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6.3 Geometric intrinsic calibration 295

6.3.5 Radial distortion

When images are taken with wide-angle lenses, it is often necessary to model lens distor-
tions such as radial distortion. As discussed in Section 2.1.6, the radial distortion model
says that coordinates in the observed images are displaced away from (barrel distortion) or
towards (pincushion distortion) the image center by an amount proportional to their radial
distance (Figure 2.13a–b). The simplest radial distortion models use low-order polynomials
(c.f. Equation (2.78)),
2
4
ˆ x = x(1 + κ 1 r + κ 2 r )
4
2
ˆ y = y(1 + κ 1 r + κ 2 r ), (6.59)
2
2
2
where r = x + y and κ 1 and κ 2 are called the radial distortion parameters (Brown 1971;
Slama 1980). 13
A variety of techniques can be used to estimate the radial distortion parameters for a
given lens. 14 One of the simplest and most useful is to take an image of a scene with a lot
of straight lines, especially lines aligned with and near the edges of the image. The radial
distortion parameters can then be adjusted until all of the lines in the image are straight,
which is commonly called the plumb-line method (Brown 1971; Kang 2001; El-Melegy and
Farag 2003). Exercise 6.10 gives some more details on how to implement such a technique.
Another approach is to use several overlapping images and to combine the estimation
of the radial distortion parameters with the image alignment process, i.e., by extending the
pipeline used for stitching in Section 9.2.1. Sawhney and Kumar (1999) use a hierarchy
of motion models (translation, affine, projective) in a coarse-to-fine strategy coupled with
a quadratic radial distortion correction term. They use direct (intensity-based) minimiza-
tion to compute the alignment. Stein (1997) uses a feature-based approach combined with
a general 3D motion model (and quadratic radial distortion), which requires more matches
than a parallax-free rotational panorama but is potentially more general. More recent ap-
proaches sometimes simultaneously compute both the unknown intrinsic parameters and the
radial distortion coefficients, which may include higher-order terms or more complex rational
or non-parametric forms (Claus and Fitzgibbon 2005; Sturm 2005; Thirthala and Pollefeys
2005; Barreto and Daniilidis 2005; Hartley and Kang 2005; Steele and Jaynes 2006; Tardif,
Sturm, Trudeau et al. 2009).
When a known calibration target is being used (Figure 6.8), the radial distortion estima-
tion can be folded into the estimation of the other intrinsic and extrinsic parameters (Zhang
2000; Hartley and Kang 2007; Tardif, Sturm, Trudeau et al. 2009). This can be viewed as
adding another stage to the general non-linear minimization pipeline shown in Figure 6.5
between the intrinsic parameter multiplication box f C and the perspective division box f .
P
(See Exercise 6.11 on more details for the case of a planar calibration target.)
Of course, as discussed in Section 2.1.6, more general models of lens distortion, such as
fisheye and non-central projection, may sometimes be required. While the parameterization
of such lenses may be more complicated (Section 2.1.6), the general approach of either us-
ing calibration rigs with known 3D positions or self-calibration through the use of multiple
13
Sometimes the relationship between x and ˆx is expressed the other way around, i.e., using primed (final)
2
4
coordinates on the right-hand side, x =ˆx(1 + κ 1 ˆr + κ 2 ˆr ). This is convenient if we map image pixels into
(warped) rays and then undistort the rays to obtain 3D rays in space, i.e., if we are using inverse warping.
14 Some of today’s digital cameras are starting to remove radial distortion using software in the camera itself.