Homography – Part II: useful facts
In order to prove the second largest singular value of Homography matrix is 1, or decompose the Homography matrix, the following facts are very important.
Fact 1: Let and . A vector makes if and only if or .
Hence implies .
People usually confuse Fact 1 and Fact 2. Just note for a symmetric matrix, we can not have implies . Compare the Fact 1 and Fact 2.
Fact 2: Let and . A vector makes if and only if and .
Proof: Sufficiency is apparent. Next prove necessity:
case 1: and are collinear. Then , hence . In general, . So it is easy to show has two zeros eigenvalues , and associating eivenvectors are perpendicular to .
case 2: and are not collinear.
a) if , then , . Easy
b) There exists but no perpendicular to . Then
c) There exists but no perpendicular to . Then
So if , then . Hence the eigenspace associating zero eigenvalue is one dimensional. So implies is collinear with .
In summary, if and only if and .
Fact 3: The homography matrix is . Let is a vector. Then if and only if .
Proof: is equivalent to . That is
where . According to the Fact 1, we have .