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# Theory and Design

nalysis

Principal point

Principal plane

Pinhole (origin)

Principal plane

Image plane

P(u,v)

P(x,y,z)

Optical axis (Z)

u

d

i

z

x

Z

U

## X

Image distance (di)

U

f

(a)

(b)

Figure 3.1 (a) The pinhole camera model. All light passes through the infinitely small pinhole forming an image on the image plane. (b) The thin lens approximation. Light passes through the c o n v e x l e n s f o r m i n g a n i m a g e a t a d i s t a n c e d i f r o m t h e p r i n c i p a l p o i n t .

camera’s image plane. This relationship is

u

ï£¹

ï£°

v

ï£»

=

ï£°

xdi z

ï£¹

ydi z

ï£»

,

(3.1)

w h e r e d i i s t h e d i s t a n c e b e t w e e n t h e i m a g e a n d t h e p r i n c i p a l p l a n e . I n r e a l w o r l d c a m e r a s

this pinhole is replaced with a lens (Figure 3.1b).

convex lens can be modeled using a

thin lens approximation.

# The thin lens equation is

1

=

1

+

1

f

i d

z

(3.2)

a n d r e l a t e s t h e f o c a l l e n g t h ( f ) , t h e i m a g e d i s t a n c e ( d i ) , a n d t h e o b j e c t d i s t a n c e ( z ) . T h e f o c a length is a property of the lens and is specified by the lens manufacturer. If the distance to the object is much greater than the focal length then Equation 3.2 can be approximated by l

d i f

(3.3)

which states that the image plane is at a distance f from the centre of the lens aperture. Due to the magnitudes of the numbers involved in this system (a focal length of 6 mm was

used with a minimum marker distance of 0.5 m) this approximation is used and Equation

3.1 can be written as

u

ï£¹

ï£°

v

ï£»

=

ï£°

xf z

yf z

ï£¹ ï£»

.

(3.4)

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