- #1

tworitdash

- 107

- 26

I want to determine the velocity field in 3D for this image. I have several measurements of the images in time. Usually, in the literature I have seen people using a continuity of the brightness itself (image) in time and space assuming that the reflection is conserved.

However, it is usually done in polar co-ordinates. It is due to the fact that the radial velocities (one component of the flow field in the polar coordinates) are also measured through a radar with Doppler effect.

I was wondering if it is possible to formulate it in a very generic way where I consider a conical coordinate system instead such that I can make use of all the cones? Or, is it a bad exercise ?

Furthermore, it is usually assumed that the energy is conserved so the following cost function is always imposed in optimizing for the flow fields:

$$ J = \left(\frac{\partial \eta}{\partial t} + V_r \frac{\partial \eta}{\partial r} + V_{\theta} \frac{\partial \eta}{r \partial \theta} + V_{\phi} \frac{\partial }{r \sin(\theta) } \frac{\partial\eta}{\partial \phi} \right)^2 $$.

However, if the total reflection is not conserved, how can I optimize for the flow field? Do I have to estimate a constant [itex]A_0[/itex] such that,

$$\frac{\partial \eta}{\partial t} + V_r \frac{\partial \eta}{\partial r} + V_{\theta} \frac{\partial \eta}{r \partial \theta} + V_{\phi} \frac{\partial }{r \sin(\theta) } \frac{\partial\eta}{\partial \phi} = A_0$$ ?, instead of 0 ?

The reflection or image is [itex]\eta[/itex], and the flow field is [itex]\vec{V}[/itex].