Textbook Error? “, … and singling out the line at infinity in *the image* or the plane at infinity in...
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Page 3 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following:
In computer vision problems, projective space is used as a convenient way of representing the real 3D world, by extending it to the 3-dimensional (3D) projective space. Similarly images, usually formed by projecting the world onto a 2-dimensional representation, are for convenience extended to be thought of as lying in the 2-dimensional projective space. In reality, the real world, and images of it do not contain points at infinity, and we need to keep our finger on which are the fictitious points, namely the line at infinity in the image and the plane at infinity in the world. For this reason, although we usually work with the projective spaces, we are aware that the line and plane at infinity are in some way special. This goes against the spirit of pure projective geometry, but makes it useful for our practical problems. Generally we try to have it both ways by treating all points in projective space as equals when it suits us, and singling out the line at infinity in space or the plane at infinity in the image when that becomes necessary.
The aforementioned section of the textbook is available freely here.
Shouldn't the last part
, ... and singling out the line at infinity in space or the plane at infinity in the image when that becomes necessary.
actually be
, ... and singling out the line at infinity in the image or the plane at infinity in space when that becomes necessary.
?
After all, in previous mentions, the line at infinity is always referred to in the context of the image (2-dimensional space), and the plane at infinity is always referred to in the context of space (3-dimensional space), as is done here:
In reality, the real world, and images of it do not contain points at infinity, and we need to keep our finger on which are the fictitious points, namely the line at infinity in the image and the plane at infinity in the world.
I would greatly appreciate it if people could please take the time to clarify this.
linear-algebra geometry projective-geometry projective-space computer-vision
asked Nov 25 at 17:14
The Pointer
2,59521333
add a comment |
up vote
2
down vote
favorite
Page 3 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following:
In computer vision problems, projective space is used as a convenient way of representing the real 3D world, by extending it to the 3-dimensional (3D) projective space. Similarly images, usually formed by projecting the world onto a 2-dimensional representation, are for convenience extended to be thought of as lying in the 2-dimensional projective space. In reality, the real world, and images of it do not contain points at infinity, and we need to keep our finger on which are the fictitious points, namely the line at infinity in the image and the plane at infinity in the world. For this reason, although we usually work with the projective spaces, we are aware that the line and plane at infinity are in some way special. This goes against the spirit of pure projective geometry, but makes it useful for our practical problems. Generally we try to have it both ways by treating all points in projective space as equals when it suits us, and singling out the line at infinity in space or the plane at infinity in the image when that becomes necessary.
The aforementioned section of the textbook is available freely here.
Shouldn't the last part
, ... and singling out the line at infinity in space or the plane at infinity in the image when that becomes necessary.
actually be
, ... and singling out the line at infinity in the image or the plane at infinity in space when that becomes necessary.
?
After all, in previous mentions, the line at infinity is always referred to in the context of the image (2-dimensional space), and the plane at infinity is always referred to in the context of space (3-dimensional space), as is done here:
In reality, the real world, and images of it do not contain points at infinity, and we need to keep our finger on which are the fictitious points, namely the line at infinity in the image and the plane at infinity in the world.
I would greatly appreciate it if people could please take the time to clarify this.
linear-algebra geometry projective-geometry projective-space computer-vision
asked Nov 25 at 17:14
The Pointer
2,59521333
1
Sure. Looks like a simple typo to me. It’s not among the errata for the second edition, though, as far as I can see.
– amd
Nov 25 at 21:53
@amd Thanks for the confirmation.
– The Pointer
Nov 26 at 5:33
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Page 3 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following:
In computer vision problems, projective space is used as a convenient way of representing the real 3D world, by extending it to the 3-dimensional (3D) projective space. Similarly images, usually formed by projecting the world onto a 2-dimensional representation, are for convenience extended to be thought of as lying in the 2-dimensional projective space. In reality, the real world, and images of it do not contain points at infinity, and we need to keep our finger on which are the fictitious points, namely the line at infinity in the image and the plane at infinity in the world. For this reason, although we usually work with the projective spaces, we are aware that the line and plane at infinity are in some way special. This goes against the spirit of pure projective geometry, but makes it useful for our practical problems. Generally we try to have it both ways by treating all points in projective space as equals when it suits us, and singling out the line at infinity in space or the plane at infinity in the image when that becomes necessary.
The aforementioned section of the textbook is available freely here.
Shouldn't the last part
, ... and singling out the line at infinity in space or the plane at infinity in the image when that becomes necessary.
actually be
, ... and singling out the line at infinity in the image or the plane at infinity in space when that becomes necessary.
?
After all, in previous mentions, the line at infinity is always referred to in the context of the image (2-dimensional space), and the plane at infinity is always referred to in the context of space (3-dimensional space), as is done here:
In reality, the real world, and images of it do not contain points at infinity, and we need to keep our finger on which are the fictitious points, namely the line at infinity in the image and the plane at infinity in the world.
I would greatly appreciate it if people could please take the time to clarify this.
linear-algebra geometry projective-geometry projective-space computer-vision
asked Nov 25 at 17:14
The Pointer
2,59521333
Page 3 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following:
In computer vision problems, projective space is used as a convenient way of representing the real 3D world, by extending it to the 3-dimensional (3D) projective space. Similarly images, usually formed by projecting the world onto a 2-dimensional representation, are for convenience extended to be thought of as lying in the 2-dimensional projective space. In reality, the real world, and images of it do not contain points at infinity, and we need to keep our finger on which are the fictitious points, namely the line at infinity in the image and the plane at infinity in the world. For this reason, although we usually work with the projective spaces, we are aware that the line and plane at infinity are in some way special. This goes against the spirit of pure projective geometry, but makes it useful for our practical problems. Generally we try to have it both ways by treating all points in projective space as equals when it suits us, and singling out the line at infinity in space or the plane at infinity in the image when that becomes necessary.
The aforementioned section of the textbook is available freely here.
Shouldn't the last part
, ... and singling out the line at infinity in space or the plane at infinity in the image when that becomes necessary.
actually be
, ... and singling out the line at infinity in the image or the plane at infinity in space when that becomes necessary.
?
After all, in previous mentions, the line at infinity is always referred to in the context of the image (2-dimensional space), and the plane at infinity is always referred to in the context of space (3-dimensional space), as is done here:
In reality, the real world, and images of it do not contain points at infinity, and we need to keep our finger on which are the fictitious points, namely the line at infinity in the image and the plane at infinity in the world.
I would greatly appreciate it if people could please take the time to clarify this.
linear-algebra geometry projective-geometry projective-space computer-vision
linear-algebra geometry projective-geometry projective-space computer-vision
asked Nov 25 at 17:14
The Pointer
2,59521333
asked Nov 25 at 17:14
The Pointer
2,59521333
asked Nov 25 at 17:14
The Pointer
2,59521333
asked Nov 25 at 17:14
The Pointer
2,59521333
asked Nov 25 at 17:14
The Pointer
2,59521333
2,59521333
1
Sure. Looks like a simple typo to me. It’s not among the errata for the second edition, though, as far as I can see.
– amd
Nov 25 at 21:53
@amd Thanks for the confirmation.
– The Pointer
Nov 26 at 5:33
add a comment |
1
Sure. Looks like a simple typo to me. It’s not among the errata for the second edition, though, as far as I can see.
– amd
Nov 25 at 21:53
@amd Thanks for the confirmation.
– The Pointer
Nov 26 at 5:33
1
1
Sure. Looks like a simple typo to me. It’s not among the errata for the second edition, though, as far as I can see.
– amd
Nov 25 at 21:53
Sure. Looks like a simple typo to me. It’s not among the errata for the second edition, though, as far as I can see.
– amd
Nov 25 at 21:53
@amd Thanks for the confirmation.
– The Pointer
Nov 26 at 5:33
@amd Thanks for the confirmation.
– The Pointer
Nov 26 at 5:33
add a comment |
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1
Sure. Looks like a simple typo to me. It’s not among the errata for the second edition, though, as far as I can see.
– amd
Nov 25 at 21:53
@amd Thanks for the confirmation.
– The Pointer
Nov 26 at 5:33