Demystifying resolution and depth in 3D lenticular pictures

Yitzhak Weissman
Nov 6
4 min read

Updated: Nov 9

How to use the picture's resolution for its depth design? Read this post and find out!

Resolution in lenticular pictures

It is common knowledge that resolution is an essential characteristic of lenticular pictures, because it determines the depth-blur trade-off; the higher the resolution, the greater the depth that can be displayed without visual defects. However, “resolution” is an elusive concept and can be defined in many ways. This can be confusing, leading to design errors and unsatisfactory visual quality in pictures. There is a practical need for a resolution concept that is unambiguously defined, measurable, and useful for the depth design.

Many mechanisms affect the resolution of a lenticular picture. These mechanisms can be divided into two groups: printing (printer resolution, calibration, resampling, etc.) and optical (lens aberrations, focus, etc.). In inkjet lenticular pictures, the printing mechanisms usually dominate. In offset-printed pictures, the printing and optical mechanisms are of the same order of magnitude. This implies that a universal resolution definition for lenticular pictures should account for multiple resolution-affecting mechanisms and cannot rely solely on the printer resolution.

The number of resolvable images

The most prominent visual quality of a lenticular picture is its ability to display different images from different viewing positions. However, the number of images that a given picture can distinctly display is finite. When the sequence length exceeds a specific limit, the displayed images start to blend with their neighbors and eventually lose their identity completely.

The maximal number of images that a picture can display distinctly is called “Number of Resolvable Images,” or, in short, “NRI.” This concept was introduced in my book “Lenticular Imaging” back in 2018 and provides a practical definition of lenticular picture resolution. It is universal because it accounts for all mechanisms that affect it. As such, it applies to both inkjet- and offset-produced pictures. In addition, it can be experimentally measured, making it an objective, well-defined physical quantity.

Measuring the NRI

The general way to measure the NRI is to produce a series of lenticular pictures with increasing numbers of images and inspect them as proposed in my book cited above. A photo of a picture ("resolution photograph") used for such a measurement is shown below.

*Measurement of NRI by inspection of the picture ("resolution photograph")*

This photograph shows a stack of five lenticular pictures, each with a different sequence length (as annotated in the photograph). In this case, the sequence images display a single black bar on a white background in a different position, as illustrated in the figure below ("bar sequence"):

If the sequence is resolved, the picture should display single black bars at different positions corresponding to different viewing points. In practice, several bars are displayed due to ghosting. We will consider the sequence resolved if there is a single bar significantly darker than the others. The displays of the pictures in the resolution photograph are synchronized; in the central view, when any one of them shows the black bar, all do. The resolution photograph shows such a view.

Applying this criterion to the resolution photograph, we conclude that in this case, NRI = 10.

Only pictures corresponding to sequence lengths 6, 8, and 10 display a black bar which is significantly darker than the rest. The bands representing sequence lengths 12 and 16 show two bars with equal grey levels, indicating a loss of resolution. The photographed picture was printed with an inkjet printer (Epson 9890) and a 40 lpi lens.

This measurement procedure is costly and laborious. However, if all the dominant resolution-limiting mechanisms are from the printing group, the NRI measurement can be significantly simplified. In such a case, the NRI can be measured by analyzing a special printed pattern like the one shown below:

*Pattern for NRI measurement (Grape 10)*

The pattern contains six bands. Only the bottom five bands are used for the analysis. Each represents an interlace of a bar sequence with a different number of images. This pattern is printed and scanned at high resolution and then analyzed by software. A result of such an analysis is shown below:

The pattern in this example was printed with an Epson 9890 printer and designed for a 40 lpi lens; the results are in good agreement with the photograph test described above. This proves that, in inkjet-printed pictures, the dominant resolution mechanism is the printing process.

The tools for measuring printer-limited NRI are included in Grape 10 software. These tools are used to generate the pattern and analyze its scan, as illustrated here.

NRI and depth design

The central role of the NRI parameter in depth design is manifested in the maximal deviation formula:

This formula gives the maximal deviation of a point from the picture plane that can be displayed both sharply and smoothly. Any point beyond this limit will suffer from either blur or stepping, or from both. The maximal continuous and sharp displayable depth is twice the maximal deviation:

For example, we have seen that the number of resolvable images in a 40lpi picture printed with an inkjet printer is ~10. Common parameters for such a lens are 2mm and 1.5 for thickness and index of refraction, respectively. It follows that for such a picture, the maximal continuously and sharply displayable depth is ~27 mm. The following formula can be used to determine the maximal parallax that is allowed in a sequence to guarantee a smooth and sharp display: