计算3D世界点

Question

I am beginner in OpenCV and I want to find 3D point from 2D (projected on image) and known two out of three world coordinates of a point.

What I have:

1. calibrated camera (known: matrix of intrinsic parameters, vector of distortion coefficients)
2. rotation vector and translation vector
3. (u, v) - coordinates of a point on image
4. two known world coordinates of a point and one unknown

Problem variant #1 Known: X, Y Unknown: Z

Problem variant #2 Known: X, Z Unknown: Y

Problem variant #3 Known: Y, Z Unknown: X

How can I find third unknown coordinate? Is it possible?

Answer 1

Yes it is possible. Consider the simple case of absent nonlinear distortion. Let Ki be the inverse of the camera matrix, and camera center at the world origin (ie no rotation nor translation). Let p=(u, v, 1) be the homogeneous pixel coordinate. Then the ray through the pixel is:

s * P = Ki * p

where s > 0 is an unknown scale. But s * P = [X, Y, Z] , so if you know any one of X, Y or Z you can solve for s and find the missing coordinates.

For non-zero roto-translation, replace Ki with the inverse of the projection matrix. For non-zero distortion, replace the simple multiplication by Ki with the complete reprojection equation.