next up previous
Next: Epipolar geometry Up: Camera model and epipolar Previous: Camera model and epipolar

Camera model

The camera is modelled by its optical centre tex2html_wrap_inline1099 and its retinal plane (or image plane) tex2html_wrap_inline1101 . In each camera, a 3-D point tex2html_wrap_inline1103 in world coordinates (where the world coordinate frame is fixed arbitrarily) is projected into an image point tex2html_wrap_inline1105 in camera coordinates, where tex2html_wrap_inline1107 is the intersection of tex2html_wrap_inline1101 with the line containing tex2html_wrap_inline1111 and tex2html_wrap_inline1099 . In projective (or homogeneous) coordinates, the transformation from tex2html_wrap_inline1111 to tex2html_wrap_inline1107 is modelled by the linear transformation tex2html_wrap_inline1119





The points tex2html_wrap_inline1111 for which S=0 define the focal plane and are projected to infinity.

Figure 1: Pinhole camera model.

Each pinhole camera is therefore modelled by its perspective projection matrix (PPM) tex2html_wrap_inline1119 , which can be decomposed into the product


The matrix tex2html_wrap_inline1127 gathers the intrinsic parameters of the camera, and has the following form:


where tex2html_wrap_inline1129 are the focal lengths in vertical and horizontal pixels, respectively, and tex2html_wrap_inline1131 are the coordinates of the principal point. The matrix tex2html_wrap_inline1133 is composed by a tex2html_wrap_inline1135 rotation matrix tex2html_wrap_inline1137 and a vector tex2html_wrap_inline1139 , encoding the camera position and orientation (extrinsic parameters) in the world reference frame, respectively:


Let us write the PPM as


The plane tex2html_wrap_inline1141 (S=0) is the focal plane, and the two planes tex2html_wrap_inline1145 and tex2html_wrap_inline1147 intersect the retinal plane in the vertical (U=0) and horizontal (V=0) axis of the retinal coordinates, respectively.

The optical centre tex2html_wrap_inline1099 is the intersection of the three planes introduced in the previous paragraph; therefore




The optical ray associated to an image point tex2html_wrap_inline1107 is the line tex2html_wrap_inline1157 , i.e. the set of points tex2html_wrap_inline1159 . The equation of this ray can be written in parametric form as


with tex2html_wrap_inline1161 an arbitrary real number.

Andrea Fusiello
Tue Feb 3 17:18:41 MET 1998