Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations

Evaluation of amplitude encoded fringe patterns usingthe bidimensional empirical mode decomposition

and the 2D Hilbert transform generalizations

M. Wielgus* and K. PatorskiInstitute of Micromechanics and Photonics, Warsaw University of Technology,

8 Sw. A. Boboli Street, 02-525 Warsaw, Poland

*Corresponding author: [email protected]

Received 12 May 2011; revised 14 July 2011; accepted 14 July 2011;posted 14 July 2011 (Doc. ID 147227); published 30 September 2011

We propose an application for a bidimensional empirical mode decomposition and a Hilbert transformalgorithm (BEMD-HT) in processing amplitude modulated fringe patterns. In numerical studies we in-vestigate the influence of parameters of the algorithm and a fringe pattern under study on the demo-dulation results to optimize the procedure. A spiral phase method and the angle-oriented partial Hilberttransform are introduced to the BEMD-HT and tested. A postprocessing filtration method for BEMD-HTis proposed. Results of processing experimental data, such as vibration mode patterns obtained by time-average interferometry, correspond richly with numerical findings. They compare very well with the re-sults of our previous investigations using the temporal phase-shifting (TPS) method and the continuouswavelet transform (CWT). Not needing to perform phase-shifting represents significant simplification ofthe experimental procedure in comparison with the TPS method. © 2011 Optical Society of AmericaOCIS codes: 120.2650, 120.3180, 120.4120, 120.7280, 100.2650.

1. Introduction

The indisputable attractiveness and success ofoptical testing methods in contemporary scienceand technology result from their noninvasive andcontactless character, adjustable sensitivity range,high measurement accuracy, and relatively easyautomation of themeasurement process. In a group ofso-called full-field measurement techniques withparallel data acquisition and processing for all pointsof the object under test, the output is delivered as afringe pattern, e.g., an interferogram, moirégram,elastooptic fringe pattern, and raster or gratingprojection. The automated fringe pattern analysis(AFPA) makes optical testing methods user friendly,reliable, and highly accurate. Although AFPA hasover 20 years of history [1–4], it is still under dy-namic development dictated by new needs and chal-lenges. On the other hand, several novel information

processing concepts developed in various fields ofphysics, engineering, medicine, etc. are beingimported to the modern fringe pattern processingtechniques.

In the overwhelming majority of fringe patternanalyses the main task is to extract the measurandinformation (some physical quantity) encoded in thepattern phase, i.e., the shape, spacing, and orienta-tion of fringes. Nevertheless, this information may beencoded in contrast or modulation spatial changesover the fringe pattern. The following examplescan be quoted: time-averaged fringe patterns (ob-tained in vibration testing of large and medium scaleengineering objects as well as silicon microelementsof microelectro-mechanical systems (MEMS) andmicro-optoelectro-mechanical systems (MOEMS)systems—see, for example, [5–9] and referencestherein), additive type superposition moiré and thereflection moiré patterns [10–12], photoelastic fringepatternswith simultaneous presence of isopachic andisochromatic fringes [13–15], three-beam interfero-grams [16], and optical vortex fork interferograms

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[17]. The common task in the mentioned applicationsis to find the fringe modulation envelope. For thispurpose several multi- and single-frame (image)AFPA techniqueswere proposed. For example, in caseof time-average interferometry for vibration testing,the applications of temporal phase-shifting (TPS)[7,8,18,19], Fourier-transformation (FT) [20], spatialcarrier phase-shifting (SCPS) [21], and continuouswavelet transformation (CWT) [22] were reported.

In AFPA, the empirical mode decompositionmethod (EMD) [23] was firstly proposed for noise re-duction and variable bias removal to improve phaseevaluation in digital speckle pattern interferometry.Applications include conventional and temporalspeckle interferometry employing 1D EMD with1D HT [24–29] and 1D ensemble empirical modedecomposition (EEMD) with 1D HT [30]. The appli-cations of the 2D variant of EMD, bidimensional em-pirical mode decomposition (BEMD), in AFPA dealwith noise reduction in digital speckle interferome-try [31], fringe pattern normalization [32], and phasemeasurement in temporal speckle interferometry[33]. In [32] the use of the partial Hilbert transform(PHT) was reported; therefore [32,33] are examplesof what we refer to as bidimensional empirical modedecomposition and a Hilbert transform algorithm(BEMD-HT).

In this paper, we indicate the extraction of thefringe pattern modulation distribution as a new ap-plication of the BEMD-HT algorithm. We also showthat the demodulation results can be significantlyimproved by using more sophisticated 2D demodula-tion methods rather than the PHT, e.g., monogenicsignal, spiral phase transform, or an angle-orientedPHT. The algorithm parameters’ values influence isdiscussed and the optimal set of parameters is eval-uated using computer simulations. To compare theprocessing results obtained using different algorithmparameters, the normalized root square error (NRS)is used. Among BEMD parameters, the most impor-tant ones include the extremum condition (EXT),number of IMFs used for the reconstruction, and thesifting process stop condition (standard deviationcondition, SD). The object features which influencethe demodulation quality include the density offringes, noise level, and modulation function fre-quency variations. We also propose BEMD-baseddenoising in postprocessing for reduction of a high-frequency noise component.

Experimentally obtained fringe patterns beingprocessed contain considerable period variationswith several zero value modulation bands. Two-beaminterference fringes with amplitude modulation pro-portional to the zero order Bessel function (siliconactive micromembrane testing using time-averagedinterferometry) and additive-type moiré patternscan serve as the examples. The comparison with pre-vious processing results obtained using TPS andthe 2D CWT method is presented. The BEMD-HTapproach yields results comparable with the onesof the TPS method, especially in case of high density

(spatial frequency) fringe patterns. Being single-frame methods, the results of the BEMD-HT ap-proach are free, however, of parasitic fringes gener-ated by possible TPS experimental errors, such asunequal phase steps and average intensities of thecomponent TPS images.

2. Method Details

A. Original EMD

EMD is a signal processing tool developed in [23].It is adaptive and data-driven. Unlike in the Fourieror wavelet transform methods, no predefined decom-position basis is used; it is rather derived from thesignal itself. EMD is capable of dealing with non-linear and nonstationary data. In the EMD method,an initial signal is decomposed into a series of zero-mean oscillatory functions, intrinsic mode functions(IMFs), and a single residue function rN

sðxÞ ¼ rNðxÞ þXNk¼1

IMFkðxÞ: ð1Þ

The decomposition is obtained in the so-calledsifting process, including the following steps:

1. h11 ¼ s, s—initial signal, i ¼ 1, j ¼ 1;2. identify sets of minima and maxima of hij, if

there are none, save hij as a residue ri and finishthe algorithm;

3. define upper and lower envelopes of hij byinterpolating between corresponding extrema andenvelopes arithmetic mean mij;

4. put H ¼ hij, T ¼ hij −mij;5. if the result of the subtraction meets the IMF

condition, save T as an IMFi, i ¼ iþ 1, j ¼ 1, and goback to (2) with hij ¼ H − T; else j ¼ jþ 1, hij ¼ T andgo back to (2).

Aforementioned IMF condition takes the form of

SD ¼Xx∈Ω

jhi;j−1ðxÞ − hi;jðxÞj2h2i;j−1ðxÞ

< C; ð2Þ

where C is a small positive value, typically 0.25 for1D algorithm. We expect to find high-frequency com-ponents in first IMFs and low-frequency components,such as a backround function, in later IMFs and inthe residue. Therefore, we can perform detrendingoperations by reconstructing signals from only thefirst few IMFs.

B. Ensemble EMD

EEMD is an algorithm introduced in [34] to avoidmodemixing phenomenon. It is an example of a noiseassisted data analysis (NADA). Instead of decompos-ing a single signal, a set of signals is created by add-ing different white noise realization to the initialdata. Each signal is then treated with EMD andcorresponding IMFs are averaged to obtain a result.It is a powerful improvement of the basic EMD

5514 APPLIED OPTICS / Vol. 50, No. 28 / 1 October 2011

algorithm, however, computing EEMD is much moretime consuming, as there are usually several dozenelements in the ensemble.

C. Bidimensional EMD

Several approaches were developed to adapt thetechnique to the analysis of images, two-dimensionalsignals. Single direction EMD (SDEMD) is thesimplest one, applying EMD in just one predefineddirection (in our case row after row) and ignoringthe correlation between subsequent lines. One canaverage SDEMD calculated for rows and columns toimprove the results, which we refer to as averagedEMD, AEMD. Directional EMD (DEMD) calculatesthe decomposition depending on the direction of theimage structure [35]. Finally, the bidimensionalEMD (BEMD) is a fully two-dimensional method,which interpolates envelopes with functions such asbidimensional cubic splines [36,37] or radial basedfunctions (RBFs) [38]. Similarly as in 1D case, it ispossible to employ NADA approach and define thebidimensional ensemble empirical mode decomposi-tion (BEEMD) which was recently applied in [33].

D. Partial Hilbert Transform

Hilbert transform (HT) and the analytic signal arewell-established tools of signal processing in 1D case.However, the extension to a higher dimensional caseis not obvious. For BEMD-HT algorithm the use ofaveraged partial HT was noted. PHT in the xdirection is defined as

sxHðx; yÞ ¼1π PV

ZR

sðu; yÞx − u

du: ð3Þ

PV denotes the Cauchy principal value. ThereforeEq. (3) is just a classic 1D HT, applied to each row ofthe processed signal. By denoting a Fourier trans-form with F and the spectral domain coordinateswith ðζ1; ζ2Þ, we can rewrite Eq. (3) in a spectraldomain as

FfsxHðx; yÞg ¼ −isignðζ1ÞFfsðx; yÞg; ð4Þ

which is essential for the FFT-based implementa-tion. Analytic signal corresponding to the PHT in xdirection is defined as

sxAðx; yÞ ¼ sðx; yÞ þ isxHðx; yÞ: ð5Þ

It is similar for the PHT in y direction. Obviously,single direction HT will not give satisfying resultsif the image structure direction varies from pointto point. Better results can be obtained by averagingabsolute values calculated with transforms appliedin perpendicular directions

jAðx; yÞj ¼ jsxAðx; yÞj þ jsyAðx; yÞj2

: ð6Þ

The effective demodulation using any methodbased on the idea of the HT demands that the datadoes not involve an additive trend function and is ofthe local zero-mean value [23,28]. This is why EMDis used as a preprocessing detrending tool.

3. Our Approach

Proposed scheme for processing amplitude modu-lated fringe pattern consists of the following steps:

1. BEMD detrending by reconstructing theimage from the first several IMFs;

2. HT-based demodulation;3. BEMD filtration in postprocessing by perform-

ing BEMD again on the demodulation result andsubtracting the first few IMFs.

A. Detrending

In this work we use bicubic splines for envelope in-terpolation in the BEMD algorithm. For an imagewith many local extrema this approach results in amuch faster algorithm than the one based on RBFs.We divide the domain into interpolation subsets withthe Delaunay triangulation procedure and interpo-late piecewise using so-obtained triangular grid(see [36,37]). Parameters of this algorithm are theSD condition value, Eq. (2), defaultly set to SD <0:9 and EXT, defaultly set to EXT ¼ 6. The EXT (ex-tremum) parameter is used to define the extremumfor the EMD algorithm purposes: we assume thatthere is a local EMD-minimum in a chosen pixel ifin its closest square neighborhood there are at leastEXT pixels with the value larger than the value forthe chosen pixel. Similarly, the local EMD-maximumcan be defined. As data is effectively interpolatedonly inside a convex hull of an extrema set, thedomain diminishes with the algorithm progress. Toreduce this effect, the initial data is extended by mir-roring the external parts of the image. We focus onthe basic BEMD, but some results obtained withBEEMD are given as well.

B. Demodulation

Below, we describe several attempts to perform am-plitude demodulation of a fringe pattern in 2D. Someof them are our ideas (oriented and local orientedPHT, two-frame oriented PHT) while some were de-veloped previously (monogenic signal, spiral phase).Nevertheless, none of these was combined withBEMD or applied to the analysis of a fringe patternamplitude distribution.

Most of the basic attempts to define a two-dimensional analogue of the analytic signal give poordemodulation results, as they locally prefer certaindirections over the different ones (anisotropy). Thisis also the case with the PHT method describedabove. For example, for pure vertical fringe structureit combines meaningful row HT with a meaninglesscolumn HT, leading to a large error in such a region.One of the ways to deal with this problem is to aver-age perpendicular partial HTs with respect to the


local orientation angle. We propose using theweighted average of the partial HTs, Eq. (7):

jAðx; yÞj ¼ jsxAðx; yÞj cos2 β þ jsyAðx; yÞj sin2 β: ð7Þ

We refer to this approach as the oriented partialHilbert transform (OPHT). β ¼ βðx; yÞ denotes the lo-cal fringe orientation angle. It is determined with therobust algorithm proposed in [39] which gives veryaccurate results outside the regions with modulationvalues close to zero. Note that, for horizontal fringesit is the pure column PHT, for vertical fringes thepure row PHT, and for orientation angle �π=4 theaforementioned averaged PHT is encountered. Possi-bly, even better results could be obtained if the anglewas determined for the unmodulated image (whichcan be recorded in metrological techniques such astime-average interferometry with a nonvibratingobject). The negative effect of modulation influenceon the fringe orientation angle determination qualitycan be avoided in this way. We denote this two-frameapproach as OPHT 2F.

Second idea is to use locally angle-adaptive HT.The so-called partial HT with respect to the angleβ was defined in the spectral domain in [40] as

FfsHðx;yÞg¼−isignðζ1 cosβþζ2 sinβÞFfsðx;yÞg: ð8ÞWe use this method locally with an angle varying

from point to point, a square spectral mask for HTcalculation is applied. This approach is referred toas the local oriented PHT (LOPHT).

An idea different than using a calculated angle or-ientation map is to design an isotropic 2D quadratictransform. Possible solution, named monogenic sig-nal, was proposed in [41]. The method is in fact basedon the Riesz transform, being, however, often consid-ered to be a multidimensional generalization of theHT. So-called monogenic signal (the 2D analogueof the analytic signal) corresponding to the signalsðx; yÞ is defined as

sMðx; yÞ ¼ sðx; yÞ þ iR1fsðx; yÞg þ jR2fsðx; yÞg; ð9Þ

where i and j denote the first and the second imagin-ary units of a quaternion algebra and R1 and R2 aredefined in the spectral domain as

FfRσfsðx; yÞgg ¼ −iζσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ζ21 þ ζ22q Ffsðx; yÞg; ð10Þ

for σ ∈ f1; 2g. Amplitude is calculated as a quater-nion modulus

jAðx;yÞj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2ðx;yÞþR2

1fsðx;yÞgþR22fsðx;yÞg

q: ð11Þ

Sign on the right-hand side of Eq. (10) differs fromthe original definition in [41]; we have decided to use

the minus sign for a correspondence with the defini-tion of HT, given previously.

Another method of this type was developed in [42],the spiral phase method [also referred to as a vortextransform (VT)]. They define the spiral phasefunction in the spectral domain

Pðζ1; ζ2Þ ¼ζ1 þ iζ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiζ21 þ ζ22

q ; ð12Þ

and the 2D HT analogue as

sHðx; yÞ ¼ −i expð−iβÞF−1fPðζ1; ζ2ÞFfsðx; yÞgg: ð13ÞNote that, for amplitude demodulation the orienta-

tion angle β, which is formally present in Eq. (13),does not need to be determined, as

jAðx;yÞj¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2ðx;yÞþjF−1fPðζ1;ζ2ÞFfsðx;yÞggj2

q: ð14Þ

We denote the spiral phase method with HS(Hilbert spiral). Spiral phase and monogenic signalmethods differ a lot as the former gives a complexsignal and the signal obtained with the second oneis quaternionic. Nevertheless, it takes rather ele-mentary calculations to show that the real nonnega-tive amplitudes calculated with Eqs. (14) and (11) areexactly the same. In numerical tests, therefore, weonly give the HS results. Although applications ofboth methods to the fringe pattern phase analysiswere discussed [43,44], no efficient preprocessingthat would enable satisfactory processing of metrolo-gical data was proposed. Additionally, applications inthe fringe pattern amplitude modulation evaluationwere not indicated. These applications of HT arelimited by the conditions under which HT generatesthe exact quadrature component for the signal of theform aðxÞ cos½ϕðxÞ�, i.e., aðxÞ sin½ϕðxÞ�. These limita-tions are mostly a consequence of a Bedrosian theo-rem and were widely treated, e.g., in [45] and morerecently in a multidimensional case in [46]. In case ofthe monogenic signal and the spiral phase method,their quadrature properties can be derived with thestationary phase method, see the discussion in [47].General clue is that all these methods are usefulwhen the carrier frequency is higher than any spec-tral component of the modulation function. This con-dition is sufficiently satisfied by the analyzed data.

C. Denoising in Postprocessing

Effective denoising of a fringe pattern with EMD/BEMD can be usually performed by subtracting thefirst IMFs, containing a high frequency noise compo-nent [31]. However, for the analyzed type of data,even the first IMF contains a large part of an impor-tant signal component, i.e., the modulated carrierand therefore can not be subtracted. This happensbecause of a relatively high carrier frequency andmode mixing between the noise component and the


modulated carrier. In our approach, modulation dis-tribution is firstly obtained with BEMD-HT withoutany filtration. Because of the presence of unremovednoise, the spoiled modulation distribution results.It is then treated as an input data for the BEMD de-noising algorithm. Such an image is decomposed intoIMFs and the first few of them are subtracted fromthe data. This approach ensures the reduction of ahigh-frequency noise component.

4. Numerical Experiments

The quality of the results is evaluated using the NRS

NRS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

ðx;yÞ∈Ω½vðx; yÞ − v̂ðx; yÞ�2Pðx;yÞ∈Ω v2ðx; yÞ

s; ð15Þ

where vðx; yÞ is a known exact value and v̂ðx; yÞ is avalue calculated with the algorithm. The data fornumerical studies was chosen to resemble a time-average interferogram of a vibrating object (e.g.,silicon micromembrane) and corresponds to its inten-sity distribution description (see [7,8])

IvðrÞ¼KðrÞ�1þVstðrÞJ0

�4πλ a0ðrÞ

�cosφvðrÞ

�; ð16Þ

where J0 denotes Bessel function of the first kindand zeroth order. Modulation of a carrier cosφvðrÞencodes vibration amplitude distribution a0ðrÞ. KðrÞrepresents the background function, which needs tobe removed with a BEMD. VstðrÞ denotes the visibi-lity (contrast) distribution in the static object two-beam interference pattern. The bold font is used todenote the position vector r ¼ ðx; yÞ. In contraryr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p. The data can be written in the form

IðrÞ ¼ BðrÞ þ CðrÞMðrÞ þGN: ð17Þ

The domain is r ∈ ½−5; 5� × ½−5; 5�, resolution 500×500, the nontrivial background function BðrÞ ¼ 3þexp½−ðxþ 3Þ2=25 − ðy − 4Þ2=45� þ exp½−ðx − 5Þ2=25�,the radial carrier

CðrÞ ¼ cosð30rÞ; ð18Þand the modulation

MðrÞ ¼ J0

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9x2 þ y2

q �; ð19Þ

GN stands for an additive Gaussian noise. In thiscase the noise standard deviation parameter wasσ ¼ 0:03. The artificial interferogram used for thenumerical experiments is presented in Fig. 1(a). InFigs. 1(b)–1(d), the first three IMFs extracted from(a) with EXT ¼ 6 and SD < 0:9 are shown. Note thatthe carrier component is visible in each of the IMFs,mostly in the second one. Figure 1(e) presents theabsolute value of the modulation used in the simula-tions, Eq. (19).

A. The Comparison of Detrending Algorithms

The detrending NRS errors for different EMD meth-ods and signal reconstruction from the first n IMFsare presented in Fig. 2. The BEEMD ensemble sizewas 100 elements with Gaussian white-noise withthe standard deviation σ ¼ 0:03 added. As BEMDand BEEMD clearly outperform directional methods,it is justified to use only a bidimensional approach infurther tests.

B. BEMD Detrending Parameters

We compare detrending accuracy of a BEMD algo-rithm for different values of EXTand SD parameters,signal reconstructed from first n IMFs. There are twopossibilities leading to good results—a strong SDcondition (small SD parameter value) and strong ex-tremum requirement (EXT ¼ 8), see Fig. 3, or a weak

Fig. 1. Top: (a) synthetic interferogram; (b), (c), (d) first three IMFs. Bottom: (e) modulation function modulus, (f) results of thedetrending-demodulation procedure using PHT, (g) OPHT, and (h) HS algorithms.


SD and EXT condition, as in Fig. 4. It is also possibleto use the algorithm with no SD condition at all(SD < ∞), but then the interpolation quality breaksafter a few IMFs. Monotonic and decreasing errorvalue suggest that there is no mode mixing betweenthe modulated carrier and the background function.

C. 2D HT Demodulation Efficiency

The results of demodulation for four different algo-rithms based on the HT generalization are comparedin Table 1. NRS is evaluated according to Eq. (15)with vðx; yÞ being a modulation function modulusjMðx; yÞj. The data used was just a modulated carrierwith no background function or additive noise.The carrier from Eq. (18) was used, with modula-tions M1 from Eq. (19), M2ðx; yÞ ¼ xJ0ð2rÞ andM3ðx; yÞ ¼ expð−r2=40Þ.

In Figs. 1(f)–1(h) the results of demodulation ofBEMD-detrended synthetic interferogram (a) areshown, comparing three aforementioned approaches.BEMD parameters used were: first 10 IMFs,EXT ¼ 6, SD < 0:9. The results show that the spiralphase method (or the monogenic signal method)gives a much better demodulation accuracy thanPHT. Also OPHT and its two-frame variant give theNRS error much smaller than PHT, in certain casesalmost as small as the spiral phase method. TheLOPHT method was calculated locally for the neigh-borhood size 25 × 25 pixels. Its performance wassimilar to the PHT, while it was much more time con-suming than any other method. Therefore we did notuse the LOPHT in further experiments.

D. BEMD Denoising in Postprocessing

The results of formerly suggested postprocessing de-noising are shown in Fig. 5. Clearly, results accuracycan be improved by using BEMD as a down-passfilter and subtraction of the first IMFs. Note thatthis is an opposite BEMD application than detrend-ing in which we reconstructed the signal from thefirst IMFs (equivalent to subtracting the trend andthe last IMFs). Even more accurate results can be

1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

n

NR

SSDEMDAEMDBEMDBEEMD

Fig. 2. (Color online) Detrending efficiency comparison of fourEMD variants.

2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

n

NR

S

SD < ∞SD < 0.9SD < 0.6SD < 0.3

Fig. 3. (Color online) BEMD detrending NRS error comparisonfor different SD values, EXT ¼ 8.

2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

n

NR

S

SD < ∞SD < 0.9SD < 0.6SD < 0.3

Fig. 4. (Color online) BEMD detrending NRS error comparisonfor different SD values, EXT ¼ 6.

0 1 2 3 4 5 6 7 8 9 100

0,1

0,2

0,3

0,028

n

NR

S

σ = 0.02σ = 0.04σ = 0.06σ = 0.08no noise

Fig. 5. (Color online) BEMD denoising NRS error as a function ofthe number n of subtracted IMFs.

Table 1. Demodulation NRS Error Values

Algorithm PHT LOPHT OPHT OPHT 2F HS

NRS M1 0.1195 0.0955 0.0647 0.0604 0.0231NRS M2 0.0763 0.0780 0.0493 0.0491 0.0465NRS M3 0.0929 0.0883 0.0384 0.0384 0.0189


possibly obtained by combining EMD-HT processingalgorithm with different filtration preprocessing(e.g., applying wavelet filtration, diffusion PDE fil-tration, or statistical filters). This issue, however,will not be discussed further in this paper.

E. Carrier Frequency Influence

For noiseless data detrended by the reconstruc-tion from the first 15 IMFs, the NRS detrending-demodulation error is compared for PHT, OPHT, andHS algorithms, see Fig. 6. The two-frame variant ofOPHT gives, for this data, the results very similar tothe basic OPHT. Tested carriers were in the form ofmodified CðrÞ of Eq. (18), CωðrÞ ¼ cosωr.F. Modulation Function Frequency Influence

Detrending-demodulation algorithm was tested fordifferent modulation functions MðrÞ, Eq. (19), fre-quencies ξ, MξðrÞ ¼ MðξrÞ. Figure 7 shows that forhigh modulation function frequencies, the 2F OPHTmethod gives much better results than standardOPHT. This is because of the fact that for high mod-ulation frequencies the fringe orientation determina-tion accuracy deteriorates and an additional error is

generated. This effect is avoided if the fringe orienta-tion is determined with an unmodulated image, as inthe 2F OPHT algorithm. For very high modulationfrequencies, the 2F OPHT algorithm can competewith the HS algorithm.

5. Processing the Metrological Data

The BEMD-HT approach was tested using severalreal fringe patterns such as vibrating silicon micro-membrane time-average interferograms. Figure 8(b)presents an exemplary amplitude modulated pattern(vibrating membrane), while Fig. 8(a) presents anunmodulated pattern of a static membrane (to beused with the 2F OPHT algorithm). Similarly, as forsynthetic data, in the case of the interferogram pre-sented in Fig. 8(b) a large part of the carrier is foundin the first IMF, Fig. 8(c). The second IMF (d) con-tains a carrier as well. The first IMF does not sepa-rate the noise from the carrier.

A comparison between HT variants gave results si-milar to those obtained for artificial data. In Fig. 9,the difference between HS and PHT demodulationcan be observed. In particular, parasitic fringes canbe observed along vertical and horizontal axes of thePHT result, Fig. 9(b), similarly to Fig. 1(f). Figure 9(c)shows the relative difference between Figs. 9(a) and9(b) in the marked region (to the right of the mem-brane center along the horizontal axis). It reaches50% of a local intensity value. This oscillatory erroris generated by the PHT because of averaging per-formed in two directions, while PHT calculated par-allelly to the fringe structure is meaningless. Thereis no such problem in demodulation with the HSmethod. OPHT and 2F OPHT results were visuallyindistinguishable from those obtained with HS.

If there was no mode mixing in the BEMDmethod,we would expect to obtain a high-frequency noise

10 15 20 25 30 35 400

0,1

0,2

0,3

ω

NR

SPHTOPHTHS

Fig. 6. (Color online) Demodulation accuracy comparison fordifferent carrier frequencies ω and different demodulationalgorithms.

1 2 3 4 5 60

0,1

0,2

ξ

NR

S

PHTOPHT2F OPHTHS

Fig. 7. (Color online) Demodulation accuracy comparison fordifferent modulation frequencies and different demodulationalgorithms.

Fig. 8. Time-average interferograms of a (a) nonvibrating and(b) vibrating circular silicon micromembrane (resonance frequency833kHz). The first two IMFs shown in (c) and (d) were extractedfrom (b).


component in the first IMF, then the whole modu-lated carrier in the second IMF and the backgroundin the following IMFs. Nevertheless, this does nothappen even for the synthetic data, where we ob-tained a monotonically decreasing detrending errorfor reconstructions from 1 to 20 first IMFs (Figs. 3and 4), indicating that all these IMFs represent thecarrier component. This was not the same for the realdata. In Fig. 10 the results of HS demodulation froma single first IMF (a) and the first 15 IMFs (b) arecompared. In the first case, not the whole domainis represented as the image is incomplete in the cen-ter and close to the borders. This is because of an in-sufficient number of IMFs used for reconstruction.With 15 IMFs a delicate pattern of a carrier appearsin the demodulation result, indicating that alreadytoo many IMFs were used and a part of a backgroundappeared in the reconstructed signal because ofmode mixing between the carrier and the back-ground component. To obtain best demodulation

results one should balance between these oppositeeffects and choose the optimum number of IMFs usedfor the reconstruction. We tested an approach todetermine the optimum number of IMFs based ondetermination of the dominant carrier frequency by

finding a value ωD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiζ21 þ ζ22

qfor which the spectral

power density has a maximum and thresholding therelative value of this frequency in a spectrum of sub-sequent IMFs. For example, we used nth IMF for thereconstruction if at least 50% of its spectral power

was inside a ring ð1þ δÞωD ≥

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiζ21 þ ζ22

q≥ ð1 − δÞωD.

Another method is to estimate the size of a regionwith reconstructed carrier and stop adding IMFswhen it is acceptably large in comparison with awhole picture. For the real data, these methods sug-gest processing a small number of IMFs, between 2and 5.

Figures 10(c) and 10(d) demonstrate the postpro-cessing BEMD filtration of Fig. 10(b). In this case,BEMD acts like a down-pass filter by removing thefirst few IMFs from analyzed signal. This may en-hance the image, however subtracting too manyIMFs results in a blurred image, see Fig. 10(d). Astop criterion for the EMD filtration was presentedin [48]. It is easilly adaptable to this application.

Figure 11 presents the results of applying theBEMD-HT algorithm to the additive moiré patternsobtained by real-time optical shearing of in-planedisplacement fringes in the moiré interferometrytechnique [49] (implementation of the so-called me-chanical differentiation or the moiré of the moirémethod). This shows that despite being optimized fora time-average interferograms processing, the algo-rithm works fine with a different data, in particularof nonradial carrier.

Fig. 10. Top: demodulation results for resonance frequency833kHz obtained with (a) one IMFand (b) 15 IMFs. Bottom: resultof postprocessing filtration by subtracting (c) two IMFs and (d) fiveIMFs from (b).

Fig. 11. (a) and (b) Exemplary additive moiré patterns and (c)and (d) the corresponding BEMD-HT demodulation results.

(a) (b)

(c)0.10.20.30.4

Fig. 9. Demodulation result for resonance frequency 724kHzobtained with (a) HS and (b) PHT algorithms. Relative differencebetween (a) and (b) in the marked region is displayed in (c).


The results of demodulation with the proposedBEMD-HT algorithm are compared with those ob-tained with TPS and CWT algorithms in Figs. 12and 13. For TPS demodulation we used a phase stepequal to π=2 and performed calculations with a 5Lalgorithm (see [18,50]). Obtained modulation mapslook similar. The CWT method filters noise betterthan BEMD-HT, but performs less effectively thanBEMD-HT in the regions close to the membraneedge, which is because of a modulation “leakage”effect, see [22].

In Fig. 14 the demodulation results for horizontalaxial cross section of Figs. 13(a) and 13(c) are pre-sented (row number 288 from the data of size768 × 576). Postprocessing BEMD filtration was usedfor this example (2 IMFs subtracted). Clearly,BEMD-HT gives similar modulation values as theTPS algorithm. The largest difference between thesetwo is present in the central region of the image witha high fringe curvature.

6. Conclusions

An approach to amplitude demodulation of a fringepattern based on the bidimensional mode decom-position and 2D demodulation algorithms was pro-posed. By using such techniques asmonogenic signal,spiral phase, or oriented PHT we managed to obtaindemodulation accuracy much higher than with apreviously used PHT method. We also proposed aBEMD-based denoising in postprocessing, which isadequate for demodulation of patterns with high car-rier frequency. Parameters of the BEMD algorithmand demodulation method variants were discussedand optimized. The comparison of our method withTPS and CWT shows that the proposed BEMD-HTalgorithm provides the demodulation quality similarto the ones of aforementioned methods. Unlike TPS,BEMD-HT demodulation can be performed with asingle recording. Possibly less complicated setup andno need of performing phase-shifting favors BEMD-HT over TPS for AFPA applications. The presentedmethod can also be directly applied to the fringepattern normalization problem.

All the implementations and simulations pre-sented in this paper were conducted in the MATLAB/Octave environment. For better time performance,demodulation algorithms were implemented withFFT (or 2D FFT), in the spectral domain. Interpola-tion was based on the fast Quickhull algorithm devel-oped in [36]. Nevertheless, triangulation and splineinterpolation are the most expensive parts of the al-gorithm. For a medium class PC it lasts 5–10 secondsto extract a single IMF from analyzed size of data.The procedure of extrema selection and any FFT-based demodulation (PHT, HS, OPHT) take muchless than a single second in every analyzed case.There is no need to perform a full decomposition forthis application as we always need just a first few

Fig. 12. (a) Demodulation results for resonance frequency 256kHz, obtained using the TPS, (b) CWT [22], and (c) BEMD-HT methods.

Fig. 13. Demodulation results for resonance frequency 172kHz obtained using the (a) TPS, (b) CWT [22], and (c) BEMD-HT.

100 200 300 400 500 6000

20

40

60TPSBEMD−HT

Fig. 14. (Color online) Comparison of TPS and BEMD-HT resultsfor a single row from experimental data.


IMFs. The time performance is therefore satisfactoryfor a BEMD-HTand enables performing full detrend-ing-demodulation-denoising procedure in less than aminute. Despite of better results, Fig. 2, the applica-tions of BEEMD are limited by its time performance,as in our current implementation decomposing ahundred images lasts simply hundred times longerthan decomposing a single signal.

The experimental work described in [8], fromwhich some data were taken, was cosupported bythe European Union (EU) Project OCMMM (a partof the experiments under this project was performedat Laboratoire d’Optique P. M. Duffieux, Universitéde Franche-Comté, Besançon, France). This workwas supported, in part, by the Ministry of Scienceand Higher Education, grant N N505 464 238, andstatutory activity funds.

References

1. J. Schwider, “Advanced evaluation techniques in inter-ferometry,” in Progress in Optics, E. Wolf ed. (Elsevier, 1990),Vol. 28, pp. 271–359.

2. J. E. Greivenkamp and J. H. Brunning, “Phase shifting inter-ferometry,” in Optical Shop Testing, D. Malacara ed. (Wiley,1992), pp. 501–598.

3. D. Robinson and G. Read, eds., Interferogram Analysis: DigitalFringe Pattern Measurement (Institute of Physics, 1993).

4. D. Malacara, M. Servin, and Z. Malacara, InterferogramAnalysis for Optical Testing (Marcel Dekker, 1998).

5. H. Osterberg, “An interferometer method of studying thevibrations of an oscillating quartz plate,” J. Opt. Soc. Am.22, 19–35 (1932).

6. G. Rosvold, “Video-based vibration analysis using projectedfringes,” Appl. Opt. 33, 775–786 (1994).

7. A. Bosseboeuf and S. Petitgrand, “Application of microscopicinterferometry in the MEMS field,” Proc. SPIE 5145, 1–16(2003).

8. L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki,A. Jacobelli, and T. Dean, “Active micro-elements testing byinterferometry using time-average and quasi-stroboscopictechniques,” Proc. SPIE 5145, 23–32 (2003).

9. M. Ragulskis, R. Maskeliunas, and V. Turla, “Investigation ofdynamic displacements of lithographic press rubber roller bytime average geometric moiré,” Opt. Lasers Eng. 43, 951–962(2005).

10. J. D. Hovanesian and Y. Hung, “Moiré contour-sum, contourdifference and vibration analysis of arbitrary objects,” Appl.Opt. 10, 2734–2738 (1971).

11. O. Bryngdahl, “Characteristics of superposed patterns inoptics,” J. Opt. Soc. Am. 66, 87–94 (1976).

12. K. Patorski, Handbook of the Moiré Fringe Technique(Elsevier, 1993).

13. M. Nishida and H. Saito, “A new interferometric method oftwo-dimensional stress analysis,” Exp. Mech. 4, 366–376(1964).

14. R. J. Sanford and A. J. Durelli, “Interpretation of fringes instress-holo-interferometry,” Exp. Mech. 11, 161–166 (1971).

15. B. Chatelain, “Holographic photo-elasticity: independentobservation of the isochromatic and isopachic fringes for asingle model subjected to only one process,” Opt. LaserTechnol. 5, 201–204 (1973).

16. A. Styk and K. Patorski, “Fizeau interferometer for quasi par-allel optical plate testing,” Proc. SPIE 7063, 70630P (2008).

17. K. Patorski and K. Pokorski, “Examination of singular scalarfields using wavelet processing of fork fringes,” Appl. Opt. 50,773–781 (2011).

18. K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting methodcontrast calculations in time-averaged interferometry: erroranalysis,” Opt. Eng. 44, 065601 (2005).

19. K. Patorski and A. Styk, “Interferogram intensity modulationcalculations using temporal phase shifting: error analysis,”Opt. Eng. 45, 085602 (2006).

20. S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie,“Quantitative time-averaged microscopic interferometry formicromechanical device vibration mode characterization,”Proc. SPIE 4400, 51–60 (2001).

21. A. Styk and K. Patorski, “Analysis of systematic errors inspatial carrier phase shifting applied to interferogram inten-sity modulation determination,” Appl. Opt. 46, 4613–4624(2007).

22. K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using thecontinuous wavelet transform,” Appl. Opt. 49, 3640–3651(2010).

23. N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih,Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empiricalmode decomposition and the Hilbert spectrum for non-linearand non-stationary time series analysis,” Proc. R. Soc. A 454,903–995 (1998).

24. M. B. Bernini, G. E. Galizzi, A. Federico, and G. H. Kaufmann,“Evaluation of the 1D empirical mode decomposition methodto smooth digital speckle pattern interferometry fringes,”Opt.Commun. 45, 723–729 (2007).

25. F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann,“Phase measurement improvement in temporal speckle pat-tern interferometry using empirical mode decomposition,”Opt. Commun. 275, 38–41 (2007).

26. F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann,“Hilbert transform analysis of a time series of speckle inter-ferograms with a temporal carrier,” Appl. Opt. 47, 1310–1316(2008).

27. A. Federico and G. H. Kaufmann, “Phase recovery in tem-poral speckle pattern interferometry using the generalizedS-transform,” Opt. Lett. 33, 866–868 (2008).

28. S. Equis and P. Jacquot, “Phase extraction in dynamicspeckle interferometry with empirical mode decompositionand Hilbert transform,” Strain 46, 550–558 (2010).

29. S. Equis and P. Jacquot, “The empirical mode decomposition: amust have tool in speckle interferometry?” Opt. Express 17,611–623 (2009).

30. X. Zhou, H. Zhao, and T. Jiang, “Adaptive analysis of opticalfringe patterns using ensemble empirical mode decompositionalgorithm,” Opt. Lett. 34, 2033–2035 (2009).

31. M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noisereduction in digital speckle pattern interferometry usingbidimensional empirical mode decomposition,” Appl. Opt.47, 2592–2598 (2008).

32. M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normaliza-tion of fringe patterns using the bidimensional empiricalmode decomposition and the Hilbert transform,” Appl. Opt.48, 6862–6869 (2009).

33. M. Bernini, A. Federico, and G. Kaufmann, “Phase measure-ment in temporal speckle pattern interferometry signalspresenting low modulated regions by means of the bidimen-sional empirical mode decomposition,” Appl. Opt. 50, 641–647(2011).

34. Z. Wu and N. Huang, “Ensemble empirical mode de-composition: a noise assisted data analysis method,” Tech.Rep. No. 193, Centre for Ocean-Land-Atmosphere Studies(2005).


35. Z. Liu, H. Wang, and S. Peng, “Texture classification throughempirical mode decomposition,” in Proceedings of the 17th In-ternational Conference on Pattern Recognition (ICPR 2004)(IEEE, 2004), pp. 803–806.

36. C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm forbidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704(2005).

37. C. Barber, D. Dobkin, and H. Huhdanpaa, “The Quickhullalgorithm for convex hulls,” ACM Trans. Math. Softw. 22,469–483 (1996).

38. J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and .Ph.Bunel, “Image analysis by bidimensional empirical mode de-composition,” Image Vision Comput. 21, 1019–1026 (2003).

39. X. Yang, Q. Yu, and S. Fu, “A combined method for obtainingfringe orientations of ESPI,” Opt. Commun. 273, 60–66(2007).

40. T. Bülow and G. Sommer, “Hypercomplex signals—a novel ex-tension of the analytic signal to the multidimensional case,”IEEE Trans. Signal Process. 49, 2844–2852 (2001).

41. M. Felsberg and G. Sommer, “The monogenic signal,” IEEETrans. Signal Process. 49, 3136–3144 (2001).

42. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Naturaldemodulation of two-dimensional fringe patterns. I. Generalbackground of the spiral phase quadrature transform,”J. Opt. Soc. Am. 18, 1862–1870 (2001).

43. R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processingof interferogram using a two-dimensional discrete Hilbert

transform,” in Fringe 2005, the 5th International Workshopon Automatic Processing of Fringe Patterns, W. Osten ed.(Springer, 2006), pp. 82–89.

44. T. Bülow, D. Pallek, and G. Sommer, “Riesz transformsfor the isotropic estimation of the local phase of moiréinterferograms,” presented at the 22nd DAGM SymposiumMustererkennung, (Heidelberg, 2000).

45. B. Boashash, “Estimating and interpreting the instant-aneous frequency of a signal,” Proc. IEEE 80, 520–568(1992).

46. M. Venouziou and H. Zhang, “Characterizing the Hilberttransform by the Bedrosian theorem,” J. Math. Anal. Appl.338, 1477–1481 (2008).

47. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demo-dulation of two-dimensional fringe patterns. II. Stationaryphase analysis of the spiral phase quadrature transform,”J. Opt. Soc. Am. 18, 1871–18810 (2001).

48. Q. He, R. Gao, and P. Freedson, “Midpoint-based empiricaldecomposition for nonlinear trend estimation,” in Proceedingsof IEEE Engineering in Medicine and Biology Society Confer-ence 2009 (IEEE, 2009), pp. 2228–2231.

49. K. Patorski, D. Post, R. Czarnek, and Y. Guo, “Real-timeoptical differentiation for moire interferometry,” Appl. Opt.26, 1977–1982 (1987).

50. K. G. Larkin, “Efficient nonlinear algorithm for envelopedetection in white light interferometry,” J. Opt. Soc. Am. A13, 832–843 (1996).


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Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations