Revised Nov. 15, 2020
Introduction
The 3D lenticular print method can be used to print stereo photographs and present them as true 3D pictures in full color and without any visual aids. However, not every stereo photograph can be printed as a 3D lenticular picture. Stereo photography must be specially staged for such an application.
Printing of stereo-pair consists of two steps:
Computation of intermediate frames,
Lenticular printing.
The main challenge is to achieve success in the first step. The success of this step cannot be guaranteed because it involves "guessing" of missing information. Moreover, it is impossible to formulate general guidelines because the success depends on the scene characteristics, not only on the photography. Therefore, in many cases, an experimental "try-and-error" loop may be inevitable. Such a procedure is suggested below.
The second step, lenticular printing, imposes additional constraints. The design rules in this step are well established, and if followed, a high-quality lenticular picture can be produced with certainty.
This post relates to a stereo photograph of an object place against a background. We use an example of a portrait stereo photograph (figure 1) to present the various considerations involved. The general principles can be deduced, to a certain extent, from this discussion.
The photograph shown below was taken from 1m with a camera separation of 15cm.
Figure 1: Stereo portrait used as an example in this post
Computation of intermediate images
A stereo photograph consists of two images only, which are projections of a scene from two adjacent points. The lenticular picture needs more than two images, typically 10. This set of images is called "lenticular sequence."
The first step in producing a lenticular picture from a stereo pair is to generate the additional images required for the lenticular sequence. These images are projections of the scene from a series of equally spaced points on the line connecting the two shooting points of the stereo pair.
The computation of the intermediate images involves "guessing" of missing information, and, therefore, its success cannot be guaranteed. Understanding the failure mechanisms will help the photographer reduce their effect and increase the chances of success. The good news is that the success of the computation does not depend on the sequence length. If it succeeds for a certain number of images, it will succeed with any number of images.
There are two main failure mechanisms: excessive differences and occultations.
The success of the lenticular sequence computation depends critically on the differences between the two stereo images. For small differences, the success probability is high, and it diminishes as these differences increase. The parallax concept is used to quantify these differences (see below).
Normally, any given object point is present in both images of the stereo pair. But sometimes, a certain point in one image has no corresponding point in the other image. Such a condition is called "occultation." Occultations lead to visual distortions and artifacts in the computed lenticular sequence. In some cases, these defects may be acceptable; they may render the whole sequence useless in others.
Avoiding occultations
The image window is a common cause of occultations. Objects near the image edge may be seen in one image but not in the other and vice-versa. In the figure below, the window occultation regions in the background are denoted by vertical green bands. These occultations may cause defects in the computed lenticular sequence.
Figure 2: Window occultations in the stereo images
In the present example, the background is a vertical plane. In such a case, the window occulted regions are vertical rectangles adjacent to the stereo pair images edges (figure 2). Such obscurations can be easily cropped out from the sequence. For this reason, a vertical plane background is preferred.
Scene occultations occur due to the scene geometry can be avoided only by modifying the subject or the photography angle. In the present example, scene occultations are present in the background near the subject's face. These occultations can also cause disturbing defects in the computed frames. Although the defects may be barely visible, they may be conspicuous in the 3D picture.
A simple way to avoid the effects of scene occultations in the background is to use uniform or almost uniform backgrounds, in which occultations defects will be unnoticed both in the images and in the 3D picture. The stereo image shown in figure 1 has such a background.
A stereo photograph with a uniform color background can be used to create occultation-free sequences with any background using the chroma-key method. This requires two stereo photographs: one of the background only (without the subject) and the other of the subject with a uniform color background. Next, both stereo pairs are converted to lenticular sequences, and the two sequences are combined into a single sequence using the chroma-key method.
The parallaxes of a stereo image pair
The parallax concept quantifies the differences between a pair of stereo images. Each point on the photographed object has an associated parallax. The parallax is the distance between the given object's positions in the two images (when they are superimposed one on the top of the other).
Let us consider a pair of stereo images shown in figure 1. The pixel width of the images is 800 each.
The parallaxes can be conveniently measured using Stereo Photo Maker (SPM). The SPM parallax adjustment window for this photograph is shown below:
Figure 3: Parallax adjustment window of SPM
Of special interest are the parallaxes of the extreme points of the scene. These are the closest and farthest points from the camera (distance is measured along the camera axis). The parallaxes corresponding to these points are called "front parallax" and "back parallax." In this case, the nearest point is the model nose tip. Its parallax can be measured by moving the 'H Position' slider at the top until the nose tip appears grey. The result is shown in the figure below:
Figure 4: Measurement of the front parallax
The front parallax value, in this case, is 24 pixels. To measure the back parallax, the horizontal slider is moved until the background appears grey. Since, in the present example, the background is almost uniform, this method cannot be used. Instead, one can rely on the furthest viewable objects of the subject, which are, in this case, the shoulder and the neck silhouettes. The result is shown in the figure below:
Figure 5: Measurement of back parallax
The back parallax, in this case, is -19 pixels. Note that the back parallax is smaller than the front parallax, both in magnitude and in (algebraic) value. According to the SPM sign convention, the algebraic value of the back parallax is always smaller if the left image (of the SPM side-by-side display, shown in figure 1) corresponds to the left view.
The front and the back parallaxes are a quantitative measure of the differences between the two stereo images. Smaller parallaxes magnitudes increase the success rate of the lenticular sequence computation.
Balancing of stereo photographs
The collection of scene points with zero parallaxes lies on a plane called the "zero-parallax plane" or "stereo window plane." This plane's position can be easily adjusted using SPM, but it involves a small horizontal cropping amount.
Adjustment of the zero-parallax plane's position affects the values of the front and the back parallaxes. This adjustment affects both parallaxes identically, either increasing or decreasing their values. The algebraic difference between the front and back parallax values (43 in this case) reflects the scene depth and is not affected.
In our example, the front parallax is 24 pixels, and the back parallax is -19 pixels. By moving forward the zero-parallax plane by 3 pixels, it is possible to reduce the front parallax to 21 pixels and the back parallax to -22 pixels. This operation will decrease the maximal parallax magnitude from 24 to 22 pixels.
A stereo pair in which the front and the back parallaxes' magnitudes are equal is called "balanced." Thus, the success probability of the lenticular sequence computation can be increased by balancing the stereo pair. In many cases, the balanced stereo pair may produce a good sequence, while its original version does not. In this example, the change in the maximal parallax magnitude (from 24 to 22) is not significant, and its balancing serves only as an illustration.
In addition to the beneficial effect of balancing on the sequence computation, it also allows the printing of larger pictures (see below).
Example of a lenticular sequence computation
A lenticular sequence of five images derived from the stereo pair of figure 1 is shown in the figure below.
Figure 6: Lenticular sequence computed from the stereo pair
A stereo pair with a transparent background is shown in the figure below. Such a pair can be used to add backgrounds without occultation effects.
Figure 7: Stereo image with transparent background
An example of a sequence computed from the pair shown in figure 7 with an inserted background is shown in the figure below. Note that a properly inserted background usually must have a parallax to not "collide" with the subject.
Figure 8: Stereo sequence with inserted background
Lenticular print restrictions
The pixel parallax values of a stereo pair are not characteristic of the photography alone; they scale linearly with the image pixel width. To factor out this dependence, it is convenient to introduce the concept of relative parallax, given by
For a given photography scenario, the relative parallax does not depend on the image pixel width. For example, in the photograph of figure 1, the relative parallaxes are:
Table 1: Parallax values in the present example
The pixel values were derived from figures 4 and 5, and the relative values were derived by dividing the pixel values by 800, which is the image width in pixels. Since the relative parallax values are typically small, it is convenient to represent them as a percent. The important parameter for lenticular print is the maximal relative parallax magnitude, which in this example is 3%. Let us denote it by p.
The lenticular picture is made from a lenticular sheet, a dense array of cylindrical lenses called "lenticules." The number N of lenticules in the picture sets a lower limit on the required sequence length n:
The lenticular picture, as a visual instrument, possesses a finite resolution. This is manifested in the fact that it can resolve only a certain limited number of images Nr. Feeding the lenticular printing engine with sequences of length exceeding Nr will introduce blurring in the picture. The blurring of a given object point increases with its deviation from the zero-parallax plane, thus introducing a visual effect similar to the familiar depth-of-field effect in photography. To avoid blurring, n must be smaller or equal to Nr, a condition which leads to the following limitation on p:
Equation 1: The lenticular print parallax limitation for a blur-free picture
The number of resolvable images Nr depends on the printer and the lenticules density. For example, for Epson Stylus inkjet printers, it is widely accepted that
Equation 2: Estimation of the number of resolvable images for Epson Stylus printers
where LPI is the lenticular sheet lens density in units of 1/inch.
Finally, let us apply these rules for printing a 10" wide picture using a 60LPI sheet and an Epson Stylus printer. The number of lenticules in this picture will be N = 10*60 = 600, and by equation 2, the number of resolvable images is 12. According to equation 1, for a blur-free picture, p must be smaller than 3.7%. Since the value of p in the present example is smaller (3%), the present photograph can be printed in this size without blurring.
Adjusting the sequence relative parallax to the picture size
Sequences with reduced values of relative parallax can be derived from a given stereo pair.
Suppose you wish to reduce the parallax of a stereo image pair by a factor of two. You can use the original stereo pair to compute a lenticular sequence with 5 images. If the computation is successful, you take images 2 and 4 from this sequence and use them as a stereo pair to compute a new sequence. The relative parallax of the new pair will be half the relative parallax of the original pair.
Using this method, one can adjust the maximal relative parallax magnitude to satisfy equation 1 for arbitrary N. In other words, if you can compute a lenticular sequence for a given picture size, you can always use it to compute sequences for larger images.
The fSeq software
The fSeq software from Pop3DArt is designed for stereo photographers who wish to print their photographs as 3D lenticular pictures. This program computes the lenticular sequence from a given stereo pair and gives its relative parallax values, which can be used to compute the lenticular picture's maximal size.
The fSeq software has a utility to automatically add given backgrounds to a stereo pair with a transparent background. This feature can be used to create distortion-free sequences with any background. The sequences shown in figures 6 and 8 were computed with fSeq.
Photography try-and-error loop
It is impossible to foretell whether a given stereo pair will yield an acceptable lenticular sequence. However, it is known that if the parallax is reduced below a certain threshold, the computation will succeed. The depth displayed by the picture is proportional to the stereo pair's parallax, so normally, one wants to use a sequence with the largest possible parallax value. This sequence can be determined only by an experimental try-and-error loop, as suggested below.
Choose the picture size and the lenticular sheet.
Determine the maximal allowed relative parallax values from equation 1.
Determine the photography scenario which yields a stereo pair with the required maximal parallax magnitude by adjusting the stereo base or the shooting distance.
Make a series of stereo photographs with decreasing parallax values, starting with the determined scenario.
Compute lenticular sequences for each of the stereo photographs and inspect them.
Choose the stereo photograph with the largest parallax magnitudes, which gives an acceptable lenticular sequence. (If you are lucky, this will be the one corresponding to the first scenario.)
Multi-camera arrays
Figure 9: Eight cameras array for lenticular photography (Pop3DArt)
Lenticular sequences are best photographed with a camera array. If there are enough cameras in the array, no extra frames need to be computed, and all the limitations concerning these computations are relieved. If the number of cameras in the array is not sufficient, then one can use available algorithms to compute extra frames, introducing the limitations discussed here—however, the maximal parallax value which can be treated increases linearly with the number of cameras. For instance, a setup with three cameras can produce a sequence with twice the stereo pair's parallax value.