The moiré effect was considered at a finite distance. Formulas for the wavenumber and phase of the visible moiré patterns were found with displacements of the observer and of gratings taken into account. The computer simulation and physical experiment confirm the theory. The method of measurement of the wavevectors of plane waves was proposed basing on the Radon and Fourier transformations.
©2012 Optical Society of America
For visual displays, it is important that the visible moiré patterns do not have a relation to an image displayed in the screen. The visible moiré effect only depends on the structure of layers and their relative position. This dissonance can affect perception negatively because the observer sees two independent overlapped pictures simultaneously, a useful image and moiré patterns. Therefore, the moiré phenomenon in visual displays is an undesirable effect which needs to be eliminated or, at least, minimized.
It is a common practice to design autostereoscopic three-dimensional displays of several parallel layers . Various kinds of layers are in use in autostereoscopic displays which extends much further than LCD panels, LED arrays, lenticular plates, patterned retarders, parallax barriers, microlens arrays, etc [2,3]. As a rule, the layers have a regular, periodic structure. Moreover, the periods of two layers in autostereoscopic displays are often multiple as, e.g., the period of pixels in LCD panel and the period of the corresponding parallax barrier. The periodicity and multiplicity give raise for the moiré effect, which “occurs when repetitive structures (such as screens, grids or gratings) are superposed or viewed against each other” .
When estimating the influence of moiré patterns to the visible image, it is important to know how the moiré patterns appear for an observer. In investigation of the moiré effect, the coplanar layers are traditionally considered; this (effectively) moves the observer to the infinity and the effect does not depend on the actual observer’s location. However, in the autostereoscopic displays, the layers are not coplanar rather installed at a certain non-zero distance. In this case, the position of an observer is essential. This makes the visual moiré patterns to vary, when an observer moves. The elimination of the moiré patterns is therefore an especially important issue in 3D displaying [5–7]. In some cases sometimes, it may even appear that the vivid fringes live their own, unpredictable life. Fortunately, such behavior of the moiré patterns may look attractive from the esthetic perspective; this gave rise to a unique branch of visual arts, the optical art (Op Art), which is essentially three-dimensional in many cases, see, e.g .
Under these circumstances, it becomes necessary to consider a special case of the moiré effect with the observer at a finite distance and layers at a non-zero distance.
At the same time, the non-zero distance between layers is also important in measurement techniques such as, e.g., the determination of shape by shadow moiré , the contour lines in moiré topography , and more. (The layers are not necessarily planar in these cases.) For measurements, the moiré patterns produced by light sources and grid lying in separate planes are studied ; it would be relevant here to mention that the geometry of these measurements is quite similar to the geometry of the autostereoscopic displays. The image of the moiré patterns produced by identical gratings with a non-zero gap is analyzed in .
In our study, the transparent sinusoidal gratings of the unity amplitude are mostly involved; fortunately, this approximation seems to be very helpful in describing the many basic features and particular issues of the moiré effect. In particular, it allowed us analyzing the spectra of moiré patterns . The finite-distance moiré effect for identical gratings was studied in .
In the current paper, the analytical study concerns the parallel gratings. Under these circumstances, the moiré patterns are parallel to the lines (axes) of one-dimensional gratings and the displacement along the x-axis can be referred to as the lateral displacement. We found the analytical expressions for visible moiré patterns and made the computer simulation together with the related physical experiments. The special experiment (wedge) was made to confirm the case of the parallel identical gratings. The simulation of the improved, two-dimensional layout corresponding to an autostereoscopic display was also made. Our computer simulation is based on the Fourier transformation; the visual perception is modeled by the visibility circle  in the spectral domain. The measurements are based on the technique which involves the Radon and Fourier transformations.
2. Theoretical background
Two parallel plain layers of regular structure (gratings) with a non-zero distance between them are modeled. The gratings have the symmetric cosinusoidal transparence functions. In this paper we only consider wavelength and phase with no respect to the amplitude; thus the unity amplitude is assumed. The wavelengths of the gratings are connected by the dimensionless ratio ρ as follows: λ2 = ρ λ1. Under these assumptions (implying that k = 2π /λ by definition), the equations for transmittance functions of gratings look like
The similar geometry of gratings (but not the observer location though) is considered in .
The geometric relationships can be described with using the projection transformations. For the center of projection (observer or camera) located at C = (xc, yc, zc) and the screen at z = 0 as shown in Fig. 1 , the homogeneous transformation matrix is found in ,
For any location (x, y, z), its projection onto the screen is a product of the matrix Eq. (3) and a general column vector P = (x, y, z, 1)T. In the current paper, we consider the one-dimensional lateral displacement only; thus, yc ≡ 0 can be taken.
2.2 Wavenumber and wavelength
The matrix Eq. (3) describes the transformation of coordinates only. The meaningful physical quantities like a wavenumber, phase, etc. should be derived on their own.
The first grating lie in the xy-plane (z = 0), the second grating in the parallel plane z = -d. The transformation of the first grating G1 is the identity transformation, and the grating remains unchanged. A transformation of the second grating G2 can be obtained by multiplication of the matrix Eq. (3) and the column vector P = (x, 0, -d, 1)T. The projected wavelength of the second grating can be found basing on the symmetry of gratings. For the observer at C0, the corresponding coordinate vector is P = (λ2, 0, -d, 1)T, and one can obtain
If the displacement of the observer is non-zero (xc ≠ 0), the projected second grating is generally non-symmetric, and its displacement from the origin can be obtained by transforming the point P = (0, 0, -d, 1)T by Eq. (3):
The dimensionless phase of the projected (and displaced) second grating is equal to zero without displacement and to 2π when the displacement is equal to the wavelength. This can be expressed as a ratio of Eqs. (4) and (6) with the proper coefficient,
The formula Eq. (8) should be used instead of Eq. (2) in further calculations of the moiré effect. After that, the wavelength and phase of the visible moiré patterns can be found by known formulas. The wavenumber (the spatial frequency) of the moiré wave is the modulus of the difference between the wavenumbers of the gratings Eqs. (1) and (8) can be found by Eq. (2).11) in . Here, it is enough to deal with wavenumbers, because the observed one-dimensional moiré wave has a “pre-defined” direction, and k is effectively a scalar. Thus, the wavenumber and the wavelength of the moiré patterns are
In particular, Eq. (10) gives the infinitely long moiré wave for ρ = 1 and d = 0 (i.e., when s = 1) which corresponds to the singular moiré-free state (1,0) . For ρ = 1, Eq. (10) is simplified (as in ),
2.3 Phase and displacement
The phase of the visible moiré pattern can be found from phases of the gratings by Eq. (7).27) in , which indicates that in the case of two gratings, the phase of the moiré wave is equal to the difference of phases of the gratings. If one of these two grating remains in place (so as its phase can be effectively considered as zero), the dimensionless phases of the moiré wave and of the moved grating are equal. The dimensionless phase is the ratio of the displacement to the wavelength. In other words,
The equation of the visible moiré patterns can be written using the wavenumber Eq. (9) and the phase Eq. (7) caused by the observer displacement and the corresponding displacement of the projected grating,
The ratio of φm and km can be obtained formally from Eq. (13) as the ratio of the constant term to the coefficient of x as follows, However, this expression cannot be considered as a final equation for the displacement. In contrast to the wavelength, the phase is a signed quantity since the displacement takes place along the corresponding wavevector in one of two opposite directions (“to” or “from”); with that, the displacements for ρ < 1 < s and for ρ > s are opposite. It means that the formula for xm has to include the sign depending on ρ.
In order to obtain the sign basing on the mentioned ratio, the following should be taken into account. If ρ = 1, the visual displacement of the moiré wave is exactly equal to the lateral displacement of the observer. This is experimentally confirmed in  and can be explained as follows. Let the visible maximum of a pattern produced by identical gratings be at the origin x = 0, when the observer is at C0 (Fig. 1). When the observer moves from C0 to C, the visible maximum goes to x = xc exactly “below” him. This imaginary experiment shows that for ρ = 1, the displacements of moiré patterns and of the observer are identical and have the same sign. Additionally recalling the definition of the distance coefficient Eq. (5), from which the numerator of the ratio of φm and km from Eq. (13) can be expressed as and the displacement can be written as
In Eq. (14), the parenthesized superscript is added, because two more displacements will be introduced in the next subsection. For the identical gratings, the visible displacement of moiré patterns derived from Eq. (14) is equal to the lateral displacement of the observer for any gap and distance:
2.4 Non-symmetric layout
Besides the observer displacement, there is another reason of the visual displacement of moiré patterns: the displacement of the grating itself. This factor is considered in  in determination of the position of a grating. In this case, the layout in not symmetric, and the displacement can be found from Eq. (12) in the similar manner as Eq. (14) was obtained, keeping in mind that the influence of displacement of G2 is opposite to that of G1:
3. Experimental technique
3.1 Physical experiments
The physical experiments were made basing on the layout Fig. 1 with the upper grating G1 printed on transparency and the lower G2 on the paper. The gratings were installed between the cover glasses with the controllable distance between them. The lateral displacement of gratings was also controlled. In the experiments, the period of the first grating was 0.0762 cm or 0.1 cm, the period of the second changed from 0.07 cm to 0.14 cm, and the distance between gratings from 2.2 cm to 4.25 cm. The moiré patterns of each particular configuration were photographed, i.e. the photograph was taken by a digital camera each time when a parameter (period, distance, gap, etc.) changed. The geometric characteristics of moiré patterns (wavelength and displacement) were measured from digital images. In relation to the measurement procedure, this is virtually the same as measurement of printed images by a metal ruler.
3.2 Computer simulation
In the paper, the computer simulation for sinusoidal gratings is presented. The simulation program is based on the multiplicative model  the two-dimensional Fourier transformation and the concept of the visibility circle . The images of projected gratings are superimposed (multiplied). Then, the Fourier transformation, the higher spectral components are rejected by applying the visibility circle in the spectral domain. After that, the inverse transformation yields in the visible moiré pattern. In simulation, the distance and displacement of the observer are considered as given values whereas the wavelength and displacement of moiré patterns are measured.
Important is that the program does not rely on equations for the wavelength and displacement obtained in Sec. 2; it always performs the Fourier transformation followed by measurements in an equal manner for any source (e.g., simulated) image. Therefore the simulation can be considered as an independent tool to prove or disprove the theory.
From the image processing perspective, the problem is how to measure the wavevectors in the images of plain waves, i.e. their wavelength and direction. The proposed measurement technique is based on the Radon transformation  of two-dimensional images. In the current study, the technique was applied to the simulated images, but clearly, it can be equally applied to images (photographs) of plane waves of other nature. The characteristics (incl. direction) of the visible moiré patterns for various locations of the observer can be measured at least for two dominant waves. In the next section examples of application of this technique to the one-dimensional case (when the angle is not essential) will be given.
The program recognizes the plain waves in images and measures their characteristics as follows. Firstly, the Radon transformation is applied to the symmetrized image, secondly, the columnwise Fourier transformation is made which is followed by one more Radon transformation, and finally the cross-sections of the power spectrum give the numerical result.
The procedure is explained below, basing on the example. Although in this paper the measurement of the lateral (one-dimensional) displacement is only presented, the procedure allows measuring the angle as well. Therefore the example is given for two dimensions. Let’s consider, for instance, the moiré patterns between the coplanar grid and grating, ρ = 1.2, α = 20°, when two moiré waves are directed to approx. 67° and 112°. In this case, the simulated pattern looks like in Fig. 2(a) . In order to exclude the influence of the circumscribed square, the shape was made angular-independent by applying a rotation symmetric circular window, as shown in Fig. 2. In the symmetrized image S(x, y), all angles are treated equally.
The series of Radon transformations with angles from 0 to 180° is applied to the symmetrized image. The result of multiple Radon transformations is shown in Fig. 3 as a matrix R(i, j) in which the Radon transformations for different angles j are located by columns: R(i, j) = Radon(S, j). It is significant that in this matrix, some periodic vertical structures exist. They appear when the current angle of the transformation is orthogonal to the wavevector of a moiré wave. In Fig. 3, these two angles j0 are indicated by arrows.
In order to recognize the periodic structures, the series of one-dimensional Fourier transformations is applied to the columns of the Radon transformation R(i, j) as follows, F(k, j) = Fourier(R(i, j)). In the resulting power spectrum, the periodic structures appear as pairs of the peaks at the corresponding angles, see Fig. 4 where the result of DFT is shown.
The locations of these pairs can be found, e.g., by an orthogonal projection, an additional Radon transformation with the fixed angle, or a cross-section along the corresponding row and column. This is shown in Fig. 5(a) ; basing on it, the angles j0 of the peaks RF(j) corresponding to the periodic structures in R(i, j) can be determined as abscissas of the maxima. Once the angles j0 are measured, one can find the frequencies of the moiré waves from the corresponding columns of the Fourier transformation (Fig. 4); one of columns F(k, j0) is shown in Fig. 5(b). This is the final operation after which, the wavevector of the moiré wave is known, including the magnitude and direction.
To summarize. In this example, a combination of two plain waves was analyzed. The angle = 74° (Fig. 5(a)) and the wavelength = 53 arb. units (Fig. 5(b)) of the first wave were recognized and measured automatically. Then, after the first peak was excluded, the same procedure was repeated for the second wave, and the angle 103° and wavelength 74 arb. units were determined. After proper calibration, the arbitrary units can be transformed in the conventional linear units (e.g., centimeters).
4. Experimental results
In this section, the results of the numerical simulation and the physical experiments are presented. The three experiments were performed: 1) the measurements of the wavelength and displacement of moiré patterns produced by identical gratings, 2) the special wedge experiment to confirm the equal lateral displacement for any gap and distance, 3) the measurements of the wavelength and displacement for ρ other than one.
4.1 Identical gratings
The results of the experiments with identical gratings are reported in  for gaps 2.75 cm and 4.25 cm and distances between 41 cm and 118 cm. It is important that the displacement of moiré patterns vs. the observer displacement does not depend on the gap . The experiment and simulation conform to the theory (RMS error is within 1% - 3%). Examples of experimental photographs for identical gratings are shown in Fig. 6 .
To illustrate the usage of the proposed measurement technique (Sec. 3.2), it was applied to experimental photographs like Fig. 6. Additionally, to demonstrate the stability of results, the images were rotated by arbitrary angles 37° and 78°. This was possible because in this experiment, the angle is known in advance and therefore finding it is unnecessary. Particular examples of the images (gap 2.75 cm) corresponding to Fig. 3 (the Radon transformations with symmetrized window) and to Fig. 5(a) (the Radon transformation with the angle 90° of Fourier transforms of Fourier-Radon transforms) are shown below in Figs. 7 and 8 , resp.
The measured wavelength is 7 pixels, the same value in both cases; the proper calibration yields 2.1 cm, which is close to the value 2.17 cm measured directly with using the ruler.
4.2 Wedge experiment
The wedge experiment with inclined gratings was performed in order to confirm the formula Eq. (15) for the identical gratings directly. In this experiment, two gratings were not parallel, rather installed as a wedge; the upper grating was horizontal, while the lower was inclined, see Fig. 9 . This layout is similar to the layout Fig. 1, but the second grating G2 is inclined; consequently, the axes of the gratings are not parallel. The camera is located at zc = 110 cm approx.
The wavelength of the visible moiré patterns depends on the gap (the gap is shown in Fig. 9(a) by double-headed arrows) which, in this experiment, depends on the y-coordinate. The lateral displacement, however, is the same for any y and thus, for any gap. A visual proof is in the presence of the wide dark band, which remains vertical in all photographs, as shown by arrows in Fig. 10 .
The measurements along two lines y = const with the constant gap (lines l1 and l2 in Figs. 9(a) and 10(b)) demonstrate the identical (and equal to the displacement of the camera) displacement of the visible moiré patterns for various heights of the camera along the corresponding lines in the visual field (near the upper and lower edges, where the gap is 0.5 cm and 2.75 cm), see Fig. 11 .
To summarize. For the gap 0.3 – 3 cm and the distance to the camera 100 – 125 cm, this experiment confirms the formula Eq. (15) with the RMS error 2.5%.
4.3 Non-identical gratings
In the case of ρ ≠ 1, the period and displacement of the moiré patterns were measured experimentally for distance coefficients between 1.017 and 1.054 which correspond to the gaps 2.2 and 2.75 cm with distances between 103 cm and 129 cm. The period and the relative displacement are shown in Fig. 12 together with the experimental data for ρ = 1 (Sec. 4.1) which were recalculated here in order to be represented in the same graph with the data for various ρ. Since the distance coefficient s depends on particular distance and gap Eq. (5), and varies in two experimental series, the dashed vertical line indicating s in Fig. 12 is intentionally blurred.
In the graph of the wavelength (Fig. 12(a)), the equivalent of the visibility circle  is a horizontal line; it shows that the waves with the wavelength below it (i.e. shorter than 0.1 cm) are practically unrecognizable from the distance longer than 100 cm approx.
In Fig. 12(b), the bold dot shows the identical gratings  ρ = xm/xc = 1. It can be observed in both graphs Fig. 12 that the period and displacement of moiré patterns tend to be infinitely large when ρ → s, i.e. s can be qualified as a special (critical) value of ρ.
The experimental data Fig. 12 fit the theoretical values calculated by formulas Eq. (10) and Eq. (14) with the normalized RMS deviation 4% for the wavelength, and 5% for the displacement. In the experiments, is was observed that the displacement of the moiré patterns has the same direction as the direction of the observer (camera) for ρ ≤ 1, being opposite for ρ > 1. This confirms the sign in Eq. (14). The simulation data follow all theoretical curves (although not shown in Fig. 12).
4.4 Comparison with real display
Additionally, the moiré effect in a particular autostereoscopic display with the slanted parallax barrier was studied. For that purpose, the computer simulation was made for the rotated gratings with the rectangle transparency function, simulating a real device. Although this layout is not longer a one-dimensional as the layout Fig. 1, its two differences from the basic layout are that the layer G2 in the square grid and the layer G1 is rotated around the z-axis. The following device parameters were simulated: the number of view images 12, the inclination angle 18.43°, the ratio of periods ρ = 1.057 (in fact, this value is the product of two numbers, the distance coefficient s = 1.003 and the angular factor 1/cos 18.43° = 1.054).
The experimental photograph is shown in Fig. 13(a) . The measured values are the period 2.88 periods of pixels and the angle 96.7°. The simulation showed that the corresponding moiré wave has the wavelength 3.03 simulated pixels and its wavevector is directed to 100°, see Fig. 13(b). (N.B. In this simulation, the coplanar grids with ρ = s = 1.003 were used, which corresponds to the perfectly aligned, rotated non-coplanar grids with a given gap and a proper ρ at the optimal distance.)
The comparison of the numerical values demonstrates a good agreement (within 2 – 4%) with the simulation.
5. Discussion and conclusion
In both Figs. 12(a) and 12(b), two regions can be recognized, the near region (ρ ≈s) and the far region (ρ < s, ρ > s). In the former, both wavelength and the displacement are changed essentially with a small change of ρ (and other parameters too). In the latter, the change of the moiré patterns is much smaller. It means that the regions are different in respect of stability, and in minimization the latter region seems to be preferable, at least because the moiré patterns are less changeable (and besides less visible) in it.
The measurement technique not only allows measuring the wavelength (which is the only necessary for the current study), but also the angles. Although, the technique was demonstrated here for two waves only, it seems that it can be applied for larger number of plain waves. The displacement was not measured here because it does not affect the power spectrum; nevertheless, it can be measured through the ratio of the real and imaginary components of F(i, j). In order to improve the accuracy of the wavelength measurements, fitting a parabola to three points was applied.
At the same time, it should be mentioned that for the 3D display, two moiré waves (branches) were uncovered in simulation, the high-frequency wave at the optimal distance and the low-frequency wave at other distances. The former is confirmed in experiments as reported in Sec. 4. The latter, however, seems to be out of scope for a while. Out task, therefore, is to improve the theory and simulation so as both branches could be efficiently described together.
An extension of one-dimensional gratings, the square grids with parallel axes do not seem to produce some more visual effects except for the additional (and orthogonal) visible net of moiré patterns. The net obeys the same rules in two directions. This means that the formulas similar to Eqs. (10) and (14) can be equally applied for both x- and y-axes independently. In particular, for the y-axis,
Generally, the visible movement of moiré patterns does not follow a pattern for other objects normally displayed in a 3D display. Based on its motion parallax, the moiré wave probably can be treated as lying in a separate depth plain (though no other depth cue would confirm that). This may produce a visual gap between the image and the moiré patterns which may affect the binocular perception because the perceived plane of moiré patterns is not directly controlled and could exceed reasonable limits. Also, the visual behavior of moiré patterns for an observer approaching the screen may be unclear in respect to the perspective, because, e.g., the decreased wavelength of moiré patterns (their “size”) can be sometimes treated as a more distant location. In this situation, there could be confusion in perception of moiré patterns because the motion parallax and the perspective show different distances. Such ambiguous and uncertain visual behavior might probably destroy the normal stereoscopic perception.
Also, it is interesting that the moiré patterns observed from a finite distance can be pictorially interpreted as an “image” of an observer in the mirror. For identical gratings, the visual displacement of moiré patterns is equal to the lateral displacement of an observer Eq. (15). In this case, the moiré patterns seem to follow the reflection of the observer in a (non-existent) plane mirror located at the half-way to the display device, see Fig. 14 . This “reflection” repeats all movements of the observer. The distance to this reflection can be considered as estimation based on the motion parallax.
The current study describes the finite-distance moiré patterns, especially their wavelength and the lateral displacement. The method to measure the wavevector of plane waves is proposed. The special experiment was made to prove out the particular case. There is a good agreement between all three general methodologies of investigation presented in this article: the theory, the computer simulation (modeling), and the physical experiments; it is within 2% - 5% for the theory and the physical experiment and less than 1% for the theory and the computer simulation.
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