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ENHANCEMENT OF MAMMOGRAPHIC IMAGES FOR
DETECTION OF MICROCALCIFICATIONS
! " & ( 1
Department of Electronic Systems and Information
Processing,Faculty of Electrical Engineering and Computing,
University of Zagreb,
Vukovar Avenue 39, CROATIA
Tel: +385 1 6129973; fax: +385 1 6129652e-mail:
[email protected], [email protected]
ABSTRACT
A novel approach to image enhancement of digital
mammography images is introduced, for more accuratedetection of
microcalcification clusters. In originalmammographic images
obtained by X-ray radiography,
most of the information is hidden to the human observer.The
method is based on redundant discrete wavelettransform due to its
good properties: shift invariance and
numeric robustness. The procedure consists of three
steps:low-frequency tissue density component removal,
noisefiltering, and microcalcification enhancement. The
experimental results have shown good properties of theproposed
method.
1. INTRODUCTIONMany authors deal with the problem of
automaticsegmentation of microcalcification clusters in
digitalmammography. Presence of microcalcifications and skin
thickening is an indirect sign of malignant
masses.Unfortunately, mammograms (obtained by breastradiography) as
normally viewed, display only about 3%of the information they
detect [Laine, Fan, and Yang,1995]. Main obstacle lays in low
contrast between normaland malignant glandular tissues, especially
in younger
women. On the other hand, calcifications have highattenuation
properties, which is a good visibility property.The problem is in
their very small size, especially in the
early stage of tumor development, making them extremelydifficult
to view.
A number of digital image processing techniques havebeen applied
to mammography, to address the mentionedproblems. Several authors
used adaptive neighborhoodimage processing techniques to enhance
mammographicfeatures while reducing noise [Gorden and
Rangayyan,1984, Dhawan and Le Royer, 1988, etc.], or
spatialfiltering [Tahoces et al. 1991]. Recent discoveries show
that a multiresolution approach exists in human visionsystem,
thus leading to an idea of using wavelet basedmultiresolution
analysis for mammographic image
processing. Wavelet approach has been used by[Strickland and
Hahn, 1997] for detection ofmicrocalcifications, while [Qian et al.
1993] used wavelets
and tree-structured nonlinear filtering formicrocalcification
segmentation. [Laine, Fan, and Yang,1995] used wavelets for
contrast enhancement in digital
mammography, as well as many other authors.
Microcalcifications usually come in clusters, havingvery sharp
edges, and usually irregular shape of very small
size. Due to their high attenuation properties, they appearas
white (or high intensity) spots on mammograms.
There are two goals of this work: enhancement of
mammographic images to achieve better visibility of theobserved
phenomena to the human observer (radiologist),and processing of
mammograms to enable automatic
detection of micro-calcifications, as a first step to
the"automated second-opinion" procedure. To achieve bothgoals, we
first used redundant wavelet transform applied to
suspicious cutouts of mammograms.
2. A METHOD FOR MAMMOGRAMENHANCEMENTIn this work, we developed a
specific wavelet-basedscheme for image enhancement and compared
differentwavelet choices, as well as different filtering
procedures
applied to wavelet coefficients.A quality measure is developed
for comparison of the
processed image to the binary, human made drawing of
microcalcifications. The measure is based on relativeenergy
comparison between microcalcification area andits complement.
2.1 Non-decimated wavelet transform
Among other linear transforms, wavelets have anumber of useful
properties: they can successfully
represent smooth functions, as well as singularities;expansion
functions are local - so the algorithms based on
wavelet coefficients are adaptive to inhomogeneities;wavelets
are computationally inexpensive and near optimalfor statistical
estimation, signal recovery and data
compression.Discrete time wavelet transform expands analyzed
signals into components with different shifts and scales,
where scales are usually chosen from a dyadic set:
Xm,n = ,
m,n=2-1/2
(2-m x - n). (1)
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Hence, m,n are shifted, expanded or shrank versions of a
mother wavelet . Mother wavelet function is typicallychosen to
achieve desired localization properties in time
and frequency domain. There are many choices ofwavelets that
lead to orthogonal or biorthogonalexpansions, some of them
realizable by FIR perfect
reconstruction wavelet filter banks. The number ofcalculated
wavelet coefficients decreases with enlargedscale, which
corresponds to decimation in wavelet filter
banks:
Figure 1 Decimated analysis wavelet filter bank
H and L filters are related to mother wavelet and its
associated scaling function respectively, while
"detail"coefficients d1, d2, dn, ... correspond to the
waveletcoefficients in different scales (m=0, 1, ...). The set
of
wavelet coefficients for orthogonal or
biorthogonaldecompositions is minimum size, equal to the length
ofanalyzed signal x. The number of necessary calculations is
O(N), which is computationally very inexpensive whencompared to
other linear transforms. On the other hand,such wavelet
coefficients are shift dependent, in the sense
that time shifts of the original signal result in different
setsof wavelet coefficients, with different statistical
properties.That fact has been noticed as a significant
drawback,
especially in detection problems, as well as in de-noising
and compression applications. All-shifts DWT expansionis
redundant, but shift-invariant in the previously
mentioned sense:
m,n=2-1/2(2
-m (x - n)). (2)By calculating all shifts, orthogonal expansions
turn toframes, withholding reconstruction properties. Frames(due to
their redundancy) bring numerical robustness
[Daubechies, 1992], which will show its value in non-linear
wavelet coefficient processing. [Beylkin, 1992] hasshown that the
order of computation can be reduced to
O(N log N) operations using corresponding non-decimatedwavelet
filter bank, instead of O(N
2) operations, whichfollows from equation (2).
Figure 2 Non-decimated analysis wavelet filter bank
H(z2), H(z4), H(z8), ... can be easily realized by
insertingzeros between samples of h(k) in the time domain
("algorithm trous"). Most of numerical simulationsoftware tools
spend processor's time even formultiplications by zero, so we
rather use recursive
subsampling-upsampling structure illustrated in thefollowing
figure:
Figure 3 Recursive subsampling-upsampling structure
The last branch filters in recursion structure are H(z),applied
to l-times decimated input. In the linear phase
case, symmetry of coefficients can be used to reduce thenumber
of multiplications (by factor of 2 or 4, for 1-D or2-D case,
respectively).
Shift-invariant wavelet expansion can be easilyextended to the
two-dimensional (2-D) case:
Figure 4 Filter bank implementation of the 2-D non-
decimated wavelet decomposition using 1-Dfilters. Both analysis
and reconstruction sides
in a level l are shown.
The single level expansion results in 3 "details" images:
dHH, dHL, and dLH, (shorter: HH, HL, LH) coveringindependent
bands in the frequency domain. The"approximation" aLL (or LL) is a
low-pass component,
which is passed to the next level of decomposition.
2.2 Application to mammograms
At first, 5-levels redundant wavelet decomposition of
theoriginal mammogram cutout is performed. Mammogramimages were
obtained by scanning the X-ray images in
30m x 30m resolution, 12 bits per pixel. Typical size
ofmicrocalcification varies from 0.1 mm to more than 1 mm,
which corresponds to the range from the smallest 3 x 3pixel
round objects to more than 30 pixels wide irregularshapes. The
5-octaves analysis is taken to cover the whole
range.
H
L H
L
x d1
d2
s2
H(z)
L(z) H(z2)
L(z2)
x d1
d2
a2
( )H zl2 ( )H z
l2
1
( )H zl
21
z-1
z
( )H zl
2
aLLl
rows
dLHl+1
aLLl+1
dHHl+1
dHLl+1
columns
( )L zl
2
( )H zl
2
( )L zl
2
( )H zl
2
( )L zl
2
( )~H zl
2
aLLl
columns
dLH
l+1
aLLl+1
dHHl+1
dHLl+1
rows
( )~L zl
2
( )~H zl
2
( )~L zl
2
( )~H zl
2
( )~L zl
2
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Density of the breast tissue varies across different parts ofthe
mammogram, thus increasing the dynamic range of
the image. Fine breast tissue structure andmicrocalcifications
are almost invisible in dense parts ofthe original image,
especially if gray-value does not cover
the necessary dynamic range.
Several wavelet choices were taken in consideration,but B-spline
wavelets yield the best results, due to theirlinear phase and
symmetry, as well as some similarity to
observed calcifications (which complies to [Strickland andHahn,
1997] ).
Visual inspection of wavelet coefficient images showsthat
first-level detail coefficients (HH, HL and LH) containmostly
noise. Detail coefficients in levels 2-5 contain fine
breast structure and microcalcifications (together withsome
noise). Finally, level 5 approximation coefficients(LL) contain low
frequency background, which
corresponds to the tissue density.Reconstructed sub-images
(after applying
reconstruction part of filter bank) are additive componentsof
the original image, so the reconstructed details HHr, HLr
and LHr at observed level were added in a singlerepresentation
Dr.
To enhance the image for a human observer, severalactions have
been taken. Subtraction of the reminder A5
r
shrinks the dynamic range of the image and makes the fine
structure more visible, as well as microcalcifications. But,the
image is still noised, and small microcalcifications arehardly
visible.
[Donoho and Johnstone, 1994] suggest a denoisingscheme by
killing and shrinking wavelet coefficients. If weassume additive
noise in the form:
xi = si + n ni, i = 1, ..., N; (3)where signal si is corrupted
by zero mean, Gaussian noise
ni with standard deviation n, then the risk (l2 measure oferror
between estimated and original signal s) of the socalled soft -
thresholding scheme:
( )
( )
( )
X DWT x
Xsign X X thr X thr
otherwise
s DWT X
=
=
=
,
( ) ,
,
,
0
1
(4)
is within a logarithmic factor log N of ideal minimum risk.
A good choice for threshold thr is:
thr N N n
= log , (5)
where n is standard deviation of noise, and N is number
of wavelet coefficients. We used a robust estimation of
n,calculated from detail wavelet coefficients of an
additionaldecomposition of x:
( ) 6745.0/dmedian1n
= (6)
Such estimation is insensitive to presence of strong outliers(as
the microcalcifications are). Denoising schemeconfirms that
decomposition at level 1 contains "pure"
noise, and should be killed. Notice that our sampling
interval was 30 m, and if the same decomposition would
have been applied to the images sampled in 100mresolution, (like
University Hospital Nijmegen images),level 1 decomposition would
contain signal information aswell. Applied to other levels,
denoising enhances the
reconstructed images, especially in higher frequency sub-bands
(level 2 and 3). Results of denoising in level 2 are
visible in figure (5).
Figure 5 Denoissed detail reconstruction D2r image
Finally, we would like to amplify the microcalcifications.
[Strickland and Hahn, 1997] have shown that redundant
wavelet transform by itself act as a multiscale matchedfilter.
B-spline redundant wavelets closely approximate the
prewhitening matched filter for detecting Gaussian objectsin
Markov noise. Small microcalcifications are blob-likeobjects that
fit in the assumed scheme, and the background
can be modeled as a combination of separable and non-separable
Markov noise. Microcalcifications are wellrepresented by the
non-decimated B-spline wavelet
decomposition. If the scales match, coefficients show ahuge peak
at the locations of calcifications.[Burley and Darnell, 1997]
analyze the suppression of
impulse noise using wavelets, and suggest a kind ofreversed
scheme of [Donoho, 1995]. If the signal is
corrupted by additive Gaussian noise and impulse noisemi:
xi = si + n ni + mi, i = 1, ..., N; (7)they propose shrinking of
wavelet coefficients larger then
3.3 n to the Donoho level. The procedure is
eliminatingcoefficients who belong to the impulse noise, and
preserveGaussian signal which is under the threshold.
Wavelet coefficients that correspond to micro-calcifications
have huge peaks in all scales, thus behavingsimilarly as they were
impulse noise. We used a reversed
scheme to amplify their contribution to the final image.We
estimated the variance of the background signal usingour robust
estimator (insensitive to peaks), and then
calculated upper threshold. Almost all coefficientsbelonging to
the fine tissue structure are bellow the upperthreshold. The
"upper" images (soft thresholded images
using upper threshold) in all scales are good candidates forthe
feature vector representation for detection ofcalcifications. If
the upper sub-images are added to
denoised sub-images, a visible enhancement ofcalcification areas
will be done.
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3. RESULTS AND DISCUSSIONAnalysis has been taken on a number of
mammograms,
all containing microcalcifications showing presence oftumors
(either benign or malign).
Figure 6 Original mammogram cutout
Figure 7 Enhanced image, reconstructed from levels 2-5
Figure 8 Microcalcifications, marked by human
Figure 9 Upper image, reconstructed from levels 2-5
It is clearly visible that "upper" image is nearly a detectorof
microcalcifications. We convert the upper image to thebinary form,
and estimate the similarity to human drawn
calcifications, by calculating the energy of the difference.The
similarity is higher for B-spline (linear phase)wavelets, and
somewhat less for a simple Haardecomposition.
4. CONCLUSIONA new method for enhancement of mammogram images
ispresented in the paper. The method has been applied to a
number of mammogram images and has shown goodresults.
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