![](bg3.png)
The authors originally designed this algorithm on
standard mammography images. We eliminated the
use of the shape coefficient they propose since the
masses used in our database were of numerous shapes
and sizes and the research of fuzzy contours implies
contours of different radii. Concretely, the image to
segment is first converted in the polar domain. The
rows and the columns of the new image represent re-
spectively the distance to the lesion center and the
angle around it. This representation is illustrated in
Figure 2. A cost matrix is then defined on the same
domain leading to the definition of a cost function for
any path linking the first to the last column:
C (c) =
∑
θ∈[[0;θ
max
]]
M(θ,c(θ))
with M a cost matrix, θ
max
the index corresponding
to the maximum angle (2π), and c a path, which as-
sociates any angle in [0;2π[ with a radius. In our im-
plementation we used the measures based on the im-
age gradient along the radial direction and the optimal
gray value of the contour, which were proposed in the
original approach (Timp and Karssemeijer, 2004).
Now to introduce some smoothness in the con-
tours we may want to consider, we can restrict the
search space to the following set of contours:
P = {c/∀θ ∈ [[0;θ
max
]] |c(θ) − c(θ − 1)| ≤ f}
with f ∈ N the parameter allowing to tune the contour
smoothness.
The segmentation problem is then solved by
searching for a contour ˆc such that:
ˆc = arg
c∈P
minC (c)
This method was initially designed to segment cir-
cumscribed lesions in standard mammography. Our
implementation, which is similar as the original one
except for the lesion size prior, has been tested on
spiculated and circumscribed lesions in DBT as it will
be presented in Section 5.
We will now present our main contribution over
the original work by proposing means to extract fuzzy
contours from an image.
4.2 Penalization
To obtain several contours, the first idea was to pre-
vent the algorithm from using the pixels that have al-
ready been crossed by a path in the cost matrix by
setting their costs to an infinite value. This operation
can be iterated while non infinite cost paths exist in
the cost matrix.
When looking for a path in the cost matrix, the
DP-based algorithm is constrained by the parameter
f along the vertical axis whose role was previously
detailed. Thus, if we consider a pixel (θ,r) in the
cost matrix, the path can reach 2f + 1 pixels from
one column to another. This parameter allows radial
variations in the polar coordinates system described in
(Timp and Karssemeijer, 2004). Thus, a simple penal-
ization of the pixels through which goes the first path
does not prevent contours from crossing each other.
Contours obtained with such a method are often
similar to the first contour. To force finding a contour
with a significantly different shape, it is necessary to
generate a lot of contours, which might be hard to
handle in further processing. The information con-
veyed by the repetition of quasi-identical contours is
related to the imprecision contained in the lesion we
want to segment. This correlation has some mean-
ing but we may miss information related to the uncer-
tainty implicitly present in some images.
In order to satisfy the inclusion criterion required
by Definition 1, it is possible to penalize a band
nearby the path found in the cost matrix. Instead of
setting a one-pixel band to an infinite value, we pe-
nalize a band of 2f + 1 pixels. Thus, the algorithm
forces the contours to be nested.
4.3 Contour Selection and Elimination
In this section, we present an approach to model un-
certainty rather than imprecision. The goal is to ob-
tain a set of contours different from each other. To
guarantee that the contours will be different, we de-
fine a distance on the space of contours represented
in polar coordinates. Let c
1
(θ) and c
2
(θ) be two con-
tours, we define the distance d(r
1
,r
2
) as follows:
d(c
1
,c
2
) = max
θ
(|c
1
(θ) − c
2
(θ)|)
We can now extract representative contours us-
ing the penalization technique previously presented
with no need to use forbidden bands of pixels but
skipping contours that are too close from each other.
This is done iteratively by discarding all the contours
whose distance with the former representativecontour
is less or equal to a given threshold. This approach al-
lows modeling uncertainty and produces contours that
are not nested but that exhibit differences for at least
few points. Segmentation results obtained with this
method are presented in Section 5.
FUZZY SEGMENTATION OF MASSES IN DIGITAL BREAST TOMOSYNTHESIS IMAGES BASED ON DYNAMIC
PROGRAMMING
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