FUZZY SEGMENTATION OF MASSES IN DIGITAL BREAST

TOMOSYNTHESIS IMAGES BASED ON DYNAMIC

PROGRAMMING

Louis Apffel

†

, Giovanni Palma

‡§

, Serge Muller

‡

and Isabelle Bloch

†

Ecole Centrale Paris, Grande Voie des Vignes F-92 295 Chˆatenay-Malabry Cedex, France

‡

GE Healthcare, 283, rue de la Mini`ere, 78530 Buc, France

T´el´ecom ParisTech - CNRS LTCI, 46, rue Barrault, 75013 Paris, France

Keywords:

Fuzzy segmentation, Fuzzy contours, Dynamic programming, Digital breast tomosynthesis, Computer aided

detection.

Abstract:

In this paper we propose a new fuzzy segmentation method to segment lesions in Digital Breast Tomosynthesis

(DBT) datasets. In the proposed approach we model a contour as a path in the image. The optimal contour

is deﬁned as the path associated with a minimal cost, which is derived from the image content. Using this

formalism we present several ways to alter this cost in order to extract several relevant contours from a single

image. The set of contours is then used in the fuzzy contour framework to perform mass detection. The

method has been tested on synthetic data as well as images containing lesions and provides promising results.

1 INTRODUCTION

Digital Breast Tomosynthesis (DBT) is a new 3D

imaging technique (Dobbins III and Godfrey, 2003;

Wu et al., 2003) that may potentially overcome some

limitations of standard mammography such as tissue

superimposition (Gennaro et al., 2008). Such an im-

provement is gained at the expense of an increased

amount of data to be reviewed by the radiologist. In

this context the design of a Computer Aided Detec-

tion (CAD) system may help the radiologist to keep

a high sensitivity in his detection and characterization

task. Segmentation is usually a crucial step in CAD

systems, and a wrong segmentation can have a disas-

trous impact on the whole detection scheme. The use

of fuzzy sets avoids these drawbacks (Peters, 2007).

This can be done by using the fuzzy contour frame-

work, which is suitable to represent several possible

contours for one speciﬁc structure. Unfortunately,

only a small amount of techniques exist in the litera-

ture (Bothorel et al., 1997; Peters, 2007; Palma et al.,

2008) to extract such fuzzy contours and they suffer

from various limitations. In this paper, we introduce a

new fuzzy contour extraction procedure by extending

a crisp segmentation method originally proposed for

standard mammography images, which models con-

tours as paths in an image (Timp and Karssemeijer,

2004).

First, in Section 2 we present the limitations of

crisp segmentation methods. Then in Section 3, we

recall the formalism of fuzzy contours. In Section 4,

we describe the new proposed algorithm and we dis-

cuss two aspects: uncertainty and imprecision, which

are both handled by our approach. Finally we illus-

trate results on both synthetic and clinical data.

2 LIMITATIONS OF CRISP

SEGMENTATION

In breast imaging, crisp segmentation has many lim-

itations. Indeed, segmenting some masses may be

particularly difﬁcult when the contrast between lesion

and backgroundof the image is low. Radiologists may

have different and various opinions on the exact loca-

tion of a contour. A segmentation algorithm that pro-

poses a single answer to this kind of problem is usu-

ally taking an arbitrary decision on the nature of the

contour. Since the shape of the contour is suggestive

of the nature of the lesion (benign or malignant), an-

ticipating its nature from a single and inaccurate con-

tour could lead to a classiﬁcation error.

The problem is illustrated in Figure 1 where a

Apffel L., Palma G., Muller S. and Bloch I. (2010).

FUZZY SEGMENTATION OF MASSES IN DIGITAL BREAST TOMOSYNTHESIS IMAGES BASED ON DYNAMIC PROGRAMMING.

In Proceedings of the International Conference on Imaging Theory and Applications and International Conference on Information Visualization Theory

and Applications, pages 7-13

DOI: 10.5220/0002826300070013

 SciTePress

(a) (b)

Figure 1: Crisp segmentation problem: several contours can

ﬁt a given lesion and may lead to different conclusions on

its class.

spiculated mass is presented with two different seg-

mentation results.

3 FUZZY SEGMENTATION

The fuzzy set framework seems to be suitable to deal

with the mass segmentationproblem (Peters, 2007). It

allows deﬁning the concept of fuzzy contours, which

can overcome several limitations of regular segmen-

tation techniques by allowing not to make a decision

too early.

Deﬁnition 1. A fuzzy contour is a set of nested crisp

contours where each contour is associated with a

membership degree to the class contour.

3.1 Imprecision and Uncertainty

In DBT images, masses may present two types of im-

perfections. First, a lesion may be hard to deﬁne (i.e.

there are several possible contours that can ﬁt the le-

sion boundaries) even if we are sure of its presence.

This is called imprecision. Secondly, it may some-

times be difﬁcult to detect and locate a mass. In this

case we are dealing with uncertainty. A single fuzzy

contour can handle the notion of imprecision because

it considers several potential contours. On the other

hand, uncertainty can be handled by considering sev-

eral fuzzy contours.

3.2 Existing Schemes to Extract Fuzzy

Contours

In the literature, several techniques to extract fuzzy

contours have been proposed. Originally, a multi-

thresholding approach was designed in order to de-

tect microcalciﬁcations in mammography (Bothorel

et al., 1997). This approach has also been used in

order to segment masses in projected images used to

reconstruct DBT volumes (Peters, 2007). Unfortu-

nately, because of potential superimpositions of tis-

sues in projection images, this approach is not well

suited for masses in 2D projection images.

More recently, a segmentation scheme based on

the level set framework (Osher and Fedkiw, 2002) has

been proposed (Peters, 2007; Palma et al., 2008). This

approach is taking advantage of the side effects in-

troduced on the Lipschitz function, which implicitly

represents a contour, during the minimization stage

of the contour energy. The idea is then to slice this

function at several levels after convergence to build a

fuzzy contour.

Unfortunately, tuning parameters of this segmen-

tation approach is a pretty difﬁcult task. In fact, the

problem comes from the large variability of structures

that may be encountered in mammography. Further-

more, slicing the function representing the contour at

several levels is something that may be hard to inter-

pret.

4 DYNAMIC PROGRAMMING

AND FUZZY SEGMENTATION

4.1 Original Approach

Recently, an alternative to conventional approaches

was proposed (Timp and Karssemeijer, 2004). In this

approach, a contour is modeled as a path in an image

converted into the polar domain. This path is associ-

ated with a cost that depends on the image content.

This formulation allows using well known dynamic

programming (DP) techniques in order to efﬁciently

solve the problem. Several improvements have been

proposed based on this approach (Rojas and Nandi,

2007).

(a)

(b)

Figure 2: Polar coordinates representation (b) of a circum-

scribed lesion (a).

IMAGAPP 2010 - International Conference on Imaging Theory and Applications

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

(θ) and c

(θ) be two con-

tours, we deﬁne the distance d(r

) as follows:

d(c

) = max

(|c

(θ) − c

(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

4.4 Extraction of Several Fuzzy

Contours

In order to capture both uncertainty and imprecision

contained in a given image, we can combine the band

penalization and the distance-based approaches. Ob-

viously, we can extract for each contour c obtained

with the second approach, a fuzzy contour by apply-

ing the ﬁrst method on a matrix M

′

= M where the

path c is set to ∞ on a large band. Doing this enables

us to have a set (uncertainty) of nested contours (im-

precision).

4.5 Membership Degree

To complete the extraction of fuzzy contours, we need

to assign a membership degree to each contour consti-

tuting the fuzzy set. This membership degree should

be representative of the conﬁdence we can have in the

contour.

The dynamic programming algorithm provides us

with a direct measurement of conﬁdence linked to the

values of the gradient along the contour and its gray

levels: the total cost of the path (C ). This cost can be

used to derive a membership value for the contour if

we normalize it considering the whole set of contours

obtained for an image. For instance, if we obtain n

contours with the associated costs C

,i ∈ [[1;n]], we

can deﬁne the membership degree µ

as follows:

max

−C

max

−C

min

(1)

where C

max

and C

min

are max

} and min

}, re-

spectively. Thus, we get a value ranging between 0

and 1, with 1 being the smallest cost (hence highest

membership value) and 0 being the highest cost.

Another type of membership degree can be de-

ﬁned. Since we extract a lot of contours to construct

a pool from which we extract the contour with the

distance method, we are able to create a map of con-

tours density. Each time a path is derived in polar

coordinates (in the cost matrix), we set to 1 the corre-

sponding pixels in the contour map. A pixel cannot be

crossed twice because of the penalization. So, we ob-

tain a map as represented in Figure 3(a). By averaging

the image vertically with a Gaussian ﬁlter, we obtain

a blurred version as represented in Figure 3(b). This

step allows us to take into account the spatial den-

sity of the contours, thus the resulting density map is

representative of the possibility of having a contour

crossing given pixels. Summing up the values of the

density map D along a path enables us to obtain a rep-

resentative value of the conﬁdence we can have in the

contour. The higher this value, the more conﬁdent we

can be in the contour. This total density can also be

normalized to create a membership degree:

′

min

′

∈C

max

∑

θ=0

D(θ,c

′

(θ)) −

max

∑

θ=0

D(θ,c

(θ))

max

′

∈C

max

∑

θ=0

D(θ,c

′

(θ)) − min

′

∈C

max

∑

θ=0

D(θ,c

′

(θ))

(2)

with C the set of extracted contours and c

the consid-

ered contour.

The behavior of both approaches will be discussed

in Section 5.

(a)

(b)

Figure 3: Computation of a contour density map. (a) Con-

tour map extracted from an image. 100 contours have been

marked on the image. (b) Density map after smoothing.

5 RESULTS

The database we used for evaluation contains 52 re-

gions of interest extracted from DBT slices. Sixteen

of them contain a circumscribed mass, while remain-

ing ones contain a spiculated lesion. The algorithm

was ﬁrst validated on these images for crisp segmen-

tation. The results are presented in Section 5.1. In

Section 5.2 we interpret the results obtained with our

new fuzzy segmentation approach. We focus on the

results obtained with both the penalization and the

distance and discuss the combination of the two ap-

proaches. Finally we compare the two ways of con-

structing membership degrees.

5.1 Crisp Segmentation on DBT Slices

In order to validate the approach, we performed an

assessment of the crisp approach on two subsets of

IMAGAPP 2010 - International Conference on Imaging Theory and Applications

lesions (spiculated and circumscribed). This assess-

ment was done using ground truth contours built by a

human reader and a criterion P, which is the average

of 3 measures (Rojas and Nandi, 2007):

|A∩R|

|A∪R|

similarity

= 1 −

|R\(A∩R)|

|R|

under-segmentation

= 1 −

|A\(A∩R)|

|A|

over-segmentation

where R and A are the reference contour and the one

to be evaluated, respectively.

The evaluation was done using the leave-one-out

method in order to learn without bias the algorithm

parameters. The best results were obtained for the

segmentation of circumscribed lesion with a learn-

ing step using the same type of lesions (P = 0.87).

The performance obtained for spiculated lesions with

a learning step using the same population was a bit

lower (P = 0.71). Finally testing the approach on cir-

cumscribed lesions while learning on spiculated data

results in a small performance decrease compared to

the ideal case.

5.2 Fuzzy Segmentation

Figure 4 illustrates an example of contour extraction

using the penalization of a band on each side of the

contours to force them to be nested. Let us notice

that three of the contours are really close to each

other. This could be interesting because this models

the imprecision we have on the edges. Let us also

remark that it may be interesting to get a common

portion between several contours since imprecision

may take place only in a part of the lesion. Unfor-

tunately, the penalization method prevents such a be-

havior. Nonetheless, with a high enough radius quan-

tization step in the polar domain, the obtained con-

tours can be close enough to ﬁnally overlap when ex-

pressed in the Cartesian domain.

(a) (b)

Figure 4: Fuzzy segmentation (b) of a cicumscribed le-

sion (a) using the penalization method.

We now present the results obtained using the

distance-based contour selection method. Figure 5 il-

lustrates a clinical example with contours extracted

using the distance method. As we can see, the ad-

vantage of this method is that it captures the uncer-

tainty: there is a lot of variations in the proposed con-

tours. Let us remark that most of these contours make

sense according to the lesion to segment. Nonetheless

some are less likely to represent the lesion. However

this is not an issue if we can associate those contours

with a low membership degree. In this example, the

gray levels of the contours are proportional to their

membership degree: the lower the membership de-

gree (contour gray levels), the less possible the con-

tour.

(a) (b)

Figure 5: Fuzzy segmentation (b) of a spiculated lesion (a)

using the distance-based contour selection method.

Figure 6 shows how the fuzzy contours extraction

method relying on the two contours selection method

performs on a synthetic image. The distance based

method allows retrieving four contours representing

the uncertainty in the image. Those contours appear

in black boxes, corresponding to the two disks, their

union and intersection. For each of those contours, a

set of nested contours is then extracted using the band

penalization method allowing to build a fuzzy con-

tour. Here we have 4 fuzzy contours in Figure 6(b),

6(c), 6(d) and 6(e). Note that for the sake of clar-

ity, slight variations of the presented contours (part of

the imprecision) that are also extracted are not shown.

Let us also remark that fuzzy contours 6(b) and 6(e)

are almost the same. Such a redundancy may ap-

pear because we want to extract as much main con-

tours as possible in the ﬁrst stage. Nonetheless such

a behavior should not be a problem in real applica-

tions like automatic detection/characterization of can-

cer because no information is discarded.

5.3 Comments on the Membership

Degree Deﬁnition

The comparison between the two techniques of mem-

bership degree computation is illustrated in Figures 7

and 8. These ﬁgures present the ranking of the con-

tours in a fuzzy contour according to their member-

ship degrees. It is interesting to notice the position of

FUZZY SEGMENTATION OF MASSES IN DIGITAL BREAST TOMOSYNTHESIS IMAGES BASED ON DYNAMIC

PROGRAMMING

(a) (b) (c)

(d) (e)

Figure 6: Fuzzy contours extraction. (a) Synthetic image

to segment. (b) First fuzzy contour. (c) Second fuzzy con-

tour. (d) Third fuzzy contour. (e) Fourth fuzzy contour. For

each sub-ﬁgure, the black box designates the main contour

retrieved by the distance based method, while the remaining

contours constitute the fuzzy contour.

the smallest contour correspondingto the kernel of the

lesion. When we use Equation 1 to compute the mem-

bership degrees, this contour is the worst one while

when Equation 2 is used, this contour is the best. Ac-

tually, the membership degrees of the three best con-

tours according to Equation 2 are almost the same,

which is coherent with the visual quality of these con-

tours. Let us also remark that this method could still

be improved because the second and third contours

may appear more relevant, but again since the dif-

ference between the membership degrees is low, this

would be a minor improvement.

(a) (b)

(e) (f)

(g) (h)

Figure 7: Membership degree computation using Equa-

tion 1. (a) Fuzzy contour to be considered. The remain-

ing images are ranked in decreasing order according to the

membership degree of the contour they contain. Due to this

ranking, the ﬁrst contour also corresponds to the result of a

crisp segmentation.

6 CONCLUSIONS

We have proposed a new technique to extract fuzzy

contours from images containing breast masses. This

method relies on previous works that model a contour

as a path in the image to segment, which has been

converted in the polar domain (Timp and Karssemei-

IMAGAPP 2010 - International Conference on Imaging Theory and Applications

(a) (b)

(e) (f)

(g) (h)

Figure 8: Membership degree computation using Equa-

tion 2. (a) Fuzzy contour to be considered. The remain-

ing images are ranked in decreasing order according to the

membership degree of the contour they contain.

jer, 2004; Rojas and Nandi, 2007). We extended this

method in order to capture imprecision as well as un-

certainty of lesions in mammography images or DBT

slices. This is performed through two distinct contour

extraction schemes, which we ﬁnally combine in or-

der to get several fuzzy contours for a given structure

to be segmented.

We also proposed a new technique to compute the

membership degree of each contour contained in a

fuzzy contour. This approach relies on the idea that

location where we can extract a lot of contours are

likely to represent good contours. Using a contour

density map, we thus derived a way to compute such

degrees.

Because building a ground truth database for

fuzzy contours is a pretty difﬁcult task, as of today we

only evaluated our approach visually on the lesions of

our whole database. One next step may be to work

with clinical experts to review images in order to get

several contours for these lesions. Of course to be rel-

evant all the segmentations for each lesion should be

performed independently.

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FUZZY SEGMENTATION OF MASSES IN DIGITAL BREAST TOMOSYNTHESIS IMAGES BASED ON DYNAMIC

PROGRAMMING