The probability of the moiré effect in LCD displays is estimated as a function of angle based on the experimental data; a theoretical function (node spacing) is proposed basing on the distance between nodes. Both functions are close to each other. The connection between the probability of the moiré effect and the Thomae’s function is also found. The function proposed in this paper can be used in the minimization of the moiré effect in visual displays, especially in autostereoscopic 3D displays.
© 2015 Optical Society of America
The moiré phenomenon relates to an optical interaction between semi-transparent layers . The layers are typical for autostereoscopic displays ; one layer usually is the pixelated LCD screen, another is a lenticular or barrier plate; the ratio of the periods of layers is commonly integer. The layered structure with the integer ratio gives rise to noticeable moiré patterns which reduce the visual contrast of a displayed image. To improve the image quality in the autostereoscopic three-dimensional displays, it is important to reduce (minimize) the moiré effect.
A variation of the angle between the LCD panel and the barrier plate changes the period and the orientation of the moiré patterns . When the period of the moiré patterns reaches a local maximum, the visibility of the patterns become highest among neighboring angles (this is based on the visibility assumption ). The graph of the experimentally measured the moiré period as a function of angle is shown in Fig. 1 for two display devices with the same lenticular plate 50 lpi (period 0.508 mm) but the different period of pixels (0.18 mm and 0.266 mm). In this experiment (similarly to ), the moiré patterns with the period shorter than 0.8 mm were not observed. Correspondingly, the angular ranges with no moiré patterns in both devices are indicated in Fig. 1 by dots at the horizontal axis.
There are 5 maxima of different height and width in each device of this experiment; however this picture may look different in other devices, for instance, the number of maxima is often different (see, e.g., ), as well as the relative height of the peaks. Typically, the peaks 0 and 45° are the widest, and the moiré patterns around these angles were detected within the angular regions approx. ± 16° and ± 6°, resp. Three other peaks are essentially narrower and their half-width is between 1° and 2.5° (see Fig. 1).
We refer the angle with the maximal period of the moiré patterns to as the moiré angle. In most LCD panels, the aspect ratio of square pixels equals 1; such pixels represent rational cells. In general, the moiré angles in rational cells with the rational ratio of sides are the rational angles ,Fig. 1 the rational angles are shown by triangles; in this experiment, both m and n are not higher than 3.
At the moiré angles (i.e., when the period reaches a local maximum), the moiré patterns are parallel to the axis of the rotated grating. This can be explained as follows in the spectral domain. The wavevector of a moiré pattern is the ray from the origin to the spectral component under consideration. The change of the angle α1 between the grating and the plate (see Fig. 2) makes the rotation of the spectral component around the center A. This rotation can be treated as the movement of the component along a trajectory (circular arc l in Fig. 2). On this movement, the moiré period is changed together with the orientation of the moiré patterns (the angle α2), as described in .
While the trajectory l passes the neighborhood of the origin, the period reaches a maximum. This happens at point B, where the trajectory l crosses the line OA connecting the origin O and the center A of the trajectory. In this case, the angles α1 and α2 are equal, and the moiré patterns are parallel to the axis of the grating.
In respect to the moiré effect, the rational angles have different significance, since it was experimentally observed in  that at different angles the moiré effect happens more or less frequently. In displays with rational cells, the moiré angles can be classified into two categories at least, “often” and “rare” angles . Besides, it was mentioned in the discussion to  that the moiré patterns in lenticular displays also appear at the rational angles. Correspondingly, in the current paper we consider the moiré effect in periodic layers with the rational grid cells in terms of probability.
Previously, the probability function was applied to the elimination of the moiré patterns . The probability of the time-averaged moiré fringes was considered in . The probability was mentioned in  for the moiré superstructures. In this connection, it is also worth to mention that the moiré patterns in periodic gratings can comprise non-periodic structures  which could be considered from statistical point of view. In this case, the non-periodicity is caused by the well-known mathematical fact that the plane can be filled with regular polygons of very few kinds (namely, triangles, squares, and hexagons only), see, e.g [11,12]. The moiré effect in aperiodic layers (so called Glass patterns) is considered in  and in the references therein, although statistical considerations are not taken into account. Thus, the usage of statistics for studying the moiré phenomenon is not common. To the authors' knowledge, a statistics-based moiré method was applied to the characterization of the moiré structures for the first time in .
In the current paper, we deal with the moiré angles (the local maxima of the moiré period). We analyze the moiré occurrences in double layer devices  from the statistical point of view and build the corresponding experimental probability function similarly to the previous report . We also build the experimental probability function for the moiré effect in autostereoscopic LCD displays with lenticular plate which was preliminary presented in . The experimental probability function  is based on the previous experiments  with the barrier displays, while the function  is measured in lenticular displays. The previously published conference papers [15,16] related to this topic consider only few details of the minimization, and did not study the barrier and lenticular displays together, as we do in the current paper, where we propose a combined function for both types of the autostereoscopic displays.
Another difference between the current paper and the mentioned conference papers is in the complex approach which allowed us to build the theoretical function basing on the geometry of the square grid and to make a detailed comparison with a related known function. We compare the experimental and theoretical data in the autostereoscopic displays of barrier and lenticular types and build the composite function. For this paper, we additionally found an extra function with similar behavior. At the same time, the previously known functions were not applied to the minimization of the moiré effect.
The current paper is based on the experimental results with several display devices described above (Fig. 1). One more advantage is that the current paper presents the technique to find the maximal period of the moiré patterns basing on their angle relative to the rotated grating (Fig. 2).
2.1 Analysis of the experimental probability for barrier displays
In the experiments , a continuous angular range was tested for the moiré effect in LCD panels (square cells), as well as in printed rectangular grids (rectangular cells of various aspect ratios) printed on the paper and transparencies. In these experiments, the barrier plates were applied to the panels and samples. A static image was displayed, a black (opaque) rectangular grid on the while (transparent) background.
The visual observation was made from the distance 50 cm; the photographs were taken from the same distance. The examples of the photographed moiré patterns are shown in Fig. 3 (in both photographs, the angle between the gratings is 0°).
The angle between the static grid and the barrier plate was changed from 0 to 90°. The period of the moiré patterns was estimated and the local maxima of the period were recorded as the moiré angles. There was a variety of combinations of cells and plates in that experiment; the gratings with 13 different values of the aspect ratio were used.
The experimental data  show that within the angular range 0 – 90°, the moiré effect has maxima at 25 angles between the axis of the plate and the vertical axis, and there were typically from 3 to 8 moiré angles for each particular σ. In the rational cells (either LCD or printed), the detected moiré angles are the rational angles by Eq. (1). The experimentally observed rational angles satisfy the condition m, n ≤ 11; although for the great majority of them (more than 70%), the involved numbers are considerably smaller, i.e., m, n ≤ 5.
Using the experimental data , the probability of the moiré effect can be estimated basing on the amount of the observed moiré cases for all screens combined with different barrier plates, as in . The number of successful attempts (when the moiré effect was observed) is counted for each angle with no regards to σ. Then, the relative count of the successful cases represents the probability.
The resulting function for barrier displays is shown in Fig. 4. The probability is shown by vertical bars. This is an experimental function for the probability of the moiré effect observed in experiments, which shows an expectation of the moiré occurrence versus angle. The function models the maxima of the moiré period only, and therefore the value of function between the peaks is zero.
2.2 Experimental probability for lenticular displays
The experimental probability function in the lenticular displays  was measured in the manner similar to the barrier displays as described in Sec. 2.1. Experiments were made with 8 displays (the screen size from 7 in. to 24 in., 1 in. = 2.354 cm; the pixel pitch from 0.06 mm to 0.27 mm) and 5 lenticular plates (15 – 75 lpi, i.e. the period from 0.39 mm to 1.69 mm) in arbitrary combinations; for details about the experimental equipment, the reader is referred to Sec. 5. Examples of the observed moiré patterns are shown in Fig. 5.
Similarly to the barrier case, the current experiment with the lenticular plates confirms the rational angles in LCD displays, as it was expected in . Experimentally observed were 9 rational angles. The lenticular probability function built exactly as the barrier probability function in Sec. 2.1 is given in Fig. 6.
2.3 Lenticular and barrier displays together
It can be noticed that the probability function for the lenticular displays looks similar to that of the barrier displays, although the number of the moiré angles is different. The locations and heights of the peaks of both functions in Figs. 4 and 6 nearly coincide.
Therefore, a composite probability function for both barrier and lenticular displays together can be built basing on two experimental data sets. It merges the weighted barrier and lenticular functions. The graph is shown in Fig. 7.
In this section we consider the angles as themselves, without any relation to the moiré effect. The angles can be can be characterized in variety of ways, i.e., by grade of slope, by number of turns, by circular arc, by trigonometric functions, etc. In particular, the rational angles satisfying Eq. (1) can be characterized by pairs of non-negative integer numbers (m, n), as in .
The rational angles on the square grid are illustrated in Fig. 8. The circles indicate the points of intersection of the nodes of the grid and the rays from the origin, while the numbers near the sides of the graph are tangents of some angles. The non-rational angles lie somewhere between the rational ones.
The rays with the rational slopes repeatedly cross the nodes (jm, jn), j integer (their tangent equals m/n); whereas the non-rational rays do not cross nodes at all (and the tangent cannot be represented as a ratio of any integer numbers). A numerical characteristic of a ray crossing nodes could be the distance between nodes along rays in the square grid (i.e., spacing). For example, the ray m = 0, n = 1 has the unit distance between the nodes, while the ray m = n = 1 has the distance.
Formally, the spacing can be defined by the number of crossed nodes per length unit. It can be calculated basing on the 1st intersection point (which corresponds to the coprime m and n). The non-rational rays never cross a node; therefore on such rays the distance is infinite and the spacing is exactly zero.
We define the spacing as a reciprocal distance between the nodes
The spacing is shown in Fig. 9 as a function of an angle. This is a theoretical mathematical function which describes the angles in connection to the nodes of the square grid.
4. Experiment vs. theory
The experimental and theoretical functions presented in the previous sections of this paper seem to be close to each other. In particular, almost all peaks from Fig. 7 have their counterpart in Fig. 9. The locations of the corresponding peaks of both functions practically coincide; their heights are close too.
Generally, the similarity of functions can be estimated basing on the distance between them. In order to make a comparison, we define the distance between two functions F1 and F2 basing on the root-mean-square (RMS) deviation,
Defined as above, the calculated distance between the functions of the probability and the spacing (i.e. between the experimentally measured probability and the theoretical spacing of nodes) for the barrier displays is 11.8%. The probability for lenticular displays can be similarly compared to the theory; in this case, the distance between the experimental and theoretical functions is 15.5%. Similarly, the distance between the composed probability function and the theory is 14.2%.
All three above distances are about 15% or less. Therefore, the spacing function defined by Eq. (2) can be considered as a reasonable mathematical approximation for the experimental probability of the moiré effect.
In this section we describe the experimental equipment of the current and previous experiments, briefly discuss the difference between the moiré effect in lenticular and barrier displays, and consider the related functions.
The characteristics of the experimental equipment are as follows. The periods of the barrier plates for experiment  were 0.44 mm, 0.45 mm, and 0.57 mm; while the period of printed samples varied between 0.24 mm and 2.4 mm; the aspect ratio of these samples was between 0.33 and 3. The lenticular plates for discussion to  and for the current experiment are listed in Table 1. The LCD panels of both experiments are listed in Table 2.
There is a difference between the moiré effect in displays with the barrier and with the lenticular plates. For the barrier displays, the distance from the LCD panel to the barrier plate as well as the distance from the display to the observer does not affect the observed moiré angles. The moiré patterns in the barrier displays appear at the same angles with no respect to the mentioned distances. However in the lenticular case, there is the distance between the LCD panel and the lenticular plate which provides the strongest moiré effect. Similarly, the distance to the observer affects the visibility of the moiré patterns. Therefore, the moiré angles In a LCD panel may depend on the mentioned distances. In this paper, the distance to the plate is near the focal distance of the lenticules (according to the well-known thin lens equation), otherwise, the moiré patterns could not be observed at all. In both lenticular and barrier cases, the distance to the camera is about 50 cm.
One of the related functions is the known Thomae’s function  which characterize the rational numbers, i.e., essentially the tangents of angles. The Thomae’s function is defined as follows (see also Fig. 10),
The locations of peaks of Thomae’s function (after certain renormalization of its abscissas, because the Thomae’s function is defined on the segment between 0 and 1 while the spacing function for the angles between 0° and 90°) coincide with the peaks of the spacing function, although the heights are slightly different.
The distance between the experimental probability and Thomae’s function can be found basing on RMS Eq. (3). For the barrier case it is 12.5%, for the lenticular 18.7%, and for composed function 15.2%. These percentage values are very close to the percentage differences of the spacing function (the difference is less than 3%). However, basing on such small difference, the preferable theoretical function for an analytical estimation of the moiré probability cannot be unambiguously chosen, because the distance between the functions themselves is not less than 8%.
The relationship between the spacing and Thomae’s functions defined by Eqs. (2) and (4) can be described like follows. The ratio of these functions for the rational angles is
For m/n ≤ 1, the square root in Eq. (5) can be replaced by the infinite series for (m/n)2
When m/n < 1/2, the second term in Eq. (6) is approximately one order less than the first and therefore can be neglected; however when the ratio m/n is increased, the two terms become comparable. Therefore, for smaller proportions m/n (less than 1/2, i.e., the angles < 27°), the two functions nearly coincide, but for larger ratios (the angles > 27°), the functions become essentially different.
Another function, the quality function  looks very similar to the Thomae’s function (and therefore to the experimental probability of the moiré effect too). In Fig. 11, the maxima of the quality function are shown in relative coordinates across one period of the function in the projective space. The quality function estimates the image quality of an autostereoscopic display ; the function varies from 0 (lowest quality) to 1 (the highest quality). Despite of the completely different nature of this function, the locations and the heights of peaks of both functions are very similar. Also, it is interesting in the current content that the “envelope” line (similar to the dashed line from point (0, 0) to point (1, 1) in Fig. 11) can be clearly recognized in all cases previously mentioned in the current paper: the probability, the spacing, and the Thomae’s functions.
The probability of the moiré effect is calculated basing on the experimental data as the function of the angle. The probability functions for lenticular and barrier displays are built separately and additionally merged into a composite function.
A mathematical description of the probability of the moiré effect in visual displays with the rational cells is proposed. Two mathematical functions can be used for the estimation of the probability: the spacing function Eq. (2) and the known Thomae’s function. Although the formal comparison of the proposed and the known functions does not give a definite answer, the spacing function looks more efficient for the practical minimization of the moiré effect, because the proposed function is defined in the wider angular range and includes the variables for two dimensions.
The results of the current research can be used in the minimization of the moiré effect by angle. The proposed function makes the minimization simpler. Basing on the experimental measurements and statistical calculations presented in the current paper, there could be two following options to minimize the moiré effect: either to use non-rational angles (where the moiré patterns were not observed experimentally and the probability is zero), or to select a rational angle with a low probability of the moiré effect. In any case, one should keep in mind that the width of the moiré peaks is finite and does not equal to zero even for the narrower peaks, as shows the experimental graph Fig. 1. This is especially important for improving the image quality in the autostereoscopic 3D displays, where the moiré effect may essentially reduce the visual contrast of the image.
This work was supported by 'The Cross-Ministry Giga KOREA Project' grant from the Ministry of Science, ICT and Future Planning, South Korea.
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