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I'd like if someone could elaborate on the definition of the following formula, X(λ) = pinv(P)x + λC.

According to Zisserman book (multiple view geometry) page 162 (6.13), and to my understanding (probably am wrong...), this formula defines the back-projected ray of the image point x. And it's derived from 2 points on that ray: the camera center C and the point pinv(P)xץ

Where X is the ray, or a line in other words(in homogenous cords in 4D), pinv(P) is the pseudo-inverse of P, which maps x - a point in the image plane, to a ray X in the world coordinates. C is the camera center in world coordinates (C is derived from PC=0) and λ is just a constant.

Now i just don't understand how the explanation above, forms the line X exactly, because if that is the case, so from the formula, C is the direction of that ray, but C is really just a point on that ray, not it's direction, just like pinv(P)x is also a point on that ray.

Thanks in advance.

Nir Yakobovits
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1 Answers1

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Same question was posted here:

Equation of a ray passing through the camera center and an image point

And actually I was very, very confused by this. Main thing is the ray is actually a euclidean ray but in homogeneous coordinates (confusing). And the coordinates C and pinv(P)x are also in homogeneous coordinates. Adding homogeneous coordinates is not the same as adding euclidean coordinates, because you have to divide by the last coordinate for the homogenous case to bring the result back to euclidean coordinates.

A more verbose answer (which I hope is correct...) with simple example is shown here:

https://math.stackexchange.com/a/4304298/39581

JustinBlaber
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