Where is the focus point on a picture with full parallax?

Question

Let's assume I have a theoretical display that can show a different flat picture depending on the viewer's location. It would provide different images to the left and right eyes of a human observer. A classic 3D display that could be constructed using lenticular sheets perhaps.

Now, the display shows a very simple image: a single white pixel. The $(x_1,y_1)$ coordinate of the white pixel shown to the left eye matches the $(x1,y1)$ coordinate of the left eye, and the $(x2,y2)$ coordinate of the white pixel shown to the right image matches the $(x2,y2)$ coordinate of the right eye.

The user's eyes would converge on the dot, and the brain reconstructs a dot seemingly infinitely far away, like a star in the sky. However, unlike the sky, the lens of his eye would need to focus on the screen and not infinity, since the eyes are still looking at a flat screen. If the user had only one eye, this eye would only see a regular screen showing a regular pixel, there would be no parallax information and the focus is definitely on the screen.

Now, let's assume this theoretical screen can support an infinite number of viewers, or eyes, all at the same time, creating a full parallax in both horizontal and vertical direction.

This theoretical screen has now turned into a parallel light source similar to a bright star, casting a hard shadow on objects placed in front of it.

Question: If one were to look at the apparent dot, what distance would one need to focus. Is it still the screen's distance or infinity? What are the conditions?

I have two trains of thought.

• I would argue that in the limit and a screen with infinite resolution, the focus point has now moved to infinity. Each pixel emits an infinitesimal narrow beam of light. I should be able to expect to put a lens in front of it and focus the beam to its focus point.

• When resolution is limited though, each pixel would send out a narrow light cone, like a narrow spotlight that spreads with distance. The light cones of neighboring pixels would start overlapping at some distance and the point source would be imperfect. In this case, I'm not sure what to expect. If I'm close enough to the screen, I would not expect that I can focus at infinity and still get a sharp picture.

Why I'm asking: There is a Hologram printing service where I can submit (flat) pictures of an object from up to 150'000 viewing angles. They will calculate and print a diffraction pattern at 250um. I'm wondering whether this will only produce a parallax effect, or whether I'll be able to actually focus different depths of the apparent object, get a depth of field effect, and so forth. I am thinking I cannot expect that to happen. I would argue this Hologram cannot be proper in principle. Points that float above or below the surface of the holographic plate will show ghosting artifacts.

Answer 1

I'll argue that the blurring of an image that comes from defocusing is actually a form of parallax. A pinhole camera produces an image in perfect focus on a screen infinitely far behind it (neglecting diffraction). Replacing the pinhole with a finite aperture produces an image in imperfect focus, essentially because of parallax: treat each point inside the aperture as an imaginary pinhole. All the pinholes are seeing different images due to parallax, so the image produced by the entire aperture, being the sum of all the images due to all the pinholes, contains blurriness. (Placing the screen infinitely far from the aperture lets us neglect another source of blurriness due to parallax between a finite aperture and the screen. Consider the screen to have pixels larger than the aperture, so each pixel integrates over the whole aperture. In a practical camera/eye, a lens magnifies the screen so it doesn't actually have to be huge.)

In short: defocusing in an imaging system comes from parallax between different points of the aperture.

This gives a good condition for your theoretical screen. If it can change the image it displays with resolution better than the size of the aperture that will be looking at it, then it will replicate the focusing characteristics of whatever it is displaying. I.e. if the "cone of light" associated with each viewing direction is significantly thinner than your pupil, so that your pupil takes in many of the distinct viewing directions, the screen will fool you. If the screen doesn't achieve this condition, then you will have a mismatch between binocular parallax/vergence depth and accomodation (focusing) depth.

In formulas: let $\delta\theta$ be the angular resolution of your display. This means the display can show one image in one direction and and a different image in another direction if the two directions differ by at least $\delta\theta.$ Let $a$ be the aperture size, i.e. the diameter of your pupil. Let $d$ be the viewing distance, and assume $a\ll d.$ Then the angle subtended by your pupil, when viewed from the screen, is $\theta\approx a/d.$ The screen will be a good mimic if $\delta\theta\ll\theta.$

In the specific case of the hologram, let $\Theta$ be the total FOV (in one dimension) covered by your viewpoints, and let $N$ be the number of viewpoints (again, in one dimension). Then $\delta\theta=\Theta/N.$ To put some numbers into this, take $a=4\;\mathrm{mm},$ $\Theta=30^\circ$ (a wild guess, smaller is more charitable), and $N=390\approx\sqrt{1500000}$ (assuming the full 150000 will be spread in two dimensions). Then the condition on viewing distance is $d\ll3\;\mathrm m,$ which is rather reasonable.