Camera matrix

In <a href="/facts/Computer_vision/Tl2Yyk66">computer vision</a> a camera matrix or (camera) projection matrix is a 
 
 
 
 3
 ×
 4
 
 
 {\displaystyle 3\times 4}
 
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> which describes the mapping of a <a href="/facts/Pinhole_camera/8JmYNdeL">pinhole camera</a> 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 <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a> (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

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

where 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbf {C} }
 
 is the camera matrix and the 
 
 
 
 
 ∼
 
 
 {\displaystyle \,\sim }
 
 sign implies that the left and right hand sides are equal <a href="/facts/Modulo_(mathematics)/lg6V4FF9">except</a> for a multiplication by a non-zero scalar 
 
 
 
 k
 ≠
 0
 
 
 {\displaystyle k\neq 0}
 
:

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

Since the camera matrix 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbf {C} }
 
 is involved in the mapping between elements of two <a href="/facts/Projective_space/7MSpA9cM">projective spaces</a>, 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.

Camera matrix open-in-new

Camera matrix