Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path Miguel Arevallilo Herra´ez, David R. Burton, Michael J. Lalor, and Munther A. Gdeisat
We describe what is to our knowledge a novel technique for phase unwrapping. Several algorithms based on unwrapping the most-reliable pixels first have been proposed. These were restricted to continuous paths and were subject to difficulties in defining a starting pixel. The technique described here uses a different type of reliability function and does not follow a continuous path to perform the unwrapping operation. The technique is explained in detail and illustrated with a number of examples. © 2002 Optical Society of America OCIS codes: 100.0100, 120.0120, 070.0070.
1. Introduction
It is commonly the case in fringe-pattern analysis that the phase is obtained by means of an expression involving the arctangent function. This mathematical function returns values that are known between the limits ⫹ and ⫺. Hence the result is given modulo 2 and discontinuities with values near to 2 appear in the phase distribution. Unwrapping is the procedure by which these discontinuities are resolved, the result is converted into the desired continuous phase function.1 The unwrapping problem has been a major topic of research for over a decade.2 Many different algorithms do exist, but a correct solution is by no means guaranteed, and long execution times are often involved. At first sight, phase unwrapping seems a relatively simple problem but, when working with real images, procedures have to be designed to deal with many different issues. Discontinuities in the phase wraps, under sampling in local areas, high local variations of signal-to-noise ratios, and masked M. A. Herra´ez is with the Computing School, Mediterranean University of Science and Technology, Edificio Galileo Galilei, Avda. de los Naranjos s兾n. Valencia, Spain. E-mail: miguel@ must-es.com. D. R. Burton, M. J. Lalor, and M. A. Gdeisat are with the Coherent and Electro-Optics Research Group, School of Engineering, Liverpool John Moores University, Byrom Street, Liverpool, L3 3AF, United Kingdom. E-mail: d.r.burton@ livjm.ac.uk,
[email protected],
[email protected]. Received 2 May 2002; revised manuscript received 31 July 2002. 0003-6935兾02兾357437-08$15.00兾0 © 2002 Optical Society of America
areas are some of the problems that an unwrapper has to overcome. Many phase-unwrapping algorithms have been proposed in the past to improve the noise immunity of the unwrapping algorithms. These algorithms can be classified as follows:3 共1兲 Global algorithms, 共2兲 region algorithms, and 共3兲 path-following algorithms. The global algorithms formulate the unwrapping algorithm in terms of minimization of a global function.4 All the algorithms in this class are known to be robust but computationally intensive.3,5 The region algorithms divide the image into regions. Each region is processed individually, and then the processed regions are unwrapped with respect to each other to form larger regions until the whole image is processed. The region algorithms can be subclassified into two groups: the tilebased3,6,7 algorithms and the region-based5,8,9 algorithms. The region algorithms provide a compromise between the robustness and the computational intensity.9 The path-following algorithms class can be subclassified into three groups.3 共1兲 path-dependent methods, 共2兲 residue-compensation methods, and 共3兲 quality guided path methods. The simplest algorithm between the pathdependent methods is the Schafer and Oppenheim’s unwrapper.10 This group includes spiral and multiple scan direction methods.1 These methods detect the position of edges or abrupt phase steps in an image and use this information to calculate running phase offsets. These algorithms are usually fast but 10 December 2002 兾 Vol. 41, No. 35 兾 APPLIED OPTICS
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are unable to handle noisy images because of the fixed data evaluation order. The residue compensation algorithms search for residues in an image and generate cuts between positive and negative residues.11 These algorithms are generally computationally efficient but not robust.5 The quality guided path algorithms depend on unwrapping the highest-quality pixels with the highestreliability values first and the lowest-quality pixels that have the lowest-reliability values last to prevent error propagation. The unwrapping path is determined by use of the pixels’ reliability.12–19 Even though some unwrapping errors can remain undetected and propagate in a manner that depends on the unwrapping path, the algorithms are surprisingly robust in practice.2 Also, these algorithms are generally computationally efficient.5 The proposed algorithm falls into this category. The proposed unwrapping algorithm follows discrete paths as will be explained in Section 2, while the unwrapping algorithms in Refs. 12–19 tend to follow a continuous path. These unwrapping algorithms tend to unwrap pixels with the best reliability values that are neighbors to the integrator pixel. 2. Algorithm
Fig. 1. Calculation of the second differences in an image.
j ⫹ 1兲, and 共i ⫹ 1, j ⫹ 1兲 pixels are called diagonal neighboring pixels. The second difference D of an 共i, j兲 pixel can be calculated by the equation: D共i, j兲 ⫽ 关H 2 共i, j兲 ⫹ V 2共i, j兲 ⫹ D 12共i, j兲 ⫹ D 22 共i, j兲兴 1兾2 , where H共i, j兲 ⫽ ␥关共i ⫺ 1, j兲 ⫺ 共i, j兲兴
In quality guided path unwrapping algorithms there are two main issues: the choice of the reliability function and the design of the unwrapping path. These two issues are explained from the perspective of the proposed algorithm. A.
V共i, j兲 ⫽ ␥关共i, j ⫺ 1兲 ⫺ 共i, j兲兴 ⫺ ␥关共i, j兲 ⫺ 共i, j ⫹ 1兲兴
Reliability Function
Quality guided path unwrapping algorithms use different criterion to determine the reliability of a point. This criterion is usually based upon the gradients or differences between a pixel and its neighbors. Those points with the lowest module 2 gradients with respect to their neighbors are determined to be the best points and, therefore, are processed first.15 The use of the absolute value of the gradients in the reliability function has a number of disadvantages. If a high carrier value is present 共compared with the modulation of the fringe pattern兲, the carrier becomes the major modulation component. However, a low carrier value would increase or decrease the values of the gradients depending on the pixel of the image, and produce a nonappropriate measurement for reliability. Second differences would provide a measurement for the degree of concavity兾convexity of the phase function. By use of second differences a better detection of possible inconsistencies in the phase map is provided.16 The calculation of second differences for pixels in an image can be explained with the aid of Fig. 1. To calculate the second difference for a pixel in an image, the values of its orthogonal and diagonal neighbors’ in a 3 ⫻ 3 window are required. The pixels 共i, j ⫺ 1兲, 共i, j ⫹ 1兲, 共i ⫺ 1, j兲, and 共i ⫹ 1, j 兲 that are neighbors to the 共i, j兲 pixels are called orthogonal neighboring pixels. Whereas 共i ⫺ 1, j ⫺ 1兲, 共i ⫹ 1, j ⫺ 1兲, 共i ⫺ 1, 7438
⫺ ␥关共i, j兲 ⫺ 共i ⫹ 1, j兲兴
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D 1共i, j兲 ⫽ ␥关共i ⫺ 1, j ⫺ 1兲 ⫺ 共i, j兲兴 ⫺ ␥关共i, j兲 ⫺ 共i ⫹ 1, j ⫹ 1兲兴 D 2共i, j兲 ⫽ ␥关共i ⫺ 1, j ⫹ 1兲 ⫺ 共i, j兲兴 ⫺ ␥关共i, j兲 ⫺ 共i ⫹ 1, j ⫺ 1兲兴 ,
(1)
where ␥共兲 is a simple unwrapping operation to remove any 2 steps between two consecutive pixels. The second differences can be calculated for all pixels in an image except the borders. The second differences D of the pixels at the borders of the image are set to infinity to be resolved last as will be explained in the following subsection. The reliability R of a pixel is defined as R⫽
1 . D
(2)
Consequently, pixels are more reliable as their second differences are lower. B.
Unwrapping Path
An edge is an intersection of two pixels that are connected horizontally or vertically. Any pixel with its left hand side, right hand side, upper, or lower neighboring pixel can construct an edge. Figure 2共a兲 shows a part of an image whose pixels’ reliabilities are shown in the figure. Every two orthogonal neighboring pixels can produce an edge as shown in
Fig. 2. Numerical example of the proposed algorithm. (Continues on next page.)
Fig. 2共b兲. The reliability of an edge is defined as the summation of the reliabilities of the two pixels that the edge connects as illustrated in Fig. 2共b兲. Edges can be classified as horizontal edges that appear in a green color in the figure and vertical edges, which are colored in red. An unwrapping path cannot be defined relative to the reliability of the pixels. Instead, it is defined by looking at the value of the reliability of the edges. The definition of the unwrapping path is relatively simple: Those edges with higher reliability are unwrapped first. The edges are stored in an array and sorted by value of reliability. The edges with a higher reliability are resolved first. Figures 2共c兲–2共j兲 illustrate the principle of the proposed algorithm. Suppose that we want to unwrap
the pixels whose reliability values are shown in Fig. 2共a兲 using the proposed algorithm. The edges that can be constructed and their reliability values are shown in Fig. 2共b兲. Initially all the pixels are considered not belonging to any group. In Fig. 2共c兲, pixels f and g are unwrapped first with respect to each other because the edge that connects them has the highest reliability value. Both pixels belong to the same group, and this is illustrated by coloring them the same color. Then the unwrapping operation of pixels a and b is carried out, and both pixels construct a second group. The unwrapping of pixels i and j follows, and they construct a third group. Figures 2共c兲 and 2共d兲 show that the edge that connects pixels a and e has the fourth highest value, and 10 December 2002 兾 Vol. 41, No. 35 兾 APPLIED OPTICS
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Fig. 2. (Continued.)
the two pixels that construct the edge should be unwrapped with respect to each other. But pixel a is already unwrapped with respect to pixel b, and both pixels belong to the same group. To unwrap pixel e, the 2 multiples required to be added兾subtracted to unwrap pixel e is calculated and added兾subtracted from pixel e. Pixels a, b, and e are now considered to belong to the same group, and they are given the same color as shown in Fig. 2共d兲. Figures 2共e兲 and 2共f 兲 illustrate the unwrapping of two groups with respect to each other. The first group consists of the pixels a, b, and e; whereas the second group contains i and j pixels. Both groups are connected by a vertical edge whose reliability value is higher than or equal to the reliability values of the other unprocessed edges. Both groups should be unwrapped with respect to each other. The second group contains a smaller number of pixels, so the multiples of 2 of the difference between one of the pixels in the first group and one of the pixels in the second group is calculated. The 2multiples value is then added to or subtracted from all the pixels in the group that contains the smaller number of pixels. The five pixels are joined together and are now considered to belong to the same group. The five pixels are given the same color as shown in Fig. 2共f 兲. The proposed algorithm proceeds as shown in Fig. 2 until all the pixels are unwrapped. The unwrapping path is discrete and therefore, the 7440
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unwrapping operation is local. This is a major difference when compared to the Quiroga et al. algorithms.15,16 This can be demonstrated in Fig. 2共d兲, where the pixels l and q are unwrapped with respect to each other, and they are disconnected from the rest of the unwrapped pixels 共a, b, e, f, g, i, and j pixels兲. In the unwrapping process, three cases are possible: 1. Both pixels have not been unwrapped before. The pixels are unwrapped with respect to each other and gathered into a single group of unwrapped pixels. 2. One of the pixels has been processed before 共and therefore belongs to a group of unwrapped pixels兲 but the other has not 共does not belong to any group兲. The pixel that has not been processed before is unwrapped with respect to the other pixel and added to the second pixel’s group. 3. Both pixels have been processed before. If they do not belong to the same group, the two groups need to be unwrapped with respect to each other. The smallest group is unwrapped with respect to the largest group. This operation involves unwrapping the pixel belonging to the smallest group with respect to the other pixel and adding the same constant that is added to the pixel to the rest of the smallest group. Finally, both groups are joined together into a single new group.
Fig. 3. Flow chart of the proposed algorithm.
A flow chart that describes the operation of the proposed algorithm is shown in Fig. 3. The reliability function may be modified, and lowreliability points may cause problems in the final result. It may be better to ignore intersections with very low reliability values and interpolate these after unwrapping is complete.
3. Simulated and Experimental Results A.
Simulated Results
The proposed algorithm has been tested by use of simulated images. All images depicting wrapped phases in the range 关⫺, 兴 are scaled between black and white for display. Unwrapped phase images 10 December 2002 兾 Vol. 41, No. 35 兾 APPLIED OPTICS
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Fig. 4. Unwrapping of a simple simulated wrapped phase distribution: 共a兲 wrapped phase distribution 共b兲 unwrapped phase distribution.
are also scaled between black and white to cover the full dynamic range. Figure 4共a兲 shows a very high quality wrapped phase distribution with no noise and no physical discontinuities present. This image has been used to test the proposed algorithm under ideal conditions. Figure 4共b兲 shows the resulting unwrapped phase. Figure 5共a兲 shows a simulated wrapped phase distribution in which there is no noise in the body of the phase, but there is a small central area that contains only random noise. The purpose of this data set is to test the unwrapper’s ability to isolate this noise and prevent it from corrupting the unwrapping of the good data. Figure 5共b兲 shows the resulting unwrapped phase map. B.
Experimental Results
Figure 6共a兲 shows a wrapped phase map resulting from the analysis of a real fringe pattern. The wrapped phase map contains corrupted areas that result from a shadow of the projected fringes. The wrapped phase map has been unwrapped by use of the proposed algorithm. The unwrapped phase map is shown in Fig. 6共b兲, which indicates that the algorithm was capable of unwrapping the unreliable re-
gions last to prevent error propagation. The algorithm copes well with these problems and successfully unwraps the image. Figure 7共a兲 presents a noisy wrapped phase map that is corrupted by salt-and-pepper noise. The image has been unwrapped by use of the proposed algorithm, and the unwrapped phase map is shown in Fig. 7共b兲. A three-dimensional representation of the unwrapped phase map is shown in Fig. 7共c兲. The wrapped phase map shown in Fig. 7共a兲 has been unwrapped by use of Schafer and Oppenheim’s unwrapping algorithm,10 and the unwrapped phase map is shown in Fig. 7共d兲. The dynamic range of the unwrapped phase map can be much larger than the dynamic range of the wrapped phase map. The displaying of the unwrapped phase map by scaling it between black and white is not always indicative of the unwrapping process being performed correctly. Therefore following Ghiglia and Romero’s method,4 we rewrapped the unwrapped phase map to permit a direct comparison with the wrapped image. This rewrapping is visually convincing that the unwrapping is qualitatively correct. Figure 8 shows the rewrapped phase map of the unwrapped phase map shown in Fig. 7共b兲. Figs. 7共a兲 and 8 show a good agreement between the wrapped and the rewrapped phase maps, which demonstrates that the proposed algorithm unwrapped the wrapped phase map successfully. The execution time of the proposed algorithm varies from image to image and depends on the particular phase distribution being analyzed. The tested images are 512 ⫻ 512 pixels in size. The proposed algorithm has been executed on a PC system. The PC contains dual Pentium III processors that run at 1 GHz clock speed. The memory on this PC is 512 Mbyte Rambus RAM. The execution time is in the order of half a second on average. The execution
Fig. 5. Unwrapping of a simple simulated wrapped phase distribution with a noisy area: 共a兲 wrapped phase distribution, 共b兲 unwrapped phase distribution. 7442
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Fig. 6. Unwrapping of a wrapped phase distribution obtained from a real fringe pattern. The region crossing the image is a product of its shadow: 共a兲 wrapped phase distribution. 共b兲 unwrapped phase distribution.
Fig. 7. Unwrapping of a wrapped phase distribution obtained from the measurement of a human face: 共a兲 wrapped phase distribution, 共b兲 unwrapped phase distribution, 共c兲 three dimensional representation of the resulting phase, 共d兲 result obtained with the linear scanning procedure. 10 December 2002 兾 Vol. 41, No. 35 兾 APPLIED OPTICS
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5.
6.
7.
8.
9.
10. Fig. 8. Rewrapped phase map.
time to unwrap the image shown in Fig. 7共a兲 is 0.65 seconds.
11. 12. 13.
4. Conclusions
A fast, reliable two-dimensional unwrapper has been proposed, described, and tested. The algorithm, based on unwrapping points with higher reliability values first following a noncontinuous path, produces consistent results when discontinuities or noisy areas are present.
14.
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16.
References 1. D. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W. R. Robinson and G. T. Reid, eds., 共Institute of Physics Publishing, Bristol and Philadelphia, 1993兲, pp. 194 –229. 2. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software 共Wiley, New York, 1998兲. 3. A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187–1194 共2001兲. 4. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional
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17.
18.
19.
weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 共1994兲. K. M. Hung and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 共1998兲. M. Arevallilo Herra´ ez, D. R. Burton, and D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 共1996兲. P. Stephenson, D. R. Burton, and M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 共1994兲. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 818, 2–9 共1987兲. P. G. Charette, and I. W. Hunter, “Robust phase-unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506 –3513 共1996兲. R. W. Schafer, and A. V. Oppenheim, Digital Signal Processing 共Prentice-Hall, Englewood Cliffs, N.J., 1975兲, pp. 507–511. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268 –3270 共1989兲. O. Y. Kwon, D. M. Shough, R. A. Williams, “Stroboscopic phase-shifting interferometry,” Opt. Lett. 12, 855– 857 共1987兲. J. Scho¨ rner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, and P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 共1991兲. H. A. Vrooman and A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636 –1641 共1991兲. J. A. Quiroga and E. Bernabeu, “Phase unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725– 6731 共1994兲. J. A. Quiroga, A. Gonzalez-Cano, and E. Bernabeu, “Phase unwrapping algorithm based on adaptive criterion,” Appl. Opt. 34, 2560 –2563 共1995兲. Y. Xu and C. Ai, “Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis, Proc. SPIE 30, 254 –263 共1991兲. H. Lim, W. Xu, and X. Huang, “Two new practical methods for phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’95 33, Firenze, Italy, 196 –198 共1995兲. W. Xu, and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’96 34, Lincoln, Nebraska, USA, 2044 –2046 共1996兲.
