7.4 Pose estimation using quaternions
Figure 7.6 With the world axes and camera axes aligned, the intersection of the lines occur at the camera’s focal point in world coordinates. In this figure the markers are shown as red circles, the camera as a blue circle, and the vectors pointing towards the camera’s focal point are shown a s b l u e a r r o w s e x t e n d i n g f r o m t h e m a r k e r s . T h e a x e s X c , Y c r e p r e s e n t t h e c a m e r a ’ s a x e s a n d a r e a l i g n e d w i t h t h e w o r l d a x e s X w , Y w . T h e v e c t o r s a r e d e fi n e d i n c a m e r a c o o r d i n a t e s a n d t markers’ positions are defined in world coordinates. The reason that there is a single intersection point is because the camera and world axes are aligned. h e
each centroid in the direction of the camera’s origin. These vectors are defined in the cam- era’s coordinate space. Now project lines from the markers in world coordinates using each line’s corresponding vector. This time the lines will not intersect at one point (Figure 7.7). This is because points and vectors have been used from different coordinate frames. For every pair of lines their intersection point is used to construct the above mentioned point-cloud. The premise is that the size of this cloud is a measure of how close the quater- nion q is to the true camera orientation (with respect to the world frame). For example, when the axes are aligned as in the first example the cloud is reduced to a single point.
Extension to 3D
The examples given in Section 7.4.1 have been in 2D. In 2D two lines will always intersect unless they are parallel to each other. In 3D this is not the case. It is unlikely that two lines will intersect. Noise and number precision are two examples of problems that could stop even two specially constructed lines from intersecting at a single point. To overcome this, in 3D, the closest two points on each line are found and these two points are used as the
intersection points. This doubles the number of intersection points giving N = 2 ×M intersection points for M markers1.