Last update: September 6, 2012
Contents for Stereoscopic Perceived Depth (Roundness)Conclusions and Comments on Roundness ( 3dphoto forum on roundness ) Equations for perceived depth: S&S: 2 John Hart's equation for roundness, including clarifying comments by Michael Verity. 
Stereoscopic paradoxes.If a round ball is moved further away in real life, it looks smaller, but it is still round. In a stereoscopic image, a round ball only looks round from one viewing distance. If the image is enlarged or reduced, the ball is still round, but only if you keep the same magical viewing distance. Viewing a stereoscopic image of a ball from further away makes the ball look deeper, so that a round basket ball becomes an elongated rugby ball. The picture, with everything in it, including the ball, looks smaller. By using floating windows to sink it into the screen, the ball stays round, despite seeming further away. But it looks bigger, even though the image has not been magnified. Read on to discover why this magic happens. 
"Roundness" is an unfamiliar term to amateur stereo photographers. I first heard it when talking to professional 3D movie makers. I liked "roundness" but I did not understand exactly what they meant. Roundness was not an absolute thing. You could have different degrees of roundness. # 
While photographing Other circular things came out oval, notably Charles Darwin's table discussed previously. 
A sphere looks perfectly round in a stereo picture when the perceived depth is equal to the diameter of the sphere.
A soccer ball should look round. If there is not enough roundness in a 3D view, the spherical ball will flatten towards a hockey puck. Too much roundness and the soccer ball elongates into a rugby ball. This is the" sporting code effect" on roundness and becomes obvious when the above anaglyphs are viewed in stereo.
It emerged that four things have to be balanced for perfect stereoscopic roundness: #Viewing factors:
Photographic factors
V / n = i / B V = n i / B B = n i / V n = B V / i 
n, the distance from the camera lens to the nearest object, can be hard to measure in closeup photography. It is only approximately the distance from the front of the glass to the object. That is good enough for a simple magnifying glass, but a camera lens has lots of elements and is thick. The point we want to measure from is inside the lens at the first nodal point. How you find that point will emerge in the macro and panorama stereo photography sections. Beware, do not get confused between roundness and Maximum Acceptable Deviation (MAD) in a stereo picture. Roundness and MAD are often in conflict, as subsequent pages will show. Stereo base computed for perfect roundness is often NOT the most comfortable base for 3D photography. A computer screen or book page, viewed (V) from 1 meter or less, is not always the best distance to see good 3D. However, 1 meter is a very good viewing distance for closeup or macro 3D. 2 meters has been the usual viewing distance to aim for in amateur stereo photography. Projection onto a screen, 3DTV, or lens stereoscope viewing, give an excellent result (V around 2 meters). But now that amateurs see their results on computers, 3D images made for stereoscopes suddenly look flat. "Roundness" explains why this happens. 
Previously investigating orthostereoscopy, I found a disc shape worked well as a subject to photograph, because the correct viewing distance was well defined by moving back and forth until the disc looked round. Flickr, Aug 2009 Applying discs to "roundness" did not seem right. After hunting for a suitable sphere (and finding a globe not so useful, as I could not see through it) a wire art work was stumbled upon and gleefully purchased. The wire sphere was put on a grid and photographed with the camera on a horizontal bar. The lens was 1 meter from the grid and the stereo base was varied: (The squares are 10mm on a 22 inch monitor running at 1680x1050 pixels. They may not be 10mm on your screen). The stereo window was set on the grid so that the sphere, as it "inflated and deflated" under the influence of a varying stereo base, did so on the screen surface. If the window were set at the front of the sphere, the grid would be jumping backwards and forwards. To keep the camera geometry consistent with the viewing method, the distance, n, was from the first nodal point of the lens to the grid. (The first nodal point had been found earlier while setting up panorama photography.) n measured this way, during photography, is consistent with measuring the viewing distance (V) from your eye to the screen surface. This measurement of V is easy to do.
Anaglyphs were made and can be checked for the correct viewing distance to provide perfect roundness, as described below with the images.

Roundness in 3D is very sensitive to how far you are from the viewing screen.Try moving backwards and forwards, wearing your anaglyph goggles, until the sphere is spherical, just like a ball . A sphere should appear at 1000mm from the screen, if your eyes are 65mm apart, which is average. The grid has 10mm squares. If they are 10mm on your screen, then the scene is orthosteroscopic. That means the stereo anaglyph is as close as possible to reality. You may need a big computer screen to see this orthostereo Undistorted viewingTrue roundness does not need to be orthostereoscopic. As long as the ball looks round, it is not distorted and its size is not important. Once the stereo pair have been photographed, roundness persists at any magnification, but only if you keep viewing from the same distance of 1 meter .

Roundness in 3D is very sensitive to how far the camera was from the ball and the stereo base

Flattened SphereYou are probably still looking at the screen from 1 meter but the sphere now seems deflated. Move back until it is a ball again and you will be sitting at near enough to 1600 mm from the screen. The magnification is the same as the 1000mm view example. Check by measuring the grid. Now the scene is not orthostereoscopic. The cameras have not reproduced the distance between your eyes, but are closer together, at 40mm, than even a child's eyes. You have to sit 1.625 times further away (65/40) . StretchBy moving back you have stretched the sphere in the depth plane (z) to make it round. SqueezeSitting closer, at a more usual distance, the sphere is flattened.

Photography factors
Persistence of perspective andPersistence of perceived stereo depth,Despite a change in magnification.The experiment shows roundness, like perspective, is not changed by magnification of the image. M = v/u M: Magnification on the receptor (film or CCD) v = F + E F : Focal length of the lens M = (F + E) / n (Only true if the nearest object has been focused on, but works pretty well in macro photography, which is the situation you really care about magnification.) Final magnification on your computer screen depends on how much the receptor image is further magnified, (frame magnification) which gets complex if the image is both cropped and resized. 
This Column For Specialists. Avoid on your first reading:Relative Parallax Invariance with MagnificationNo matter what magnification you view the sphere at, perfect roundness persists, as long as your distance to the screen (V) does not change. If you zoom into a stereo picture while viewing it (as you can easily do in a program like StereoPhotoMaker) roundness does not vary. Just as roundness has not varied here with magnification change. Perspective InvariancePerspective behaves the same way. The relative magnification of objects to each other at different distances in the scene does not change when you magnify the whole picture. A chair half the height of a table stays half the height as the image size changes. That is why a tiny digital camera can take a picture just as realistic as a large view camera photographing from the same position. Except the small camera result is more noisy, less sharp and shows a bigger depth of field. Magnification in photography is determined by focal length of the camera lens (combined with the extension needed to focus it), plus any change in picture size made during postprocessing of the image. (Photographic magnification multiplied by frame magnification). The experiment shows roundness, like perspective, is not changed by magnification. That is the same as saying the focal length of the camera lens has no effect on these two indicators of depth. But it only works as long as the distance from camera to nearest object, n, is kept constant as the lens is zoomed. You need to ponder that because it is counterintuitive. We are used to thinking that a telephoto lens flattens perspective. But that is only because you zoom in to magnify distant objects, which means you are photographing from further away than usual (n is bigger). We say that telephoto stereo causes cardboarding (which is the same as saying roundness has decreased). Again that happens because you usually take telephoto shots from further away (n has increased). If you use telephoto from further away, you must increase the stereo base. Twice as far away needs twice as much base to retain roundness (or no card boarding). BUT beware, because increasing the base can cause excessive stereo deviation, as we will see later. 
Football Code DeformityIf you are back at 2 meters, the sphere has become an ellipsoid, like an egg pointing at you. A soccer ball changes code to become a rugby ball. Getting it back to spherical means moving closer, to just over 800mm away from the screen. Base over nearest object.B / n Here B/n is 1/12.5 B/n is the rule you can use while taking a photograph to get a perfect shape in a stereo picture, but it is not a constant because it must be set to equal: Interocular over screen distancei/V Here i/V is 65/813 = 1/12 .5 But only if your interocular distance is 65mm. My interocular distance is 68mm, so for perfect roundness I need to sit at

Photography factors

Correction attempt with floating windowInstead of the observer moving back, the stereoscopic image itself can be moved further away with floating windows. 
Photographic factors

To my eye, the floating window experiment has failed to correct the lack of roundness. However, the ball looks bigger because I think it is further away, where it should look smaller. In truth, the image of the ball has not changed size and it is still sitting on the flat screen. Perceptually, the ball has retreated behind the screen and now looks bigger, because we think it is further away. But it has stayed just as flat as ever and measurement reveals it has not changed size. This is a good example of a stereoscopic illusion. It shows our perception of depth does not follow Euclid's laws of geometry. Depth perception is a learned experience and since it is an illusion, it is hard to measure.

So how about trying to make the roundness worse by floating a sphere, good for viewing close up on the screen, forwards to where it might flatten? The sphere is correct for 813 mm view distance, an average sort of computer screen viewing distance 
Floating the sphere forward should make the spherical deformity worse but to my eye it is no different. The back of the sphere has moved forwards, balanced by the same for the front of the sphere. Moving the stereo window forward or back does not help or hinder roundness. 
There seemed to be a pattern to the viewing distance required to make spheres look round. As a guess I thought the viewing geometry should be in proportion to the taking geometry. I decided to compare human measurements with conceptually similar camera measurements, by ratios. By rearranging the ratios, I attempted to predict the correct viewing distance from the camera geometry, by making the ratios equal.

V/n = i/B V: Viewing Distance n: Lens distance to the grid i: Interocular distance: taken to be the human average of 65mm B: Stereo Base 
The viewing distances, V, predicted from this formula for correct roundness are shown on the top left of each image. 
V= i n / B = 65N/B 
Now anybody can check how far their eyes should be from the screen (V) to get the impression of a round sphere. The distances marked on the top left should be correct, if your eyes are 65mm apart. If your interocular distance (i) is much different from 65mm, it is necessary to run the formula to see if that gives you a better personal viewing distance. This experiment did not measure stereoscopic roundness, it is based on a binocular visual assessment.
Geometry of perceived depthSubsequently, I sorted out the mathematics for perceived depth and found everything stated here, based on perception, works out in formal geometry too. I can measure depth now. Geometry explains the way magnification (zooming) makes no difference to perceived depth and the way objects are stretched and squeezed as the viewing distance changes. See stretch and squeeze 2. I bet somebody else has worked all this out, I just cannot find where he published it! 
ExamplesI have a Zalman 3D computer screen and I like to view it from 1 meter (V), because closer than that the interlace lines become too obvious.
This result is more interesting than you may realise. Digital cameras set up side by side cannot get the lenses closer than about 130 to 140mm. Some experts say this is no good for 3D, because the cameras are too far apart. Ah, but, they have not thought about roundness and viewing the results on a computer monitor. It turns out the 135mm stereo base for a pair of Sony V3 cameras is just right for computer stereo. BUT the stereo pairs for a computer or book cannot be projected without causing too much stereo deviation. (Maximum Acceptable Deviation, MAD, is exceeded.) SO computer users and projectionists end up having a fight! What is the roundness if we view a 3D picture, designed for computer viewing, on a 3D television 2 meters away? New V = 2000mm Roundness = [Stereo Base Used] * V / n i Roundness = 130 * 2000 / (2000 * 65 ) So; things looking perfectly round on a computer monitor, or when printed in a book, look twice as round as they should on a 3D television 2 meters away. If the TV is viewed from 3 meters, the roundness is 3, which is beginning to look ridiculous.
There is a rule in photography, which some people are madly insistent on, called the 1/30 rule for setting stereo base (B/n = 1/30) OH NO it seems 1/30 rule is going to give you flat pictures, if they are to be seen on a computer screen! On a movie screen the 3D depth will be really excessive, if the 1/30 rule is followed. So the 1/30 rule is not always correct. The 1/15 rule for macro photography was derived in a very different way by Frank Di Marzio in the first edition of this web site. It was based on MAD and not roundness considerations. Now I want to view a 3D picture of my family on a television 2 meters away, instead of on my computer. The stereo base required to get perfect roundness is smaller. In fact it is exactly the same as the viewer's interocular distance.
So the 1/30 rule is correct if you are viewing the stereo image from 2 meters  which was a very common distance before computers. Using a lens stereoscope, such as a Holmes viewer, people adjust the virtual optical distance to the stereo pair by changing the focus. They usually end up at a virtual viewing distance of around 2 meters, so the 1/30 rule has a wide application. 
Enhanced stereoscopic perceived depth or excess roundness has its place in scientific stereoscopy. I displayed solar spicules on the middle of the sun's disc by showing them, proprtionally, about twice as deep as they really are here.
Next page: Conclusions and comments on roundness
http://nzphoto.tripod.com/3d/315roundness.html