Phase recovery from a single interferogramwith closed fringes by phase unwrapping
Jesús Muñoz-Maciel,1,* Francisco J. Casillas-Rodríguez,1 Miguel Mora-González,1
Francisco G. Peña-Lecona,1 Víctor M. Duran-Ramírez,1
and Gilberto Gómez-Rosas2
1Universidad de Guadalajara, Centro Universitario de los Lagos, Enrique Díaz de León 1144,C. P. 47460, Lagos de Moreno, Jalisco, México
2Universidad de Guadalajara, Centro Universitario de Ciencias Exactas e Ingenierías,Boulevard Marcelino García Barragán 1421, C. P. 44430, Guadalajara, Jalisco, México
Fringe analysis methods are required to reconstructthree-dimensional phase information from two-dimensional intensity distributions as obtained inseveral optical measurements. Phase informationis encoded in the argument of a cosine function re-sulting in fringes whose distribution and frequencyneed to be interpreted. Often, phase shifting techni-ques [1,2] are the first option when environmentalconditions are free of vibrations and air turbulencesand the event to be registered remains unchangedwhile the frames are acquired. However, when thoseconditions are not met, phase shifting techniques aredifficult to implement because special hardware re-quirements arise. An alternative is the analysis of asingle interferogram. In many experimental setups,it is possible to introduce a carrier function in the in-terferogram being analyzed (usually by tilting the re-
ference or the test beam). If it is done, the phase froma single interferogram can be recovered with theFourier method [3], spatial phase shifting [4,5], andphase locked loops [6,7] among other methods. Theamount of the introduced tilt must be enough to forcemonotonic behavior of the phase without infringingthe sampling theorem [8,9]. This limits the dynamicrange of the phase that can be measured aboutone-half as compared with temporal phase shiftingtechniques.
An active field, and the subject of extensive re-search, is the phase recovery from an interferogramwith closed fringes. If a priori knowledge of the phaseis available, the task of recovering becomes easier, asis demonstrated with the spatial synchronous meth-od [10]. Otherwise, one may use algorithms such asthe phase tracking [11–13] and spiral quadraturetransform methods [14,15]. The first method esti-mates the phase at each point, adjusting a planein a window using a minimization process. The sec-ond one estimates the signal quadrature (the sine of
the phase) using a spiral phase spectral operator inthe frequency domain and an orientational phasespatial operator. This last operation is achieved bycalculating the fringe orientation from intensityderivatives. Even though the last two mentioned pro-cedures are robust, they require extensive computa-tional calculations. Regularized phase trackingthroughout the minimization process and the spiralquadrature transform in the estimation of the orien-tation of the fringes.
In this work, we implement two orthogonal band-pass filters of one-half of the spectrum in the Fourierdomain, obtaining two wrapped phases with abruptsign changes as described by Kreis [16]. However,since any specific procedure is mentioned aboutphase unfolding for such wrapped phase maps, weaddress this task requiring continuity in the phaseand its derivatives. This approach has been provedto be useful in the recovery of the phase [17,18].The phase unwrapping procedure is carried out, tak-ing into account the two wrapped maps inspectingthe value of the local derivatives at each pixel beingevaluated. The value of the phase is chosen from thetwo wrapped maps that best fit the continuity criter-ia with respect to an adjacent pixel already evalu-ated. Finally, the feasibility of this approach istested in simulated and experimental data.
2. Theory
The two-dimensional intensity distribution of afringe pattern may be written as
Iðx; yÞ ¼ Ibðx; yÞ þ Imðx; yÞ cos½ϕðx; yÞ�; ð1Þwhere Ibðx; yÞ is the illumination background,Imðx; yÞ is the modulation, and ϕðx; yÞ is the phaseto be determined. The illumination and the modula-tion terms are supposed to vary slowly with respectto the cosine of the phase. The Fourier transform ofthe above equation gives
~Iðη; ξÞ ¼ IfIðx; yÞg ¼ ~Ibðη; ξÞ þGðη; ξÞ þG�ðη; ξÞ: ð2ÞHere, ~Ibðη; ξÞ is represented as a peak at the center inthe Fourier spectrum and Gðη; ξÞ and G�ðη; ξÞ arecomplex conjugated functions that contain the mod-ulation and the phase information in the frequencydomain. If a tilt term is aggregated to the phase,the terms Gðη; ξÞ and G�ðη; ξÞ will be located symme-trically with respect to the center of the Fourier spec-trum. Using a bandpass filter of one-half of thespectrum, it is possible to isolate G�ðη; ξÞ or Gðη; ξÞ,considering that they do not overlap. Then thewrapped phase is found as
ϕwðx; yÞ ¼ arctan�Im½IKðx; yÞ�Re½IKðx; yÞ�
�;
where
IKðx; yÞ ¼ I−1f~IKðη; ξÞg: ð3Þ
The term ~IKðη; ξÞ represents the filtered spectrum,IKðx; yÞ is its inverse transform, and ϕwðx; yÞ is thewrapped phase. The introduced tilt term forces thewhole phase to behave monotonically, increasing ordecreasing as required with the Takeda method.
If the interferogram that is analyzed does not con-tain a tilt term (or not enough), then the complex in-tensities G�ðη; ξÞ or Gðη; ξÞ cannot be isolated and theFourier method is not applicable as stated above.However, a bandpass filter of one-half of the spec-trum is still possible. In such a case, the wrappedphase found is related to the actual one as
ϕwkðx; yÞ ¼ �ϕwðx; yÞ: ð4Þ
The process of applying the Takeda method to an in-terferogram without a carrier function is illustratedin Fig. 1. An interferogram containing closed fringesis seen in Fig. 1(a), and the filtered Fourier trans-forms in the x and y directions are shown inFigs. 1(b) and 1(d), respectively. The found wrappedphases from the filtered spectra are graphed in
Fig. 1. Results of applying the Fourier method to a circularfringe pattern: (a) interferogram with closed fringes, (b) filteredFourier spectrum and (c) the resulting wrapped phase (x direction),(d) filtered Fourier spectrum and (e) the resulting wrapped phase(y direction), and (f) the actual wrapped phase.
Figs. 1(c) and 1(e). Finally, the actual wrapped phasecan be appreciated in Fig. 1(f).
3. Continuity in the Phase and Its Derivatives
Clearly, thewrapped phases seen in Figs. 1(c) and 1(e)contain the correct phase information except for thesign changes. Thus, the task of phase recovery fromsuch wrapped maps is to determine where thoseabrupt changes occur. From Fig. 1(c) it is easy toobserve that the sign change occurs along an imag-inary central horizontal line through the lower re-gion of the wrapped phase map. Just by multiplyingthis part by −1, the wrapped map would be cor-rected. However, the given example is possibly theeasiest problem that one may be faced with. In amore complex interferogram, this same task wouldbecome very difficult to be implemented by visualinspection, especially in the presence of noisymeasurements.
We describe a procedure for phase unwrappingusing the information of the wrapped phases ob-tained with the Fourier method. The sign of thephase maps is determined by monitoring the valueof the pixel and their derivatives with respect to anadjacent pixel already verified. In order to limit theextension of the subsequent equations, we willchange the spatial dependence ðx; yÞ to the subscriptnotation ði; jÞ for the rest of this work. Let us denoteas ϕw1
i;j and ϕw2i;j the phase maps shown in Figs. 1(c)
and 1(e), respectively. We can calculate the phasederivatives at the location ði; jÞ averaging the phasedifferences in a 3 × 3 window in the following way:
ϕx1i;j ¼
ϕxi;j þ ϕx
iþ1;j þ ϕxi;j−1 þ ϕx
iþ1;j−1 þ ϕxi;jþ1 þ ϕx
iþ1;jþ1
6;
ð6Þ
ϕy1i;j ¼
ϕyi;j þ ϕy
i;jþ1 þ ϕyiþ1;j þ ϕy
iþ1;jþ1 þ ϕyi−1;j þ ϕy
i−1;jþ1
6;
ð7Þwhere
ϕxi;j ¼ arctan
�sinðϕwi;j − ϕw
i−1;jÞcosðϕw
i;j − ϕwi−1;jÞ
�; ð8Þ
ϕyi;j ¼ arctan
�sinðϕwi;j − ϕw
i;j−1Þcosðϕw
i;j − ϕwi;j−1Þ
�: ð9Þ
Once the phase derivatives are available, the un-wrapping procedure is stated as follows:
1. We select an initial pixel ði; jÞ inside the pupiland test the conditions
if ½jϕw1i;j − ϕw2
i;j j ≤ t and jϕy1i;j − ϕy2
i;j j ≤ t and
jϕx1i;j − ϕx2
i;j j ≤ t� then ϕri;j ¼
ϕw1i;j þ ϕw2
i;j
2;
ϕyri;j ¼
ϕy1i;j þ ϕy2
i;j
2; ϕxr
i;j ¼ϕx1i;j þ ϕx2
i;j
2; ð10Þ
where t is a threshold value. If the above condition istrue, ϕr
i;j, ϕyri;j, and ϕxr
i;j represent the values of the op-timized phase and phase derivatives at the locationði; jÞ. In order to indicate that the pixel has been suc-cessfully recovered, a mask function is set to one,pi;j ¼ 1. Otherwise, the mask function is set to zero,pi;j ¼ 0, and another location that meets with thecondition given in Eq. (10) is searched.
2. Then, once a starting pixel has been found, thealgorithm searches for a pixel not optimized with anadjacent pixel already recovered. New values of ϕy1
i;j ,
ϕx1i;j , ϕ
y2i;j , and ϕx2
i;j are calculated at the current loca-tion. The phase and their derivatives are foundif any of the following conditions and equations aresatisfied:
if f½ðjϕy1i;j j ≥ t0Þ or ðjϕx1
i;j j ≥ t0Þ� and ⌊jϕy1i;j − ϕ̂
yrj ≤ t1and jϕx1
i;j − ϕ̂xrj ≤ t1⌋g then
ϕri;j ¼ ϕ̂rþarctan
�sinðϕw1i;j − ϕ̂
rÞcosðϕw1
i;j − ϕ̂rÞ
�;
ϕyri;j ¼ϕy1
i;j ϕxri;j ¼ϕx1
i;j ; ð11Þ
or
if f½ðjϕy2i;j j ≥ t0Þ or ðjϕx2
i;j j ≥ t0Þ� and ⌊jϕy2i;j − ϕ̂
yrj ≤ t1and jϕx2
i;j − ϕ̂xrj ≤ t1⌋g then
ϕri;j ¼ ϕ̂rþarctan
�sinðϕw2i;j − ϕ̂
rÞcosðϕw2
i;j − ϕ̂rÞ
�; ϕyr
i;j ¼ϕy2i;j ;
ϕxri;j ¼ϕx2
i;j ; ð12Þ
where ϕ̂r, ϕ̂yr, and ϕ̂xr represent the values of thephase and the phase derivatives of an adjacent pixelalready evaluated and t0 and t1 are thresholds. Thevalue of t0 discriminates low values in magnitude ofthe phase derivatives. This is necessary to avoidthose regions where the phase reaches a local max-ima or minima. In such places, the procedure givenhere may not find the correct solution. The value of t1determines the continuity condition of the phase de-rivatives among adjacent pixels. If the conditions ofEqs. (11) and (12) are not satisfied, the calculationsmust be repeated with the negative values of ϕw1 andϕw2. If a satisfactory solution is found using �ϕw1 or�ϕw2, set pi;j ¼ 1; otherwise, set pi;j ¼ 0.
3. If no pixel meets the conditions of Eqs. (11) or(12), set the thresholds t0 and t1 to lower values andrepeat step 2.
4. Interpolate over the isolated pixels or regionswhere a satisfactory solution using Eqs. (11) and (12)was not found.
In the following section, we present the numericaland experimental results concerning the describedalgorithm.
4. Experimental and Simulated Data
Here we discuss the technical issues with regard tothe describedmethod. We use the interferogramwithcircular fringes (corresponding to a defocus aberra-tion) as shown in Fig. 1(a) and its wrapped phases,shown in Figs. 1(c) and 1(e). The fringe pattern sizeis 127 × 127 pixels with an interferogram radius ofone. The starting point was selected at the pixel(i ¼ 22, j ¼ 22). We implemented the algorithm witha row-by-row scanning strategy in search of not-recovered pixels with an adjacent pixel alreadyestimated. The first scanning followed a path fromleft to right and then from top to bottom regions.The second running followed the inverse order.
The threshold values of t, t0, and t1 were all set to0:1π. Figure 2(a) shows the results of the two runsmentioned above. After that, all the threshold valueswere set to 0:01π. With these new threshold values,the phase seen in Fig. 2(b) was obtained. No addi-tional pixels were found, even with lower thresholdvalues. The final phase was found by extrapolatingover the regions where a solution was not reached,as can be seen in Fig. 2(c). Finally, the phase is shownwrapped in Fig. 2(d) in order to compare it with theactual result graphed in Fig. 1(f).
A second example considering a noisy inter-ferogram is also included. The interferogram size
is 127 × 127 pixels; the phase was obtained mixingsome optical aberrations given by the Zernike poly-nomials. The noise was aggregated to the phase as arandom number between 0.785 and −0:785 rad withuniform distribution. The noisy interferogram isseen in Fig. 3(a). The wrapped phases obtained withbandpass filters in the vertical and horizontal direc-tions are shown in Figs. 3(b) and 3(c), respectively.The result of the unwrapping algorithm after sixiterations with threshold values of t0 ¼ 0:13π andt1 ¼ 0:1π is seen in Fig. 3(d). Two more iterationswith threshold values of t0 ¼ 0:1π and t1 ¼ 0:1π wereimplemented, resulting in the phase shown inFig. 3(e). The extrapolated phase can be appreciatedin Fig. 3(f). The wrapped map of the solution reachedand the noiseless wrapped phase used in this exam-ple are shown in Figs. 3(g) and 3(h), respectively.
The last example of the phase unwrapping algo-rithm was carried out in an experimentally obtainedinterferogram. The fringe pattern was obtained inthe testing of an optical component using a commer-cial Fizeau interferometer. The interferogram size is256 × 240 pixels, and the image was multiplied witha pupil function in order to delimit the useful infor-mation of the fringes at the edges of the optical com-ponent. The real interferogram can be observed inFig. 4(a), the unwrapped phase obtained in eightiterations is seen in Fig. 4(b), and the extrapolatedphase in the complete phase field is shown inFig. 4(c). The cosine of the recovered phase is alsoshown in Fig. 4(d) in order to qualitatively observethe correspondence of the obtained solution withthe recorded experimental data. In this example,the threshold value t0 was set from 0:25π for the firstiteration to 0:1π for the last iteration, decreasing pro-portionally between each iteration. The value of t1was set to 0:1π for all iterations.
The unwrapping procedure described here worksefficiently for interferograms with absolute valuesof the phase derivatives higher than 0:1π. Lowervalues of phase derivatives mean that a maxima
Fig. 3. (a) Noisy interferogram with complex fringe distribution,(b) wrapped phases with sign changes in the x direction and (c) inthe y direction. Partial unwrapped phases after (d) six and (e) eightiterations. (f) Final continuous phase recovered over the whole in-terferogram field, (g) wrapped final phase, and (h) wrapped noise-less phase shown for comparison purposes.
Fig. 2. (a) Phase reconstruction after two iterations with thresh-old values of t ¼ t0 ¼ t1 ¼ 0:1π, (b) result with thresholds values oft0 ¼ t1 ¼ 0:01π, (c) final phase estimation after extrapolation overregions with inconsistent data, and (d) wrapped final phase.
or a minima of the phase is near and should beavoided in order to prevent incorrect results. In prac-tice, the threshold value t0 must be chosen close tothe maximum value in the magnitude of the phasederivatives of the interferogram being analyzed anddecreased to a value of 0:1π for the last iterations. Wefound that the value of t1 must be set between 0:15πand 0:08π. The lowest value for this threshold mayincrease the number of iterations required, but italso increases the reliability of the process.
After no pixel with phase derivatives that meetEqs. (11) and (12) is found, the remaining pixels mustbe extrapolated from neighboring pixels. This occursas mentioned earlier where the phase reaches a localminimum or a maximum but also happens at theedges of the interferogram because the phase deriva-tives are calculated in a 3 × 3 window. We calculatedthese pixels by fitting a low-order polynomial in asquare window using least-squares regression. An-other useful extrapolation method may be consultedin [19].
The maximum amount of noise (uniform distribu-tion) that the method could handle repeatedly in oursimulations was in the range of 1.178 to −1:178 rad.For such noise values, an additional procedure wasimplemented in step number 2. We required thatno residues be found [20] in the 3 × 3 window usedto calculate the phase derivatives. This strongly en-hances the noise robustness of the method but alsoconsiderably increments the number of calculationsrequired.
The processing times for the examples given in thissection were 12, 18, and 32 s for the interferogramsshown in Figs. 1(a), 3(a), and 4(a), respectively. Thealgorithm was implemented using MATLABsoftware in a personal computer with a PentiumIV processor at 3:0GHz and 504Mbytes of RAM.
Much lower times could be expected if the algorithmwere implemented using the C language.
5. Conclusions
We have demonstrated a procedure for phase recov-ery from a single interferogram with closed fringesusing the wrapped phases as obtained with the Four-ier method. Given a starting location, the method re-covers the phase values of the remaining pixelsunder the scheme of the well-known unwrappingtechnique using wrapped maps with sign changes.The selection of the phase map and the sign of thesame used to unwrap the remaining pixels are basedon the assumption that the phase is a smoothfunction with continuous derivatives. Those regionswhere the phase reaches a local maxima or minimaare avoided from the first calculations and are esti-mated last. This situation arises where the magni-tude of the phase gradient has values near zero.
The proposed method was implemented using twosimulated fringe patterns with closed fringes. Also,the effectiveness of the method was tested in a realinterferogram with characteristics that we considerto be difficult to demodulate. The interferogram ad-ditionally with several regions of closed fringes, someof them near the edges, also possesses very highfringe density. Only eight iterations and the extrapo-lation process were required to obtain a successfulsolution. This reveals that the developed procedureworks fast and withstands noisy measurements.
The proposed technique requires that two (ormore) closed fringe regions be connected withstraight lines or nearly straight lines with phase de-rivative values (in at least one direction) above 0:1πin magnitude in order to reliably connect such re-gions. This situation may be achieved during the ex-periment by introducing a small amount of tilt in thefringe pattern. Other limitations are due to noise andbroken fringes. These problems make the methodpath dependent because noise and broken fringesviolate the main assumption of continuity in thephase and in the phase derivatives. However, noisemay be alleviated with low-pass filtering techniques,and broken fringes may be masked out to avoid thepropagation of an erroneous solution.
Finally, the method can be applied in the recoveryof high dynamic range phases when phase shiftingtechniques are difficult to implement due to environ-mental disturbances.
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