2.5  Summary and conclusions

This chapter has introduced the projective geometry that uses homogeneous coordinates to represent the position of 2D and 3D points. Using homogeneous coordinates, the projection of 3D points onto the 2D image plane can be described using a linear projection matrix. We have seen that this projection matrix can be decomposed into two matrices. The first matrix K includes the intrinsic camera parameters, comprising internal parameters such as the focal length and the principal point. The second matrix [R|-RC] includes the extrinsic camera parameters that correspond to the position C and orientation R of the camera. Subsequently, the chapter has presented a flexible calibration technique that enables the estimation of the camera parameters. The camera calibration algorithm employs a planar pattern captured from, a least, three viewpoints. Finally, the two-view geometry that describes the geometric relationship between two images was introduced. For example, this two-view geometry is particularly useful to recover the 3D structure of a scene, which is a problem that is addressed in the next chapter.

A summary of camera parameters and important equations referring to epipolar geometry is presented in Table 2.1.



K = (            )
   f  τ   ox
(  0  ηf  oy )
   0  0   1

Intrinsic camera parameters indicating the focal length and the skew parameter.

E = [  R   - RC  ]
  0 3    1

Extrinsic camera parameters indicating the camera orientation and position.

λp = Hp′

Homography or 2D projective transformation H.

λp = [◟K-|0◝3◜]E◞MP

Projection matrix M to map a 3D point onto the 2D image plane.

(  X )
(  Y )

   Z =        -1   -1
C◟-+-λR◝◜-K----p◞ray P(λ)

Back-projection operation of a 2D point in the 3D space.

Table 2.1 Summary of the camera parameters and important equations.