Graphics Reference
In-Depth Information
observes the scene point
(
X
c
,
Y
c
,
Z
c
)
=
(
100, 200, 400
)
. Then applying Equation (
6.4
)
10
·
100
400
,
10
·
200
400
results in the image coordinate
(
x
,
y
)
=
=
(
2.5, 5
)
.
6.2.2
Lens Distortion
Of course, real cameras, especially those used for filmmaking, are much more com-
plicated than pinhole cameras, largely due to complex, multi-component lenses in
front of the aperture that gather and focus incoming light. As a result, the image we
obtain usually deviates fromwhat Equation (
6.4
) predicts due to
lens distortion
. This
distortion is usually a function of the radial distance from a center point (often near
the principal point), as illustrated in Figure
6.5
. In the resulting images, a parallel
grid either appears to bulge toward the edges (known as barrel distortion) or pinch
toward the middle (known as pincushion distortion). In modern, high-quality cam-
eras, barrel distortion is usually muchmore pronounced than pincushion distortion,
and occurs when shooting with wide-angle lenses.
We can't mathematically model lens distortion in the same convenient form
as Equation (
6.4
). Instead, we must interpose a step between Equation (
6.1
) and
Equation (
6.2
) that expresses the radial distortion in the form
x
dist
˜
˜
˜
x
˜
x
2
y
2
x
2
y
2
2
+
κ
1
(
˜
+
˜
)
+
κ
2
(
˜
+
˜
=
(
1
)
)
(6.5)
y
dist
y
affine transformation in Equation (
6.2
) is applied to the distorted parameters. That
is, the observed (distorted) pixel coordinates
where the
κ
(
x
dist
,
y
dist
)
are related to the correct
(a)
(b)
Figure 6.5.
An example of lens distortion. (a) An ideal image. (b) Barrel distortion observed using
a camerawith awide-angle lens. Note how the straight edges of the checkerboard, monitor, desk,
and light are bowed outward as a function of distance from the image center.
4
We can add coefficients on higher powers of
x
2
y
2
for a more accurate model, but one or two
terms are often sufficient for a high-quality camera. This formulation also assumes the center of
distortion is the principal point, which is usually sufficient.
(
˜
+ ˜
)
