Camera Calibration for 3D Reconstruction and View Transformation - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 85

When the plumb-line method is applied, “straight” lines, which are distorted
straight image lines, need to be extracted first, typically by means of an edge
detection technique, before the optimization on equation 16 can be performed.
Clearly, the accuracy of the extracted lines determines the accuracy of the
estimated parameters. If the calibration set-up is carefully designed so that those
“straight” lines can be located accurately, an overall accuracy in the order of
2 10 -5 can be achieved (Fryer, Clarke & Chen, 1994). However, for irregular

natural scenes, it may be difficult to locate “straight” lines very accurately. To
tackle this problem, an iterative strategy has been adopted (Devernay &
Faugeras, 2001). According to this strategy, edge detection (step 1) and
optimization (step 2) are first performed on the original images. Based on the
estimated distortion coefficients, images are corrected (undistorted), and then
steps 1 and 2 are repeated. This iterative process continues until a small deviation
ς is reached. Applying this strategy on natural scenes, a mean distortion error
of about pixel (for a 512×512 image) can be obtained (Devernay & Faugeras,
2001). Improved results can be obtained by modifying equation 16 (dividing, for
example, the function f (...) by

×

ρ i [Swaminathan & Nayer, 2000]) and by
carefully defining the “straightness” of a line (using, for example, snakes [Kang,
2000]).

Utilization of projective geometry properties

The plumb-line method explores only one invariant of the projective transforma-
tion. Other projective invariants or properties, such as converging of parallel
lines, can also be employed for estimating distortion in a fashion similar to the
plumb-line method. Some of these methods are summarized below. They make
use of:

Convergence of parallel lines: Based on the observation that a set of parallel
lines should have a common unique vanishing point through linear projective
projection, the distortion is estimated by minimizing the dispersion of all
possible candidate vanishing points (Becker & Bove, 1995).
Invariance of cross ratio: Since the cross ratio is still an invariant even when
radial distortion is present, it is employed to recover the distortion center
first, followed by the use of preservation of linearity of the projective
geometry to calibrate the distortion coefficients and the aspect ratio (Wei
& Ma, 1994).
Linear projection matrix P : Line intersections are accurately detected in the
image. Four of them are selected to define a projective basis for the plane.
The others are re-expressed in this frame and perturbed so that they are