Talk:Homography

message: how do you add the matrix R and the scalar (t \cdot \frac{n^T}{d})?
Only guessing: column vector by row vector: http://en.wikipedia.org/wiki/Dyadic_product. 84.163.99.62 17:28, 2 August 2007 (UTC)

message: Is a homography just an affine transformation? It appears to be using homogeneous coordinates... —Ben FrantzDale 03:54, 24 October 2007 (UTC)
thread_title:  Affine transformation? 

message: I have a question about the 3D plane to plane equations:
[quoted from article]:
: H_{ba} = R - \frac{t n^T}{d} 
R is the rotation matrix by which b is rotated in relation to a; t is the translation vector from a to b; n and d are the normal vector of the plane and the distance to the plane respectively.
[end quoted]
It seems to me that there are two distances in this problem, distance from camera a to the plane, and distance from camera b to the plane.  Which distance should be used in computing H?  Or am I misunderstanding something implicit about the distance d?  
The problem I'm working on involves the same camera viewing the same scene from two different locations, and I want to take the image from location a and transform it to appear as though it was taken from position b.  What's missing is the homography between those two locations, which this section seems to answer (save for my understanding about d).
--Pi Alpha Squared (talk) 19:12, 11 February 2010 (UTC)
thread_title: 3D plane to plane equation

message: Both of these sentences are present in the computer vision section:
:"...any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model)." 
:"If the camera motion between two images is pure rotation, with no translation, then the two images are related by a homography (assuming a pinhole camera model)."
Which is correct? The second does not technically imply that the first MUST be incorrect, but it does suggest this.
70.116.11.243 (talk) 01:07, 21 April 2011 (UTC)
thread_title:  conflicting explanations 

message: Material has been added to this article:
In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model).  This has many practical applications, such as image rectification, image registration, or computation of camera motion—rotation and translation—between two images.   Once camera rotation and translation have been extracted from an estimated homography matrix, this information may be used for navigation, or to insert models of 3D objects into an image or video, so that they are rendered with the correct perspective and appear to have been part of the original scene (see Augmented Reality).
=== 3D plane to plane equation ===
We have two cameras a and b, looking at points P_i in a plane.
Passing the projections of P_i from {}^bp_i in b to a point {}^ap_i in a:
: 
{}^ap_i = K_a \cdot H_{ba} \cdot K_b^{-1} \cdot {}^bp_i
where the homography matrix H_{ba} is
: H_{ba} = R - \frac{t n^T}{d}. 
R is the rotation matrix by which b is rotated in relation to a; t is the translation vector from a to b; n and d are the normal vector of the plane and the distance to the plane respectively.
Ka and Kb are the cameras' intrinsic parameter matrices. 
The figure shows camera b looking at the plane at distance d.
Note: From above figure, n^T P_i is the projection of vector P_i into n^T, and equal to d. So \frac{t n^T}{d} P_i = t. And we have H_{ba} P_i = R P_i - t.
This formula is only valid if camera b have no rotation and no translation. In the general case where R_a,R_b and t_a,t_b are the respective rotations and translations of camera a and b, R=R_a R_b^T and the homography matrix H_{ba} becomes
: H_{ba} = R - R_a \frac{(t_b-t_a) n^T}{d-n^T t_b}R_b^T = R_a (I-\frac{(t_b-t_a) n^T}{d-n^T t_b})R_b^T . 
The material included this reference:
*
The material has been removed since it is a nonstandard use of the term "homography". It is suggested that homography (computer vision) be created for readers and editors interested in this topic.Rgdboer (talk) 23:20, 13 August 2013 (UTC)
thread_title: Computer vision

message: We have two cameras a and b, looking at points P_i in a plane.
Passing the projections of P_i from {}^bp_i in b to a point {}^ap_i in a:
: 
{}^ap_i = K_a \cdot H_{ba} \cdot K_b^{-1} \cdot {}^bp_i
where the homography matrix H_{ba} is
: H_{ba} = R - \frac{t n^T}{d}. 
R is the rotation matrix by which b is rotated in relation to a; t is the translation vector from a to b; n and d are the normal vector of the plane and the distance to the plane respectively.
Ka and Kb are the cameras' intrinsic parameter matrices. 
The figure shows camera b looking at the plane at distance d.
Note: From above figure, n^T P_i is the projection of vector P_i into n^T, and equal to d. So \frac{t n^T}{d} P_i = t. And we have H_{ba} P_i = R P_i - t.
This formula is only valid if camera b have no rotation and no translation. In the general case where R_a,R_b and t_a,t_b are the respective rotations and translations of camera a and b, R=R_a R_b^T and the homography matrix H_{ba} becomes
: H_{ba} = R - R_a \frac{(t_b-t_a) n^T}{d-n^T t_b}R_b^T = R_a (I-\frac{(t_b-t_a) n^T}{d-n^T t_b})R_b^T . 
The material included this reference:
*
The material has been removed since it is a nonstandard use of the term "homography". It is suggested that homography (computer vision) be created for readers and editors interested in this topic.Rgdboer (talk) 23:20, 13 August 2013 (UTC)
thread_title:  3D plane to plane equation 

message: The edits by user:Wcherowi‎ are a great improvement of the article. Nevertheless there are some terminology questions that deserve to be discussed.
*The term "homography" is much older than the abstract notion of projective space. In fact, that is the study of the homographies of the Euclidean spaces that was the main motivation of introducing the projective space.Therefore, I suggest to add to the lead something like: "Homography may also refer to the restriction to a Euclidean space or more generally to an affine space of a projective (linear) transformation."
*In the preceding item, I have put "linear" between parentheses, because the article projective transformation includes in the definition of "projective transformation" not only the collineations but also the duality. IMO, "projective transformation" is a synonym of "homography", and the definition of the article projective transformation refers to the "transformations of the projective space". My suggestion here is to correct the article without changing its title.
* Above definition of the homographies in the affine space implies the notion of "projective completion" of an affine space. It seems that this notion which is referred to by several articles is not defined in Wikipedia. I have only found the definition of the projective completion of a vector space in projectivization. IMO the terminology of this article is incorrect, because, as far as I know, "projectivization" is a synonym of "projective completion".
* Before my recent edits, "homographic function" did not appear in this article and homographic function was a redirect to Möbius function, restricting unduly the notion to the complex case. I suggest to mention "homographic function" in the lead. This would avoid a redirect to an anchor.
By the way, the best modern reference for all these questions seems to be the book "Geometry" by Marcel Berger, and, in particular, its chapter 6, which is devoted to the subject of this article. Unfortunately I have not, presently, an easy access to this book. D.Lazard (talk) 10:25, 30 December 2013 (UTC)
replies: text: Agree on all counts. The problems with the projective transformation page stem from the same editor who reeked havoc here. That page had been a redirect to this one (as it should be since the terms are synonyms). I'll fix it. Bill Cherowitzo  (talk) 18:40, 30 December 2013 (UTC)
thread_title:  Terminology 

message: Currently the article states:
: Any permutation of the points of a projective line will be a collineation.
In fact, a transformation is projective iff it preserves the relation of projective harmonic conjugates. In classical projective geometry the structure of a line is expressed by a projective range. Removal of the material on cross-ratio, the homography that moves three given points to the projective line over GF(2), does not follow proper editing protocol. My mistake of writing PSL(2,A) instead of PGL(2,A) precipitated an overhaul of the article by the author of ovoid (projective geometry) who seems to misunderstand fundamentals of classical projective geometry. Since this article serves students entering the study of complex analysis, it is Top priority.Rgdboer (talk) 20:53, 3 January 2014 (UTC)
replies: text: The two statements are not in conflict. The permutation statement is obvious given the definition of a collineation, but I will provide a reference for it when I get to working on this section. A projective transformation (projectivity) is a collineation, but not vice versa (as the fundamental theorem of projective geometry tells us!). The only thing that I am confused about is why you would bring up ovoid (projective geometry). I am not the author of that article, my contributions were essentially to add missing references and there is absolutely no mention of classical projective geometry in the entire article. Bill Cherowitzo  (talk) 05:43, 4 January 2014 (UTC)
replies: text: Alright, I admit to having expertise in finite geometry, but this is of no consequence unless I am composing new material. The job of a Wikipedia editor is to report on what is available in the secondary literature and not to add to, expand upon or amend that material, so my expertise does not come into play. Bill Cherowitzo  (talk) 20:07, 5 January 2014 (UTC)
thread_title: Structure

message: I have rewritten the lead. In fact, the previous lead emphasized more on collineations than of the subject of the article. I have essentially proceeded by changing the order of the sentences. This allows also to state the fundamental theorem of projective geometry and to correct some errors (homographies were not defined as bijective).
I am not very satisfied by the second paragraph, in which, I have tried to clarify the previous explanation of the number of equivalent terms (by the way, I have removed one, which seems very rare, and put the other in bold, because two of them are targets of redirects) and the distinction between analytic and synthetic approach.
I hope that the new version is understandable by non-specialists. The old one one was certainly not. For example, it did not explain why the definition of the projective space as the set of lines in a vector space is coordinate geometry (another name of analytic geometry).
D.Lazard (talk) 17:48, 9 January 2014 (UTC)
thread_title:  New lead 

message: It appears that the disagreements in the preceding discussions come mainly from misunderstandings caused by the bad writing of the article. The first bad thing is the unnecessary emphasizing on collineations. But the main cause is the confusing use of "synthetic" and "analytic". For me, "synthetic" refers to the definition of the geometry through axioms, by opposition to its modern definition through linear algebra. On the other hand, "analytic" is an old fashioned term for "using coordinates". IMO, both term must be avoided, the former because only very few people understand it correctly, the latter because it is together ambiguous (analytic commonly refers to formal power series) and too restrictive (is coordinate–free linear algebra "analytic" or not?). This being acted, I now understand why User:Wcherowi asked to put at the top the "synthetic material". Therefore, I have rewritten the part of the section "Synthetic approach" that deserve to be just after the lead and moved it at this place under the name "Geometric motivation". The remainder of the section is strongly related to the notions of projective frames and projective coordinates. I have thus just renamed if "Simplex and homography". In the future it should be merged with the section about frames and coordinates. D.Lazard (talk) 17:45, 13 January 2014 (UTC)
thread_title:  Synthetic vs. analytic 

message: Substituted at 01:02, 12 June 2016 (UTC)
thread_title: Assessment comment

message: As a non-specialist, I have two questions/comments. First, the last sentence of the second paragraph of the "Central collineations" section begins with the word "Not". Should that be "Note"? It looks like a possible typo, but either makes at least some sense.
Second, part of this second paragraph seems confusing to me. If the axis is fixed pointwise under α, then I would think any line in the axis is also fixed pointwise. α then maps these lines to themselves. Isn't every point in the axis then also a center?  — Preceding unsigned comment added by Clay Spence (talk • contribs) 14:23, 29 June 2018 (UTC)
replies: text: I have fixed the typo. About your second question: α fixes the lines contained in the axis, but (if it is not the identity) does not fix all the other lines passing through a point of the axis that is not the center (in the case of a elation). So, there is only one center, D.Lazard (talk) 15:23, 29 June 2018 (UTC)
replies: text: Because this can be a bit confusing, let me expand a little on what Prof. Lazard has said. All the lines through a center are mapped to themselves, that is they are fixed. The lines through a fixed point are generally mapped to other lines through that fixed point, not to themselves. Once you see this distinction, the fact that there exists at most one center (for a non-identity map) follows almost immediately.--Bill Cherowitzo  (talk) 21:14, 29 June 2018 (UTC)
thread_title:  Central collineations section 

message: Hello, I find this article and the "Perspectivity" article quite confusing about what exactly is a perspectivity and what is a central collineation. Here are the conclusions I've come to, please let me know if I'm right or not.
* The term "perspectivity" should only apply to maps P \rightarrow Q between two fixed hyperplanes P and Q of a given projective space. In particular, the definition in the paragraph "Geometric motivation" is plain wrong: Given two projective spaces P and Q of dimension n, a perspectivity is a bijection from P to Q that may be obtained by embedding P and Q in a projective space R (...). At first I thought that maybe this definition is equivalent to that of central collineation given later; but actually, by this definition, any projective map is a perspectivity. (Indeed, let f \colon P \to Q be any projective map, let j : Q \rightarrow R be an embedding of Q as a hyperplane, and let i = j \circ f....) 
In particular, a perspectivity from a projective space to itself is a meaningless concept, or it's only the identity map.
By the way, I also don't understand the sentence "With these definitions, a perspectivity is only a partial function, but it becomes a bijection if extended to projective spaces.": I think it is confusing at best.
* The term "central collineation" should only apply to self-maps of a given projective space P. They can described as such: embed P as a hyperplane in a projective space R. Then f \colon P \to P is a central collineation if and only if there exists a hyperplane Q \subset R and perspectivities g \colon P \rightarrow Q and h \colon Q \rightarrow P such that f = h \circ g.
In particular, while a (superficial, I'll admit) reading of the two wikipedia articles gave me the impression that perspectivities and central collineations were more or less the same thing, they're actually completely different. If anything, the thing to remember is that a central collineation is the composition of two perspectivities. And, by the way, calling a central collineation a "perspective collineation" (as in the "Perspectivity" article) is absurd: since perspectivities are collineations, a "perspective collineation" should not mean anything different than a perspectivity).
Am I missing something? Seub (talk) 15:44, 13 July 2021 (UTC)
replies: text: I'm also confused on this. It seems what one article refers to as a perspectivity is referred to as a central collineation by the other, and one or both need to be rewritten. – NuclearDuckie (talk) 08:19, 2 June 2022 (UTC)
replies: text: Reading closer, the Perspectivity article says that perspecitvies and projectivities/homographies are bijections between lines (or subspaces), while this article and others use perspectivity and projectivity/homography to mean bijections between spaces which map lines to lines (i.e. each line undergoes a perspectivity or projectivity by the other definition). – NuclearDuckie (talk) 14:36, 3 June 2022 (UTC)
thread_title:  Confusion: what are perspectivities?