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).

Each parameter has an associated tradeoff between depth display and visual quality. This post discusses methods to achieve maximum displayable depth without introducing visual defects.

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. Here, we focus on the maximal values of these parallaxes 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 nearly equal. Therefore, one can characterize a 3D sequence with single left and right incremental parallaxes.

The incremental parallaxes limit 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 the physical sizes of the picture to the sequence images. Violating the left or the right incremental parallax limit value will create a visual defect.

The displayed depth increases with both positive and negative incremental parallaxes. Therefore, it will be maximized if both values are set to the lenticules' width.

The sequence length

The displayed depth is proportional to the sequence length for a sequence with a given incremental parallax. Therefore, increasing the sequence length could theoretically increase the displayed depth indefinitely.

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, resulting in blur and possibly other artifacts.

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:

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. This sequence is resolved if the picture can display a single bar from any viewing point. Otherwise, it is unresolved. To determine N, one can make a series of pictures with increasing sequence lengths.

Remarks

1. Scaling of the picture size

The incremental parallax value of a given sequence increases with the picture size. Therefore, the sequence incremental 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

The table below presents 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	18	24
30	2	24	64
20	3	36	144

3. Increasing the displayed depth

If visual defects are allowed, the displayed depth can be significantly increased. The method for such design is explained briefly in another post.

Lenticular photography

Those interested in photographed 3D sequences can consult our presentation on the subject. This presentation explains how to photograph 3D sequences optimized for depth display and visual quality.