And the answer is: the number of distinct images that your picture can resolve.
The number of resolved images
Lenticular pictures stand out by their ability to display several distinct images depending on the angle of view. In real pictures there will always be a limit on the number of distinct images that can be displayed, regardless of the technology being used. we call this limit "Number of resolvable images," and denote it "Nres." This characteristic may be regarded as the resolution of the lenticular picture.
As long as the number of images displayed by the picture is smaller than its resolution, the images are displayed distinctly. However, as the sequence length approaches the picture resolution, neighboring images in the sequence start to blend into each other. When the sequence length exceeds the picture resolution, clean view of the individual sequence images is no longer possible.
The picture resolution determines the primary visual assets of the picture. In dynamic pictures, it sets the limit to the number of frames which can be displayed distinctly. In 3D pictures it determines the overall depth D which can be displayed without blur or discretization. This depth is given by the formula:
----------------> D = 2xNresxt/n
where t and n are the thickness and the refraction index of the lenticular sheet respectively.
Measurement of the number of resolved images
To determine the picture resolution one can simply produce a sequence of pictures with increasing number of displayed images (with all other parameters intact), and observe each one of them visually. The picture with the longest sequence which still displays each image distinctly determines the resolution.
This procedure is very inefficient. It is costly, and the resolution estimation is inaccurate. It is much better to develop a custom pattern, which can be used to determine the resolution in a single picture and with a high precision. We call this pattern "resolution pattern."
The resolution pattern is a vertical array of horizontal bands, much like the common pitch calibration pattern. Each band is an interlaced image of a special sequence, like the one shown below:
Each image in the sequence displays a vertical black bar on a white background. The position of the bar in a given image is shifted by a constant step with respect to its position in the previous image. The step is larger than the bar width.
A picture made from such sequence will normally display more than one bar from a random viewing angle and range. When the picture is observed from the calibration distance, there may be certain viewing angles in which one bar will appear significantly darker than the others. In such case the sequence will be regarded as resolved.
The bands in the resolution pattern are made from sequences with increasing lengths. All bands must be interlaced with the pitch corresponding to the lens being examined. If the sequence length of the last band exceeds the picture resolution, there will be a band such that all bands after it are unresolved. This band determines the resolution.
Example
As an example, we show below the resolution pattern for a 60lpi lens. Epson Stylus 7900 printer was used to print the pattern with a pixel density of 720ppi, and all the sequences were interlaced with a pitch of 60lpi. The hand-written numbers indicate the sequence length of each band.
Probably all inspectors will agree that sequences 4 through 12 are well resolved. Sequence 14 is barely resolved, and the sequences below it are clearly unresolved.
The optical parameters of this lens are t = 1.2mm and n = 1.58. If we use the estimate of Nres = 12, then the estimate of the displayable depth for this lens will be
-------------------> D ~ 18mm ~ 0.7".
The chart displays bands corresponding to sequences with even number of images only, to allow alignment of the bars in the vertical direction. This analysis can be carried out also with a chart showing odd sequences; however, the conclusions are very similar.
The pattern used in this example can be downloaded from https://www.sugarsync.com/pf/D6147637_08838767_7818681
If you want to use it for testing your printer, you will need a 60lpi lens with a known pitch. Before printing, you will need to resample the pattern in order to match its pitch to your lens pitch. The pattern was interlaced with 60lpi pitch and was designed to be printed with a horizontal pixel density of 720ppi and vertical pixel density of 120ppi.
Photography
The picture below shows the setup with which the resolution patterns were photographed. A micro-meter controlled rotation stage was used for fine-tuning the photography angle.
Comments and discussion
Nowadays, the number of resolved images is determined primarily by the printing technology. Different printers may yield different values of Nres for the same lens. Offset printers are probably capable of achieving significantly higher values of Nres compared to digital printers. It will be of great interest to see measured values of Nres for different printers and different printing technologies.
Once Nres is measured for a given lens, its value (for the same printer) can be estimated for lenses with other lpi values by the formula:
--------------------------> Nres(lpi) ~ Nres' x(lpi' /lpi)
where Nres' is the measured value for lpi'. However, the safe procedure is to measure Nres for each lens individually.
The number of resolved images may depend on the printing density used to print the resolution pattern. In the example we have used 720ppi, but practitioners may try different densities, like 360ppi, 1440ppi, etc. It may also depend on other printing characteristics, and also on the RIP software and its settings (in case that it is used to print the picture).
The pitch used to interlace the sequences may also affect the value of Nres. Thus, it is possible to have different values of Nres for different lenses even if both are quoted by the manufacturer to have the same pitch (like "60lpi"). Same is true for pitch modifications due to different viewing distances for the same lens.
Although there is a certain ambiguity in the estimation of Nres, the resolution pattern can be used to compare unambiguously different printing technologies.