## Abstract

Unified interpretation for the real and pseudo moiré phenomena using the concept of biased and unbiased frequency pairs in the Fourier spectrum is given. Intensity modulations are responsible for pseudo moiré appearance in the image plane rather than average intensity variations dominating real moiré. Detection of pseudo moiré necessitates resolving superimposed structures in the image plane. In the case of the product type superimposition generating both real and pseudo moiré, our interpretation utilizes the Fourier domain information only. The moiré pattern characteristics such as an effective carrier, modulation and bias intensity distributions can be readily predicted. We corroborate them using two-dimensional continuous wavelet transform and fast adaptive bidimensional empirical mode decomposition methods as complementary image processing tools.

©2011 Optical Society of America

## 1. Introduction

The moiré fringe technique is a well established tool for optical metrology [1, 2]. It consists in comparing a quasi-periodic structure with deformed lines (information carrying structure) with the reference one. Depending on the illumination used, one can distinguish two modes of superimposition. In the first mode, overlapping structures are incoherently illuminated or else superimposition is by photographic or video techniques. In the second mode coherent moiré fringes are formed by illuminating the structures coherently and filtering out double diffraction orders or by overlapping real time interference fringes with a recorded pattern.

In this study we deal with the incoherently superposed structures. Three types of incoherent superposition can be distinguished: additive, subtractive and multiplicative [2, 3]. The optical realization and appearance of the resulting moiré pattern differ for the three types of superposition. In majority of applications two structures are usually superimposed. Multiplicative and subtractive superimpositions yield directly visible low spatial frequency beat pattern fringes and are most frequently used. Direct appearance of moiré fringes in the multiplicative and subtractive cases can be readily explained by their simple mathematical description – the lowest spatial frequency beat term (moiré) appears independently in the formula describing the resulting intensity distribution [2, 3]. Component structures do not have to be resolved in the image. On the other hand, linear additive superimposition results in moiré fringes in the form of the amplitude modulation of a carrier. The beat pattern (the fringe pattern macrostructure) does not occur in isolation and can be observed only when the microstructure (carrier pattern) is resolved [2, 3]. Correspondingly, higher resolution imaging system is required in this case as compared with product moiré imaging. This carrier type modulation moiré of this type is sometimes called pseudo moiré [4].

Moiré fringes are most commonly observed when spatial periods of multiplicatively superimposed amplitude structures are the same or very close and their lines are parallel or almost parallel. Such fringes are sometimes called real moiré patterns. Average intensity variations in the image are responsible for the moiré phenomenon in this case. On the other hand the so-called pseudo moiré fringes [4, 5] can be observed as detectable modulations in the image in two cases: a) conventional additive superimposition of amplitude structures of quasi-equal period, and b) multiplicative (product) superimposition of two structures with spatial frequency ratio 2:1 and slightly mutually inclined lines.

Our goal is to propose the moiré phenomenon interpretation based on the Fourier theory [2–9] and applicable to both real and pseudo moiré patterns. We indicate the frequency pairs present in the pseudo moiré spectrum responsible for detectable intensity modulations. Our concept of biased (real moiré) and unbiased (pseudo moiré) frequency pairs, generating the lowest beat frequencies, provides natural explanation of encountered phenomenon. No other arguments such as nonlinearities have to be used to interpret it. The location of these frequency pairs with respect to the so-called visibility circle is re-examined and clarified.

A common method to extract the lowest spatial frequency pattern in the moiré image plane is to perform low pass filtering by using various spatial averaging filters or to apply slight defocus to smear out higher frequency terms. Although these approaches are quite simple and frequently used in multiplicative and subtractive moiré techniques, they are useless in the case of additive moiré and the product type pseudo moiré fringes. In this study we corroborate the moiré image intensity distribution features, following from our pseudo moiré phenomenon interpretation, using two modern and powerful image processing tools: two-dimensional continuous wavelet transformation (2D CWT, below abbreviated as CWT) [10–13] and bidimensional empirical mode decomposition (BEMD) [14–17]. To shorten the computation time of the latter we apply fast adaptive bidimensional empirical mode decomposition (FABEMD) method [18, 19]. A simple modification of FABEMD algorithm is proposed providing very good correspondence between moiré patterns and decomposed bidimensional modes. CWT and BEMD methods are ideally suited to analyze carrier-type fringe patterns [10–19].

Numerical studies conducted with simulated cosine gratings fully corroborate the macro and microstructures of pseudo moiré fringes and their interpretation. At the same time, considerable potential of the CWT and FABEMD methods as complementary image processing tools is shown. Our investigations are the first ones, to the best of authors’ knowledge, dealing with fringe pattern analysis using the FABEMD method.

## 2. Heuristic interpretation

For the sake of simplicity and clarity of presentation of our interpretation the cosinusoidal type amplitude gratings are used as structures being superimposed. This fact in no way restricts the generality of the interpretation proposed. Figure 1 shows computer generated amplitude type cosine gratings. The first grating (a) with vertical lines will be used to generate all three moiré patterns under study. The second grating (b), of the same spatial period as the first one but with inclined lines, will be used to form real moiré in the case of their multiplicative superimposition and conventional additive moiré in the case of additive superimposition. The third structure (c) with half spatial frequency and lines identically inclined as the second one will be used for the product type pseudo moiré generation.

Figure 2 shows, from top to bottom: real moiré fringes (product superimposition of gratings (a) and (b), Fig. 1); pseudo-moiré fringes (product superimposition of gratings (a) and (c), Fig. 1); and the additive type moiré pattern (additive superimposition of gratings (a) and (b), Fig. 1). Intensity distribution diagonal cross-sections are shown on the right.

In the case of pseudo moiré, Fig. 2(b), intensity variation bands can be visually discerned; they run as real moiré fringes, Fig. 2(a). Their intensity distribution is, however, of different character as compared to the case of real moiré fringes with average intensity variations. Modulation variations are responsible for the appearance of pseudo moiré fringes, similarly to the case of conventional additive moiré, Fig. 2(c). Correspondingly, the modulated carrier structure should be resolved in the image plane to follow the modulation envelope. The case of conventional additive moiré fringe formation is well known [2–9]. The carrier fringe frequency is equal to half the sum of component grating frequencies; carrier fringes run perpendicularly to the direction bisecting the directions of lines of superimposed structures.

Now let us interpret in detail the most involved case of pseudo moiré shown in Fig. 2(b). Figure 3 shows the Fourier spectrum plane for all three cases indicated in Fig. 2.

Pseudo-moiré fringes are generated by spatial beating in two frequency pairs, Fig. 3(b). One pair denoted by [ +f_{2}; ( +f_{1}, –f_{2})] comprises plus first harmonic f_{2} of the second (half frequency) grating and the beat frequency formed by +1 harmonic f_{1} of the first (denser) grating and −1 harmonic -f_{2} of the second grating. Second pair is denoted as [-f_{2}; (-f_{1}, +f_{2})] and comprises relevant frequency –f_{2} of the second grating and the beat frequency (-f_{1}, +f_{2}). Each frequency pair generates a difference frequency determining the period and orientation of modulation bands. Whole information, therefore, is contained in the Fourier domain plane. Note that the zero order (bias) frequency, common to both gratings, does not appear along the direction passing through just mentioned frequency pairs. Correspondingly, we can interpret pseudo-moiré fringes (modulation bands) as generated by unbiased frequency pairs forming the lowest beat frequency in the moiré configuration under consideration.

The same interpretation is valid for additive moiré where frequency pairs comprise the same sign first harmonics of the two gratings, Fig. 2(c). On the other hand, the situation is quite different for real moiré fringes generated by biased beat frequency pairs comprising +1 harmonic of one grating and −1 harmonic of the second grating and vice versa, Fig. 2(a). The (+/−)1 and (-/+)1 beat frequencies in this case are symmetrically located with respect the zero order (bias) frequency peak and lie along the same direction in Fourier space [2–9]. In result a variable average intensity distribution is obtained and can be observed without resolving a carrier pattern.

The questions concerning the carrier patterns in both cases of pseudo moiré, Figs. 2(b) and 2(c), arise in a natural way. In the case of additive moiré, Fig. 2(c), Fourier plane does not provide any information on the carrier, see Fig. 3(c). Nevertheless it can be readily assessed in the image domain either directly, see Fig. 2(c), or by applying image processing methods such as CWT or BEMD, see the description of numerical studies below. Carrier fringes run perpendicularly to modulation (moiré) bands and have the spatial frequency equal to half the sum of frequencies of superimposed structures [2–4]. The situation is much more involved in the case of the product type pseudo moiré, Figs. 2(b) and 3(b). Note in Fig. 3(b) that direction of the line joining the unbiased frequency pair, e.g., [ +f_{2}; ( +f_{1},-f_{2})] is perpendicular to two mutually parallel directions formed by connecting locations of the frequencies in the two following pairs: [ +f_{2}; (-f_{1},-f_{2})] and [-f_{2}; ( +f_{1}, + f_{2})]. Additionally, taking into consideration the above discussion on formation of modulation bands we conclude that it is necessary and enough to resolve both superimposed gratings with frequencies f_{1} and f_{2}.

Our simple interpretation does not resort to introducing fictitious, zero amplitude Fourier spectrum peaks in the case of pseudo moiré as proposed by Amidror and Hersch [4]. We point out the necessity of resolving the lines of both superimposed structures in the image plane. In other words, +/−1 harmonics of both gratings have to fall within the visibility circle, not outside it as indicated in Figs. 4
, 6
and 9
of [4]. Resolution requirements for the pseudo moiré imaging system are just higher than for real moiré imaging. In the case of product superimposition pseudo moiré, Figs. 2(b) and 3(b), the value of beat frequencies formed by frequency pairs [ + f_{2}; ( + f_{1}, –f_{2})] and [-f_{2}; (-f_{1}, + f_{2})] should be smaller than frequencies of both component gratings. Last statement is universal for all superimposition type moiré phenomena.

## 3. Numerical experiments

Our Fourier domain interpretation of real and pseudo moiré formation processes has been corroborated by numerical simulations by showing most important and characteristic features of real and pseudo moiré patterns. These are, correspondingly, average intensity and intensity modulation variations. As indicated in the heuristic interpretation part of the paper average intensity variations arise because the relevant spatial frequency pair (generating lowest difference frequency) is biased; in the case it is unbiased the modulation envelope of a carrier results in pseudo moiré. These two basic descriptors of moiré intensity distributions in the signal (moiré) plane are calculated to corroborate our Fourier space interpretation of the moiré formation process. They are readily found using two modern, complementary image processing tools, i.e., two-dimensional continuous wavelet transform (2D CWT) and fast adaptive bidimensional empirical mode decomposition (FABEMD). The complementary character of the two methods is expressed by the fact that CWT eliminates the bias whereas BEMD allows its unique determination (in the BEMD nomenclature frequently the name “trend” instead of “bias” is used). More detailed comparison of the two methods used in our paper with, for example, the Fourier transform approach, can be found in the references quoted.

#### 3.1 Continuous wavelet transform processing

Excellent properties of the continuous wavelet transformation (CWT) method gaining considerable interest in fringe pattern processing are well known [10–13]. We adopt our recently developed technique for fringe pattern modulation determination using 2D CWT [12,13] to corroborate the properties of various moiré intensity distributions. We will not recall here the CWT principles, they can be found in quoted references. Our CWT processing of fringe patterns shown in Fig. 2 consists of the following steps.

- 1. 2D CWT ridge is extracted for every point of the analyzed pattern (coefficient with highest absolute value is selected). Thus two dimensional complex ridge map is obtained.
- 2. CWT image filtration. 2D CWT of the fringe pattern is computed using a complex, directional wavelet (in our case, modified Morlet wavelet) and selected values of scale and angle parameters. Each image point real value of the ridge map is computed, giving a filtered image. Adequate selection of parameters ensures noise filtration and bias suppression since CWT acts as a bandpass filter.
- 3. CWT image modulation determination. This step processes analogically to the previous one, but, instead of the real value of the ridge map, the absolute value is taken. Computing it for every image point provides the fringe pattern modulation distribution.
- 4. Background distribution is determined by Gaussian filter convolution averaging.

Proper choice of parameter *m* controlling the Morlet wavelet Gaussian window width (with no influence on the plane wave frequency) is crucial for the fringe pattern modulation determination [10–13]. In our studies Morlet wavelet with parameter *m* = 1 was used. The CWT normalization parameter was set to 1.98 to ensure uniform modulation reference for the whole image as suggested in [12]. The range of possible wavelet scale and angle parameters was set to cover all carrier fringe scales and angles present in an analyzed image. Employing the Direct Maximum CWT ridge detection algorithm [11, 12] allows us to select wavelet scales and angles automatically. Since, as mentioned above, the CWT method does not provide an information on the bias (average) intensity distribution we have applied Gaussian filtering for this purpose as a supplementary tool at this calculation stage. The Gaussian filter size was set to ensure robust carrier fringe filtering. This approach results in low spatial resolution of the background intensity determination. It can be compared, however, with the trend distribution determined by FABEMD and presented in the following part of the paper.

Figure 4 presents the results of CWT processing of the real moiré pattern, Fig. 2(a). Figure 5 shows the results for the product superimposition pseudo moiré, Fig. 2(b), and Fig. 6 presents the results for conventional additive moiré fringes, Fig. 2(c). Significant differences in the modulation and bias distributions are readily observed.

In the case of real moiré filtered image, Fig. 4(a), because of bias removal, the filtered pattern appears similar to the additive moiré pattern, Fig. 2(c) or its filtered version, see Fig. 6(a). In the case of product moiré strong average intensity variations in the bias decide about the pattern visibility, see Figs. 4(c) and 4(d).

On the other hand, the filtered pseudo moiré image shown in Fig. 5(a) provides an interesting novel result, i.e., the effective carrier-like structure corresponding to the first component grating with dense vertical lines. Note that lines of this carrier-like structure are not perpendicular to the pseudo-moiré modulation bands (the perpendicularity condition between the carrier and modulation directions holds for conventional additive moiré, Fig. 6, as well as real product type moiré, Fig. 4). The result revealed in Fig. 5(a) has been possible due to the property of CWT processing – detecting the wavelet transform ridge with modulus maximum values. The calculated modulation distribution shown in Fig, 5(b), in spite of its complex appearance, corroborates the modulation character of observable pseudo moiré bands – they can be discerned in the image in the presence of component (superimposed) structures. The bias, Figs. 5(c) and 5(d) has a uniform distribution – a property of pseudo moiré patterns. The dominant role of intensity modulation effects in pseudo moiré is evident.

CWT processing of the conventional additive superimposition, Fig. 6, reveals the moiré fringe visualization by calculating the modulation distribution, Fig. 6(b) [12, 13, 17]. The similarity of Figs. 6(b) and 4(b) as well as their diagonal cross-sections is to be noted (the calculated cross-section of Fig. 4(b), not included in Fig. 4, almost coincides with the one of Fig. 6(c)). The uniform bias distribution, Fig. 6(d), is characteristic to pseudo moiré patterns.

#### 3.2 Processing using FABEMD approach

The one-dimensional version of the empirical mode decomposition method was firstly developed in [20]. It constitutes a part of the adaptive data analysis technique, so-called Hilbert-Huang transform (HHT). In HHT analysis a signal is first decomposed into a series of components (so-called intrinsic mode functions, IMFs, of oscillatory character with zero mean value, and a monotonic residue) using the EMD sifting procedure. IMFs are further HT processed using Hilbert spectral analysis (HSA). EMD signal decomposition is very different from the one based on calculating inner product of the signal under test with a set of basis functions (like in Fourier or wavelet transform). Instead, EMD is adaptive and fully data-driven, no fixed basis is used. Its unique properties enable analyzing nonstationary and nonlinear data.

The bidimensional EMD (BEMD) is a fully two-dimensional method, which interpolates envelopes of the corresponding data extrema with functions as bidimensional cubic splines [21, 22] or radial based functions [23]. Reported applications of BEMD in fringe pattern analysis deal with noise reduction in digital speckle interferometry [14], fringe pattern normalization [15], phase measurement in temporal speckle interferometry [16] and evaluation of amplitude encoded fringe patterns [17]. Practical usefulness of BEMD is dictated by the calculation time – triangulation and spline interpolation are the most expensive parts of the algorithm. To overcome this limitation FABEMD (Fast Adaptive BEMD) approach was recently proposed [18, 19] in which the envelope determination of conventional BEMD is modified by replacing the 2D surface interpolation by an order-statistics-based filtering followed by a smoothing operation. Variable window size selection for extrema and smoothing filters enable image different decompositions; it results in the approach adaptivity. Beside significantly shortening the computation time, more accurate estimation of the bidimensional intrinsic mode functions (BIMFs), representing image features at various spatial scales, is obtained in many cases. We introduce the FABEMD method to obtain BIMFs giving additional insight into the pseudo moiré intensity distribution features. Making use of very good spatial scale resolution of FABEMD we undertake the task of extracting characteristic intensity distribution terms of pseudo moiré patterns.

For that purpose we propose the modified FABEMD method to provide a continuous decrease of the number of extrema at subsequent decompositions stages, without artificially increasing the filter mask window width. Order-statistics filter width (OSFW) is determined by its type, number of detected local extrema points and their locations via calculation of Euclidean distance between adjacent minima/maxima (for details, see [18] and [19]). Correct decomposition is characterized by: (1) a decreasing number of extrema points, and (2) increasing the order-statistics and smoothing filter mask window width obtained at consecutive decomposition stages. Extrema detection executed by a sliding window of the same size for each BIMF often results in unfulfilling the conditions (1) and (2). In result one may extract erroneous BIMFs. In [18] authors suggest to artificially increase the filter mask window width (multiplication by a predefined factor) when the currently calculated one is not greater than previously used size of the filter mask. This approach limits adaptivity of FABEMD algorithm and causes phase shift between extracted BIMF and source image. In order to ensure correct decomposition conditions (1) and (2) we propose a simple modification of FABEMD algorithm. The adjustment step for the extrema detector window width is added at the beginning of calculating each BIMF. We find a proper sliding window size, which provides detection of extrema maps meeting conditions (1) and (2). Having proper extrema maps we continue decomposition as described in [18] and [19]. For generation of first BIMF 3x3 extrema detector window size is applied. In result the BIMFs obtained correspond very well to moiré patterns and their component intensity distributions.

Detailed studies of FABEMD properties in application to spectral analysis of moiré patterns will be published in a separate paper. It is interesting to note, however, that because of the spatial frequency beating phenomenon we have not been able to extract, separately, the gratings being superimposed (except for the special case of pseudo moiré, see Fig. 9 below). This task of main interest in texture analyses (see [24] and references therein), where moiré effects are not encountered. This property, however, is not a deficiency for us since we are mainly interested in the beat patterns.

The results of the FABEMD method are presented in Figs. 7 , 8 , 9 and 10 showing decompositions of real and pseudo moiré patterns presented in Figs. 2(a), 2(b) and 2(c). The parameters of each decomposition (extrema detection window width, detected extrema number and applied filter mask width) are listed in Tables 1 , 2 and 3 showing that decomposition conditions (1) and (2) had been met. Figure 7(f) can be considered as the lowest frequency real moiré pattern [2, 3] (cosinusoidal average intensity variations). It is cophasal with the input moiré, Fig. 2(a). Additionally, similarity between intensity distributions of Fig. 7(f) and Fig. 4(c) can be observed; it was checked by calculating diagonal intensity cross-section of Fig. 7(f) and compared with Fig. 4(d) (the contrast obtained is higher in the case of CWT processing). Detailed studies of BIMFs of the product moiré fringes and their decompositions will be given in a separate paper. Note some boundary effects visible in higher number BIMFs in Figs. 7, 8 and 10. Their analysis, however, is out of scope of this paper.

In Fig. 8(d) BIMF4 of the decomposition of the multiplicative superimposition pseudo moiré shows modulation-like intensity variations instead of average intensity changes. Note some similarities between Figs. 8(d) and 5(b). Direct comparison of FABEMD results with the CWT one (Fig. 5(b)) would require calculating the modulation (amplitude) distributions from Fig. 8(c) as proposed in our recent paper by using the spiral Hilbert Transform [17]. Because of many factors influencing the result we will discuss the issue in a separate paper. BIMF4 in Fig. 8(d) is completely different from Fig. 7(f) of the real moiré decomposition. Further decompositions, not presented here, of all five BIMFs shown in Fig. 8 did not provide an intensity distribution similar to the one of Fig. 7(f). This fact is an additional argument that the dominating role in pseudo moiré perception is played by the intensity modulation effect.

To corroborate our important finding presented in Fig. 5(a), i.e., the effective carrier-like structure corresponding to the first component grating dense vertical lines, we have made the attempt to obtain this result by applying FABEMD analysis. The success is illustrated in Fig. 9 showing carrier fringes together with horizontal cross-section. BIMF3 of Fig. 8(c) was decomposed 4 times, each subsequent decomposition used BIMF2 from the preceding one as the starting image pattern. This multiple stage decomposition decided about the resulting very low amplitude of the final pattern.

Since the Fourier spectrum of the product type pseudo moiré, Fig. 3(b), is more rich in beat frequencies than the one of real moiré, Fig. 3(a), more complicated BIMF intensity distributions are obtained, see Fig. 8. In general, individual BIMFs represent intensity patterns formed at different scales (formed by different dominating frequencies). For example, BIMF1 and BIMF2 shown in parts (a) and (b) contain carrier fringes formed by biased beat frequency pair [(-f_{1},-f_{2}); ( +f_{1}, +f_{2})], see Fig. 3(b). These carrier fringes are coarser than carrier fringes in the case of real moiré, Figs, 2(a) and 3(a). The modulation bands are formed by frequency beating in three groups: [(-f_{1},-f_{2}); -f_{1}; (-f_{1}, +f_{2})]; [-f_{2}, 0, +f_{2}] and [( +f_{1},−f_{2}), + f_{1}, ( +f_{1}, +f_{2})]. In all these three groups beating frequencies lie along mutually parallel directions and are mutually equidistant. Although exemplary BIMFs just discussed could be obtained by sophisticated Fourier or wavelet band pass filtering, the adaptive FABEMD approach is simpler and more straightforward. Complementary character of CWT and FABEMD methods can be readily apprehended.

Figure 10 shows first four BIMFs and the residual part of the conventional additive moiré decomposition. As mentioned above, the uniform bias distribution obtained is characteristic to pseudo moiré fringes. One of first three BIMFs can be used for further CWT or Hilbert transform processing to visualize the modulation distribution.

## 4. Conclusions

Average intensity variations and amplitude modulations play dominant role in the so-called real and pseudo moiré patterns, respectively. Their origin can be readily traced in the Fourier spectrum of moiré images. Our interpretation of real and pseudo moiré formation processes is based on identifying, as we call them, biased and unbiased frequency pairs. Lowest moiré beat frequencies are generated by these pairs in the case of pseudo moiré, whereas real moiré pattern frequency is formed by the zero order (bias) term and generated beat frequencies. In the case of pseudo moiré both superimposed structures should be resolved in the image to perceive relevant modulation patterns; this fact imposes higher resolution demands on the optical imaging system than in the case of real moiré.

It has been shown that basic information on moiré, i.e., frequencies contributing them, is just present in the Fourier domain. There is no necessity to introduce additional, non-existing, zero amplitude frequencies to aid moiré phenomenon interpretation. Basic real and pseudo moiré intensity distribution descriptors such as modulation and average intensity distributions and an effective carrier structure have been evaluated using two modern, complementary image processing tools, i.e., continuous wavelet transformation (CWT) and bidimensional empirical mode decomposition (BEMD). A simple modification of the fast adaptive BEMD (FABEMD) algorithm was proposed to obtain good correspondence between obtained modes (BIMFs) and moiré pattern intensity distributions. Numerical experiments fully corroborate findings of our interpretation method to be used both for real and pseudo moiré images.

Although our interpretation has been worked out using amplitude cosinusoidal gratings as structures being superimposed, it is directly applicable to Ronchi type binary gratings (with opening ratio equal to 0.5) frequently met in practice.

## Acknowledgements

This work was supported, in part, by the Ministry of Science and Higher Education, grant N N505 464 238, the European Union in the framework of the European Social Fund through the Warsaw University of Technology Development Programme, and statutory activity funds.

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