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I am trying to determine the depth of view for a hypercatadioptric camera (camera lens system and a hyperbolic mirror) based on 1.

The following illustration seems pretty clear. For an image point $p$, we are looking for a virtual point $p_u$, given the parameters of the optical system.

Image from [1]

I have troubles finding the right equations in the paper, though. There is the distribution of a virtual point which seems to be connected to $p_u$, but is not defined anywhere.

My goal is to replicate the diagrams they have later in the paper, like e.g. this one: enter image description here

Which for a mirror (blue) gives the virtual image points of the scene (red). I would like to calculate the depth of view, so the area at which the image blur is below a threshold.

1 Zhang, S., Zenou, E.: Optical approach of a hypercatadioptric system depth of field. In: 10th International Conference on Information Sciences, Signal Processing and their Applications

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