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52 2 Image formation

To see this more clearly, let η z = t −1 and s = η z f. We can then re-write the above
z
equations as
r x · p + t x
x s = s + c x (2.75)
1+ η z r z · p
r y · p + t y
= s (2.76)
y s + c y
1+ η z r z · p
(Szeliski and Kang 1994; Pighin, Hecker, Lischinski et al. 1998). The scale of the projection
s can be reliably estimated if we are looking at a known object (i.e., the 3D coordinates p
are known). The inverse distance η z is now mostly decoupled from the estimates of s and
can be estimated from the amount of foreshortening as the object rotates. Furthermore, as
the lens becomes longer, i.e., the projection model becomes orthographic, there is no need to
replace a perspective imaging model with an orthographic one, since the same equation can
be used, with η z → 0 (as opposed to f and t z both going to infinity). This allows us to form
a natural link between orthographic reconstruction techniques such as factorization and their
projective/perspective counterparts (Section 7.3).

2.1.6 Lens distortions

The above imaging models all assume that cameras obey a linear projection model where
straight lines in the world result in straight lines in the image. (This follows as a natural
consequence of linear matrix operations being applied to homogeneous coordinates.) Unfor-
tunately, many wide-angle lenses have noticeable radial distortion, which manifests itself as
a visible curvature in the projection of straight lines. (See Section 2.2.3 for a more detailed
discussion of lens optics, including chromatic aberration.) Unless this distortion is taken into
account, it becomes impossible to create highly accurate photorealistic reconstructions. For
example, image mosaics constructed without taking radial distortion into account will often
exhibit blurring due to the mis-registration of corresponding features before pixel blending
(Chapter 9).
Fortunately, compensating for radial distortion is not that difficult in practice. For most
lenses, a simple quartic model of distortion can produce good results. Let (x c ,y c ) be the
pixel coordinates obtained after perspective division but before scaling by focal length f and
shifting by the optical center (c x ,c y ), i.e.,
r x · p + t x
x c =
r z · p + t z
r y · p + t y
= . (2.77)
y c
r z · p + t z
The radial distortion model says that coordinates in the observed images are displaced away
(barrel distortion) or towards (pincushion distortion) the image center by an amount propor-
tional to their radial distance (Figure 2.13a–b). 3 The simplest radial distortion models use
low-order polynomials, e.g.,
4
2
= x c (1 + κ 1 r + κ 2 r )
ˆ x c
c c
2
4
ˆ y c = y c (1 + κ 1 r + κ 2 r ), (2.78)
c
c
3
Anamorphic lenses, which are widely used in feature film production, do not follow this radial distortion model.
Instead, they can be thought of, to a first approximation, as inducing different vertical and horizontal scalings, i.e.,
non-square pixels.