3.1 Supporting theory
The lens aperture determines how much light enters the lens. This is normally specified with a quantity named ‘f-number’ and sometimes written f/#. This number is a ratio of focal length to the clear aperture ‘φ0 (the effective lens diameter ) as shown by
large f-number corresponds to a small aperture and therefore lets less light enter the
small f-number gives a large aperture and lets more light into the lens.
discussion on optics is given in Born and Wolf .
Quaternions are used in this research to describe the orientation of coordinate frames. Quaternions are the outcome of research by W.R. Hamilton in the 18th century . They are hyper-complex numbers and form a 4 dimensional vector space over the real numbers.
The four components of a quaternion q can be written
q = w + xi + yj + zk.
Here w represents a scalar and x, y, z are often combined into a vector v. The terms i, j, and k are similar to the imaginary term i in a complex number and are orthogonal with respect to each other.
Quaternions can be used to elegantly describe rotations. Other representations such as Eu- ler ngles, rotation matrices, and the angle and axis representation exist [41–43]. However, quaternions do not suffer from a degenerative problem known as gimbal lock [41–43]. This
can affect the composition of rotations using the Euler
ngle and rotation matrix represen-
tation of rotations resulting in a loss of one degree of freedom at some orientations.
quaternion of unit length represents a rotation. The length or norm of a quaternion is similar to the standard vector definition, i.e.,
|q| = pw2 + x2 + y2 + z2.
Quaternion multiplication, unlike multiplication of the real numbers, does not commute, i . e . , f o r t w o q u a t e r n i o n s q 1 , q 2 , q 1 q 2 6 = q 2 q 1 . T o d e t e r m i n e t h e p r o d u c t o f t h e t w o q u a nions the components of each quaternion are multiplied term-by-term. The multiplication t e r -