Planar Homographies as a Warp
Recall that a planar homography is an warp operation (which is a mapping from pixel
coordinates from one camera frame to another) that makes a fundamental assumption of the
points lying on a plane in the real world. Under this particular assumption, pixel coordinates
in one view of the points on the plane can be directly mapped to pixel coordinates in another
camera view of the same points.
Figure 1: A homography H links all points xπ lying in plane π between two
camera views x and x
in cameras C and C
respectively such that x
0 = Hx.
[From Hartley and Zisserman]
Q1.1 Homography (5 points)
Prove that there exists a homography H that satisfies equation 1 given two 3×4 camera
projection matrices P1 and P2 corresponding to the two cameras and a plane Π. You
do not need to produce an actual algebraic expression for H. All we are asking for is
a proof of the existence of H.
x1 ≡ Hx2 (1)
The ≡ symbol stands for identical to. The points x1 and x2 are in homogenous
coordinates, which means they have an additional dimension. If x1 is a 3D vector
xi yi zi
, it represents the 2D point
(called inhomogenous coordinates).
This additional dimension is a mathematical convenience to represent transformations
(like translation, rotation, scaling, etc) in a concise matrix form. The ≡ means that
the equation is correct to a scaling factor.
Note: A degenerate case happens when the plane Π contains both cameras’ centers,
in which case there are infinite choices of H satisfying equation 1. You can ignore this
special case in your answer.
The Direct Linear Transform
A very common problem in projective geometry is often of the form x ≡ Ay, where x and y
are known vectors, and A is a matrix which contains unknowns to be solved. Given matching
points in two images, our homography relationship clearly is an instance of such a problem.
Note that the equality holds only up to scale (which means that the set of equations are of
the form x = λHx0
), which is why we cannot use an ordinary least squares solution such as
what you may have used in the past to solve simultaneous equations. A standard approach
to solve these kinds of problems is called the Direct Linear Transform, where we rewrite
the equation as proper homogeneous equations which are then solved in the standard least