Statistically distributed developmental asymmetries appearing in paired body structures such as breasts in women are usually related to unhealthy biological conditions. In particular, there is evidence that breast cancer patients show more breast dimensional asymmetries and larger breasts than age-matched healthy women. It was also recently reported the application of Gabor nonorthogonal wavelets filtering for analyzing asymmetries of the directional structures appearing in mammograms corresponding to healthy and breast cancer patients. In this paper the Tree-Structured Wavelet Transform, which uses orthogonal wavelet bases, is applied for the first time in order to quantify texture asymmetries between left- and right- mammograms in women. In order to assess the suitability of the method, it was applied to characterize breast asymmetries of mammograms corresponding to a small population of 63 healthy women with ages ranging between 30 and 70 years. The obtained results justify further improvements of this method and its application for correlating asymmetries and the predisposition to breast cancer.Eje: Aplicaciones biomédicasRed de Universidades con Carreras en Informática (RedUNCI

Cairo, L. A.

Caselli, Eduardo

Gutierrez, L.P.

Santiago, Martin

Sorensen, E.

SEDICI - Repositorio de la UNLP

Breast Asymmetry Analysis employing Tree-Structured Wavelet Transform
E. Sorensen (a), L. A. Cairo (a), M. Santiago (a, b), E. Caselli (a, c),  and  L.P. Gutierrez (a)
(a) Instituto de Física Arroyo Seco, Universidad Nacional del Centro de la Provincia de Buenos
Aires, Pinto 399, 7000, Tandil, Argentina
(b) Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)
(c) Comisión de Investigaciones Científicas de la Provincia de Buenos Aires (CICPBA),
ecaselli@exa.unicen.edu.ar
Keywords: breast asymmetry, wavelet transform, digital mammography, computer aided diagnosis
Abstract
Statistically distributed developmental asymmetries appearing in paired body structures such as
breasts in women are usually related to unhealthy biological conditions. In particular, there is
evidence that breast cancer patients show more breast dimensional asymmetries and larger breasts
than age-matched healthy women. It was also recently reported the application of Gabor non-
orthogonal wavelets filtering for analyzing asymmetries of the directional structures appearing in
mammograms corresponding to healthy and breast cancer patients. In this paper the Tree-Structured
Wavelet Transform, which uses orthogonal wavelet bases, is applied for the first time in order to
quantify texture asymmetries between left- and right- mammograms in women. In order to assess
the suitability of the method, it was applied to characterize breast asymmetries of mammograms
corresponding to a small population of 63 healthy women with ages ranging between 30 and 70
years. The obtained results justify further improvements of this method and its application for
correlating asymmetries and the predisposition to breast cancer.
1. Introduction
It is generally accepted that symmetrical development of paired body structures is an indicator of
good health in bilaterally symmetrical organisms (Rasmusen, 1960). Deviations from symmetry
appear either as major errors, such as conspicuous birth defects, or as minor randomly-distributed
deviations (Ludwig, 1932), which are dubbed fluctuating asymmetry (FA). High FA has been found
in several physical and mental diseases, such as Down's syndrome (Townsend, 1983), trisomy
(Colasino, 1978), cleft lip and palate (Adams, 1967), autism (Campbell, 1978), schizophrenia
(Markow, 1986), etc. Since breasts are also symmetrical body structures, investigations were carried
out in order to determine whether cancer predisposition is related to breast asymmetry. In particular,
it was found that dimensional (length, width and volume) and shape breast asymmetries are an
indicator of the risk of developing breast cancer (Scutt, 1997). Apart from the physical features of
breasts, mammography has become and important tool for early detection of breast cancer.
Mammography has been applied successfully in screening programs on large populations of women
and much effort has been given to the investigation of computer-aided methods for digital filtering
of mammograms in order to detect incipient breast tumors (Gulsrud, 1996). Computer aided
diagnosis (CAD) is by now a useful complementary tool which can be added to routine diagnosis
protocols to improve radiologist's performance for detecting subtle, small, malignant masses in
breasts. Although most of the reported methods for digital filtering relay on the isolation of
punctual-shaped formations associated to microcalcifications and circumscribed masses, there has
recently been an increased interest in the use of texture analysis to identify suspicious extended
masses in mammograms. For instance, Gupta et al. applied a segmentation filter based on the Laws
filter masks to improve the detection of stellate and regular masses (Gupta, 1995). Gulsrud et al.
presented a method for texture segmentation of digital mammograms that uses a single filter which
is optimal with respect to the Fisher criterion, instead of a filter bank approach (Gulsrud, 1996).
Among the algorithms developed so far for characterization and classification of general textures,
there has been an increasing interest in those based on wavelet transform. Wavelet transform uses a
multiresolution approach for image decomposition, which allows for the information contained not
only in the low-frequency channels, but also in the medium-frequency channels which often bear
most of the information contained in textures (Chang, 1993). On this trend, Ferrari et al. recently
reported the investigation of asymmetries in mammograms by employing wavelet-based directional
filtering (Ferrari, 2001). In that work, non-orthogonal Gabor wavelets were used to compare
directional structures appearing in the left- and right-breast mammograms in order to detect cancer-
related abnormalities. Although the performance of the method is acceptable, it has the drawback of
using a non-orthogonal basis, which generates redundant information through the decomposition
process, and further the method is reported to be time-consuming.
In this paper the Tree-Structured Wavelet Transform (TSWT) put forward by Chang and
Kuo for texture classification (Chang, 1993) is used for the first time for the quantification of the
asymmetry between the texture of left and right mammograms. TSWT is based on orthogonal
wavelet bases, which allows image decomposition without redundant information production. The
method basically quantifies the asymmetry between the texture of two user-selectable bilaterally
corresponding windows on each mammogram. In this sense, it comes to be a complementary tool
which adds information to that the radiologist obtains by the routine visual comparison of the left-
and right-mammogram. In order to assess whether the method is suitable for quantifying breast
texture asymmetry, it was applied to the measurement of the asymmetry of a population of 63
healthy women. The obtained results are encouraging and justify further investigation and
improvement of the method. In particular, it will be applied to the analysis of large databases of
mammograms corresponding to healthy and cancer-diagnosed cases in order to determine the
relationship between texture asymmetry and breast cancer predisposition.
In what follows, the TSWT is briefly explained, then its application to a practical case is
presented, and, finally, the general results arising from the analysis of the aforementioned women
population are shown and discussed.
2. Tree-Structured Wavelet Transform
The 2-D wavelet transform algorithm consists basically in the iterative decomposition of a 2n × 2n
pixels image by means of a 2-D wavelet basis. A 2-D wavelet basis is built up by the tensor product
of two 1-D wavelet basis along the horizontal and vertical directions (Chang, 1993). As a final
result of a decomposition iteration of the original image, four 2 n−1 × 2 n−1 sub-images are obtained.
The first one (HH or high-high channel) contains the information corresponding to the high-
frequency components in both axes and the second one (LL or low-low channel) the information of
the low-frequency components also in both axes. The two remaining sub-images bear the
information of those components lying into the middle-frequency region, and are termed either LH
or HL depending on the axis the low- and the high-frequency channels correspond to. Actually, as
result of a one-iteration decomposition four sets of wavelet coefficients are obtained. Each set
corresponds to each frequency channel, and from them the four mentioned sub-images can be later
reconstructed.
The usual wavelet decomposition, known as pyramid-structured wavelet transform (PSWT),
continues the process only on the LL sub-images, decomposing the original image into a set of
frequency channels that have narrower bandwidths in the lower frequency region. This transform is
suitable for images consisting primarily of smooth components. In contrast to the PSWT algorithm,
the TSWT follows through the decomposition process on those channels that are significant, say, on
those having the most of the information contained in the original image. This discrimination step is
achieved by calculating the energy e of each sub-image. In this work we resorted to the averaged l1-
norm for this purpose:
( )∑∑
= =
=
N
i
M
j
jix
NM
e
1 1
,1  ,
where N and M are the dimensions of the sub-image, and x stands for its wavelet coefficients.
Although there exist other norms, the l1-norm was chosen due to its simplicity and good
performance. Once the energy of each sub-image has been calculated, the next decomposition
iteration is performed only on those channels which satisfy the condition e > CeMAX, where eMAX
stands for the higher energy value, and C is a discrimination constant less than 1. Practically, the
size of the smallest sub-images should serve as a criterion to stop the decomposition process. For
practical reasons the size of the smallest sub-images should not be less than 16 × 16 pixels.
Consequently, if the input image size is 256 × 256, a 4-level tree-structured wavelet transform is
appropriate. By taking into account these considerations, when the decomposition procedure is
exhausted a tree-shaped structured is obtained. This can be seen in Fig. 1, where an example of a 4-
level decomposition is shown schematically as a quadtree representation. For sake of simplicity, the
four channels have been renamed as follows: LL→A, HL→B, LH→D, and HH→D. In this way,
the ABBB-chain, for example, represents the channel bearing the information that arises from a first
LL decomposition and three subsequent HL wavelet transforms. The number of channels appearing
in the deepest decomposition level will depend on the value of the discrimination constant C. If we
order the deepest-level channels according to their energy values, namely, according to their
significance, a vector is obtained, which is called the energy map of the image. Finally, the J first
energy values of this vector of ordered energies are taken as the features of the image. In the
example shown in Fig. 1 the most energetic channel ABBB were denoted by a bold line, and its
energy value will appear at first place in the energy map of the decomposed image. The other 4-
level channels appearing in the energy map will be ABBA, ABBC and ABBD, each at its
corresponding position in the vector.
Fig.1: Example of the quadtree representation of a 4-level
TSWT of an arbitrary image.
3. Measurement of breast symmetry
Chang and Kuo give three very sophisticated TSWT-based methods for classifying textures
by using adaptive algorithms along with extensive databases (Chang, 1993). In the present work we
adapted their algorithms for the purpose of measuring the asymmetry between the textures of two
mammograms. Our method is straightforward and consists of the following steps: 1) digitalized
images of the right and left mammograms must be obtained, and the constant C and the number J of
features must be chosen. 2) The left mammogram is inverted along the vertical axis. In this way,
both mammograms are oriented in the same direction. 3) A 2n × 2n pixels region of interest (ROI) is
selected on the left mammogram and a TSWT is performed in order to obtain its energy map. 4) At
this point, it is necessary to select on the right mammogram that part of the image that actually
corresponds to the selected ROI. In other words, what we have to compare are the images related to
equivalent positions in both breasts. Because of slight differences in breast compression and
positioning during image capture, equivalent regions are not necessarily imaged in the
mammograms at symmetrical positions. In order to guarantee that the region corresponding to the
ROI is included a region of larger size than the ROI must be selected. Next, once the right region
has been selected, it is scanned up-down and from left to right by a 2n × 2n window which moves at
2-pixel steps. At each position of the window, a TSWT is performed on the contained sub-image
and its energy map is obtained. In Fig. 2 (a) the left-breast mammogram is shown, from which a
128 × 128 ROI has been selected. On the right-breast mammogram (Fig. 2 (b)) a larger scanning
region has been also selected. The scanning window at a given position is shown in green. At this
step, the energy maps of both the left-ROI and the scanning window are compared. To do this, the
energy of the first J dominant channels of the ROI are picked up as features. Let us denote this
feature set by e = (e1, …, eJ),   e1 > … > eJ . Then the energies mi (i = 1, …, J) of the same channels
are taken from the energy map of the scanning window (of course, they are not necessarily the first
J channels of the window-energy map) and the distance D between both images is calculated
according to:
( )∑
=
−=
J
j
jj mxD
1
2
The distance D is an indicator of the symmetry between the left ROI and that specific scanning
window. Low values for D will correspond to breast symmetry, and high values to breast
asymmetry. It is worth mentioning that if both images are quite different and the constant C is high
enough not all the J more significant channels of the ROI will be present in the energy map of the
scanning window. In this case the distance D cannot be properly evaluated and the textures of both
images can be regarded as non-coincident.
Fig. 2: An example of the application of the method is shown.
Further explanations can be found in the text.
After the scanning window has mapped the entire selected right region, usually many
coincident regions are found, each with its own D value. Then, either the lowest D value or an
average of all the D values can be taken as a measure of the texture symmetry.
4. An application
The method described so far for evaluating the symmetry between mammogram textures, was
applied to the left- and right- CC mammograms belonging to a population of 63 healthy women
with ages ranging from 30 to 70 years. We used the DAUB16 Daubechies functions as wavelet
basis because they bring about higher energy values for the dominant channels than the other
Daubechies functions. On the other hand, it has been reported that Daubechies functions are well-
conditioned for wavelet decomposition of mammograms, since they reproduce efficiently at
different scales the breast structure (Lange, 1999). A value of J = 5 was selected for the number of
features and a discrimination constant C = 0.15. For the left mammograms of each patient, three
128 × 128 ROIs were selected and analyzed separately. For each ROI a 256 × 256 region was
scanned on the right mammogram and all the coincident subimages were determined. Finally, all
distances less than 1200 (maximum recognition distance) were averaged to obtain an estimate of the
symmetry between both mammograms of each patient. In Table 1 the different cases were grouped
by age and distance. In the sixth column those cases whose lowest distance was higher than the
maximum recognition distance were grouped.
Average distance
Age [year] 0 - 300 301 - 600 601 - 900 901 – 1200 1201 or
larger
Occurrence
number
30 - 40 1 3 3 5 1 13
40 - 50 0 5 5 6 4 20
50 - 60 2 2 3 3 6 16
60 - 70 1 4 2 3 4 14
Occurrence
number
4 14 13 17 15 63
Table 1: The results of the analysis of the texture breast
symmetry of the mammograms belonging to 63 healthy women.
Although it can be said that the number of high asymmetries slightly increases whit age, no
general conclusions can be extracted from the results shown in Table 1. That is not surprising for
several reasons. First, the analyzed population is too small for allowing a reliable statistics. Second,
breast tissue does not have a well-established pattern. It not only changes drastically from women to
women, but it also changes in time for the same woman. This characteristic of breast tissue
introduces large dispersions among the mammograms of different patients, and, consequently, a
serious difficulty for the analysis of mammograms. However, this example demonstrates that a
quantification of the texture symmetry of a patient's mammograms is possible, besides the above
reported breast asymmetry measurements, based mainly on dimensional analysis.
5. Summary
In this paper an additional method for quantifying the asymmetry between the left- and right-
mammograms of a patient is reported. It is based on the algorithm developed by Chang and Kuo for
characterizing and classifying quite general textures. The method reported in this paper was
successfully applied to a small population of healthy women for the analysis of their breast texture
asymmetries, showing the feasibility of the method. The study of larger populations of women in
order to correlate breast asymmetries with the predisposition to develop breast cancer can be
envisaged.
Bibliography
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Campbell M, Geller B, Small AM, Petti Ta, and Ferris SH, Minor physical anomalies in young
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Chang T and Kuo J, Texture analysis and classification with Tree-Structured Wavelet Transform,
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Colacino SC, Petterson JC, Analysis of the gross anatomical variations found in four cases of
trisomy, Am. J. Med. Genet. 2 (1978) 31.
Ferrari RJ, Rangayyan RM, Desautels JEL, and Frère AF, Analysis of Asymmetry in Mammograms
via Directional Filtering With Gabor Wavelets, IEEE Trans. Med. Image 20 (2001) 953.
Gulsrud TO and Kjøde S, Optimal filter for detection of stellate lesions and circumscribed masses
in mammograms. In SPIE's Visual Communications and Image Processing 1996, Orlando, FL,
March 1996, pp. 430-440.
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mammography, Physics in Medicine & Biology 40 (1995) 835.
Lange R, Gamba P, Saccomano C, Lester M, and Caselli E, Computer assisted diagnosis of
mammograms employing wavelets, VIII RPIC Anales 2 (1999) 7-5.
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English

Breast asymmetry analysis employing tree-structured wavelet transform

Servicio de Difusión de la Creación Intelectual

Breast Asymmetry Analysis employing Tree-Structured Wavelet TransformE. Sorensen (a), L. A. Cairo (a), M. Santiago (a, b), E. Caselli (a, c),  and  L.P. Gutierrez (a)(a) Instituto de Física Arroyo Seco, Universidad Nacional del Centro de la Provincia de BuenosAires, Pinto 399, 7000, Tandil, Argentina(b) Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)(c) Comisión de Investigaciones Científicas de la Provincia de Buenos Aires (CICPBA),ecaselli@exa.unicen.edu.arKeywords: breast asymmetry, wavelet transform, digital mammography, computer aided diagnosisAbstractStatistically distributed developmental asymmetries appearing in paired body structures such asbreasts in women are usually related to unhealthy biological conditions. In particular, there isevidence that breast cancer patients show more breast dimensional asymmetries and larger breaststhan age-matched healthy women. It was also recently reported the application of Gabor non-orthogonal wavelets filtering for analyzing asymmetries of the directional structures appearing inmammograms corresponding to healthy and breast cancer patients. In this paper the Tree-StructuredWavelet Transform, which uses orthogonal wavelet bases, is applied for the first time in order toquantify texture asymmetries between left- and right- mammograms in women. In order to assessthe suitability of the method, it was applied to characterize breast asymmetries of mammogramscorresponding to a small population of 63 healthy women with ages ranging between 30 and 70years. The obtained results justify further improvements of this method and its application forcorrelating asymmetries and the predisposition to breast cancer.1. IntroductionIt is generally accepted that symmetrical development of paired body structures is an indicator ofgood health in bilaterally symmetrical organisms (Rasmusen, 1960). Deviations from symmetryappear either as major errors, such as conspicuous birth defects, or as minor randomly-distributeddeviations (Ludwig, 1932), which are dubbed fluctuating asymmetry (FA). High FA has been foundin several physical and mental diseases, such as Down's syndrome (Townsend, 1983), trisomy(Colasino, 1978), cleft lip and palate (Adams, 1967), autism (Campbell, 1978), schizophrenia(Markow, 1986), etc. Since breasts are also symmetrical body structures, investigations were carriedout in order to determine whether cancer predisposition is related to breast asymmetry. In particular,it was found that dimensional (length, width and volume) and shape breast asymmetries are anindicator of the risk of developing breast cancer (Scutt, 1997). Apart from the physical features ofbreasts, mammography has become and important tool for early detection of breast cancer.Mammography has been applied successfully in screening programs on large populations of womenand much effort has been given to the investigation of computer-aided methods for digital filteringof mammograms in order to detect incipient breast tumors (Gulsrud, 1996). Computer aideddiagnosis (CAD) is by now a useful complementary tool which can be added to routine diagnosisprotocols to improve radiologist's performance for detecting subtle, small, malignant masses inbreasts. Although most of the reported methods for digital filtering relay on the isolation ofpunctual-shaped formations associated to microcalcifications and circumscribed masses, there hasrecently been an increased interest in the use of texture analysis to identify suspicious extendedmasses in mammograms. For instance, Gupta et al. applied a segmentation filter based on the Lawsfilter masks to improve the detection of stellate and regular masses (Gupta, 1995). Gulsrud et al.presented a method for texture segmentation of digital mammograms that uses a single filter whichis optimal with respect to the Fisher criterion, instead of a filter bank approach (Gulsrud, 1996).Among the algorithms developed so far for characterization and classification of general textures,there has been an increasing interest in those based on wavelet transform. Wavelet transform uses amultiresolution approach for image decomposition, which allows for the information contained notonly in the low-frequency channels, but also in the medium-frequency channels which often bearmost of the information contained in textures (Chang, 1993). On this trend, Ferrari et al. recentlyreported the investigation of asymmetries in mammograms by employing wavelet-based directionalfiltering (Ferrari, 2001). In that work, non-orthogonal Gabor wavelets were used to comparedirectional structures appearing in the left- and right-breast mammograms in order to detect cancer-related abnormalities. Although the performance of the method is acceptable, it has the drawback ofusing a non-orthogonal basis, which generates redundant information through the decompositionprocess, and further the method is reported to be time-consuming.In this paper the Tree-Structured Wavelet Transform (TSWT) put forward by Chang andKuo for texture classification (Chang, 1993) is used for the first time for the quantification of theasymmetry between the texture of left and right mammograms. TSWT is based on orthogonalwavelet bases, which allows image decomposition without redundant information production. Themethod basically quantifies the asymmetry between the texture of two user-selectable bilaterallycorresponding windows on each mammogram. In this sense, it comes to be a complementary toolwhich adds information to that the radiologist obtains by the routine visual comparison of the left-and right-mammogram. In order to assess whether the method is suitable for quantifying breasttexture asymmetry, it was applied to the measurement of the asymmetry of a population of 63healthy women. The obtained results are encouraging and justify further investigation andimprovement of the method. In particular, it will be applied to the analysis of large databases ofmammograms corresponding to healthy and cancer-diagnosed cases in order to determine therelationship between texture asymmetry and breast cancer predisposition.In what follows, the TSWT is briefly explained, then its application to a practical case ispresented, and, finally, the general results arising from the analysis of the aforementioned womenpopulation are shown and discussed.2. Tree-Structured Wavelet TransformThe 2-D wavelet transform algorithm consists basically in the iterative decomposition of a 2n × 2npixels image by means of a 2-D wavelet basis. A 2-D wavelet basis is built up by the tensor productof two 1-D wavelet basis along the horizontal and vertical directions (Chang, 1993). As a finalresult of a decomposition iteration of the original image, four 2 n−1 × 2 n−1 sub-images are obtained.The first one (HH or high-high channel) contains the information corresponding to the high-frequency components in both axes and the second one (LL or low-low channel) the information ofthe low-frequency components also in both axes. The two remaining sub-images bear theinformation of those components lying into the middle-frequency region, and are termed either LHor HL depending on the axis the low- and the high-frequency channels correspond to. Actually, asresult of a one-iteration decomposition four sets of wavelet coefficients are obtained. Each setcorresponds to each frequency channel, and from them the four mentioned sub-images can be laterreconstructed.The usual wavelet decomposition, known as pyramid-structured wavelet transform (PSWT),continues the process only on the LL sub-images, decomposing the original image into a set offrequency channels that have narrower bandwidths in the lower frequency region. This transform issuitable for images consisting primarily of smooth components. In contrast to the PSWT algorithm,the TSWT follows through the decomposition process on those channels that are significant, say, onthose having the most of the information contained in the original image. This discrimination step isachieved by calculating the energy e of each sub-image. In this work we resorted to the averaged l1-norm for this purpose:( )∑∑= ==NiMjjixNMe1 1,1  ,where N and M are the dimensions of the sub-image, and x stands for its wavelet coefficients.Although there exist other norms, the l1-norm was chosen due to its simplicity and goodperformance. Once the energy of each sub-image has been calculated, the next decompositioniteration is performed only on those channels which satisfy the condition e > CeMAX, where eMAXstands for the higher energy value, and C is a discrimination constant less than 1. Practically, thesize of the smallest sub-images should serve as a criterion to stop the decomposition process. Forpractical reasons the size of the smallest sub-images should not be less than 16 × 16 pixels.Consequently, if the input image size is 256 × 256, a 4-level tree-structured wavelet transform isappropriate. By taking into account these considerations, when the decomposition procedure isexhausted a tree-shaped structured is obtained. This can be seen in Fig. 1, where an example of a 4-level decomposition is shown schematically as a quadtree representation. For sake of simplicity, thefour channels have been renamed as follows: LL→A, HL→B, LH→D, and HH→D. In this way,the ABBB-chain, for example, represents the channel bearing the information that arises from a firstLL decomposition and three subsequent HL wavelet transforms. The number of channels appearingin the deepest decomposition level will depend on the value of the discrimination constant C. If weorder the deepest-level channels according to their energy values, namely, according to theirsignificance, a vector is obtained, which is called the energy map of the image. Finally, the J firstenergy values of this vector of ordered energies are taken as the features of the image. In theexample shown in Fig. 1 the most energetic channel ABBB were denoted by a bold line, and itsenergy value will appear at first place in the energy map of the decomposed image. The other 4-level channels appearing in the energy map will be ABBA, ABBC and ABBD, each at itscorresponding position in the vector.Fig.1: Example of the quadtree representation of a 4-levelTSWT of an arbitrary image.3. Measurement of breast symmetryChang and Kuo give three very sophisticated TSWT-based methods for classifying texturesby using adaptive algorithms along with extensive databases (Chang, 1993). In the present work weadapted their algorithms for the purpose of measuring the asymmetry between the textures of twomammograms. Our method is straightforward and consists of the following steps: 1) digitalizedimages of the right and left mammograms must be obtained, and the constant C and the number J offeatures must be chosen. 2) The left mammogram is inverted along the vertical axis. In this way,both mammograms are oriented in the same direction. 3) A 2n × 2n pixels region of interest (ROI) isselected on the left mammogram and a TSWT is performed in order to obtain its energy map. 4) Atthis point, it is necessary to select on the right mammogram that part of the image that actuallycorresponds to the selected ROI. In other words, what we have to compare are the images related toequivalent positions in both breasts. Because of slight differences in breast compression andpositioning during image capture, equivalent regions are not necessarily imaged in themammograms at symmetrical positions. In order to guarantee that the region corresponding to theROI is included a region of larger size than the ROI must be selected. Next, once the right regionhas been selected, it is scanned up-down and from left to right by a 2n × 2n window which moves at2-pixel steps. At each position of the window, a TSWT is performed on the contained sub-imageand its energy map is obtained. In Fig. 2 (a) the left-breast mammogram is shown, from which a128 × 128 ROI has been selected. On the right-breast mammogram (Fig. 2 (b)) a larger scanningregion has been also selected. The scanning window at a given position is shown in green. At thisstep, the energy maps of both the left-ROI and the scanning window are compared. To do this, theenergy of the first J dominant channels of the ROI are picked up as features. Let us denote thisfeature set by e = (e1, …, eJ),   e1 > … > eJ . Then the energies mi (i = 1, …, J) of the same channelsare taken from the energy map of the scanning window (of course, they are not necessarily the firstJ channels of the window-energy map) and the distance D between both images is calculatedaccording to:( )∑=−=Jjjj mxD12The distance D is an indicator of the symmetry between the left ROI and that specific scanningwindow. Low values for D will correspond to breast symmetry, and high values to breastasymmetry. It is worth mentioning that if both images are quite different and the constant C is highenough not all the J more significant channels of the ROI will be present in the energy map of thescanning window. In this case the distance D cannot be properly evaluated and the textures of bothimages can be regarded as non-coincident.Fig. 2: An example of the application of the method is shown.Further explanations can be found in the text.After the scanning window has mapped the entire selected right region, usually manycoincident regions are found, each with its own D value. Then, either the lowest D value or anaverage of all the D values can be taken as a measure of the texture symmetry.4. An applicationThe method described so far for evaluating the symmetry between mammogram textures, wasapplied to the left- and right- CC mammograms belonging to a population of 63 healthy womenwith ages ranging from 30 to 70 years. We used the DAUB16 Daubechies functions as waveletbasis because they bring about higher energy values for the dominant channels than the otherDaubechies functions. On the other hand, it has been reported that Daubechies functions are well-conditioned for wavelet decomposition of mammograms, since they reproduce efficiently atdifferent scales the breast structure (Lange, 1999). A value of J = 5 was selected for the number offeatures and a discrimination constant C = 0.15. For the left mammograms of each patient, three128 × 128 ROIs were selected and analyzed separately. For each ROI a 256 × 256 region wasscanned on the right mammogram and all the coincident subimages were determined. Finally, alldistances less than 1200 (maximum recognition distance) were averaged to obtain an estimate of thesymmetry between both mammograms of each patient. In Table 1 the different cases were groupedby age and distance. In the sixth column those cases whose lowest distance was higher than themaximum recognition distance were grouped.Average distanceAge [year] 0 - 300 301 - 600 601 - 900 901 – 1200 1201 orlargerOccurrencenumber30 - 40 1 3 3 5 1 1340 - 50 0 5 5 6 4 2050 - 60 2 2 3 3 6 1660 - 70 1 4 2 3 4 14Occurrencenumber4 14 13 17 15 63Table 1: The results of the analysis of the texture breastsymmetry of the mammograms belonging to 63 healthy women.Although it can be said that the number of high asymmetries slightly increases whit age, nogeneral conclusions can be extracted from the results shown in Table 1. That is not surprising forseveral reasons. First, the analyzed population is too small for allowing a reliable statistics. Second,breast tissue does not have a well-established pattern. It not only changes drastically from women towomen, but it also changes in time for the same woman. This characteristic of breast tissueintroduces large dispersions among the mammograms of different patients, and, consequently, aserious difficulty for the analysis of mammograms. However, this example demonstrates that aquantification of the texture symmetry of a patient's mammograms is possible, besides the abovereported breast asymmetry measurements, based mainly on dimensional analysis.5. SummaryIn this paper an additional method for quantifying the asymmetry between the left- and right-mammograms of a patient is reported. It is based on the algorithm developed by Chang and Kuo forcharacterizing and classifying quite general textures. The method reported in this paper wassuccessfully applied to a small population of healthy women for the analysis of their breast textureasymmetries, showing the feasibility of the method. The study of larger populations of women inorder to correlate breast asymmetries with the predisposition to develop breast cancer can beenvisaged.BibliographyAdams MS and Niswander JD, Developmental noise and a congenital malformation, Genet. Res.Cambridge 10 (1967) 313.Campbell M, Geller B, Small AM, Petti Ta, and Ferris SH, Minor physical anomalies in youngpsychotic children, Am. J. Psychiat. 135 (1978)Chang T and Kuo J, Texture analysis and classification with Tree-Structured Wavelet Transform,IEEE Transactions on Image Processing Vol. 2 (1993) 429.Colacino SC, Petterson JC, Analysis of the gross anatomical variations found in four cases oftrisomy, Am. J. Med. Genet. 2 (1978) 31.Ferrari RJ, Rangayyan RM, Desautels JEL, and Frère AF, Analysis of Asymmetry in Mammogramsvia Directional Filtering With Gabor Wavelets, IEEE Trans. Med. Image 20 (2001) 953.Gulsrud TO and Kjøde S, Optimal filter for detection of stellate lesions and circumscribed massesin mammograms. In SPIE's Visual Communications and Image Processing 1996, Orlando, FL,March 1996, pp. 430-440.Gupta R and Undrill PE, The use of texture analysis to delineate suspicious masses inmammography, Physics in Medicine & Biology 40 (1995) 835.Lange R, Gamba P, Saccomano C, Lester M, and Caselli E, Computer assisted diagnosis ofmammograms employing wavelets, VIII RPIC Anales 2 (1999) 7-5.Ludwig W, Das Rechts-Links Problem in Tierreich und beim Menschen, Springer, Berlin, 1932.Markow TA, Wandler K, fluctuating dermatoglyphic asymmetry and the genetics of liability toschizophrenia, Psychiatry Res. 19 (1986) 323.Rasmusen M, Frequency of morphological deviants as a criterion of developmental stability,Hereditas 46 (1960) 511.Scutt D, Manning JT, Whitehouse GH, Leinster SJ, and Massey CP, The relationship betweenbreast asymmetry, breast size and the occurrence of cancer, The British Journal of Radiology 70(1997) 1017.Townsend G, Fluctuating dental asymmetry in Down's syndrome, Aust. Dent. 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Breast asymmetry analysis employing tree-structured wavelet transform

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