3.1 Colony Counting
The simplest approach is to correlate the contour of
the colonies with a circular pattern, assuming that
the maxima of correlation is located at the centers
of colonies. This has been the first method we have
tested (Schier, 2009a). During experiments, we have
experienced poor performance in the case of big clus-
ters and sensitivity to mismatch between the radius
of the circular pattern diameter and the radius of the
colony (and, hence, sensitivity to dispersion of colony
radii).
An interesting approach to the counting of bac-
teria colonies has been described in (Marotz et al.,
2001): to find colonies, a number of shape and struc-
ture criteria are used on the image pixels, including
e.g. mean object radius, roundness of an object, com-
pactness and asymmetry, radial monotony of fall-off
in intensity, etc. These criteria are combined into
shape and structure quality parameters and evaluated
using fuzzy logic. The method works best under the
assumptions of well-defined circular shape and mono-
tone intensity fall-off of the colonies, which are well
satisfied for bacteria (the case treated in the paper),
but not necessarily in our case of the yeast colonies.
A popular method to detect circular objects is the
circular Hough transform (for definition see e.g. (Bal-
lard and Brown, 2003)). Based on our experiments, if
multiple yeast colonies are overlapping or touching,
it tends to detect (incorrectly) adjacent colonies as a
single object. Also, following the results presented
in (Loy and Zelinsky, 2003), the output of the Hough
transform may be rather noisy.
Another possibility is to use the radial symme-
try-based methods: the radial symmetry transform
has been introduced in the work of Reisfeld (Reis-
feld et al., 1995). Fast radial transform – a modi-
fied version of the radial symmetry transform with
improved computational complexity – has been pre-
sented in (Loy and Zelinsky, 2003). Since this trans-
form is the method that is currently employed in our
tool, it will be reviewed in the next section.
3.1.1 Fast Radial Transform
Fast Radial Transform is a transform that maps the
original image to the transformed image according to
its contribution to radial symmetry of the gradients at
distance n ∈ N (N is the set of radii) away from each
point. For full details of the method see the original
description in the paper of Loy and Zelinsky.
First, image gradient g
i,j
at each point (i, j) is cal-
culated. Then, the positively- and negatively-affected
pixels are determined:. The affected pixel is defined as
the point in the direction of the gradient vector g
i, j
is
(for the positively-affected ones) or counter the direc-
tion of the gradient vector (for the negatively-affected
ones), at a distance n pixels away from the point at
coordinates (i, j). The coordinates of affected pixels
are given by
p
+
(i, j) = (i, j) + round
g
i, j
kg
i, j
k
n
p
−
(i, j) = (i, j) − round
g
i, j
kg
i, j
k
n
The gradient matrix g, together with the coordi-
nates of affected pixels, is used to determine the ori-
entation and magnitude projection images O
n
and M
n
for the given radius n:
O
n
(p
+
(i, j)) = O
n
(p
+
(i, j)) + 1,
O
n
(p
−
(i, j)) = O
n
(p
−
(i, j)) − 1,
M
n
(p
+
(i, j)) = M
n
(p
+
(i, j)) + kg
i, j
k,
M
n
(p
−
(i, j)) = M
n
(p
−
(i, j)) − kg
i, j
k.
The radial symmetry at radius n is defined by con-
volution
S
n
= F
n
∗ A
n
,
where
F
n
(i, j) =
M
n
(i, j)
k
n
|
˜
O
n
(i, j)|
k
n
α
,
and
˜
O
n
(i, j) =
(
O
n
(i, j) i f O
n
(i, j) < k
n
k
n
otherwise.
A
n
is a two-dimensional Gaussian, α is the radial
strictness parameter and k
n
is a scaling factor used to
normalize M
n
and O
n
. Projection images M
n
and O
n
are initially set to zero.
An example of a symmetry map for a dish with
yeast colonies is given in Figure 3.
To estimate the colony centers, the transform must
be completed with two additional procedures: esti-
mation of the set of radii N to be tested, and by a
procedure to find local maxima (corresponding to the
centers) in the symmetry map S (map S is a weighted
sum of all symmetry maps S
n
). Both procedures will
be discussed in the following sections.
3.1.2 Colony Radius Estimation
To estimate the range of radii of the objects contained
in the image, the following steps are used:
• the equivalent diameter d and eccentricity ε of all
objects in the image is computed. The diameter
is computed from area A (number of pixels) of an
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