Up to a scale factor

Question

I am reading up on homographies and i have seen some places that it says that the homography is defined "up to a scale factor" what does this mean? Is there an upper limit for scaling the homography or what does it mean, and why?

Ennoble

I think math.stackexchange.com would've been a better place for this question. They have LaTeX-support! ;) – Mcneely

Answer 1

General Meaning

• A is unique up to Variation
• A is the same as B up to Variation
• A is equal to B up to Variation
• Statement up to Variation

Phrases of the forms above typically mean that the Statement - the part before "up to" - is true excepting some kind of Variation. It can be thought of as meaning "...up to...but no further."

Example

Two points in the plane determine a line.
One point in the plane determines a line up to rotation about the point.

Meaning with respect to Homographies

Taken from the first section of this document:

1. From 3D to 2D Coordinates

Under homography, we can write the transformation of points in 3D from camera 1 to camera 2 as:

X2 = H*X1,    X1,X2 in R^3

In the image planes, using homogeneous coordinates, we have

a*x1 = X1,    b*x2 = X2,    therefore b*x2 = H*a*x1

This means that x2 is equal to H*x1 up to a scale (due to universal scale ambiguity).

In the next section of the same document, Homography Estimation is described, wherein the z1 variable being solved for is "without loss of generality" set to 1. There is a whole set of solution homographies (with variation across scale), so a convention is made in this case to always choose the homograph that has universal scale z1 set to 1.

Numerator

I get what you are saying. But then again not totally. Maybe it is my english which does not suffice. A get that you can just scale the homography and that they just choose that given scaling. But does it litteraly mean that the homography is not valid if too big a scale is chosen? And why is that? – Ennoble

That is not what it means. You can choose whatever scale you want. "x2 is equal to H*x1 up to a scale" means that the scale is still a free variable. -- In the example I gave, "One point in the plane determines a line up to rotation about the point," the meaning is that one point actually doesn't by itself determine a line - it just determines a line up to rotation (where the rotation is still a free variable). – Numerator

aahh.. Now i think i get it. So it is like if you said, "it is up to the rotation to determine the line". So the homography is determined only if a scale is chosen (Makes sense since it has 8 degrees of freedom, right?) – Ennoble

@Ennoble Exactly. - If your question/confusion is answered/explained by my answer, please upvote and accept. :) – Numerator

@TimothyShields Can I replace "up to" with "depending on" in 'line and rotation example'? – Thunderbolt

@JaeJunLEE Usually that phrase has different meaning. For example, "The store is open or closed depending on the time of day." It suggests a functional dependency. – Numerator

Answer 2

Consider this Homography matrix, H.

[ h11  h12  h13 ]
[ h21  h22  h23 ]
[ h31  h32  h33 ]

It has 9 elements and say, I have a scale factor s, which is a scalar. Now if I were to multiply the s with H, it would still be the same planar transformation.

    [ s*h11  s*h12  s*h13 ]
    [ s*h21  s*h22  s*h23 ]
    [ s*h31  s*h32  s*h33 ]

But, say, I have to do the same transformation in reverse, I would need to know just one of these new elements(e.g.: we set h33 = 1), and now the others can adjust in value to reach the same transformation.

Lesleylesli