Depth: it's all about parallax and sequence length
Introduction
Lenticular pictures can create a compelling 3D illusion without any visual aids. Most practitioners strive to increase the displayed depth to the maximum. However, increasing the displayed depth indefinitely may create visual defects that spoil the viewing sensation. What is the correct balance between these contradicting drives? Here we present a few simple rules to maximize the 3D impact without sacrificing the visual quality.
Two parameters of the lenticular sequence are relevant to the present discussion:
Incremental parallax (the maximal parallax between successive images),
Length (the number of images).
With each one of these parameters, there is an associated tradeoff between depth display and visual quality. To achieve the maximal displayed depth, one needs to know the maximum value allowed for each.
The incremental parallax
In 3D sequences, the images differ by a horizontal displacement called 'parallax.' The parallax varies across the image; objects close to the picture plane have small parallaxes, and those with large protrusions have large parallaxes.
The parallax can be either in the right or the left direction, depending on whether its corresponding object point is imaged in front or behind the picture. We focus here on the maximal values of these parallaxes separately between successive images. They will be called left and right incremental parallaxes. In a 3D sequence, left and right incremental parallaxes between any two successive images are generally equal.
The incremental parallaxes limits for both left and right is the lenticules width. This limit is expressed in physical length units (like millimeters). Normally, images are measured in dimensionless pixels. To apply this limit, one needs to assign to the sequence images the physical sizes of the picture. Violation of either the left or the right incremental parallax limit value will create a visual defect.
Since the displayed depth increases with positive and negative incremental parallaxes, it will be maximized if both values are set to the lenticules width.
The sequence length
For a sequence with a given incremental parallax, the displayed depth is proportional to its length. Therefore, theoretically, the displayed depth could be increased indefinitely by increasing the sequence length.
There is a limit to the number of images that a given lenticular picture can resolve. We will denote this number by N. Interlacing sequences with a length greater than N will cause adjacent images to blend into each other resulting in blur.
N is determined by the printer's resolution and the lenticular sheet lenticules density (LPI). Common estimates of N for leading brands of inkjet printers are:
N = 720/LPI for Epson Stylus printers,
N = 600/LPI for Canon and HP printers.
N can also be measured using a special sequence like the following one:
Each image shows a black bar on a white background progressing to the right with the image number. The displacement of the bars between adjacent pictures should be larger than the bar width. This sequence is interlaced and printed. If the picture can display a single bar from any viewing point, then this sequence is resolved. Otherwise, it is unresolved. One can make a series of pictures with increasing sequence lengths to determine N.
Remarks
1. Scaling of the picture size
The incremental parallax value of a given sequence increases with the picture size. Therefore, the sequence parallax needs to be adjusted for the picture size, and the same sequence cannot be used for different picture sizes.
2. Quantitative estimates
The maximal protrusion (or recess) h of an imaged point from the image plane is given by
h = Nt/n
Parameter | Meaning | Units |
h | Maximal protrusion from the picture plane | length |
N | Number of resolvable images | - |
t | Sheet thickness | length |
n | Optical index of refraction | - |
The maximal displayable depth D is twice the maximal protrusion:
D = 2h
In the table below, we present a few values D for an Epson Stylus inkjet printer and an index of refraction of 1.5.
LPI | t (mm) | N | D (mm) |
75 | 0.6 | 9.6 | 7.68 |
40 | 1 |