Open Access
Add to BibSonomyAbstract
A method for demodulating fringe patterns containing contrast reversals is proposed. It consists of two steps. First, the absolute value of the fringe intensity distribution with its background removed is calculated. Then, two dimensional continuous wavelet transform with enhanced ridge extraction algorithm is applied to extract the fringe phase map. Proposed approach allows to dispose of phase jumps along the contrast reversal bands. The method requires only one image and has no special demands concerning the fringe pattern design. Method validity and robustness is confirmed using experimentally acquired time-averaged interferograms of vibrating silicon micromembranes.
© 2013 Optical Society of America
1. Introduction
Optical metrology is indispensable for quantitative studies of diverse scientific phenomena and engineering problems. Interferometry, moiré and structured light illumination methods are common representatives. As the full-field techniques they enable simultaneous acquisition of experimental data and parallel processing to evaluate both macro and microscale objects under static and dynamic loads. The measurand information is usually encoded in the shape and position of fringes, i.e., the departure from the interferogram uniform intensity distribution (null field or null-fringe detection mode) or the reference fringe departure from straightness (finite fringe detection mode). Quantitative analysis of fringe patterns is performed using computer-aided automatic fringe pattern analysis (AFPA) methods (see, for example, [1, 2]).
On the other hand, the measurand can be encoded in the contrast or intensity modulation of fringes as well. As examples, the following applications can be quoted: time-averaged studies of vibrations using two-beam interferometry [3–12], electronic/digital speckle interferometry [13], and the structured illumination technique [14–16]; surface shape measurements using structured illumination [17, 18]; surface microprofile determination using white light interferometry [19]; modulation transfer function measurement [20]; biotissue studies using optical coherence tomography (OCT) [21–23]; structured illumination microscopy [24,25]; and quasi-parallel optical plate/block testing [26–30]. To find the fringe pattern modulation envelope several multi- and single-frame AFPA techniques were proposed, for example phase shifting (TPS), Fourier transform (FT), spatial carrier phase shifting (SCPS), continuous wavelet transform (CWT), and the 2D Hilbert transform (HT) aided by the bidimensional empirical mode decomposition (BEMD). Single frame methods (FT, SCPS, CWT, BEMD+HT) [1, 2, 10–12] are much less susceptible to environmental disturbances, require simpler experimental setups, and could be useful in real-time data processing. They are, therefore, of great interest among research groups. The accuracy of single frame fringe pattern analysis methods depends on the algorithmic solutions developed.
Certainly, single frame methods require a carrier to be introduced irrespectively whether the fringe pattern phase or modulation distributions are extracted. In coherent methods finite period two-beam interference fringes serve the purpose whereas in the incoherent methods structural illumination (e.g., grating imaging or projection) is used. In some experimental cases the carrier is subject to contrast reversals (bright fringe changes into dark one and vice versa). Main two practical cases relate to the time-averaged vibration amplitude profilometry (fringe or grid pattern projection) [3–12, 14–18] and quasi-parallel optical element testing [26–30]. A single time-averaged fringe pattern carries the information on the vibration amplitude distribution (projected carrier modulation bands) and the vibrating object shape (projected carrier fringes) [3–12,14–18]. In the case of quasi-parallel plate testing one fringe family contains the information on the glass plate thickness variations and the second one carries the information on the sum of shapes of the front and back surfaces. Processing of two separated fringe patterns allows reconstructing the shape of front and rear surfaces with respect to the interferometer reference [26–30].
Since the carrier fringes and their modulation bands are mathematically described by the product of two fringe functions their separation requires satisfying well known rule: the carrier frequency has to be higher than any spectral component of the modulation function. All developed envelope demodulation methods have to satisfy this condition. On the other hand, up to the authors best knowledge, no method for the carrier extraction has been reported. The reason is that the modulating function, e.g., the cosine or the zero order Bessel functions (the latter one describes the modulation amplitude function in case of harmonic vibration testing by time-averaged interferometry) can assume both positive and negative values. In result the carrier pattern phase changes by π where the modulation fringe pattern assumes zero value. These phase jumps have prevented phase extraction from the carrier fringe family.
This paper describes, for the first time, a successful algorithm for phase retrieval from complex fringe patterns with fringe contrast reversals (discontinuous phase jumps by π). The advanced continuous wavelet transform is proposed for this purpose. The method will be presented on the example of processing time-averaged vibration interferograms to derive the shape of the vibrating silicon micromembrane. No static membrane interferogram is required so the system does not have to be stopped for the element shape information acquisition. This condition can be sometimes difficult to satisfy in engineering applications [31]. Additionally, the experimental proof that the average silicon micromembrane surface is the same for the static and vibrating conditions is included.
Because of reasonable paper length and specific character of another important application of the proposed method, i.e., the comprehensive information retrieval from the three-beam interference patterns (quasi-parallel optical plate testing), this topic is left for a separate treatment.
Section 2 presents basics of time averaged interferometry. In section 3, a short theoretical background of 2D CWT is presented followed by the description of the preprocessing path to eliminate the carrier fringe phase jumps, section 4. Section 5 presents the enhanced fringe demodulation algorithm details. In sections 6 and 7 results of demodulation of experimentally acquired time-averaged interferograms and estimation of the demodulation method accuracy are presented. The final section presents conclusions and outlines main merits of the proposed approach.
2. Time averaged interferometry
Time average interferometry consists in acquiring a two-beam interferogram of a vibrating object with acquisition time longer than vibration period. In case of a specular reflection object surface (e.g., silicon microelements) and harmonic excitation, the resulting interferogram is describd by the following formula [5]:
where K(x,y) is the interferogram bias, Cstat (x, y) is the contrast of interference fringes for nonvibrating object, J0 is the zero order Bessel function of the first kind, φvib(x, y) = (2π/λ)OPD(x,y), where OPD(x,y) is the mean optical path difference between the interfering beams, and λis the light wavelength used. Term a0(x, y) represents the vibration amplitude distribution.According to Eq. (1), the element vibration amplitude is encoded in the interferogram modulation map. Therefore, analysis of time average interferograms has been based on determination of the interferogram contrast or modulation [3–18]. Additionally, the phase of carrier fringes represents average shape of the analyzed object, just as in classic two-beam interferometry. As the J0 function has negative and positive values, the carrier fringe phase changes by π as the J0 function assumes zero values. These phase jumps, corresponding to fringe contrast reversals, combined with the carrier fringe contrast decrease, cause major difficulties in the carrier phase extraction. To the best of the authors knowledge, there has been no successful attempt to extract phase from a single time average interferogram without resorting to nonvibrating object interferogram calculations. Till now, for determining the shape of a vibrating object, vibrations had to be stopped and classic two beam interferogram had to be acquired. Robust method for simultaneous shape and vibration amplitude measurement from a single interferogram breaks a new ground for determining both parameters and checking possible mean surface changes, especially in the case of PZT active membranes with excitation electrodes added [5].
Note that references [4–12] concern the studies of MEMS devices showing great potential of the time-average technique in testing MEMS components operating at high resonance frequencies. Fast growing applications of MEMS, including actuators and sensors, introduce pioneering requirements on their design and testing to ensure product quality and reliability.
3. Two dimensional continuous wavelet transform
Two dimensional Continuous Wavelet Transform (2D CWT) has become an important tool in the field of fringe pattern processing and analysis. It was successfully used to analyze fringe pattern phase [32–34] and modulation distributions [9]. 2D CWT has modest requirements concerning fringe shape and spatial frequency, and is known for its excellent abilities to filter out high frequency noise and low frequency background signal. What is most significant for this work, 2D CWT can cope with low contrast fringes and local acquisition errors, which makes it a natural candidate for highly modulated pattern processing and analysis.
Recently, we proposed a simple 2D CWT method for demodulating time averaged interferograms [35]. However the method requires acquisition of at least two phase shifted interferograms. In this paper, utilizing our latest algorithmic developments and the proposed preprocessing path we present simple and elegant solution to the phase demodulation of fringe patterns with contrast reversals, e.g., two beam time-averaged interferograms encountered in silicon element vibration testing. Proposed algorithm enables to avoid errors resulting from high noise levels in low contrast areas. It can be used to determine the carrier fringe phase utilizing single experimentally acquired fringe pattern.
2D CWT of an image I(x) is defined as [36]:
where S2D(s, b,θ) denote the wavelet coefficients, ψ* denotes a complex conjugate of a mother wavelet function, b (b ∈ R2) is the translation parameter, s ∈ R+ is the scale parameter, η is the normalization parameter (ensuring that the phase corresponding to the wavelet coefficient with maximum magnitude provides the local fringe phase), and rθ is the rotation matrix. The 2D CWT coefficients reflect the local similarity between a given signal and the wavelet function rotated, scaled, and translated along the x and y axes.In this paper the two-dimensional modified Mexican hat mother wavelet [37] is utilized:
where k sets the plane wave frequency and direction, σ denotes the wavelet width. This wavelet combines merits of the modified Morlet and the Mexican hat wavelets. It satisfies the admissibility condition for every σ parameter value and gives accurate phase demodulation results in case of fringe patterns with significant phase gradients [37].4. Fringe jumps removal
In order to dispose of carrier fringe phase jumps we take advantage of the fact that the phase jump value is exactly equal to π, i.e., dark fringes turn into bright fringes along the modulation function zeros, and vice versa (in case of time-averaged interferograms, this is true in case of harmonic vibration testing). First, the fringe pattern background is determined using low pass Gaussian filtering. Subsequently, it is subtracted from the original image. Obtained pattern has zero mean value. Finally, absolute value of the pattern is calculated, resulting in a fringe pattern with doubled spatial frequency but without π phase jumps.
By calculating absolute value of the pattern we introduce nonlinearity. Choosing absolute value rather than, for example, squaring as a tool for fringe multiplication, we do not amplify a nonuniform fringe ridge value distribution resulting from inaccurate background calculation. Still, nonlinearity is removed in the 2D CWT processing (the first harmonic is extracted).
By doubling the fringe spatial frequency, a new pattern is constructed that preserves low modulation bands but without phase jumps along them (see Fig. 3(c)). In subsequent 2D CWT processing steps this newly constructed pattern is utilized as an input image.
5. Processing algorithm
The proposed method for demodulation of preprocessed fringes consists of the following stages, Fig. 1. First, 2D CWT of the preprocessed pattern is computed for selected values of wavelet scales and angles. This selection is arbitrary, although the general rule is that the larger number of angle and scale values the more accurate the demodulation is [38]. At the same time, increasing the number of angles and scales increases the computation time. In [38, 39] optimization methods for the wavelet scale and angle selection that can be utilized here, are presented.
Next, the ridge is extracted using the direct maximum algorithm (DM), i.e., for each image pixel, maximum value of the wavelet coefficient is selected [33]. This operation results in a two dimensional complex wavelet coefficient map. During the ridge extraction, values of angles and scales corresponding to selected wavelet coefficients are stored, creating two more maps: one stores information about wavelet angle used and the other about its scale.
In the zero contrast bands of the analyzed image, both angles and scales may be extracted erroneously using DM algorithm, as the fringes disappear there, but the noise remains. These areas have to be masked out as potentially erroneous. The most straightforward method consist in masking out the areas with modulation value below a selected threshold value. Threshold value has to be set for the particular interferogram type so not to confuse the zeros of the modulating Bessel function with other modulation drops resulting, for example, from the experimental setup flaws. Fringe modulation map is computed as an absolute value of the ridge map [9].
In this paper we employ different modulation thresholds for scale and angle maps as they are influenced differently by the fringe modulation drops. Angle map tends to be more fragile to low modulation, i.e., below a certain value the scales are detected correctly while angles are not. Using equal threshold value for both maps leads to higher demodulation errors as some actual scale values are needlessly lost. The threshold values were optimized for the utilized time-averaged interferograms and should be adjusted for different types of fringe patterns.
Subsequent processing step consist in extrapolating angle and scale maps into the masked areas. The aim of this operation is to obtain maps that can be used as coordinates for the successive ridge extraction procedure. Extrapolation is executed utilizing spring-metaphor inpainting method [40]. This technique imagines springs connecting each node with its eight adjacent neighbors and extrapolated value corresponds to the lowest energy state of the set.
In order to further improve the quality of the obtained scale and angle maps and dispose of slight errors, Gaussian smoothing is conducted. The maps are finally rounded to nearest initially selected scale and angle values, so they can be utilized in subsequent processing step as the ridge coordinates.
Final step is another 2D CWT ridge extraction procedure. This time, instead of employing the direct maximum algorithm, wavelet coefficients are indicated by extrapolated scale and angle maps. Hereby, two dimensional complex coefficient map is obtained. This map is free of errors resulting from contrast reversal bands, thanks to the employed extrapolation procedure. The phase of the preprocessed fringe pattern is extracted as arctan(ℑ [ridge]/ℜ [ridge]) [32]. The map is unwrapped using the method proposed in [41], and the simplified phase compensation φc is introduced in order to reduce errors resulting from the finite wavelet size [42]:
where c is a constant, s represents selected wavelet scale and φ is the smoothed fringe phase. Finally, as the input fringes have doubled spatial frequency, the obtained phase has to be divided by 2 to undo fringe multiplication. This completes the proposed demodulation procedure. This way, the phase of the carrier fringes of the input fringe pattern containing contrast reversals is obtained.It should be noted that the extrapolation procedure concerns only 2D CWT scale and angle coordinates, not the fringe pattern itself. While the masked out area width is comparable with the wavelet dimensions, the proposed method does not introduce additional errors to the 2D CWT itself, because 2D CWT demodulation method assumes low phase gradient under the wavelet. If, however, relatively large areas are masked out, still considerable demodulation accuracy can be achieved as slight wavelet scale and angle mismatch do not transfer to high phase demodulation inaccuracy (see for example [38]). The proposed algorithm can generate significant phase demodulation errors in the case of fringe patterns containing large areas of low modulation (larger than the wavelet width) that coincide with significant phase gradients. The proposed procedure combining applying threshold with simple extrapolation is not sufficient in this case.
6. Experiments
The proposed method performance was verified on the example of processing a two-beam time-averaged interferogram of silicone micromembrane vibrating in the resonant mode, frez = 833 kHz, Fig. 2 [5]. The interferogram was acquired in the Twyman-Green setup. Vibrating membrane of 1 mm diameter was not flat and its concavity introduced closed circular carrier fringes.
Fig. 2 Time-averaged interferogram of a vibrating circular silicon micromembrane, frez = 833 kHz [5].
Fig. 3 Preprocessing steps: (a) original interferogram, (b) its background distribution, and (c) absolute value of the original image with its background removed.
In order to clearly visualize consecutive steps of the proposed method, we limit our analysis to one quarter of the membrane area, without loss of generality. To properly demodulate closed carrier fringes of time-average interferograms our method has to be combined with the angle constriction algorithm presented in [43]. This step should be added before the scale and angle map extrapolation step, while ensuring that there are no remote islands in the masked maps. Without this operation the presented method could be used only to demodulate interferograms with fringe angle span no greater than π.
The center part of the membrane contains evident defect which, like the membrane edge, introduces 2D CWT computing errors. To focus closely on the presented processing method and its errors, not the 2D CWT errors, which usually appear in regions near the image corners (especially in the presence of low spatial frequency of carrier fringes), we mask out the regions mentioned above.
In Fig. 3 the consecutive preprocessing step results are presented. Main idea behind calculating the background map using Gaussian filtering instead of subtracting the image mean value, has been to obtain uniform fringe ridge distribution over the image. However, in case of two-beam interferometry, background variations resulting from setup imperfections are rarely significant and steep. Therefore this simple method is sufficient for the presented purpose.
In Fig. 3(c) the image after preprocessing is shown. Its carrier fringe spatial frequency is doubled in comparison with the original pattern and the original contrast decrease regions are preserved. The preprocessed image has no carrier fringe jumps along the zero contrast bands and can be further processed using the 2D CWT. Note that while the original interferogram has sufficient resolution, preprocessed images contain carrier fringes with spatial frequency approaching Nyquist frequency. Although Nyquist frequency has not been reached, dense fringe demodulation is challenging, especially in the case of considerable noise levels.
In Fig. 4 the processing steps of the 2D CWT ridge calculations are presented. Original scale and angle maps, Figs. 4(a) and 4(e), contain expected errors in the regions where the modulation of the input interferogram approaches zero value. The errors are caused by image noise. It should be noted that in the case of experimental setup capable of phase shifting, the noise can be strongly suppressed by subtracting two phase shifted interferograms, what was presented in [35]. However, this approach eliminates the advantages of a single frame method and is not utilized in this paper.
Fig. 4 Scale and angle maps preparation for the second ridge extraction procedure: (a) and (e), scales and angles extracted using the direct maximum algorithm; (b) and (f), modulation map and angle mask; (c) and (g), masked scale and angle maps; (d) and (h), scale and angle maps extrapolated, filtered and rounded.
Figure 4(b) presents modulation map calculated as the absolute value of the ridge map. In the zero contrast regions there are some slight modulation extraction errors encountered. They again are caused by noise present in the image. Wavelet detects noise when no signal is present. However, these errors have no influence on further processing, as modulation is utilized only as a tool for creating a mask for cutting out calculated scale and angle maps. Figure 4(f) presents this mask obtained by applying a threshold of 25% of maximum value to the calculated modulation map. This map is subsequently used to mask out erroneous areas of the angle map. Analogous mask is created for the scale map, but with a 18% threshold value. Masked scale and angle maps are presented in Figs. 4(c) and 4(g), respectively.
Figures 4(d) and 4(h) show scale and angle maps after extrapolation, Gaussian smoothing and rounding to the nearest allowed value (selected during first ridge extraction using DM algorithm). Note the presence of masked area near the membrane edge in Fig. 4(c). It causes, after extrapolation, slightly inaccurate (too small) scale value selection in this region. However, as the extrapolated images are just maps with indices for selecting 2D CWT parameters, these slight errors have not prevented phase extraction, but introduced some inaccuracies (Sec. 7).
In Fig. 5 the obtained phase map is presented along with reconstructed image (calculated as 1 + cosφ, where φ represents the extracted phase map) and the original one for comparison. The method properly removes the carrier fringe contrast reversal so the original interferogram reconstruction was possible. Visible errors are mostly present near the membrane edges, which is a weak point of the 2D CWT. This issue will be addressed in our further work using advanced extrapolation techniques.
Fig. 5 Proposed method carrier demodulation results: (a) original image, (b) extracted phase, (c) reconstructed image.
All the new algorithms introduced in this paper take considerably less time (approx. 1.5 sec) than the computation of the 2D CWT coefficients itself (approx. 45 sec) so the total computation time is similar to the standard 2D CWT processing of the equivalent fringe pattern without contrast reversals. All the computation times provided refer to algorithms containing testing procedures and not optimized for execution time. Calculations were conducted using standard desktop computer with 3GHz processor and the MATLAB environment.
7. Accuracy estimation
7.1. Comparison with the temporal phase shifting method
Till present, there has been no method for the time-averaged interferogram carrier fringe demodulation. Therefore, to assess the method accuracy, five phase shifted interferograms of a nonvibrating membrane were utilized to compute the membrane static profile using five frame temporal phase shifting algorithm (TPS) [5]. The TPS method is widely considered as the most accurate carrier fringe phase demodulation tool. Therefore, it was used as reference for the proposed method accuracy estimation. The error estimation was conducted under the assumption that the mean profile of the vibrating micromembrane is equal to its static profile and that the profile did not change during the time between acquiring interferograms of static and vibrating membrane. Correctness of this assumption is proven in Sec. 7.2.
A map of errors calculated as a difference between the phase calculated by the proposed method and the reference TPS phase is presented in Fig. 6(a). Root mean square (RMS) error for this map is 0.05 rad. (approx. 2.5nm). Main source of the method demodulation error is the fact that the 2D CWT is suitable only for analysis of fringe patterns with small phase gradients [32]. Therefore it misses all the tiny details of the membrane (where detail dimensions are comparable with fringe period). This effect is clearly visible in Fig. 6(a), where our method fails to demodulate correctly narrow circular scratches, situated approximately diagonally across the phase map. Another example of this effect is the short scratch in the top section of the membrane. These effects are visible in the phase map as the TPS method has higher sensitivity to small details.
Fig. 6 Proposed method accuracy evaluation results: (a) error distribution relative to the TPS method, (b) error distribution relative to the standard 2D CWT processing. In the experimental setup described in [5] the error of 1 rad corresponds to OPD of approx. 53 nm (laser diode wavelength λ = 670 nm).
A standard 2D CWT error resulting from the finite wavelet width [42], manifested in phase magnification in the areas of significant phase gradients, is present near the center of the membrane. The applied compensation (see Sec. 5) mitigates this error but does not eliminate it.
Additionally, the phase of the static membrane was computed using the standard 2D CWT processing method [37] and utilized as a reference for errors introduced purely by our new algorithm, not by the 2D CWT itself. The parameters of the 2D CWT reference method were the same as the ones used for conducting our algorithm.
The map presenting the difference between the membrane phase calculated with the proposed method and the standard 2D CWT processing is presented in Fig. 6(b). Highest error values are present in the areas of low contrast of the original image, the rest of the area is demodulated the same way with both methods. Note that the standard 2D CWT method is also not capable of detecting high phase gradients, thus the absence of narrow error bands in Fig. 6(b). It should be noted, that while our method demodulates the fringe pattern with higher spatial frequency, its results can be more accurate than the ones obtained using standard 2D CWT processing, especially near the membrane center, as the fringe curvature is most significant there.
Errors resulting purely from the extrapolation in the ridge extraction procedure are clearly visible near the membrane edges (both Figs. 6(a) and 6(b)). The information base for the extrapolation was most sparse over there. In the center of the membrane, where the inpainted areas were surrounded by properly calculated scale and angle points, extrapolation errors are smaller and obtained results are satisfactory.
Most visible error region, positioned in the center of Figs. 6(a) and 6(b), results from the combination of two mentioned error sources. The area of low contrast coincides there with the stripe of fast varying phase.
7.2. Moiré tests of the reconstructed two-beam interferograms
Another way to show the quality of processing the time-averaged two-beam interferograms by our method is to use the moiré fringe technique [44, 45]. Two approaches have been tested:
- In-registry overlapping the reconstructed pattern (without contrast reversals) with a two-beam interferogram of a static (nonvibrating) membrane. This approach corresponds to the null-field (infinite fringe) detection mode. Its result is not shown here, the uniform moiré intensity distribution has been obtained.
- The so-called mechanical differentiation technique [44, 45]. It bases on the superimposition of slightly laterally displaced two identical fringe patterns. Under the assumption of a small lateral displacement amount and slowly varying function encoded in the overlapped fringe patterns, its first derivative is displayed by generated moiré fringes.
Figure 7 shows the results of multiplicative superimposition of identically laterally displaced: (a) two identical interferograms of a static micromembrane; (b) the static micromembrane interferogram and the interferogram reconstructed using our method, and (c) two identical reconstructed interferograms. Very close shape of moiré fringes in all three cases proves very good performance of the proposed technique. This comparison complements the quantitative one presented above.
Fig. 7 Results of multiplicative superimposition of: (a) two identical interferograms of static micromembrane, (b) static micromembrane interferogram and the interferogram obtained by our method, (c) two indentical interferograms generated by our algorithm as the original interferogram reconstruction. In all cases the two overlapped images were lateraly displaced by 30 pixels, the image dimensions were 350 × 350 pixels.
Finally we would like to present the result of another very simple but instructive moiré experiment. It is well known from the experimental mechanics [46] that excited object vibration amplitude changes occur with respect to its mean static shape surface. This paradigm can be most easily proved by:
- In-registry superimposition of reconstructed pattern with a two-beam interferogram of static micromembrane (as in point 1 mentioned above in this Section);
- In-registry superimposition of static and vibrating micromembrane interferograms.
Below we show the results of the second approach using the subtractive moiré superimposition of static and vibrating micromembrane interferograms, followed by rectification. Its simplified mathematical description is given by the formula
where, for simplicity, intensity distributions of the two interferograms have the forms (1 + cos α) and (1 + J0 cos α); α represents the interferogram phase, and J0 is the zero order Bessel function, as before. The above representation corresponds to the superimposition of two identical fringe patterns, but one of them is amplitude modulated by the function J0. Because of the absolute value calculation we expect double frequency fringes modulated by (1 − J0).Figures 8 (a) and (b) show the results of superimposition of co-phasial and out-of-phase component structures. The first case is represented by the last equation whereas in the second one the carrier modulation function (1 − J0) changes into (1 + J0).
Fig. 8 Results of the rectified subtractive superimposition of co-phasial (a) and out-of-phase (b) interferograms of static and vibrating micromembranes. Double frequency carrier is modulated by functions 1 ∓ J0, respectively. No moiré fringes are generated over the membrane area, which proves the same mean shape of the vibrating and stationary membranes.
Obtained intensity patterns in Fig. 8 agree quite well with presented simplified mathematical description. Over the membrane area no moiré fringes are present so the identity of average shapes of the static and vibrating membranes can be inferred.
8. Conclusions
To the best of our knowledge, we have presented the first method for demodulating fringe patterns with contrast reversals and corroborated our findings by demodulating time-average interferogram of a vibrating silicon micromembrane. The method is automatic and requires only one time-averaged interferogram, and there is no need of performing phase shifting which represents significant simplification of the experimental procedure. Conducted accuracy evaluation shows that the method provides very satisfactory phase retrieval performance.
Possible applications of the proposed method go beyond the time averaged interferogram analysis and quasi-parallel optical element testing. The method can be used to demodulate fringe patterns containing areas of extremely high noise levels or modulation decreases. However, the images should also contain areas with low levels of noise and high contrast as a basis for the extrapolation procedure.
Acknowledgments
The authors thank Adam Styk for calculating modulation maps using the TPS method, used as a reference for our method accuracy estimation.
This work was supported, in parts, by the Dean of the Faculty of Mechatronics, Warsaw University of Technology, statutory activity funds, and the National Science Center, Poland, under the project DEC-2012/07/B/ST7/01392. The work described in [5] was co-supported by EU Project OCMMM (the part of the experiments under this project was performed at Laboratorie dOptique P. M. Duffieux, Université de Franche–Comté, Besancon, France).
References and links
1. D.W. Robinson and G. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics Publishing, 1993).
2. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
3. H. Osterberg, “An interferometer method of studying the vibrations of an oscillating quartz plate,” J. Opt. Soc. Am. 22, 19–35 (1932). [CrossRef]
4. A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry in the MEMS field,” Proc. SPIE 5145, 1–16 (2003). [CrossRef]
5. L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003). [CrossRef]
6. K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45, 085602 (2006). [CrossRef]
7. S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001). [CrossRef]
8. A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46, 4613–4624 (2007). [CrossRef] [PubMed]
9. K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49, 3640–3651 (2010). [CrossRef] [PubMed]
10. M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50, 5513–5523 (2011). [CrossRef] [PubMed]
11. K. Patorski and M. Trusiak, “Highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry using Hilbert transform two-frame processing,” Opt. Express 21, 16863–16881 (2013). [CrossRef] [PubMed]
12. M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Laser Eng., http://www.dx.doi/10/1016/j.optlasereng.2013.06.003 (2013).
13. R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” Proc. SPIE 1162, 456–467 (1990). [CrossRef]
14. J. D. Hovanesian and Y. Hung, “Moiré contour-sum, contour-difference and vibration analysis of arbitrary objects,” Appl. Opt. 10, 2743–2738 (1971). [CrossRef]
15. G. Rosvold, “Video-based vibration analysis using projected fringes,” Appl. Opt. 33, 775–786 (1994). [CrossRef] [PubMed]
16. M. Ragulskis and Z. Navickas, “Interpretation of fringes produced by time-averaged projection moiré,” Strain 45, doi: [CrossRef] (2009), and references therein.
17. P. Chavel and T. C. Strand, “Range measurement using Talbot diffraction imaging of gratings,” Appl. Opt. 23, 862–871 (1984). [CrossRef] [PubMed]
18. G. Xian-Yu Su, von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993). [CrossRef]
19. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996). [CrossRef]
20. J. E. Greivenkamp and A. E. Lowman, “Modulation transfer function measurement of sparse-array sensors using a self-calibrating fringe pattern,” Appl. Opt. 33, 5029–5036 (1994). [CrossRef] [PubMed]
21. J. G. Fujimoto, W. Drexler, U. Morgner, F. Kärtner, and E. Ippen, “Optical coherence tomography: High resolution imaging using echoes of light,” Opt. Photon. News 11, 24–31 (2000). [CrossRef]
22. J. Na, W. J. Choi, E. S. Choi, S. Y. Ryu, and B. H. Lee, “Image restoration method based on Hilbert transform for full-field optical coherence tomography,” Appl. Opt. 47, 459–466 (2008). [CrossRef] [PubMed]
23. H. M. Subhash, “Full-field and single-shot full-field optical coherence tomography: A novel technique for biomedical imaging applications,” Adv. Opt. Technol. 2012, 435408 (2012).
24. M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. 22, 1905–1907 (1997). [CrossRef]
25. T. Tkaczyk, M. Rahman, V. Mack, K. Sokolov, J. Rogers, R. Richards-Kortum, and M. Descour, “High resolution, molecular-specific, reflectance imaging in optically dense tissue phantoms with structured-illumination,” Opt. Express 12, 3745–3758 (2004). [CrossRef] [PubMed]
26. C. Ai and J. C. Wyant, “Effect of spurious reflection on phase shift interferometry,” Appl. Opt. 27, 3039–3045 (1988). [CrossRef] [PubMed]
27. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000). [CrossRef]
28. K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning fizeau interferometer,” Appl. Opt. 43, 1241–1249 (2004). [CrossRef] [PubMed]
29. A. Styk and K. Patorski, “Fizeau interferometer for quasi parallel optical plate testing,” Proc. SPIE 7063, 70630P–70630P–9 (2008). [CrossRef]
30. Z. Sunderland, K. Patorski, and K. Pokorski, “Evaluation of optical parameters of quasi-parallel plates with single-frame interferogram analysis methods,” Photonics Lett. Poland 2, 63–65 (2012).
31. J. D. R. Valera, J. D. C. Jones, and A. F. Doval, “Whole-field vibration phase measurement with electronic speckle pattern interferometry (espi),” Proc. SPIE 2248, 241–248 (1994). [CrossRef]
32. Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006). [CrossRef]
33. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006). [CrossRef] [PubMed]
34. A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120–6126 (2007). [CrossRef] [PubMed]
35. K. Pokorski and K. Patorski, “Comprehensive fringe pattern processing using continuous wavelet transform,” Fringe 2013, doi: [CrossRef] 33 pp. 225–228 (Springer-VerlagBerlin Heidelberg, 2014).
36. J.-P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2008).
37. J. Ma, Z. Wang, M. Vo, and B. Pan, “Wavelet selection in two-dimensional continuous wavelet transform technique for optical fringe pattern analysis,” J. Opt. 14, 065403 (2012). [CrossRef]
38. J. Ma, Z. Wang, M. Vo, and L. Luu, “Parameter discretization in two-dimensional continuous wavelet transform for fast fringe pattern analysis,” Appl. Opt. 50, 6399–6408 (2011). [CrossRef] [PubMed]
39. X. Wang, J. Gao, and W. Chen, “A new tiling scheme for 2-d continuous wavelet transform with different rotation parameters at different scales resulting in a tighter frame,” IEEE Signal Process. Lett. 19, 407–410 (2012). [CrossRef]
40. J. dErrico, “Inpaint nans,” http://www.mathworks.com/matlabcentral/fileexchange/4551 (2004).
41. J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007). [CrossRef] [PubMed]
42. L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010). [CrossRef]
43. J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. 50, 2425–2430 (2011). [CrossRef] [PubMed]
44. P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, 1969).
45. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, 1993).
46. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).
