Camera matrix

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In computer vision a camera matrix or (camera) projection matrix is a ${\displaystyle 3\times 4}$ matrix which describes the mapping of a pinhole camera from 3D points in the world to 2D points in an image.

Let ${\displaystyle \mathbf {x} }$ be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let ${\displaystyle \mathbf {y} }$ be a representation of the image of this point in the pinhole camera (a 3-dimensional vector). Then the following relation holds

${\displaystyle \mathbf {y} \sim \mathbf {C} \,\mathbf {x} }$

where ${\displaystyle \mathbf {C} }$ 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 ${\displaystyle k\neq 0}$:

${\displaystyle \mathbf {y} =k\,\mathbf {C} \,\mathbf {x} .}$

Since the camera matrix ${\displaystyle \mathbf {C} }$ 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.