Camera matrix

In computer vision a camera matrix or (camera) projection matrix is a 3 × 4 {displaystyle 3 imes 4} matrix which describes the mapping of a pinhole camera from 3D points in the world to 2D points in an image. In computer vision a camera matrix or (camera) projection matrix is a 3 × 4 {displaystyle 3 imes 4} matrix which describes the mapping of a pinhole camera from 3D points in the world to 2D points in an image. Let x {displaystyle mathbf {x} } be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let y {displaystyle mathbf {y} } be a representation of the image of this point in the pinhole camera (a 3-dimensional vector). 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 up to a non-zero scalar multiplication. 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 this expression 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