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64 2 Image formation

O
Į įo
f = 100mm
r o

J P Į
Į d Q
įi
I
z i=102mm z o=5m

Figure 2.22 The amount of light hitting a pixel of surface area δi depends on the square of the ratio of the
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aperture diameter d to the focal length f, as well as the fourth power of the off-axis angle α cosine, cos α.

nodal point is of interest when performing careful camera calibration, e.g., when determining
the point around which to rotate to capture a parallax-free panorama (see Section 9.1.3).
Not all lenses, however, can be modeled as having a single nodal point. In particular, very
wide-angle lenses such as fisheye lenses (Section 2.1.6) and certain catadioptric imaging
systems consisting of lenses and curved mirrors (Baker and Nayar 1999) do not have a single
point through which all of the acquired light rays pass. In such cases, it is preferable to
explicitly construct a mapping function (look-up table) between pixel coordinates and 3D
rays in space (Gremban, Thorpe, and Kanade 1988; Champleboux, Lavall´ ee, Sautot et al.
1992; Grossberg and Nayar 2001; Sturm and Ramalingam 2004; Tardif, Sturm, Trudeau et
al. 2009), as mentioned in Section 2.1.6.

Vignetting

Another property of real-world lenses is vignetting, which is the tendency for the brightness
of the image to fall off towards the edge of the image.
Two kinds of phenomena usually contribute to this effect (Ray 2002). The first is called
natural vignetting and is due to the foreshortening in the object surface, projected pixel, and
lens aperture, as shown in Figure 2.22. Consider the light leaving the object surface patch
of size δo located at an off-axis angle α. Because this patch is foreshortened with respect
to the camera lens, the amount of light reaching the lens is reduced by a factor cos α. The
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amount of light reaching the lens is also subject to the usual 1/r fall-off; in this case, the
distance r o = z o / cos α. The actual area of the aperture through which the light passes
is foreshortened by an additional factor cos α, i.e., the aperture as seen from point O is an
ellipse of dimensions d×d cos α. Putting all of these factors together, we see that the amount
of light leaving O and passing through the aperture on its way to the image pixel located at I
is proportional to
2 2
δo cos α d π d 4
π cos α = δo cos α. (2.98)
r 2 2 4 z 2
o o
Since triangles ΔOPQ and ΔIPJ are similar, the projected areas of of the object surface δo
and image pixel δi are in the same (squared) ratio as z o : z i ,
δo z 2 o
= . (2.99)
δi z 2
i