Derivation of optical flow equations for a spherical camera

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Definitions

T is a 3-vector corresponding to the camera's translational velocity.

Ω is a 3-vector corresponding to the camera's rotational velocity (the camera is rotating about the axis Ω at a rate of | | Ω | | radians per unit time).

P is a 3-vector corresponding to the coordinates of a scene point in a camera-centric coordinate frame.

||P|| = (P \cdot P)^\frac{1}{2} is the depth of scene point P (distance from the camera)

\hat{P} = \frac{P}{||P||} is the projection of P onto the image surface.

\dot{P} = \frac{d}{dt}P = -T - \Omega \times P is the instantaneous velocity of a scene point.

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Goal

Want to derive some useful formula for

U = \frac{d}{dt}\hat{P} is the optical flow for the image \hat{P} of scene point P

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Derivation

\frac{d}{dt}\hat{P} = \frac{d}{dt} \left( \frac{P}{||P||} \right)
= \frac{ ||P|| \dot{P} - P \frac{d}{dt}||P|| }{ ||P||^2 }
\frac{d}{dt}||P|| = \frac{d}{dt} \left( (P \cdot P)^{\frac{1}{2}} \right)
= \frac{1}{2}(P \cdot P)^{-\frac{1}{2}} \frac{d}{dt}(P \cdot P)
= \frac{1}{2}\frac{1}{||P||}( \dot{P} \cdot P + P \cdot \dot{P} )
= \frac{1}{2 ||P||} 2 ( ( -T - \Omega \times P ) \cdot P )
= \frac{ -T \cdot P}{||P||}
= -T \cdot \hat{P}
\frac{d}{dt}\hat{P} = \frac{ ( -T - \Omega \times P ) ||P|| - P \frac{d}{dt}||P|| }{ ||P||^2 }
= \frac{ -T ||P|| - P \frac{d}{dt}||P|| }{||P||^2} - \frac{(\Omega \times P) ||P||}{||P||^2}
= \frac{ -T - \hat{P} \frac{d}{dt}||P|| }{||P||} - \Omega \times \hat{P}
= \frac{ -T - \hat{P} ( -T \cdot \hat{P} ) }{||P||} - \Omega \times \hat{P}
= \frac{1}{||P||} ((T \cdot \hat{P}) \hat{P} - T) - \Omega \times \hat{P}
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