MAMMOGRAPHIC IMAGE ANALYSIS FOR BREAST CANCER

DETECTION USING COMPLEX WAVELET TRANSFORMS

AND MORPHOLOGICAL OPERATORS

V. Alarcon-Aquino, O. Starostenko, R. Rosas-Romero, J. Rodriguez-Asomoza, O. J. Paz-Luna

K. Vazquez-Muñoz and L. Flores-Pulido

Communications and Signal Processing Research Group, Department of Computing Electronics, and Mechatronics

Universidad de las Americas Puebla, Sta. Catarina Martir, Cholula, Puebla. 72820, Mexico

Keywords: Breast cancer, Mammography, Microcalcifications, Dual-tree complex wavelet transforms, Wavelets.

Abstract: This paper presents an approach for early diagnostic of Breast Cancer using the dual-tree complex wavelet

transform (DT-CWT), which detect micro-calcifications in digital mammograms. The approach follows four

basic strategies, namely, image denoising, band suppression, morphological transformation and inverse

complex wavelet transform. The procedure of image denoising is carried out with a thresholding algorithm

that computes recursively the optimal threshold at each level of wavelet decomposition. In order to

maximize the detection a morphological conversion is proposed and applied to the high-frequencies sub-

bands of the wavelet transformation. This procedure is applied to a set of digital mammograms from the

Mammography Image Analysis Society (MIAS) database. Experimental results show that the proposed

denoising algorithm and morphological transformation in combination with the DT-CWT procedure

performs better than previous reported approaches.

1 INTRODUCTION

A mammography exam, called a mammogram, is

used to aid in the diagnosis of breast diseases in

women. A mammogram is a specialized X-ray

exam in which a set of plates are taken from breast

tissue to detect suspect tissue and

microcalcifications (MCs). The main reason to

perform a mammogram is the detection of clinically

hidden breast cancer at early time. The early

detection of breast cancer with a mammogram is

difficult due to the fact that small tumors and MCs

are very similar to normal glandular tissue.

Recently, tools for computer-aided diagnosis have

been developed especially in the image processing

field that permits an easy visualization of

mammograms. In this way the wavelet transform

(WT) has an important merit, since it has been

employed to eliminate noise in mammogram’s

image. The results have shown an improvement of

the image, making easy the visualization of

suspicious lesions (Akay, 1997). Wavelets have

been applied to biomedical signals because they

provide an analysis of non-stationary signals that

contains a high among of complex frequencies, and

have also been applied to detect MCs in digital

mammograms. In this regard, several approaches

have been proposed. A system based on fuzzy logic

has been reported in (Cheng, 1998), a mathematical

morphologist study is reported in (Zhao, 1993), and

several methods based on wavelet transforms are

reported in (Strickland, 1996; Wang, 1998; Melloul,

2002; Sebri, 2007; Mencattinni, 2008; Jamarani,

2006; Karahaliou, 2008). Strickland (1996)

introduced a two stages method for detection and

segmentation of MCs. The first stage is based on

the use of undecimated wavelet transform and the

segmentation process is realized with matched

filters. Wang (1998) reported an approach to detect

MCs using the decimated wavelet transform so that

suppression in the low-frequencies band is

performed. The visualization of MCs is improved

using a non-linear threshold based on arctan

method. Finally, Melloul (2002) reported detection

of MCs in two steps. The first consists in total

elimination of background’s mammogram with

multi-scale morphological filtering then an optimal

threshold (entropy threshold) is applied to the

segmentation step. In this paper we present an

approach to detect microcalcifications in digital

Alarcon-Aquino V., Starostenko O., Rosas-Romero R., Rodriguez-Asomoza J., J. Paz-Luna O., Vazquez-Muñoz K. and Flores-Pulido L. (2009).

MAMMOGRAPHIC IMAGE ANALYSIS FOR BREAST CANCER DETECTION USING COMPLEX WAVELET TRANSFORMS AND MORPHOLOGICAL

OPERATORS.

In Proceedings of the International Conference on Signal Processing and Multimedia Applications, pages 79-85

DOI: 10.5220/0002236400790085

 SciTePress

Table 1: Mammogram’s information format (Suckling, 1994).

mdb209 G CALC M 647 503 87

column 2

column 3

column

Reference number from MIAS

database. The database

includes 322 mammograms.

Type of tissue:

F-Fatty,

G-Fatty-Glandular,

D-Dense-Glandular.

Class of abnormality: CALC-Calcification,

CIRC-Circumscribed masses, SPIC-Spiculated

masses, MISC-others, ill-defined masses,

ARCH - Architectural distortion, ASYM-

Asymmetry, NORM-Normal.

column 5

& 6

column 7

column

Severity of Abnormality:

B – Benign, M – Malign.

(x, y) image-coordinates of

centre of abnormality.

Approximate radius (pixels) of a circle

enclosing the abnormality.

mammograms using the dual-tree complex wavelet

transform. The approach consists of four stages:

image denoising by optimal thresholding, band

suppression of low frequencies, morphological

transformation, and inverse complex wavelet

transform. The remainder of this paper is organized

as follows. In Section 2 a brief description of MCs

and the MIAS database is presented. Section 3

presents an overview of wavelet transforms. The

proposed approach to detect microcalcifications is

presented in Section 4. Experimental results are

reported in Section 5. Conclusions and future work

are discussed in Section 6.

2 DESCRIPTION OF MCS IN

MAMMOGRAMS

Initially, the breast tissue study was performed in

radiology field by analogical images including all

kind of image modalities such as magnetic

resonance image and nuclear medicine. The basic

idea for using different image methods was to

detect and diagnose at early stage the breast cancer

tissue when the probability of cure was greater and

the treatment was less aggressive. It helped to

decide the best therapy for each lesion. Currently,

mammogram screening is the only way for

detection at a short period of time. The objective of

a mammogram is to produce detailed images of the

internal structures in breast tissue to make earlier

cancer detection. Due to the need of details, high

quality spatial images are requested because the X-

ray attenuation between normal and abnormal tissue

is very small. Conventional mammogram uses film-

screen detectors to record the photons that go

through breast tissue, and it produces an analogical

image. Due to large amount of data that need to be

stored, a piece of film is an excellent storage

medium. Unfortunately, it is not possible to perform

modifications in the image to improve the

visualization of present elements. In order to

overcome the intrinsic limitations of conventional

mammograms the use of digital mammograms is

preferred. One of the fundamentals benefits present

in a digital mammogram is the capability to modify

the information present in the image. Breast micro-

calcifications are commonly discovered in the

radiological study on asymptomatic women. These

are deposits of calcium at the thickness of

mammary tissue and are represented as little white

dots, and normally show the first sign of cancerous

process. Figure 1 shows different types of grouped

MCs.

Figure 1: Types of MCs, a) y b) Grouped or clusters MCs,

c) Linear MCs and d) Linear MCs & clustered.

In order to assess the performance of the proposed

approach the Mammography Image Analysis

Society (MIAS) database is used (Suckling, 1994).

Table 1 shows the available information at the

database for each mammogram that includes type of

tissue, class of abnormality and strictness. In this

work only mammograms classified as CALC and

NORM are analyzed. The size of each image is

1024x1024 pixels and it is centered in the matrix,

and the list of images is presented in pairs. That is,

even numbers correspond to left breast

mammogram, while odd numbers correspond to

right breast mammogram.

SIGMAP 2009 - International Conference on Signal Processing and Multimedia Applications

3 WAVELET TRANSFORMS

The Wavelet Transform (WT) is a mathematical

tool that provides building blocks with information

in scale and time of a signal (Burrus, 1998). These

building blocks are generated from a single fixed

function called mother wavelet by translation and

dilation operations. The process of wavelet

transform of a signal is called analysis, and the

inverse process to reconstruct the analyzed signal is

called synthesis. The analysis generates different

sub-band blocks (multi-resolution analysis, MRA)

(Burrus, 1998), so different levels can be generated

as the application requires. This process is also

known as sub-banding coding (Burrus, 1998).

3.1 Discrete Wavelet Transform

The Discrete Wavelet Transform (DWT) is a time-

scale representation of a digital signal, obtained

with digital filtering techniques. The signal to

analyze is passed through several filters with

different cut-frequencies at different scales. The

wavelet’s family is generated by a mother wavelet

)(x

defined by

()

⎟

⎠

⎞

⎜

⎝

⎛

−

(1)

where

denotes the scale parameter,

represents

the translation parameter, the term

j controls scale

and term k controls translation. The discrete

parameters a and b are sampled in a dyadic grid

into time-scale plane and by the sampling process

with the dyadic grid an orthonormal wavelet’s

family is obtained

()

kxx

−=

−

(2)

The DWT is thus defined by (Burrus, 1998;

Alarcon-Aquino, 2003):

()

dxkxxsd

∫

−=

−

(3)

where s(x) is the signal to be analyzed.

3.2 Two Dimensional Discrete Wavelet

Transform (2D-DWT)

The two-dimensional discrete wavelet transform

analyze digital images by separation of rows and

columns, in this way the horizontal, vertical, and

diagonal details are separated. In the first stage, the

rows of an image

NN × are filtered by one-

dimensional (1D)-DWT analysis and then the same

process is applied to the columns (Gonzalez, 2001).

The previous process generates three different

detailed sub-images HH, HL and LH. These

correspond to three different directions (diagonal,

vertical and horizontal, respectively) and a sub-

image LL, known as approximation matrix, is used

to the multi-level decomposition process. To

reconstruct the image through the sub-images

results of two-dimensional-DWT, details are

recombined with the low-pass approximation and

the up-sampling processes (Gonzalez, 2001). If

(

)

is an one-dimensional wavelet associated

with the one-dimensional scaling function

(

)

then the three two-dimensional wavelets associated

with the three sub-images are defined by

(

)

(

)

(

)

yxyLH

φψ

=→

(4)

(

)

(

)

(

)

ψψφ

=→

(5)

(

)

(

)

(

)

yxyHH

ψψ

=→

(6)

where

(

)

yx, represents height and width of the

image. Note that the DWT is the non-redundant and

compact representation of a signal in the wavelet

domain. The down-sampling process makes the

DWT time variant and produces shifting. The

stationary wavelet transform (SWT) is the

redundant, non down-sampling and full time

invariant version of WT. The SWT has the same

length of wavelet coefficients for each

decomposition level. In addition, the SWT does not

have sensibility but it is computationally complex.

The computational complexity of the SWT is O(n

)

, where n denotes the length of samples in the signal

(Alarcon-Aquino, 2003). The redundant

representation of SWT does not present shifting.

This is ideal for applications as contour detection,

noise reduction, and data fusion (Taswell, 2000).

3.3 Complex Wavelet Transform

(CWT)

The Complex Wavelet Transform is used to avoid

the limitations of DWT and to obtain phase

information. The CWT employs a complex value

filtered analytically to decompose pure real signals

and real signals with complex components into real

and imaginary parts in the wavelet domain. Real

and imaginary coefficients are used to compute

amplitude and phase information, needed to

describe precisely the energy localization of

oscillating sources. Recent investigations in the

CWT field are addressed to the design of complex

MAMMOGRAPHIC IMAGE ANALYSIS FOR BREAST CANCER DETECTION USING COMPLEX WAVELET

TRANSFORMS AND MORPHOLOGICAL OPERATORS

bank filters, in which the outputs are wavelet

coefficients (real and imaginary). It is desirable that

filters form pairs of Hilbert’s Transform on each

decomposition level.

The CWT is classified into two groups:

Redundant-CWT (RCWT) and Non-redundant-

CWT (NR-CWT), and these are a powerful tool to

image compression (Shukla, 2003). The RCWT is

presented in two variants, namely, the Dual-Tree

Complex Wavelet Transform of Kingsbury (DT-

CWT (K)) and the DT-CWT of Selesnick (DT-

CWT (S)) (Selesnick, 2005). Both of them are

redundant due to a similar bank filter structure with

the DWT, but in this case the banks operate in

parallel and in quadrature. The filter’s structure is

the same in both variants; the difference is the

method that generates the wavelet and scaling

coefficients. Both DT-CWT variations generate

phase information, are insensible to shifting, and are

directional. The CWT follows the same principle of

DWT, and at the output there are the same number

of samples n that at the input, additionally, the

computational complexity is only twice of the

DWT, O(2n) (Shukla, 2003; Selesnick, 2005).

Although, both DT-CWT have the same bank filter

structure of DWT, the difference is that real filters

are replaced by analytical filters in order to obtain

complex solutions. It is similar of two parallel bank

filter structures in the DWT (Shukla, 2003). Figure

2 shows the bank filter structure to DT-CWT

analysis at three level of decomposition in one-

dimension.

The form of the conjugated filters for one-

dimensional DT-CWT is defined by Equation (7),

where

is the set of filter

{}

, hh

and

is the

set

{}

, gg

. Filter

and

correspond to low-pass

and high-pass filter respectively for real part, in the

same way filter

and

are in the imaginary

part. The synthesis bank filter is realized with the

pairs

and

() ( )

ighns += (7)

4 DETECTION OF

MICRO-CALCIFICATIONS

In this section an approach to detect MCs in digital

mammograms using the DT-CWT(S) is proposed.

The performance of the SWT is also reported. The

DWT disadvantages decrease its efficiency in

digital image processing; in addition there is an

inconvenient using the DWT for MCs detection due

to the down-sampling process that eliminates details

in the image, especially when MCs are details in the

high-frequency band. The SWT increases

significantly MCs detection to overcome the DWT

disadvantages. However, the computational

complexity of the SWT is O(n

) (Shukla, 2003). In

order to overcome the limitations of the DTW and

the SWT we use the Dual-Tree Wavelet Complex

Transform (DT-CWT).

Figure 2: Bank filter for 1D DT-CWT analysis.

4.1 Proposed Approach

MCs are small deposits of calcium that appear as

diminutive white dots in the mammogram. Due to

microcalcification’s size, the non-homogeneous

background of mammogram (breast glandular

tissue) and noise present, detection of MCs is

difficult (Melloul, 2002). In the work reported in

this paper we propose an approach based on the

hypothesis that MCs that are present in

mammograms can be obtained using a transform

that locate image characteristics into the wavelet

transform domain. The WT allows the multi-

resolution analysis and image decomposition in

sub-band frequencies, in which the low-band

frequencies are image’s background and high-

frequencies correspond to image’s detail. MCs

correspond to the high-frequencies of mammogram

spectrum (Wang, 1998). The five steps that

conforms the method to detect MCs are as follows:

Mammogram’s Sub-band Frequency

Decomposition. The original mammogram is

decomposed into a sub-band set, each band with

different resolution and frequency contents. This

process is performed with the DT-CWT proposed

by Selesnick. There are two variants of the DT-

CWT(S), the DT-CWT (Real) and the DT-CWT

(Complex). Both of them have wavelets oriented in

six directions; the difference is that the DT-CWT

(Complex) uses two wavelets for each direction,

one interpreted as the real part and the other as the

imaginary part. Due to the complex version there

SIGMAP 2009 - International Conference on Signal Processing and Multimedia Applications

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000

580 600 620 640 660 680 700 720 740

760

780

800

820

840

860

880

900

920

(a) (b)

580 600 620 640 660 680 700 720 740

760

780

800

820

840

860

880

900

920

580 600 620 640 660 680 700 720 740

760

780

800

820

840

860

880

900

920

Figure 3: Experimental results of the mammogram mdb233 G CALC M *NOTE 3*. (a) Original mammogram, (b)

Mammogram with MCs using the DT-CWT, (c) Mammogram with MCs using the SWT, and (d) Mammogram with MCs

using the Top-Hat filtering.

are double numbers of wavelets than the DT-CWT

(Real); the DT-CWT (Complex) is 4 times

expansive and the DT-CWT (Real) is 2 times

expansive (Shukla, 2003; Selesnick, 2005). The

complex wavelet transform used in this work to

detect MCs is the DT-CWT (Real).

Mammogram’s Noise Reduction. The noise

reduction in the mammogram is realized into

transform domain by an optimal threshold

algorithm that modifies the signal’s representation

coefficients according to each decomposition level.

The method used to obtain the optimal threshold

consists in the stages of initialization, iteration, and

convergence. The main objective is to implement a

method to remove image’s noise using a non-linear

and recursive algorithm called optimal threshold

algorithm (Jansen, 1999; Azzalini, 2005) with CWT

theory. Threshold application on wavelet

coefficients is an efficient method for noise removal

in a signal (Taswell, 2000; Azzalini, 2005). A

quasi-optimal threshold method depends on

sampled signal’s length and noise’s variance that

generally is unknown.

Suppression of Bands containing Low-

frequencies. To eliminate mammogram’s

background that difficult visibility of MCs the

suppression of bands that contain mammogram’s

low-frequencies is performed. This objective is

achieved by discarding the low-frequencies

subbands from real and imaginary parts of the DT-

CWT(S) (Vazquez-Muñoz, 2006).

Dilatation of High-frequency Components. It is

necessary to stand out the sub-bands components

that contain high frequencies in which MCs are

present. This is achieved by a morphological

operation of dilatation (Melloul, 2002; Vazquez-

Muñoz, 2006).

Mammogram’s Reconstruction. Finally, DT-

CWT synthesis is applied to the filter bank and the

DT-CWT sub-bands previously processed with the

described methods of image denoising, low-

frequencies sub-band suppression and high-

frequencies components dilatation, in which is

obtained the mammogram that contains only the

MCs.

5 EXPERIMENTAL RESULTS

To evaluate the performance of the proposed

approach experimental results using the SWT and

the Top-Hat transformation are also presented. The

results after applying these methods in

mammograms from the MIAS database are

reported. In the SWT case, the fourth order

MAMMOGRAPHIC IMAGE ANALYSIS FOR BREAST CANCER DETECTION USING COMPLEX WAVELET

TRANSFORMS AND MORPHOLOGICAL OPERATORS

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000

450 500 550 600 650

350

400

450

500

550

600

650

(a) (b)

450 500 550 600 650

350

400

450

500

550

600

650

450 500 550 600 650

350

400

450

500

550

600

650

Figure 4: Experimental results of the mammogram mdb249 D CALC M 544 508 48. (a) Original mammogram, (b)

Mammogram with MCs using the DT-CWT, (c) Mammogram with MCs using the SWT, and (d) Mammogram with MCs

using the Top-Hat filtering.

Daubechies wavelet is used. Other wavelets may

also be considered. Note that the detection of MCs

using the SWT is accomplished by setting low-

frequencies subbands to zero before the

reconstruction of the image. The Top-Hat

transformation is largely employed for detail

extraction in images. There are two kinds of Top-

Hat transformation. The White Top-Hat

transformation for brighten detail’s extraction and

the Black Top-Hat transformation for dark detail’s

extraction (Melloul, 2002). Because MCs are present

as bright particles rounded by a black background,

then White Top-Hat transformation is used. The

Top-Hat transformation consists on recover the

structures eliminated in the open or closed process.

Using a structuring element with adequate shape,

size and orientation it is possible to filter the image

and eliminate particular elements of the original

image. The White Top-Hat transform is the residue

between original image and the morphological open.

The results obtained with the SWT, the proposed

approach and the White Top-Hat transformations are

reported. Figure 3 shows an original mammogram

called mdb233 G CALC M *NOTE 3*

. According

to Table 1 this mammogram corresponds to a

glandular tissue and contains a set of malign MCs.

NOTE 3 denotes that when calcifications are

present, centre locations and radii are applied to a

group of MCs rather than individually. As can be

seen in Figure 3, when using the SWT the MCs

(brighten points) are appreciable, but its visibility is

difficult because other image’s details appear (tissue

and breast glands), and the computational

complexity is high, O(n

). With the proposed

approach using the DT-CWT better results are

obtained, MCs are more visible and other objects

presented by the SWT disappear, in addition the DT-

CWT has lower computational complexity, O(2n).

The results obtained with the Top-Hat

transformation show that this is the worst method to

detect MCs. This is due to the fact that other tissues

and breast’s glands are not filtered and appear

together with MCs, which are not significantly

appreciated as in the cases of the two other

simulated methods. In the same way, results are

interpreted for the case of the mammogram mdb249

D CALC M 544 508 48 shown in Figure 4. In this

case a set of MCs are present at the approximate

center of image (544, 508). Again it is observed that

using the DT-CWT a better detection of MCs

without inherent mammogram’s characteristics is

obtained. This is not possible with the SWT because

there are not tissue and glandular filtering.

SIGMAP 2009 - International Conference on Signal Processing and Multimedia Applications

6 CONCLUSIONS AND FUTURE

WORK

In the work reported in this paper we have proposed

an approach to detect MCs in digital mammograms

using the DT-CWT. The approach consists of the

DT-CWT application to obtain a mammogram’s

subband decomposition, mammogram’s denoising

by applying an optimal threshold at each

decomposition level, suppression of mammogram’s

low-frequencies, application of morphological

operators to enhanced MCs visualization, and

finally, the reconstruction of the mammogram. The

results obtained using the DT-CWT are compared to

the results obtained using the SWT and the Top-Hat

transformations. The proposed approach shows the

best performance to detect MCs in mammograms.

The SWT detects the MCs but other details are also

observed as MCs. Another inconvenient presented

by the SWT is the computational complexity, O(n

in contrast, the computational complexity of the DT-

CWT is O(2n) only. From results obtained

morphological filtering is the worst method to detect

MCs, because MCs are not well appreciated, in

addition tissue and breast glands are presented in the

reconstructed mammogram. The approach presented

in this work can be used as a basis to develop an

automatic diagnostic system to aid the results on

mammogram’s interpretation and to get an earlier

and opportune diagnostic for breast cancer.

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MAMMOGRAPHIC IMAGE ANALYSIS FOR BREAST CANCER DETECTION USING COMPLEX WAVELET

TRANSFORMS AND MORPHOLOGICAL OPERATORS