$30.00

Category: 16720A

Description

5/5 - (5 votes)

1 Homographies

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.

2

Figure 1: A homography H links all points xπ lying in plane π between two

camera views x and x

0

in cameras C and C

0

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

T

, it represents the 2D point

xi

zi

yi

zi

T

(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

3

WhatsApp us