HOMOGRAPHY

Let’s take the case where the camera is imaging a planar scene, in this case it can be said that the $z=0\forall \stackrel{\sim }{w}$, in this scenario the $PPM$ becomes

$$k\stackrel{\sim }{m}=\stackrel{\sim }{P}\stackrel{\sim }{w}=\left[\begin{array}{cccc}{p}_{1,1}& p_{1,2}& p_{1,3}& p_{1,4}\\ p_{2,1}& p_{2,2}& p_{2,3}& p_{2,4}\\ p_{3,1}& p_{3,2}& p_{3,3}& p_{3,4}\end{array}\right]\times \left[\begin{array}{c}x\\ y\\ 0\\ 1\end{array}\right]=\left[\begin{array}{ccc}{p}_{1,1}& p_{1,2}& p_{1,4}\\ p_{2,1}& p_{2,2}& p_{2,4}\\ p_{3,1}& p_{3,2}& p_{3,4}\end{array}\right]\times \left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=H\stackrel{\sim }{M}$$

this is exploited in camera callibration with the zhang’s method.

PROPERTIES

cause $\det H\ne 0$ two distinct images of a planar scene are related by homography

$${\stackrel{\sim }{m}}_1=H_1\stackrel{\sim }{M}\Rightarrow {\stackrel{\sim }{m}}_1=H_1{H}_2^{-1}{\stackrel{\sim }{m}}_2$$ $${\stackrel{\sim }{m}}_2=H_2\stackrel{\sim }{M}\Rightarrow {\stackrel{\sim }{m}}_2=H_2{H}_1^{-1}{\stackrel{\sim }{m}}_1$$

ROTATION RELATION

Any two images taken by a camera rotating about the optical center are related by homography

INTRINSIC PARAMETER RELATION

any two images taken by two different cameras (different $A$ matrix so different intrinsic parameters) in a fixed pose are related by homography

In conclusion we can say that if the camera is imaging a planar scene we can find relations between images that are taken from different angles or with different camera sensors

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