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Continuation of the maxed out thread, with one illustration.

bobn2 wrote:

J A C S wrote:

bobn2 wrote:

J A C S wrote:

bobn2 wrote:

J A C S wrote:

ii) that the axis of perspectivity is at the same location in both images

There is no such thing as axis of perspectivity of an image. They define it for a pair of triangles; and in a given photo, or a 3D scene, there could be many of those axes.

This says you're wrong.

It does?

Yes, it does.

A counter-example trumps a theorem every single time. I can just create the example I mentioned (two copies of the perspective triangles, one rotated) and take a picture of it if you do not like drawings. There will be two centers, and two axes of perspectivity, clearly labeled.

Defining perspective of an image detached from the actual scene it represents might be possible (I can define it to be the number 0 all the time, for example) but not very helpful.

The counter=example only 'trumps a theorem' if the counter example corresponds to what the theorem is describing.

Actually in this case is the other way around. I gave an example, and you brought a theorem which I did not even read ignoring the fact that my example was a counter-example to your claim. You can call it a theorem not corresponding to what was under discussion.

Your claim is that there is 'no such thing as an axis of perspectivity of an image'. For a start, a digram with two copies of the pairs of images, one rotated, would not be consistent with the perspective transformation of a scene.

Well, you are getting close. You cannot ignore the scene.

It would be consistent with the perspective transformation of two scenes, overlaid.

Or one, of a painting of Picasso, for example or of a modern architecture.

For a second, two axes clearly does not correspond with no axes, which is what your claim is. You might have been claiming that there is not a singular axis, but that isn't what you said, and it's a claim that no-one has made.

Or, the uniqueness. You have not established existence yet.

This is what happens when I allow distractions out of goodness of my heart. At some point, I get dragged into arguing about the distractions. The definition is about two figures, which are mapped a priori. An image is not that. Q.E.D.

The one you are promoting (which is incomplete)

How is it incomplete? It's only a first thought, if you think it could be refined, be my guess.

It uses a definition of perspective figures (a priori defined and mapped to each other) without saying what they are, how they are mapped, how do we even group them. To make it short, an image is not a collection of two figures, and the definition is given in that case only.

is not very useful. If I take a frontal photo of that wiki drawing, centered, with the axis of my camera perpendicular to the screen, what will the perspective of that photo be? What if I include some surrounding elements which show "undistorted" perspective.

I don't think the definition has anything to do with that scenario. I sought only to define a similarity relation between the 'perspective' of two images. That seems to be the minimum that is necessary if you're going to state that 'perspective is the same'. In brief, if you don't know what is 'the same', you can't say that something is 'the same'. I make no claims whatsoever whether it's useful or not to be able to say something is 'the same'. It's simply that was what the article cited in the OP was doing, without first defining what 'the same' means.

Now, compare this with the definition I propose below, in 3D, where the center is the viewpoint.

Now you have an very interesting idea. You have two different scenes, with a homeomorphism between every corresponding object in those two scenes. The mathematicians will extend perspective to higher dimensionalities, but I'm not sure it's of photographic interest.

The wiki page mentions any dimension. This is automatic, no additional efforts needed. What is of photographic interest is to understand 3D.

Sure, but until you have a camera capable of photographing scenes with two or three dimensions,

Well, my camera can certainly photograph 3D scenes.

and you intend to display them in other than two dimensions, the extra dimensionality is of little interest in a practical sense.

The definition there is not about photography. We want to consider "things" in the greates generaility unless it becomes too complicated. Here, it comes for free.

Thus postulating what might happen if we fail to adhere to those constraints is unuseful.

C does indeed have (roughly) the same perspective as B. In relation to this, the statement that 'Images B and C show that changing the focal length while keeping the subject distance constant has, just like cropping, has no effect on perspective', is misleading and incorrect. As shown above, cropping does change perspective, since it changes the location of the centres of perspectivity when you display the image the same size.

If you really had two figures perspective to each other in the original, and then in the crop, then the center of the perceptivity would be where the camera was (the nodal point) and would therefore be the same.

Sorry, that's just wrong. The centres of perspectivity are very definitely not at the viewer's position.

It is not wrong because I view perspective in the 3D scene before you take the photo.

There is no perspective in a 3D scene.

There is, as a relation of the scene to the image.

That's not perspective. Those are the relations used to construct perspective in a 2D rendition of a 3D scene.

Or, the perspective from which that rendition has been drawn.

Absolutely not. There is no 'perspective' in the domain of the relation.

There is, it was even capitalzied in your favorite online dictionary.

It is the operation of the relation that imbues perspective. The scene has actual 3-D geometry, it doesn't have a rendition of it.

Given that it is already in 3 dimensions, there is no need for any relations that express its reduction to a convincing representation in fewer dimensions.

This is what happens when you work with fuzzy definitions.

I'm not the one working with fuzzy definitions here. I'm the one trying to get people to state their definitions. So far as I can see, you haven't so far done this.

I never tried but see also below. You miss my point. What I am saying that your attempt to use the math definition failed.

How did it fail?

Again, you may have different perspective figures in the image (or none at all) with different centers.

OK. My definition was not well enough formed. Here is a refinement:

i) it is image relative, that is that we perceive perspective with respect to an image that we view, not the geometry of the original scene.

You are free to define it in any way you want but the results could be surprising.

https://www.mnn.com/green-tech/research-innovations/stories/10-optical-illusions-will-blow-your-mind

Thiose two people are of approximately the same height.

ii) that the axes of perspectivity are at the same locations in both images

You need to define similar objects first, which could become tricky.

iii) That the centres of perspectivity relating to objects common to both scenes are at the same locations in both images.

See above.

If you've ever done perspective drawing, that's very evident. What happens when you crop, and then view the same size, you are scaling a portion of an image, thus all the triangles that locate the centres of perspectivity scale and the centres of perspectivity move to different locations. Under my definition of 'same perspective' above, that means different perspective. You're welcome to put forward your own equality relation for perspective, but as I said, unless we agree terms the discussion becomes futile.

My point was that (1) people often have different definitions in mind, and do not even realize that; so we agree here and (2): the definition you cited does not really agree with your intuitive definition of the "same perspective".

I don't have an intuitive definition of 'same perspective'. I keep asking people to say what they mean by 'same perspective'. I provided a definition, for better than worse.

The latter...

So explain why it's 'for worse'. All you've said so far is you like a different definition better, you haven't shown that definition is either more valid or more useful.

I said more than that and until this moment, I have not proposed a new one (see also below). The one below is useful when cropping, stitching panoramas.

I can't see how it is useful for those situations. I'm willing to be convinced.

(i) When taking panoramas, to avoid the parallax error, you want to shoot from the same, well, perspective (point). (ii) Some people think that 50mm on crop has different perspective than 80mm on FF. (iii) One can convert a fisheye image to a rectilinear one, taken from the same point. (iv) Standing at the same point and rotating (a bit) the axis of the camera will not give you are shot that you cannot get by software distortion correction.

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