Lesson plan
Camera matrix
In computer vision a camera matrix or (camera) projection matrix is a 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 \mathbf{x} be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let \mathbf{y} be a representation of the image of this point in the pinhole camera (a 3-dimensional vector). Then the following relation holds : \mathbf{y} \sim \mathbf{C} \, \mathbf{x} where \mathbf{C} is the camera matrix and the \, \sim sign implies that the left and right hand sides are equal except for a multiplication by a non-zero scalar k \neq 0 : : \mathbf{y} = k \, \mathbf{C} \, \mathbf{x} . Since the camera matrix \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.
Article summary
In computer vision a camera matrix or (camera) projection matrix is a 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 \mathbf{x} be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let \mathbf{y} be a representation of the image of this point in the pinhole camera (a 3-dimensional vector). Then the following relation holds : \mathbf{y} \sim \mathbf{C} \, \mathbf{x} where \mathbf{C} is the camera matrix and the \, \sim sign implies that the left and right hand sides are equal except for a multiplication by a non-zero scalar k \neq 0 : : \mathbf{y} = k \, \mathbf{C} \, \mathbf{x} . Since the camera matrix \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.
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