]. 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]. 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]. Also, the spinning algorithm solves the problem of the high residual noise by avoiding the bad pixels [25]. Similarly, the noise at height discontinuities is discussed by the methods of digital point array [26] 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, 13, 16, 19, 20, 25], 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]. 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}

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]. 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.

	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	×	×

	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

Abstract

1. Introduction

2. Underlying principles of detection scheme and filtering operations

2.1 Noise and phase jump detection scheme

2.2 Filter A – detection scheme combined with adaptive median filter

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

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

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

4. Simulation results

4.1 Application of Filter A to noisy wrapped phase map

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

4.2.1 Path I

4.2.2 Path II

4.2.3 Path III

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

4.2.5 Effect of Filter B on Path III reconstruction performance

5. Experimental setup and results

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

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

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

6. Conclusions

Acknowledgments

References and links

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Figures (30)

Tables (4)

Equations (1)

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