A mathematical formula of calculating the fringe periods of the color moirés appearing at the contact-type 3-D displays is derived. It is typical that the color moirés are chirped and the period of the line pattern in viewing zone forming optics is more than two times of that of the pixel pattern in the display panel. These make impossible to calculate the fringe periods of the color moirés with the conventional beat frequency formula. The derived formula work very well for any combination of two line patterns having either a same line period or different line periods. This is experimentally proved. Furthermore, it is also shown that the fringe period can be expressed in terms of the viewing distance and focal length of the viewing zone forming optics.
© 2016 Optical Society of America
The thickness, i.e., widths of the lines forming the line pattern and plate thickness of viewing zone forming optics (VZFO) in the contact-type 3-D displays , are the main sources of causing moirés in them [2,3]. In the 3-D displays, when a VZFO is superposed on the top of a flat display panel, two inseparable physical interactions appear between the pixel pattern in the panel and VZFO’s line pattern formed by elemental lenses or line slits: The first interaction is that a part of the light coming from pixels is blocked by the boundary lines between adjacent elemental lenses or the gaps between the slits. Due to this blocking, the colors appearing through the VZFO becomes different from those on the display panel and the brightness of the image through the VZFO becomes reduced in proportional to the blocking amount, in compared with that on the panel. The second interaction is the mixing between the VZFO’s line pattern and the pixel pattern. This mixing induces moirés. Since the two patterns are having different periods, the VZFO’s line pattern will either overlap or align side by side to the pixel pattern. This overlapping will make parts of the pixel pattern invisible. Hence as the width of the dark lines forming the VZFO’s line pattern increases, the invisible parts will also be widened and the gaps between the aligned lines will be reduced. These overlapping and aligning create sparse and dense regions, i.e., bright and dark regions. If the VZFO and pixel patterns are periodically drawn as line grating patterns, these densely and sparsely mixed regions appear periodically. This periodical appearance of dense and sparse regions is named as moirés. So for the creation of moirés, it is necessary to make the two patterns to be mixed together and to appear at a plane. This is why the moirés induced by the gap effect are interpreted within the image planes of visual devices . In the 3-D displays, the moirés are colored and their fringes are chirped because the VZFO has a finite thickness with a certain refractive index and the lines in its pattern have a width. These make the mixing to appear differently from the gap effect. The VZFO thickness makes the panel pattern to refract toward viewer’s eyes when a viewer watches the panel . Hence the panel’s pixel pattern virtually appears at the pattern plane of the VZFO and is mixed with the VZFO pattern to form the moirés. This also explains why the moiré fringes look floating above the image on the panel. The refraction also causes the panel pattern at the VZFO’s pattern plane to have a different form from its pattern at the panel plane. The pattern is virtually contracted by the refraction process when it passes through the VZFO. The contraction will be more as the viewing angle of the pattern increases. Hence the contracted pattern has the form of a chirp signal. The chirping starts from the normal direction of viewing the panel and the viewing angle changes with the viewing distance.By this reason, the moiré fringes are also shifting as a viewer changes his/her viewing position and the fringe periods change as the viewing distance changes. The amount of contraction is functions of viewer’s viewing angle, thickness and refractive index of VZFO. Due to this chirping, the moiré fringes are also chirped and they appear even when the periods of two patterns are the same. The colors in the moiré fringes are mainly caused by the line width because the sub-pixel colors are differently blocked by the width . Hence the thickness of the VZFO and the line width of VZFO’s line pattern are the main source of viewing position dependent color moiré fringes. The problem with these color moiré fringes is that they are harmful to the 3-D displays [5–8] because of their hostile behaviors of deteriorating the 3-D image quality. However there is not a proper way of characterizing the moiré fringes. The fringe period is the first parameter used to characterizemoirés appearing when two regular line patterns are overlapped together. However, the conventional beat frequency formula cannot describe the fringe period of the color moirés appearing at the contact-type 3-D displays. This is because the moirés are chirped and the period of VZFO line pattern is usually more than two times of that of the pixel pattern in the display panel for the multiview images. The formula can work only when the periods of the two line patterns have a small difference to initiate a beat frequency. This is only formula known so far but when the periods of the two line patterns have a large difference as the contact-type 3-D displays, it does not work. A new formula is needed to describe the 3-D displays case.
In this paper, a new mathematical formula to calculate the fringe periods of the color moirés appearing at the contact-type 3-D displays is derived. And the accuracy of the formula is shown with several VZFO line patterns having different line periods for given display panels.
2. Derivation of moiré fringe periods in the contact-type 3-D displays
The main source of causing color moirés in contact-type 3-D displays has already been mentioned in the introduction but they work differently for different types of VZFO, such as lenticular and parallax barrier. In the lenticular and parallax barrier, the line patterns are formed by the boundary lines between elemental lenses and line slits, respectively. The width and the transparency of the boundary lines are not known but it is considered as very thin compared with the pitch of the elemental lenses. However, the slit width is typically in the range of one third to one quarter of the distance between slits to maximize the image brightness in the parallax barrier. This makes the dark line width of the parallax barrier be in the range of two thirds to three quarters of the distance. The moirés are induced by the boundary lines between elemental lenses for the lenticular, but by line slits for the parallax barrier. Since the boundary lines are dark and the slit lines are bright, the moiré fringes have higher brightness for parallax barrier than for the lenticular. This means that moiré fringes in the parallax barrier can be brighter than the image on the display panel but darker in the lenticular. However, the relative brightness of the moiré fringes in the lenticular and parallax barrier depends mainly on the pitch sizes of the barriers and elemental lenses relative to the pixel pitch. Hence the effect of moirés on the quality of images in the contact-type displays having the lenticular as their VZFO will be smaller than those having parallax barrier as their VZFO. Since moirés are having periodic fringe patterns, their periods are deeply involved in determining their contrasts and visibility. As the periods become smaller, they become more visible because the intensity and color variations in moiré laden images will appear more frequently. When the pitch of a sub-pixel and the period of elemental optics in VZFO are specified as and , respectively, the conventional beat frequency formula [9,10] describes the fringe period as, . But this formula does not work for predicting the periods of the moirés appearing at the contact-type 3-D displays because the period of elemental optics in the VZFO is much larger than that of the pixels in the panel. This is obvious because the moirés are formed only by the boundary lines of the elemental optics. So the new formula shoud offset the large period difference between the VZFO line pattern and the pixel pattern. The period difference can be offset by the following way: , where is remainder andrepresents the integer value produced by dividing into .Then the moiré fringe period when the VZFO has no thickness, is expressed as,
Equation (1) is different from conventional beat frequency approach of calculating moiré fringe period by the factor. Without this factor, moiré fringe period can be smaller than when . This is contradict to the beat frequency concept. Equation (1) can be derived by the following way too: Since is the remainder, elemental optics are needed to sweep a sub-pixel width for the case when . This represents the case when the period of the bottom pattern is smaller than that of the top pattern. However, when, the number of elemental optics required to sweep out is not but . For this reason, the 2nd relationship in Eq. (2) should be used, though it forces the denominator in Eq, 1 to have a negative value. Hence this represents the case when the period of the bottom pattern is bigger than that of the top pattern. Equation (1) is obtained by substitutingeither or for based on the criterion specified in Eq. (2). With Eq. (2), the waveform of the moiré fringe in the conventional formula  should be replaced by , where is the distance along the horizontal direction of the display panel, from the point which corresponds to the normal position of viewer’s an eye on the panel. When VZFO thickness is considered, the pixel pitch will be no longer constant but it will virtually vary along the horizontal direction due to the refraction phenomenon. The graphical illustration of the pixel pitch variation is shown in Fig. 1. The physical structure of the contact-type 3-D displays consists of a display panel and a VZFO which is superposed on the panel. In this structure, the panel’s pixel array looks appearing at the VZFO’s line pattern plane due to the refraction effect and it is blocked by the line pattern to form a color moiré as explained in previous section. The refraction effect forces the pixel pitch to be contracted slightly.
The amount of the contraction depends on the thickness and the refractive index of the VZFO’s medium, and the panel distance from the viewer’s normal position of viewing the panel. As a result, the pixel pitch on the pattern plane shrinks more in the close distances from the viewing position but less in the farther distances.This means that the pixel pitch is virtually chirped. The color moiré fringes are created by the superposition of the virtually chirped pixel pattern and the VZFO’s line pattern. They are nothing but the virtually chirped pixel pattern blocked by the barriers, i.e., lines on the VZFO’s line pattern. In the gap between neighboring barrier lines, the moiréfringes represent the virtually chirped pixel pattern. Hence if the gap width is more than two times of the pixel pitch, at least a pixel will be untouched. This is the reason why the current beat frequency formula fails to calculate the periods of the moiré fringes. If the virtually varying pixel pitch is represented as , it can turn to a continuous function by an interpolation and have a chirped signal form. If is normalized as , its chirped signal phase is expressed as . In this case, the moiré fringe in a continuous waveform will be written as,
In Eq. (3), since is the function of and is when is an integer multiple of , is only determined by . This means that will have the same value when is any integer multiple of . The VZFO thickness turns visually the uniform pixel pattern into a chirped pixel pattern as increases, as explained in the previous section. In Eq. (3), will be zero when equals either or , where …. Since the periods of , is defined as the distance between values corresponding to the two phase values, they are calculated as,
In the contact-type multiview 3-D displays, the relationship between periods of pixel cell/elemental image and the elemental optic in VZFO is given as ,Eq. (5), . Equation (5) can be rewritten as,
The contact-type multiview 3-D displays are classified into two groups by their optical configurations such as parallel and radial: The parallel group satisfies the relationship and the radial group . The parallel group includes those displays having an elemental image as their basic image unit and the radial those having a pixel cell . For the case of , , i.e., . Hence Eq. (8) can be rewritten as,Eq. (9) is still valid. Equation (9) relates the moiré fringe periods with the system parameters of the contact-type multiview 3-D displays. It informs that theperiods of the moiré fringes appearing at the displays are also affected by the viewing distance and focal length of VZFO. Equation (9) indicates that as becomes smaller, i.e., the designed viewing distance increases, the moiré fringe period will be slightly increased.
3. Experimental set-up
With the equations developed at section 2, the moiré fringe periods are calculated for five different VZFO line patterns having the nominal period values of 0.4833 mm, 3.4 mm, 1.608 mm, 1.4499 mm and 1.4509 mm as shown in Table 1. VZFO line patterns are drawn on polystyrene films having thickness of 0.18 mm with a photoplotter (UCAMCO Calibr8tor NaNOII series). The manufacturer’s specification indicates that it has more than 20,000 DPI (Dot per Inch) resolution . This resolution value corresponds to near 0.001 mm. All these films, except 1.608 mm, are superposed on the 42 inch monitor having a full HD resolution (1920 X 1080). To stick the film on the glass surface of the monitor as evenly as possible, a transparent plastic plate is fixed to the monitor and the film is inserted in between the plate and the glass surface. The thickness of VZFO is set to 0.68 mm by considering the protection glass thickness of the monitor, 0.5 mm. The refractive index of the VZFO is set to 1.5412 by taking arithmetic average of the refractive indexes of the film (1.6) and the glass (1.52). The film length of 0.4833 mm and 3.4 mm line periods is 750 mm and others 830 mm. For the 1.608 mm, the film was bended to fit to the monitor size. The sub-pixel pitch of the monitor is 0.1611 µm. For the 1.608 mm, it is superposed on a 28 inch UHD monitor with the resolution 3840 X 2160. The nominal value of the pixel size of the monitor is 0.16 mm but in this paper 0.1617 mm is used by dividing its nominal width value 620.93 mm by its horizontal pixel resolution 3840. This pixel value matches well with current experiment. The pattern period values indicate that 0.4833 mm and 1.4499 mm are corresponding to integer multiple of a pixel/a sub-pixel size, i.e., 0.4833 mm a pixel (three sub-pixels) and 1.4499 mm 3 pixels (9 sub-pixels), 3.4 mm (3.4/0.1611 = 21X0.1611 + 0.0169 mm) and 1.4509 mm (1.4509/0.1611 = 9X0.1611 + 0.001 mm) represent case and 1.608 mm (1.608/0.1617 = 9X0.1617 + 0.1527 mm) case. For the cases of 0.4833 mm and 3.4 mm, two different line thicknesses of 0.0805 mm and 0.4028 mm for 0.4833 mm, and 0.3222 mm and 3.0778 mm for 3.4 mm are used. These values are to demonstrate the line thickness effects on moiré fringe colors and their contrasts. All these values are tabulated in Table 1 with the fringe periods obtained with Eq. (1) and line thickness values. In Table 2, the symbols used in this paper are summarized.
The color moiré fringes appearing when each of the films on Table 1 is superposed on its appropriate monitor are viewed with a NIKON camera (D700) equipped with a wide field of view objective (G-type AF-S zoom –Nikkor lens) . The optical axis of the camera is aligned to the normal direction of the monitor. The viewing distance, i.e., the camera distance from the pattern plane of the VZFO is set to 500 mm and 1,000 mm to show the fringe period variations with the viewing distance and the accuracy of Eq. (4). The viewing distance of the moiré fringes is defined to the pattern plane where the fringes are formed.
4. Experimental results
There are several factors of reducing the moiré fringe contrasts; 1) decrease in the line thickness, 2) the presence of extra plastic plate to hold the film tight to the monitor and 3) the camera imaging. The reduction will be more as the viewing distance increases. Furthermore, the boundary between different colors can be hardly defined with the film because it cannot be attached completely to the monitor surface for all its length. This makes the direct comparison of the moiré fringe periods with the calculated by Eq. (3) difficult. To ease this difficulty, moiré fringes appearing when the films of different line periods and thicknesses are superposed on the monitors, is simulated and displayed simultaneously with the films on the monitors. In this way, the colors and the boundary between different colors of two moiré fringes can be easily compared. The waveform of the calculated moiré fringe periods by Eq. (3) is drawn above the original simulated moiré fringes in the following figures. Both the waveform and the simulated fringes are calculated to cover two times of the monitor widths to show at least a complete moiré fringe period. The starting point of calculating each moiré fringe and its period are the center point of the monitor. A vertical line in the center of the waveform which depicts the calculated fringe periods in each of the following figures, represents the center point. If the two moiré fringes on the monitor match to each other and the waveform also matches with the simulated, it can be said that Eq. (3) is a correct formula for predicting the periods of the moiré fringes induced when two regular pattern plates having a large difference in their periods are superposed. The lengths of the films are specified by two arrow headed lines in following figures to identify clearly the moiré fringes induced by the films. The moiré fringe simulation has been done with the same technique used in reference 2.
Figure 2 compares moiré fringes between the simulated and experimentally obtained (Noted by “with film” in each figure) to verify the accuracy of the fringe periods calculated by Eq. (4) forthe case when the film with the line period 3.4 mm is superposed on the 42 inch monitor. Figures 2(a) and 2(b) is for comparing the relative brightness of the moiré fringes, i.e., the contrast of the moiré fringes for two different line thicknesses of 3.0776 mm and 0.3222 mm viewed at viewing distances of 500 mm and 1,000 mm, respectively. As shown in Figs. 2(a) and 2(b), the 0.3222 mm reveals barely visible moiré fringes but for the 3.0776 mm clearly visible moiréfringes. The simulated moiré fringes on the monitor look lost their details compared with the original and even different. However, the two moiré fringes on the monitor look matched closely in both colors and their boundaries.
Since the line thickness of 3.0776 mm brings the slit width of 0.3222 mm. and 0.3222 mm 3.0776 mm, the former can be considered as representing the parallax barrier and the latter the lenticular. Figures 2(a) and 2(b) informs that the parallax barrier can bring much more visible moirés than the lenticular, though the simulated moiré fringes on the monitor appears fainter than the original due to the photographing, The color compositions and periods of the moiré fringes from the simulated and the film, match well to each other at both viewing distances. The periods of the moiré fringes are plotted on the top of each simulatedmoiré fringes as a waveform. Each period of the waveform represents exactly one of the repeatedly appearing color groups, i.e., one of the moiré fringes. As mentioned before, the moiré fringes are calculated for two times of the monitor width (), However, since the camera’s optical axis corresponds to the normal direction of the monitor surface, the periods will be symmetric along a line drawn vertically at the centerof the monitor.The vertical line in each waveform represents the center line. The number of moiré fringes is slightly more than 21 for the viewing distance 500 mm but less than 21 for 1,000mm. As expected, the fringe periods are affected by the viewing distances but not by the line thickness. The fringe periods are slightly increased with increasing distances. In Fig. 3, the moiré fringe periods between peak points of the waveforms along the right (left) side of the center line are depicted for both viewing distances. The horizontal axis represents fringe numbers counted from the center and the vertical axis period of each fringe. Figure 3 indicates that the periods increase as away from the center line. This clearly demonstrates that the fringes are negatively chirped. The periods of the moiré fringesfor the 500 mm is shorter than those of the 1,000 mm until ,but they surpass those for the 1,000 mm for farther than the 750 mm. This is because the period increment is more for the 500 mm than the 1,000 mm. As the viewing angle increases, the viewing angle increment induced by a pixel distance affects the angle less and less. Hence the period increment will be almost saturated as the distance increases further.
Figures 4-6 also compare moiré fringes between the simulated and experimentally obtained as in Fig. 2. Figure 4 is the case when the line periods are integer multiples of the 42 inch monitor’s pixel/sub-pixel pitch. The films with the line periods of 0.4833 mm and 1.4499 mm are the case for the integer multiples as shown in Table 1. Figures 4(a) and 4(b) isthe 0.4833 mm case viewed at 500 mm and 1,000mm, respectively, and each of them contains the moiré fringes for two different line thickness (Width) of 0.4028 mm and 0.0805 mm to compare the fringe contrasts. Figures 4(c) depicts the 1.4499 mm case when the moiré fringes are viewed at both 500 mm and 1,000mm distances. These line periods should notinduceany moiré fringes when the VZFO has no thickness as indicated by Eq. (1). However, Fig. 4 shows distinctive color moiré fringes. These fringes are caused by the thickness of the VZFO. Figures 4(a) and 4(b) also shows that the line thickness difference induces the difference in color compositions and contrasts of the moiré fringes.The distinctive R. G. B colors for the line thickness of 0.4028 mm are much more visible than the sky blue, yellow and violet for the 0.0805 mm. Since these colors appear repeatedly, they are forming a moiré fringe for the given line thickness. Figure 4(c) also reveals low contrast moiré fringes due to relatively narrow lines, i.e., 0.2415 mm compared with the line period. The colors consisted of a moiré fringe are not different from the line thickness 0.0805 mm of the 0.4833 mm, except more visible color transition regions. In Fig. 4, the color compositions of the moiré fringes from the simulated and the film match well to each other at both viewing distances but the color boundaries are not well defined for the film. This results a small mismatch between the simulated and the film, especially for the 0.0805 mm of the 0.4833 mm and the 1.4499 mm cases when they are viewed at 1,000mm. Added on these mismatches, the low contrast of the moiré fringes on the monitor makes hard to recognize them, even for the simulated. These mismatches will always be there because gaps and pattern line mismatches between the film and the monitor surface can be hardly avoidable. The film does not have a perfect flatness. In this regard, it is considered that moiré fringes on the simulated and the film match well. As specified by the waveform, a period of the waveform for each case covers exactly the three colors representing a moiré fringe. The periods of the moiré fringes are given as 602.432 mm and 1,204.964 mm for the 500 mm and the 1,000mm, respectively. The fringe period of the 1,000mm is slightly more than two times of that of 500 mm and much longer than the monitor width. The fringe periods of the moirés in Fig. 4(c) are not different from those of Figs. 4(a) and 4(b) as indicated by Eq. (3). For the case of the viewing distance 500 mm, the second peak of the waveform appears at 990.771 mm from the center line. Since the distance between the 1st maximum and minimum points of the waveform is 301.216 mm, the distance between the 1stminimum and 2nd maximum points will be 689.555 mm. This value is even bigger than the first fringe period 602.432 mm. This indicates that the period of the next fringe will be longer than the first.The moiré fringes are chirped. The fringes for the 1,000mm will also be chirped. Figures 2 and 4 clearly indicate that the contrast of the moiré fringes decreases as the line thickness decreases for the given VZFO line period .
The moiré fringes for the line period of 1.4509 mm are depicted in Fig. 5 for viewing distances of 500 mm and 1,000mm. Since the 1.4509 mm is almost the same as the 1.4499 mm and the line thickness 0.2415 mm is the same as Fig. 4(c), the colors consisting of the moiré fringes are not different from those in Fig. 4(c), except their shorter periods. The moiré fringes on the monitor are not clearly visible as in Fig. 4(c) but they are still comparable. The colors and color regions of the moiré fringes match well to each other. The period of the moiré fringe in the mid area is increased to 369.87 mm (455.02 mm) from 312.46 mm (429.89 mm) at the viewing distance 500 mm (1,000mm). The periods specified by the waveforms are increasing as the viewing angle/the viewing distance increases. However, each of these periods covers exactly the three colors of sky blue, yellow and violet. This behavior of the moiré fringe is the same as that in Figs. 2 and 4. The moiré fringes for the 1.608 mm which is superposed on the UHD panel are depicted in Fig. 6 for two viewing distances of 500 mm and 1,000mm. This is the case of according to Eq. (2),since 1.608 mm = 10 X 0.1617 mm - 0.009 mm. This case is different from Figs. 2 and 5 which represent cases. The −0.009 mm means that the period of the top plate (VZFO) pattern is bigger than that of the bottom plate (Monitor) pattern.
Figure 6 shows a very visible color pattern which is appearing repeatedly. The color pattern is a moiré fringe. There are some fringe distortions due to the bending of the films at the right side and also the fringes from the film are slightly shifted to the left compared with the simulated for both distances, though 1,000 mm reveals less shifting. This shifting is hardly adjusted to minimize the distortions by the bending, even with the plastic plate to tighten the film on the monitor.
Other than this shifting, the colors and periods of the moiré fringes from the film and the simulated are closely matched to each other. Each period of the waveform of the given distance are also accurately representing the length of the color pattern. There are approximately 38 moiré fringes for the distance 500 mm and 40 for 1,000mm. This means that the periods of the moiré fringes for the 500 mm are slightly longer than those for the 1,000mm. The periods of the moiré fringes for the 500 mm and 1,000mm are compared in Fig. 7. In Fig. 7, the moiré fringe periods between peak points of the waveforms along the right (left) side of the center line are depicted for both viewing distances.The horizontal axis represents fringe numbers counted from the center and the vertical axis period of each fringe. The periods decrease as away from the center line. Hence the fringes are positively chirped. The periods of the moiré fringesfor the 500 mm is longer than those of the 1,000mm to the simulated distance range but they become closer to each other as the distance increases. The periods for the 500 mm will be smaller than those for the 1,000mm as the distance increases further. This is because the period decrement is more for the 500 mm than the 1,000mm. The fringe behaviors of Fig. 6 are completely opposite to the those of Figs. 2 and 5.
The color moiré fringes appearing in the contact-type 3-D displays, can be characterized by Eq. (4). This is because Eq. (4) canaccurately predicts the periods of the color moiré fringes. The equation informs that 1) the moiré fringes are chirped, 2) when the period of a line pattern is any integer multiple of another line pattern, the moiré fringe period is the same without regarding to the integer number and 3) the behaviors of the moiré fringes when , i.e., the remainder is less than an half of the sub-pixel pitch are opposite to those when is more than an half. Furthermore, the contrast of the moiré fringes decreases as the line thickness decreases for a given VZFO line period.
This work was supported by 'The Cross-Ministry Giga KOREA Project' grant from the Ministry of Science, ICT and Future Planning, Korea.
References and links
1. J.-Y. Son and B. Javidi, “3-dimensional imaging systems based on multiview images,” J. Disp. Technol. 1(1), 125–140 (2005). [CrossRef]
3. J. Kim, J.-Y. Son, K.-H. Lee, H. Lee, and M.-C. Park, “Behaviors of moiré fringes induced by plate thickness,” J. Opt. 17(3), 035801 (2015). [CrossRef]
4. S. Uehara, T. Hiroya, K. Shigemura, and H. Asada, “Reduction and measurement of 3D Moiré caused by lenticular sheet and backlight,” SID Symposium Digest of Technical Papers, 432–435 (2009).
5. V. V. Saveljev, J.-Y. Son, and K.-H. Cha, “About a moire-less condition in Non-Square Grids,” J. Disp. Technol. 4(3), 332–339 (2008). [CrossRef]
6. K. Nagasaki, “A 3D display with variable depth moire pattern,” in Proceeding of SICE Annual Conference (IEEE, 2008), pp. 1936–1941. [CrossRef]
7. Y. Kim, G. Park, J.-H. Jung, J. Kim, and B. Lee, “Color moiré pattern simulation and analysis in three-dimensional integral imaging for finding the moiré-reduced tilted angle of a lens array,” Appl. Opt. 48(11), 2178–2187 (2009). [CrossRef] [PubMed]
8. F. P. Chiang and B. Ranganayakamma, “Deflection measurements using moirè gap effect,” Exp. Mech. 11(4), 296–302 (1971). [CrossRef]
9. I. Amidror, The Theory of the Moiré Phenomenon (Kluwer Academic Publishers, 2000).
10. G. Oster and Y. Nishijima, “Moire patterns,” Sci. Am. 5(208), 54–63 (1963). [CrossRef]
11. A. W. Rihaczek, Principle of High Resolution Radar (McGraw Hill, 1969).
12. J.-Y. Son, V. V. Saveljev, Y.-J. Choi, J.-E. Bahn, and H.-H. Choi, “Parameters for designing autostereoscopic imaging systems based on lenticular, parallax barrier and IP plates,” Opt. Eng. 42(11), 3326–3333 (2003). [CrossRef]
13. W.-H. Son, J. Kim, M.-C. Park, B.-R. Lee, and J.-Y. Son, “The basic image cell in contact-type multiview 3-D Imaging systems,” Opt. Eng. 52(10), 103107 (2013). [CrossRef]