In the Wikipedia article about Homography, the projective transformation between picture A to picture B is defined as:
$H_{ab} = R - \frac{t n^T}{d}$
However, the formula lacks the derivation. How is it derived?
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1 Answer
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Given a point on a plane $p$, $n^T p$ for the directed unit normal vector for the plane gives you the projection of the given point onto the line of the normal vector, which gives you the distance $-d$ away from the origin on that plane (negative because the normal vector is in the opposite direction to the point vector). i.e.
$$ n^T p = -d $$
For a homography from $a$ to $b$, we want
$$ p_b = H_{ab}p_a = (R_{ab}p_a+t_{ab}) $$
for some rotation and translation. The translation term is annoying because it doesn't have $p_a$ on the right even though H should be a single multiplicative term for p. We have to split up $H$ into a single multiplicative factor for $p$ with two terms, one for the $R$ which doesn't have to change, and one for $t$ for which we need to split out so that multiplying by $p$ results in the $t$ translation.
We do this by looking back at our $n^T p = -d$ equation, dividing by $-d$ on both sides so LHS = 1. The t can now be multiplied by the LHS, which doesn't change the value but introduces the $p$ to be factored out $$ p_b = H_{ab}p_a = (R_{ab}+\frac{t_{ab}n^T}{-d})p \\ = R_{ab}p+ t_{ab}\frac{n^T p}{-d} \\ = R_{ab}P + t_{ab} $$ This recovers our expected relationship
