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Angles of visible moiré patterns are observed experimentally. Experiments were made across the angular range 0 - 90° in a wide range of parameters. Two kinds of clusterization were observed, ray and discrete. In rational cells (LCD pixels), the moiré patterns appear at a few fixed discrete angles. The list of preferable moiré-less angles is presented basing on the experimental data; preferable areas in the parameter space are found. The problem of minimization of the moiré effect is formulated as the Diophantine inequality with complex coefficients. The classification of moiré angles basing on the probability of the moiré effect can be practically useful.
© 2014 Optical Society of America
1. Introduction
The moiré effect is a visual phenomenon, which appears when several transparent repeated structures are superposed [1–3]. In visual displays, the moiré effect creates a meaningless visual image of bands (a visual noise) and therefore it is considered as an adverse effect. The elimination or at least reduction (minimization) of the moiré effect is an important issue in improving the visual quality of displays, especially the autostereoscopic three-dimensional multiview displays and integral imaging displays [4–6] which are typically designed with two optical layers of the periodical structure (one-dimensional gratings or two-dimensional grids) [7].
The gratings are not necessarily sinusoidal, as it was assumed in [8]. The profile of actual gratings contains a number of spectral components. The foundation of the spectral approach is considered in [1]. An interesting, reciprocal vector technique, is proposed in [9]. In general, we have to consider all spectral components, because it cannot be guaranteed in advance that some their combination would not approach to the origin and become a visible moiré wave. This makes difficult to solve the problem with an infinite spectrum. At the same time, in a limited spectrum each combination can be counted. This approach is based on the decay of Fourier coefficients [10] which allows neglecting the spectral components with higher numbers. Also, we considered one-dimensional elementary gratings here. For a limited spectrum, the limits for the moduli of the integer numbers of spectral components pj are as follows [11], | pj | ≤ qj, where qj ≥ 0 is the number of components with non-negative numbers, so as the total number of the spectral components of j-th grating is Qj = 2qj + 1.
The equation of spectral peaks in the complex plane [11] describes the moiré effect produced by N overlapped gratings,
where M is the result of the interaction, i is the imaginary unit, j is the index of the elementary one-dimensional grating, kj (a real number) is the fundamental wave vector of the j-th grating, αj is the rotation angle, pj are integer numbers enumerating the spectral components of j-th grating.The basic expression Eq. (1) states that the moiré wave may appear as a result of interaction between spectral components (harmonics) of gratings. Due to the symmetry with respect to the origin, it is sufficient to consider one component from each pair, or, effectively, to deal with one half of terms. The vector sums can be effectively described by combinations of N integers (p1, p2, …, pN) varying within limits Qj. When one of parameters is gradually changes, the correspondingly modified Eq. (1) describes the spectral trajectories of moiré waves.
A practically important example [12] is two layers, each of two gratings, so as the total number of the elementary gratings N is up to 4. In this case, four one-dimensional gratings are arranged in two orthogonal pairs (grids), and instead of Eq. (1), we have
where σj is the aspect ratios of the grid cells (σ = 1 means a square, otherwise rectangle), ρ is the size ratio (an inverse barrier-to-pixel ratio, the amount of screen pixels per one pitch of the barrier plate in a three-dimensional display), and p1, … p4 are integers characterizing the particular interaction.The spatial resolution of the human visual system is limited. Therefore, some of terms of Eq. (1) are visible, some not. For the moiré effect to appear visually, the corresponding points should lie within the visibility circle [1]. The visibility circle roughly models the frequency response of the human visual system [13]. In the example Fig. 1, two vector sums of four vectors as in Eq. (2) are shown (the labels A and B mark the endpoints inside and outside the visibility circle). Also, our visibility assumption [11] means that the stronger visual effect is produced by the moiré waves closer to the origin the spectral domain. This assumption is related to the frequency response too. In order to find the visual effect, one has to select the representative terms of Eq. (1) which may become recognizable moiré waves.
The representative terms of Eq. (1) can be identified by the inequality
which, after the normalization by k0 (the radius of the visibility circle), looks like followswhere all wave vectors kj are redefined as kj / k0, while the unknowns | pj | ≤ qj are still integers; the products are the complex coefficients. Equation (4) is an indeterminate polynomial inequality of the first order that allows the variables pj to be integers only; thus it is the Diophantine inequality [14, 15] with complex coefficients. Then, the problem of selection of representative terms can be formulated as follows. Given real numbers kj and complex factors, find all solutions of the Diophantine inequality Eq. (4) for N unknowns p1, p2, …, pN running within certain limits qj. Instead of the individual limits qj, a common limit q for all variables can be used for any j (| pj | ≤ q).Practically (in simulation), Eq. (4) is solved numerically. For every combination of pj within their limits, the sum of spectral components is calculated and the length of the resulting vector (the distance from the endpoint to the origin) is found. The only combinations with the distance less than a certain predefined value k0 proceed to the output. For the next angle, the procedure is repeated. This way, the combination of the integer numbers can be found for all combinations of pj, when the moiré waves may appear in 3D display devices.
Equation (4) can be reduced to a quadratic Diophantine inequality or to a linear Diophantine system. For example, before solving Eq. (4), one may try the system of two linear Diophantine inequalities with the real coefficients
The conditions Eq. (5) are easier to check than Eq. (4). However, the system Eq. (5) is not exact equivalent of Eq. (4), because in it, the unit circle is replaced by the unit square. Thus, for combinations of pj satisfying Eq. (5), the calculation of the modulus by Eq. (4) remains necessary.
The finite distance effect [16] is not considered here, because in this paper we are focused on the presence or absence of the moiré effect. Correspondingly, we do not compare the visual effect of observed moiré fringes at different angles.
2. Experiments
In order to find the moiré angles φ experimentally, we performed experiments with the printed samples (using a desktop laser printer) and with the LCD panels. In these experiments, the rotating parallax barrier plates were applied to the panels and samples. We considered typical observation conditions of a desktop display. For that purpose, the observation distance was selected, as well as the ranges of the geometric parameters ρ and σ between 0.18 and 3. The range of the running angle α selected basing on the symmetry; it was 0 - 90° in general and 0 - 45° for square cells. These parameters correspond to a typical stereoscopic display or a multiview display with subpixels. Indeed, the appearance of the moiré patterns in each particular case depends on the specific values of α, ρ and σ. However in the preliminary experiments we did not notice an essential influence of ρ over the clusterization (see below). As total, 5 LCD devices (with a fixed σ) and variety of printed samples (within a wide range of σ) were combined with 3 barrier plates in various combinations.
In experiments, a static image was displayed; it is the black rectangular grid on the white background for printed images and the grid of pixels for LCD panels (uniform white screen). The angle α between the static image (printed grid or panel) and the barrier plate was gradually changed from 0 to 90°. As the plate rotates, the angles φ are determined and recorded, where the moiré effect takes place. The visual observation was made from the distance 50 cm. This way the angles of the recognizable moiré effect (the moiré angles φ) were revealed.
Two experiments were performed: the first, with an arbitrary aspect ratio of grid cells (printed samples) including the rational aspect ratio,
and the second, with square pixels (LCD panels). In experiments with the printed samples, the aspect ratio σ changed from 0.17 to 3.0 while the size ratio ρ varied between 0.18 and 0.72; in LCD experiments, σ = 1, ρ between 1.62 and 3.0.It can be said in advance that the moiré angles φ = 0 and 90° were observed in every particular experiment with any combination of parameters σ and ρ between 0.18 and 3.
2.1 Experiment 1
It was observed in experiments with arbitrary aspect ratios (printed samples) that the moiré angle φ gradually changes when the aspect ratio gradually changes. The examples of the observed moiré patterns are shown in Fig. 2.
Fig. 2 Illustration of variety of moiré fringes in Experiment 1. (Photographs ordered be tangents: 0, 1/3, 2/3, 1.) The full set of the experimental data is graphically shown in Fig. 3 below. (a) φ = 0°, σ = 1.5, ρ = 0.72; (b) φ = 18°, σ = 0.33, ρ = 0.55; (c) φ = 33°, σ = 0.67, ρ = 0.55; (d) φ = 45°, σ = 0.5, ρ = 0.72.
The experiment was made with 27 values of σ and 2 values of ρ, which means 54 particular experiments total. In the overall picture of all experiments (all experimental data drawn in the single graph tan(φ) vs. σ) the abscissas and ordinates of the experimental points are distributed more or less uniformly (abscissas at least between 0.8 and 1.5 and ordinates between 0.17 and 1.6), see Fig. 3.However, this picture does not show the experimental points uniformly distributed across the area in the parameter space. On the contrary, there are at least 3 clearly recognizable straight lines comprised from the experimental points at the angles φ 0, 45°, and 90°.
Fig. 3 The full set of experimental data. The tangent of the moiré angle φ as a function of the aspect ratio σ for ρ = 0.55 and 0.72 (square and triangle markers). The clusterization of the experimental data along the rays. The constants of proportionality are indicated near some rays.
Figure 3 presents the most experimental data of the current paper, while Fig. 2 is only an illustration of variety. A careful look to Fig. 3 can additionally uncover that the experimental data are concentrated within narrow boundaries (clustered) around the rays with the rational slope angles. These rays with their neighborhoods cover the vast majority of the experimental data. This can be clearly seen in the graph tan(φ) vs. σ, see Fig. 3. We refer this phenomenon to as the clusterization along the rays. It can be observed that particular values of ρ do not affect the clusterization.
The normalized room-mean square (NRMS) deviation between the experimental points and the corresponding rays is 1.7%. It means that generally the tangent of the moiré angle φ increases with the aspect ratio proportionally to it. Being an experimental observation, this fact virtually excludes an existence of a preferable aspect ratio for the minimization of the moiré effect. Nevertheless, the areas somewhere between the rays in the parameter space can be used for the minimization.
2.2 Experiment 1. Small k, l
The experimental data with the rational abscissas and small k, l seem to be arranged slightly in a different manner than the general picture described above. A subset with specific k and l is graphically represented by the vertical cross-sections of Fig. 3 at fixed values of σ.
Correspondingly, while the basic feature (ray clusterization) is definitely kept, another trend was identified in the corresponding subset of the experimental data. To verify that, eight rational aspect ratios with k, l ≤ 4 were selected from the data set (namely, 1/4, 1/3, 1/2, 2/3, 1, 3/2, 2, and 3). The shape of such cells includes the square as well as the square divided in several equal parts, as in typical LCD devices. In each particular experiment, several (from 1 to 5) moiré angles were observed, so as the total number of angles in experiments with these ratios is 26 (except for the “permanent” angles 0 and ∞). The number of the observed moiré angles φ is about 3 times higher than the number of aspect ratios; thus, a wide variety of the moiré angles could be expected. However, while the abscissas are located more or less uniformly, the most ordinates are concentrated around few values. See Fig. 4, where some horizontal lines are drawn, where the experimental data are concentrated. The clusterization can be graphically seen at the vertical axis of Fig. 4, where the data values are concentrated in 5 or more groups with some empty areas between them.
Fig. 4 Discrete moiré angles φ (experimental). The clusterization of the experimental data with “small rational” aspect ratios.
More formally, the ordinates of the experimental points in this subset can be categorized into 10 groups so as the NRMS deviation from the center of each group is about 5% or less; these centers of localization are 0.26, 0.32, 0.37, 0.51, 0.68, 0.74, 0.98, 1.38, 1.48, and 1.67. These 10 listed values together with the values 2 and 3 (which cannot be characterized statistically because of very few occurrences in this experiment) are very close to the following rational numbers: 1/4, 1/3, 3/8, 1/2, 2/3, 3/4, 1, 11/8, 3/2, 5/3, 2, 3.
This set of experimental data confirms the trend for “small rational” abscissas to be clustered around “small rational” ordinates. In other words, for the aspect ratios expressed by rational numbers with small k and l, there are few moiré angles observed with their tangents being the rational numbers.
To summarize, the most (75%) of the moiré angles φ observed in this subset of Experiment 1 with k, l ≤ 4 are the rational angles
This feature can be treated as the discrete moiré angles in grids with divided square cells. It means in particular that the most angles of the noticeable moiré effect in the square cells divided vertically in 2 or 3 equal parts are the same in the square cells. This consequence may have a practical value in applications.
2.3 Experiment 2
The discreteness of the moiré angles φ (see Sec. 2.2) was experimentally observed in LCD panels (square pixels with σ = 1) for all combinations of panels (4 devices) with barrier plates of 3 types, see Fig. 5 and Table 1, where all observed moiré angles are listed. In the graph tan(φ) vs. σ (as Fig. 3), all these data would lie along the vertical line σ = 1.
Fig. 5 Illustration of variety of moiré fringes in Experiment 2 (σ = 1, by angle). (a) φ = 0°, device D, ρ = 3.00; (b) φ = 18°, device C, ρ = 2.80; (c) φ = 33°, device B, ρ = 2.67; (d) φ = 45°, device C, ρ = 2.80. The corresponding set of the experimental data is given in Table 1.
Table 1. Observed Moiré Angles φ in LCD Panels Combined with Barrier Plates
The examples of the moiré fringes in LCD panels are shown in Fig. 5.
Similarly to the experiment with the rational aspect ratios in printed samples (Sec. 2.2, small k, l), the moiré angles φ in the square LCD pixel cells are very close to the angles with the rational tangents 0, 1/3, ½, 2/3, and 1 (in experiments, NRMS deviation between the tangents of the observed angles and these rational numbers is 2.5%).
3. Rational moiré angles classified
It was also observed experimentally that among all moiré angles φ in the rational cells with 1 ≤ k, l ≤ 6, the moiré effect occurred more frequently at certain particular angles than at others. Basing on the count of cases in Experiment 1, the moiré angles φ can be classified in “rare” and “frequent” categories like follows. If the moiré effect was observed only once, it is a “rare” angle; otherwise if more than two times (up to 12 times in context of the Experiment 1), it is a “frequent” angle. There are few cases at many “rare” angles, but many moiré cases at few “frequent” angles. These are distributed as follows, 12% of “rare” cases (8 angles, 8 cases), 79% “frequent” angles (8 angles, 53 cases), and 9% remaining (3 angles, 6 cases) which do not show any special behavior and there are not many angles and not many cases. To be more specific, the tangents of the “rare” angles are the following, 1/6, 3/10, 3/8, 2/5, 3/5, 5/3, 9/4, 3; the “frequent” tangents 0, 1, 1/2, 2/3, 3/4, 3/2, 2, ∞; the intermediate tangents 1/4, 1/3, 4/3.
In other words, the moiré effect at the “rare” angles was observed only in very few cases; and mostly, the moiré effect does not happen here. On the contrary, the moiré effect was almost always occurred at the “frequent” angles. However, at the intermediate angles, the moiré effect sometimes may, sometimes may not happen.
Furthermore, the most (79%) of the “rare” angles are the rational angles (7) with one of m, n ≥ 4. On the other hand, the most (88%) of the “frequent” angles are the angles φmn with both m, n ≤ 3. For the intermediate angles, one of m, n is equal to 3 or 4.
This classification might give a key to establish preferable angles in order to avoid (minimize) the moiré effect in displays. Correspondingly, the rational angles φmn with m, n > 4 are good candidates to be moiré-less angles where the moiré effect would be exceptional. Therefore, it can be suggested that under some additional conditions (assumable, the size ratio) such rational angles can be moiré-free angles in grids with rational aspect ratios. For instance, the particular list of the tangents of the rational angles 4 < m, n < 7 looks like follows, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6 (the angles which can be reduced to smaller m, n are not included in the list). In practical minimization, the listed angles are the ones to be tried first.
4. Discussion
The clusterization of two kinds was observed: the general ray clusterization and the particular horizontal clusterization. The current paper presents the experiments only, but no simulation. The clusterization of moiré angles φ is confirmed in our additional experiment with square pixels (σ = 1). This experiment was performed with several LCD devices and lenticular plates (instead of barrier plates); namely, we used 5 LCD display panels with square pixels and 4 lenticular plates. The pixel pitch of LCDs varied from 0.1905 mm to 0.27 mm, the pitch of lenticular plates between 12 and 40 lenticular lenses per inch (lpi), i.e., from 0.64 mm to 2.17 mm. Consequently, the size ratio ρ varied between 2.5 and 8.89.
It was experimentally observed that in this experiment with the lenticular plate no visible moiré fringes appeared across the angular range 0°- 45° except for a few fixed angles; the experimental data are given in Table 2.Referring to the rational angles Eq. (7), NRMS deviation between the tangents of the observed angles and the corresponding rational numbers 0, 1/3, ½, 2/3, and 1 is 0.5%. The graphical representation of this experiment in a graph similar to Fig. 3 would be a vertical line. Although the additional experiment needs some more detailed further investigation, it already confirms the main experiment (Sec. 2).
Table 2. Observed Moiré Angles φ in LCD Panels Combined with Lenticular Plates
It is worth mentioning here that a similar behavior of moiré patterns in a display with the aspect ratio σ ≠ 1 has been observed previously by independent researchers. For example, the research paper [17] by Y.Kim et al. describes the moiré effect in the 22” 3D display. This device has a non-typical layout of pixels; its square pixels are divided in two halves. This does not affect the visual image in this screen, because both halves work synchronously. However, this affects the appearance of the moiré effect, since the vertical period of the screen structure becomes twice shorter. In this display, σ = 2, ρ = 7.90, and the moiré “patterns are vivid and more distinct when the slant angle is 0, 18°, 19°, 34°, 35°”, as mentioned in [17]. Note that the items of this list are close to the angles with the rational tangents 0, 1/3, and 2/3. This fact can be considered as an independent experimental confirmation of the discrete moiré angles φmn in displays with divided square pixels.
It is also interesting to mention here that the widely used angle 30 between the screens in color printing [1], [18] is very close to one of the angles (atan3/5) found in the current paper.
Figure 5 (as well as Fig. 2) is only given to illustrate the variety of the observed moiré fringes, while the experimental results are presented in graphs Figs. 3, 4, and in Tables 1 and 2.
Our experimental results are based on the visual estimation of the moiré patterns; this gives a binary answer about the absence or presence of the moiré patterns basing on the patterns which amplitude exceeds certain threshold. However for a future development, the semi-automatic measurement of the moiré waves is not impossible [16], as well as using perfect instruments by ELDIM, in particular the tool [19] to measure the moiré effect in 3D displays.
5. Conclusion
In this paper we found the angles of the moiré effect experimentally. We formulate the problem of minimization of the moiré effect in terms of the Diophantine inequalities with the complex coefficients. In order to determine some possibly existing preferable values of the geometric parameters, the experiments were conducted.
In the experiments, we measured the angles φ of the visible moiré patterns for parameters of typical autostereoscopic displays. We found that the moiré angles are clusterized in the parameter space; the barrier plates from 3D displays were used with the LCD panels and with the printed samples. The angular range 0 - 90° was scanned for the moiré effect. The experiments with the printed samples were made with ρ from 0.18 to 0.72, σ from 1/6 to 3; the LCD experiments with ρ from 1.62 to 3.00, σ = 1. The additional (lenticular) experiment was made with ρ from 2.5 to 8.89, σ = 1. The overall range of parameters covered in our experiments is ρ from 0.18 to 8.89, σ from 1/6 to 3. Such parameters are typical for the multiview and integral imaging displays.
The clusterization was only observed along slanted lines and along horizontal lines (depending on values of σ), but never along vertical ones. In cells with the rational aspect ratio σkl with small k, l (as in LCD panels), the moiré patterns appeared at a few discrete angles φmn with using both barrier or lenticular plates. Since the vertical clusterization was not observed in experiments, a preferable aspect ratio for the minimization of the moiré effect was not identified. At the same time, the additional experiment with using different optics confirms the features of the moiré effect in grids with divided square pixels, in particular, the discrete moiré angles. The angles found in the additional and independent experiments are very close to the results of the Experiments 1 and 2 (small k, l, Sections 2.2, 2.3), even though the different optics was in use (lenticular and barrier plates).
For minimization in the general case, the preferred areas in the parameter space are found experimentally and expressed as rational numbers (7). For the particular case of small rational aspect ratio (like in LCD panels), the list of preferred angles is proposed. This practical recommendation is based on the experiments. It can be used in practical devices.
To verify additionally the minimization of the moiré effect basing on the preferable angles and the horizontal clusterization, an estimation of the visual appearance of the moiré patterns needs a deeper investigation.
Acknowledgments
This research was supported by The Cross-Ministry Giga KOREA Project of The Ministry of Science, ICT and Future Planning, Korea [GK13D0200, Development of Super Multi-View (SMV) Display Providing Real-Time Interaction] as well as supported in part by the Korea Institute of Science and Technology under the Tangible Social Media Platform Project and in part by the IT R&D program of MKE/KEIT [KI10035337, development of interactive wide viewing zone SMV optics of 3D display].
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