The Ansel Adams Fallacy: "True perspective depends only on the camera-to-subject distance"

Of course it does!

If you bother to read Jordan Steele's article, you find an explanation of what perspective means to him in the section headed "What is Perspective".

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I have no idea what the issue is. There has been extensive research on this topic which underlies the technology of computer vision. Computer vision not only is the underlying technology of robots but also of self breaking cars, self driving cars and automatous vehicles.

The mathematical foundations of the camera is that of the projection of three dimensional space onto a two dimensional plane a.k.a. the sensor. The basis of the mathematical model of the camera was really solved in the 1800's. There is no need for heuristic arguments. The camera lives in projective space. The camera performs a projective transform defined by the location of the camera and orientation of the sensor. Projective transforms do not preserve distance nor angles or any other Euclidean concept like parallel lines.

In reality a lot of the mathematics becomes easier and more intuitive if one defines distance not in units of lens focal length.

Forgive me I forgot the link.

courses.engr.illinois.edu/cs543/sp2011/lectures/Lecture%2002%20-%20Projective%20Geometry%20and%20Camera%20Models%20-%20Vision_Spring2011.pdf

It occurred to me some time ago (going on decades now) that the best way to view a photograph taken with a wide angle lens would be with an immersive and curved display.

I agree. The mathematics has been understood for a long time. There is no need for heuristic arguments, particularly when they lead to false conclusions that contradict the mathematics that is very well tried and tested. That is the whole point of this thread.

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It is not just photography where that can get dominated by heuristic "crackpot" notions. There is a popular theory among "amateur physicists" that there is a black hole at the center of the earth to explain why the earth has a molten core. Part the the problem with perspective and photography is the definition of perspective that is not consistent. Often people are talking about different things. Projective transforms don't preserve Euclidian metrics, however, they do preserve ratios. For example the girl on the bridge example that hangs around on the internet is not really about "perspective," it is about longitudinal magnification. Define distance in the proper units, that is the units of lens focal length. That normalizes the image by scaling to focal length and the confusion goes away.

The camera does not live in Euclidian space - it lives in projective space.

photography.tutsplus.com/tutorials/exploring-how-focal-length-affects-images--photo-6508

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I presume the girl on the bridge example means the one in that article you gave a link to.

As you say, it is about magnification (I don't know what you mean by longitudinal magnification).

The size of the girl's image is determined by her actual height and her distance from the camera. If we assume that she is approximately 170cm tall, then knowing the focal lengths, it is a matter of simple geometry to work out that the camera-to-subject distance is approximately 1m for the first shot, 3m for the second, 8m for the third and 15m for the fourth. I assume that the images are uncropped.

The girl's image is approximately the same size in all four photos. So, if we view all 4 from the same distance (between viewer and image), and completely ignore the surroundings and background, the girl appears to be approximately the same distance from the camera in each (because her image is approximately the same size in each). However, the actual distance between the girl and the camera varies from 1m to 15m approx. It is not hard to deduce from this that the appearance of depth in the image is inversely proportional to the focal length.

All of this depends on the fundamental equation for perspective:

object size / object distance = image size / image distance

In geometric optics there are two types of magnification. The one most thought about is lateral magnification. That is magnification in the focal plane. Longitudinal magnification sometimes called axial magnification is in the direction of the lens axis. They are different.

spie.org/publications/fg01_p10_longitudinal_magnification?SSO=1

This difference is want causes most of the food fights in photography over "perspective."

Most people think of perspective only related to objects in the focal plane or in a plane (lateral magnification). The equation for lateral magnification involves the term (f/(u-f)), f the focal length and u the distance. Where as axial magnification involves the same factor squared. There two magnifications between planes a distance apart are different. This leads to differences in as a function of the focal length. In reality anyone saying there is a difference in perspective as a relation of focal length and anyone that says there is not are equally wrong.

physics.stackexchange.com/questions/470338/longitudinal-magnification

Difference between lateral and longitudinal magnification explains the differences in the "perspective" of the girl on the bridge.

Thanks, I see what you are talking about now: the longitudinal magnification in the three-dimensional image formed by a lens.

However, longitudinal magnification is not relevant to the perspective in the two-dimensional image captured by the sensor.

That can be seen very easily by considering a pinhole camera. It has no lens and has no longitudinal magnification, yet it produces an image with the same linear perspective as a camera with an ordinary rectilinear lens. Longitudinal magnification is irrelevant to perspective in 2-D images.

The mathematics of linear perspective is summed up in the equation:

object size / object distance = image size / image distance

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