Camera matrix

Then the following relation holds where C {\displaystyle \mathbf {C} } is the camera matrix and the ∼ {\displaystyle \,\sim } sign implies that the left and right hand sides are equal except for a multiplication by a non-zero scalar k ≠ 0 {\displaystyle k\neq 0} : Since the camera matrix C {\displaystyle \mathbf {C} } is involved in the mapping between elements of two projective spaces, it too can be regarded as a projective element. This means that it has only 11 degrees of freedom since any multiplication by a non-zero scalar results in an equivalent camera matrix. The mapping from the coordinates of a 3D point P to the 2D image coordinates of the point's projection onto the image plane, according to the pinhole camera model, is given by where ( x 1 , x 2 , x 3 ) {\displaystyle (x_{1},x_{2},x_{3})} are the 3D coordinates of P relative to a camera centered coordinate system, ( y 1 , y 2 ) {\displaystyle (y_{1},y_{2})} are the resulting image coordinates, and f is the camera's focal length for which we assume f > 0. Furthermore, we also assume that x3 > 0. To derive the camera matrix, the expression above is rewritten in terms of homogeneous coordinates. Instead of the 2D vector ( y 1 , y 2 ) {\displaystyle (y_{1},y_{2})} we consider the projective element (a 3D vector) y = ( y 1 , y 2 , 1 ) {\displaystyle \mathbf {y} =(y_{1},y_{2},1)} and instead of equality we consider equality up to scaling by a non-zero number, denoted ∼ {\displaystyle \,\sim } . First, we write the homogeneous image coordinates as expressions in the usual 3D coordinates. Finally, also the 3D coordinates are expressed in a homogeneous representation x {\displaystyle \mathbf {x} } and this is how the camera matrix appears: where C {\displaystyle \mathbf {C} } is the camera matrix, which here is given by and the corresponding camera matrix now becomes The last step is a consequence of C {\displaystyle \mathbf {C} } itself being a projective element.