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?

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.