Final answer:
Recovering internal and external parameters from a projection matrix involves normalizing the matrix, performing RQ decomposition, and extracting the intrinsic matrix K, the rotation matrix R, and the translation vector t.
Step-by-step explanation:
To recover the internal and external parameters from a projection matrix, we can employ a method based on the decomposition of the matrix. The projection matrix P in computer vision is a 3x4 matrix that maps 3D points in the world to 2D points on an image plane, and it can be decomposed into internal camera parameters (intrinsic matrix K) and external parameters (rotation matrix R and translation vector t).
Let's start by considering the projection matrix P of the form:
P = K [R | t]
Where K is the intrinsic matrix, R is the rotation matrix, and t is the translation vector. This matrix can be decomposed using the QR decomposition or the RQ decomposition to extract K, R, and t.
The RQ decomposition can be performed on the first three columns of P, which will yield matrices Q (the intrinsic parameters without the translation) and R (the rotation matrix). The translation vector t can be isolated by using the fact that it corresponds to the last column of P.
By normalizing the last element of P to be 1, we ensure that we are working with homogenous coordinates, and then we can directly extract t. External parameters are then given by R and t, and internal parameters by Q, which, after normalization, is equivalent to K.
To summarize, the steps to recover internal and external parameters from the projection matrix are first normalizing the projection matrix, then performing RQ decomposition, and finally extracting K, R, and t from the resulting matrices.