Graphics Reference
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of the most likely position of the corner as computed by the corner detector. An
ideally sharp image
I of a corner with a brightness of
1inthe
bright areas can be modelled by the product of two oriented step functions according
to
1 in the dark and
+
I(x,y)
=
δ(x cos α 1 +
y sin α 1 )δ(x cos α 2 +
y sin α 2 )
1 f t< 0
with x,y
∈ R
and
δ(t)
=
(1.85)
+
1
otherwise .
The angles α 1 and α 2 denote the directions of the normals to the black-white edges.
This notation is identical to the affine transformation of an orthogonal, infinitely
large corner. Since it is assumed that r is sufficiently small (e.g. r
9 pixels), the
affine transformation is a suitable approximation of the projective transform and the
lens distortions. Otherwise, the straight line edges may be replaced by a different
model such as cubic curves.
The ideal image I is subject to blurring by the lens. An exact description of the
PSF due to diffraction of monochromatic light at a circular aperture is given by the
radially symmetric Airy pattern A(r) ∝[ J 1 (r)/r ]
=
2 , where J 1 (r) is a Bessel function
of the first kind of first order (Pedrotti, 1993 ). Consequently, the image of a point
light source is radially symmetric and displays an intensity maximum at its centre
and concentric rings surrounding the maximum with brightnesses which decrease
with increasing ring radius. It is explained in more detail in Chap. 4 that for practical
purposes a radially symmetric Gaussian function is a reasonable approximation to
the PSF. It is thus assumed that the PSF is an ideal circular Gaussian filter G of
radius σ . Hence, the continuous image
I(x,y) of the ideal chequerboard corner
I(x,y) corresponds to
G x 2
y 2 I(x,y)
1
t 2
2 σ 2 .
2 πσ e
I(x,y)
=
+
with G(t, σ )
=
(1.86)
It is formed by convolving the step image with the circular Gaussian filter. Since
the Gaussian filter is separable, we may exchange the step function δ(t) in I(x,y)
by the step response of the Gaussian filter in I(x,y) . Hence, the observed intensity
pattern corresponds to the step response H(t,σ) of the PSF G(t, σ ) with
erf t
.
H(t,σ) =
2 σ
(1.87)
The error function erf (t) is twice the integral of the Gaussian distribution with zero
mean and variance 1 / 2, i.e. erf (t)
π 0 e s 2 ds . It is scaled such that its infimum
2
=
and supremum are
+
1 and
1, respectively. A model of the observed chequerboard
corner is then given by
I(x,y)
=
H(x cos α 1 +
y sin α 1 , σ )H (x cos α 2 +
y sin α 2 ,σ).
(1.88)
The error function erf (t) is of sigmoidal shape but cannot be expressed in closed
form. In practise it is approximated numerically and quite expensive to compute.
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