10.3 Perspective camera model
10.3.1 Homogeneous coordinates and projective geometry
Euclidean geometry is algebraically represented by the Cartesian coordinate system, in which
points are defined by tuples of numbers. Each number is related to one axis and a set of
axes determines the dimension. This representation is a very natural way of describing our
3D world and is very useful in image processing to describe pixels in two-dimensional (2D)
images. Cartesian coordinates are convenient to describe angles and lengths and they are simply
transformed by matrix algebra to represent translations, rotations and changes of scale. However,
the relationship defined by projections cannot be described with the same algebraic simplicity.
Projective geometry is algebraically represented by the homogeneous coordinate system. This
representation is a natural way of formulating how we relate camera coordinates to ‘real-world’
coordinates: the relation between image and physical space. Its major advantages are that image
transformations such as rotations, change of scale and projections become matrix multiplications.
Projections provide perspective, which corresponds to the distance of