# Optimized state transfer in systems of ultrastrongly coupled matter and
radiation

Luigi Giannelli  CNR-IMM, UoS Università, 95123 Catania, Italy  Dipartimento
di Fisica e Astronomia “Ettore Majorana”, Universitá di Catania, 95123
Catania, Italy  Jishnu Rajendran  Dipartimento di Fisica e Astronomia “Ettore
Majorana”, Universitá di Catania, 95123 Catania, Italy  INFN Sezione di
Catania, 95123 Catania, Italy  Nicola Macrì  Dipartimento di Fisica e
Astronomia “Ettore Majorana”, Universitá di Catania, 95123 Catania, Italy
INFN Sezione di Catania, 95123 Catania, Italy  Giuliano Benenti  Center for
Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia,
Università degli Studi dell’Insubria, 22100 Como, Italy  INFN Sezione di
Milano, 20133 Milano, Italy  NEST, Istituto Nanoscienze-CNR, 56126 Pisa, Italy
Simone Montangero  Dipartimento di Fisica e Astronomia “G. Galilei”,
Università degli Studi di Padova, 35131 Padova, Italy  INFN, Sezione di
Padova, 35131 Padova, Italy  Elisabetta Paladino  CNR-IMM, UoS Università,
95123 Catania, Italy  Dipartimento di Fisica e Astronomia “Ettore Majorana”,
Universitá di Catania, 95123 Catania, Italy  INFN Sezione di Catania, 95123
Catania, Italy  Giuseppe Falci  CNR-IMM, UoS Università, 95123 Catania, Italy
Dipartimento di Fisica e Astronomia “Ettore Majorana”, Universitá di Catania,
95123 Catania, Italy  INFN Sezione di Catania, 95123 Catania, Italy

###### Abstract

Ultrastrong coupling may allow faster operations for the development of
quantum technologies at the expenses of increased sensitivity to new kind of
intrinsic errors. We study state transfer in superconducting circuit QED
architectures in the ultrastrong coupling regime. Using optimal control
methods we find a protocol resilient to the main source of errors, coming from
the interplay of the dynamical Casimir effect with cavity losses.

##  1 Introduction: circuit-QED and adiabatic state transfer

Circuit-QED solid-state systems [22] made of artificial atoms (AA) and
resonator modes [15] are paradigm models for studying fundamental physics from
measurement theory [22] to quantum thermodynamics [5] and quantum
communication [2] besides being one of the most promising platforms for
quantum hardware [19].

Recently solid-state ultrastrongly coupled (USC) AA-cavity systems have been
fabricated [10, 11] where the coupling constants $g_{i}$ are comparable to the
natural fequencies of the AAs ($\epsilon^{i}$) and of the cavity
($\omega_{c}$). These structures may in principle implement ultrafast quantum
operations. The USC regime exhibits new physical effects of great fundamental
interest but detrimental for quantum processing as the highly entangled nature
of the eigenstates dressed by virtual photons [10, 11, 9] and the triggering
of photon pairs by the dynamical Casimir effect (DCE) when coupling constants
are time dependent. Multiphoton effects deteriorate the fidelity of quantum
operations [3] in USC architectures even in absence of decoherence.

###  1.1 Basic equations

To overcome this problem a communication channel implemented by an adiabatic
protocol similar to STIRAP [21] has been proposed [20]. The system of two
qubits (eigenstates
$|{\sigma^{i}}\rangle\in\\{|{g^{i}}\rangle,|{e^{i}}\rangle\\}$ for $i=1,2$),
coupled to a single resonator mode is described by the Rabi Hamiltonian [15]

$H_{\mathrm{R}}=\omega_{\mathrm{c}}a^{\dagger}a+\sum_{i=1,2}\epsilon^{i}\sigma_{+}^{i}\sigma^{i}_{-}+\sum_{i=1,2}g_{i}\left(a+a^{\dagger}\right)\left(\sigma^{i}_{-}+\sigma^{i}_{+}\right),$
(1)

where $a$ ($a^{\dagger}$) is the annihilation (creation) operator of the
resonator mode satisfying $[a,a^{\dagger}]=1$,
$\sigma^{i}_{+}=|{e^{i}}\rangle\langle{g^{i}}|$ and
$\sigma^{i}_{-}=|{g^{i}}\rangle\langle{e^{i}}|$ are the qubit rising and
lowering operators. We consider resonant subsystems,
$\omega_{\mathrm{c}}=\epsilon^{1}=\epsilon^{2}$. The Hilbert space is spanned
by the factorized basis $\\{|{n\sigma^{2}\sigma^{1}}\rangle\\}$, where
$|{n}\rangle$ are the oscillator’s number eigenstates.

The Rabi Hamiltonian is adopted when the couplings $g_{i}$ are large enough to
overcome the decoherence rates of the qubits ($\gamma^{i}$) and of the cavity
($\kappa$). If in addition $g_{i}\ll\omega_{\mathrm{c}}$ the Hamiltonian
conserves approximately the number of excitation as described by the rotating
wave approximation (RWA). In the ultrastrong USC regime, where
$0.1\omega_{\mathrm{c}}\lesssim g_{i}\lesssim\omega_{\mathrm{c}}$, this is no
longer true and the full Rabi Hamiltonian $H_{\mathrm{R}}$ has to be taken
into account.

We consider time-dependent couplings $g_{i}(t)$. In the regime where the RWA
holds a STIRAP-like process implemented by turning on $g_{2}(t)$ is before
$g_{1}(t)$: yields the state transfer [20] $|{0}\rangle|{g^{2}}\rangle|{\alpha
g^{1}+\beta
e^{1}}\rangle=|{\psi_{\mathrm{initial}}}\rangle\to|{0}\rangle|{\alpha
g^{2}+\beta e^{2}}\rangle|{g^{1}}\rangle=|{\psi_{\mathrm{target}}}\rangle$. We
seek the performance of this protocol in the USC regime, allowing in principle
much faster operations. The figure of merit is the transfer efficiency

$\mathcal{F}=\left|\langle{\psi_{\mathrm{target}}}|{\psi_{\mathrm{final}}}\rangle\right|^{2},$
(2)

where $|{\psi_{\mathrm{final}}}\rangle$ is the state of the system at the end
of the process carried with the Hamiltonian Eq. (1). As in ref. [20],
$|{\psi_{\mathrm{final}}}\rangle$ is obtained by solving the Schödinger
equation
$\mathrm{i}\partial_{t}|{\psi(t)}\rangle=\left(H_{\mathrm{R}}(t)-\frac{\mathrm{i}}{2}\kappa
a^{\dagger}a\right)|{\psi(t)}\rangle$ with
$|{\psi(-\infty)}\rangle=|{\psi_{\mathrm{initial}}}\rangle$, the extra non-
Hermitian term describing cavity losses.

Fig. 1(a) reports the transfer efficiency as a function of the inverse speed
$(\omega_{\mathrm{c}}T)^{-1}$ and the maximal coupling
$g_{0}/\omega_{\mathrm{c}}$ for Gaussian time dependence. We assume a cavity
decay $\kappa=0.005\omega_{\mathrm{c}}$, which is a large figure, compensating
the oversimplified description of decoherence sources [16, 18, 17]. The red
lines refer to what could be obtained if terms non conserving the number of
excitations were dropped, i.e. the RWA. For the Rabi model the efficiency is
still remarkably large even in the presence of cavity losses up to values
$g\sim 0.3\,\omega_{\mathrm{c}}$. Here operations are already much faster than
for standard circuit-QED architectures where $g\sim
10^{-2}\,\omega_{\mathrm{c}}$ and the RWA is applicable. The efficiency is
smaller than with the RWA in what could suggesting that an important loss
mechanism are virtual and DCE photons appearing during the protocol, which are
irreversibly lost by cavity decay.

Figure 1: (a) Transfer efficiency $\mathcal{F}$ for Gaussian pulses
$g_{1}(t)=g_{0}\mathrm{e}^{-\left[\left(t-\tau\right)/T\right]^{2}}$ and
$g_{2}(t)=g_{0}\mathrm{e}^{-\left[\left(t+\tau\right)/T\right]^{2}}$ and
$\kappa=0.005\omega_{\mathrm{c}}$. The red lines represent constant
$\mathcal{F}$ lines in the RWA where ideal STIRAP is implemented. The point
$(\omega_{\mathrm{c}}T)^{-1}=0.04$, $g_{0}/\omega_{\mathrm{c}}=0.3$ (cyan x)
is analyzed in detail in the other panels: (b) shows the optimal shape of the
couplings found by QOC; (c) shows the population histories for Gaussian pulses
($\mathcal{F}\simeq 0.95$); (d) shows the population histories obtained with
QOC ($\mathcal{F}\simeq 0.99$). Notice the different time scales in panels (c)
and (d) showing that the QOC protocol is faster by a factor of $\sim 3$.

##  2 Results by optimal control

In order to improve on the results of [20] in what follows we investigate the
optimal time-dependence of $g_{i}(t)$ using the Quantum Optimal Control (QOC)
[14, 1] tools developed in [13].

We consider the point parameters $(\omega_{\mathrm{c}}T)^{-1}=0.04$,
$g_{0}/\omega_{\mathrm{c}}=0.3$ of Fig. 1(a). The optimized
$g_{i}^{\mathrm{opt}}(t)$ are step functions shown in Fig. 1(b). The bottom
panels of Fig. 1 report the evolution of the populations for Gaussian pulses
of width $T$ (c) and for the QOC solution $g_{i}^{\mathrm{opt}}(t)$ (d). For
the former protocol the duration is $\sim 3T$ and $\mathcal{F}\simeq 0.95$
while the optimized pulses $g_{i}^{\mathrm{opt}}$ allow for $\mathcal{F}\simeq
0.99$ (even slightly better than RWA) in a time interval $T$, thus being also
$\sim 3$ times faster. Notice that the optimized pulses (Fig. 1(c)) are
counter-intuitively ordered thus this process is still STIRAP-like, with pulse
shapes performing better. For the QOC case it is apparent that the mode is
more populated during the protocol (see Fig. 1(d)) but this happens for a
shorter time so the impact of losses is reduced.

##  3 Conclusions

We have shown that QOC techniques can improve state transfer in a two-qubit
circuit-QED architecture in the USC regime. We found optimized
$g_{i}^{\mathrm{opt}}(t)$ yielding larger efficiency with a time duration
shorter than for Gaussian pulses with the same maximal strength $g_{0}$. It is
likely that QOC, machine learning techniques [13, 4] or superadiabatic driving
[12] could further improve the transfer efficiency of such operations
exploiting also modulation of detunings [8, 6, 7].

##  4 Acknowledgments

This work was supported by the QuantERA grant SiUCs (Grant No.731473), and by
University of Catania, Piano Incentivi Ricerca di Ateneo 2020-22, progtto
Q-ICT.

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# Using Machine Learning to Automate Mammogram Images Analysis

Xuejiao Tang1, Liuhua Zhang2, Wenbin Zhang3, Xin Huang3
Vasileios Iosifidis1, Zhen Liu4, Mingli Zhang5, Enza Messina6 and Ji Zhang7
1Leibniz University Hannover, Germany 2Memorial University of Newfoundland,
Canada
3University of Maryland, Baltimore County, USA 4Guangdong Pharmaceutical
University, China
5McGill University, Canada 6University of Milano-Bicocca, Italy 7University of
Southern Queensland, Australia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Breast cancer is the second leading cause of cancer-related death after lung
cancer in women. Early detection of breast cancer in X-ray mammography is
believed to have effectively reduced the mortality rate since 1989. However, a
relatively high false positive rate and a low specificity in mammography
technology still exist. In this work, a computer-aided automatic mammogram
analysis system is proposed to process the mammogram images and automatically
discriminate them as either normal or cancerous, consisting of three
consecutive image processing, feature selection, and image classification
stages. In designing the system, the discrete wavelet transforms (Daubechies
2, Daubechies 4, and Biorthogonal 6.8) and the Fourier cosine transform were
first used to parse the mammogram images and extract statistical features.
Then, an entropy-based feature selection method was implemented to reduce the
number of features. Finally, different pattern recognition methods (including
the Back-propagation Network, the Linear Discriminant Analysis, and the Naive
Bayes Classifier) and a voting classification scheme were employed. The
performance of each classification strategy was evaluated for sensitivity,
specificity, and accuracy and for general performance using the Receiver
Operating Curve. Our method is validated on the dataset from the Eastern
Health in Newfoundland and Labrador of Canada. The experimental results
demonstrated that the proposed automatic mammogram analysis system could
effectively improve the classification performances.

###### Index Terms:

Breast cancer, automated diagnostic system.

##  I Introduction

Breast cancer is the most commonly diagnosed form of cancer in women and the
second-leading cause of cancer-related death after lung cancer [1]. Statistics
from the American Cancer Society indicate that approximately 232,670 (29% of
all cancer cases) American women will be diagnosed with breast cancer, and an
estimated 40,000 (15% of all cancer cases) women will die of it in 2014 [2].
In other words, 637 American women will be diagnosed with breast cancer, and
109 women will die of it every day. Similar statistics were also found in
Canada, where approximately 23,800 (26%) women were diagnosed with breast
cancer, and 5,000 (14%) died from it in 2013 [3]. Under this circumstance,
detection and diagnosis of breast cancer has already drawn a great deal of
attention from the medical world.

Studies show that early detection, diagnosis and therapy is particularly
important to prolong lives and treat cancers [4]. If breast cancer is found
early, the five-year survival rate of patients in stage 1 could reach 90% with
effective treatment. Medical imaging technology is one of the main methods for
breast cancer detection. Commonly used medical imaging technologies include
X-ray mammography, Computer Tomography (CT), ultrasound and Magnetic Resonance
Imaging (MRI), Positron Emission Tomography (PET), and Single-Photon Emission
Computed Tomography (SPECT). Among these technologies, mammography achieves
the best results in early detection of asymptomatic breast cancer and is one
of the least expensive ones. For this reason, it has become the principal
method of breast cancer detection in clinical practice, and one of the most
effective ways for general breast cancer survey.

Masses and calcifications (including marcocalcifications and
microcalcifications) are the most common and basic symptoms of breast cancer.
Masses in mammography can be recognized as a local, high-contrast area, but
the value of contrast is not unique. It changes when imaging conditions, sizes
and backgrounds change. The X-ray absorption rates of masses are very close to
dense glandular tissue in breast and other dense tissues. In addition, the
boundaries of masses are always mixed with background structures. Therefore,
microcalcification detection remains one of the most popular topics in medical
image processing research [5].

Modern equipment has improved the technical aspects of mammography. However,
there still exist a relatively high false positive rate and a low specificity,
due to fundamental physical limitations such as unobvious lesions, as well as
controllable factors like radiologists’ inexperience in reading mammograms.
This latter issue has been addressed using double reading, where two
radiologists make their own judgments independently based on the same
mammogram, and then combine and discuss both opinions. However, the solution
is expensive and highly relying on radiologists’ experience. To improve the
accuracy of reading mammograms, interest in Computer-Aided Diagnosis (CADx)
solutions has emerged [4].

In this paper, we focus on breast cancer detection using a computer-aided
automatic mammogram analysis system to improve the accuracy of diagnosis. In
proposed analysis system, entropy-based method was implemented for feature
selection as well as different pattern recognition methods, including the
Back-propagation (BP) Network, the Linear Discriminant Analysis (LDA), the
Naive Bayes (NB) Classifier, and voting schemes were employed for image
classification.

The rest of the paper is organized as follows. Section II presents background
regarding breast cancer detection and relevant work. Section III proposes the
automatic mammogram images analysis system. Subsequently, the experimental
results are presented and analyzed in Section IV. Finally, Section V draws the
conclusion and discusses lines of future work.

##  II RELATED WORK

Calcification detection and classification has been an important research
target of the automatic mammogram analysis systems. A wide variety of
approaches have so far been developed to improve the detection performance.
However, it’s still challenging for the following reasons: 1)
microcalcifications occur in various sizes, shapes, and distributions; 2)
microcalcifications have low contrast in the region of interest (ROI); 3)
dense tissue and/or skin thickness make suspicious lesion areas difficult to
detect (especially in young women); 4) the dense tissue is easily
misunderstood as microcalcification, which results in high false positive
rates among most existing algorithms.

Feature selection is crucial for detection and classification. A number of
efforts have been devoted to employing numerous different features in the
application of mammogram analysis. For example, the geometrical and
statistical features in [17, 18] are used for classification. But these
methods depend on the image translation, scaling and rotation heavily, which
results in misclassification. Consequently, these approaches can be improved
by using a robust factor, such as Fourier transform, which is certainly
translation invariant [19]. In the data time-frequency analysis, the Fourier
transform is traditionally applied, which is a global transform between the
time and frequency domain. Therefore, the Fourier transform cannot express the
local properties of signals in the time and frequency domains simultaneously.
However, these local properties are the key characteristics of non-stationary
signals in some circumstances. To analyze and process non-stationary signals,
wavelet transforms is one of the methods to retain important temporal
information in the frequency domain or partial frequency information in the
time domain. In this work, both the Fourier transform and wavelet transforms
are applied at the data transform stage to extract the features. However, not
all of the extracted features benefit classification performance. For a
feature to be useful in classification, it should be closely and uniquely
associated with a certain class [13]. Ideally, the feature will correlate with
the desired class independent of the presence of other classes. If these
conditions are met, the feature reduction (selection) problem can be addressed
by measuring the correlation with that class then establishing a pass
threshold. The pass threshold eliminates features that correlate poorly. There
are two common approaches used to measure the correlation between two random
variables, in this case between feature and class [14]. The first is the
linear correlation, where the variation in a feature value is compared to the
variation in class value. The second approach and the one adopted for our
study is Information Gain, a concept based on the reduction of entropy in the
dataset.

A target range for the number of features was determined from the work of, Lei
and Huan [15], they proposed a fast correlation based filter approach and
conducted an efficient way of analyzing feature redundancy. Their new feature
selection algorithm was implemented and evaluated through extensive
experiments comparing with other related feature selection algorithms based on
ten different kinds of feature types. The number of features ranged from 57 to
650, and the sample size of feature types ranged from 32 to 9338. At the end
of the experiment, they recorded the running time of the proposed system and
the number of features selected for each algorithm. The results showed that
the average selected number of features was 15 for the five compared feature
selection algorithms, and the selected features could lead to classification
accuracy to around 89%. In this research, we chose a threshold of information
gain which could lead to around 15 features left. Entropy is a measure of the
uncertainty of a random variable [15].

##  III System Design

The proposed automatic mammogram analysis system comprised of three
consecutive stages, including the image processing stage, the feature
selection stage and the image classification stage. Fig. 1 visualizes the
framework of the proposed system with each components being detailed in the
following sections.

Figure 1: Framework of automatic mammogram analysis system.

###  III-A Image Processing

In the first image processing stage, a set of scalar features were extracted
from an original image. This stage consisted of two steps: image pre-
processing and data transform (including wavelet and Fourier transforms). In
the image pre-processing step, the original digitized mammogram image is
flipped, de-noised, and scaled to a common maximum value. In the data
transform step, the normalized images are decomposed by three wavelet
transforms with different bases (Daubechies db2, Daubechies db4, and
Biorthogonal bior6.8) and the Fourier transform separately.

Multiple levels of decomposition were used, and four images are produced at
each level of the decomposition. Finally, four statistic features, including
the mean, standard deviation, skewness and kurtosis of the image intensities,
were calculated.

####  III-A1 Mammogram Image Pre-processing

for the automatic mammogram analysis system, the original images are different
in size and directions. Furthermore, artefacts and noise may also exist in
some mammograms, which would generate wrong or poor analysis result. Thus,
several mammogram pre-processing steps were implemented to regularize the
appearance of the images, and remove unnecessary artefacts and noise. Based on
the studies [20], the steps taken in this work are as below:

Figure 2: An example of MLO view mammogram: A.Right side; B.Left side; C.the
image after orientation matching of A.

  1. i)

Orientation Matching: our study only involved the MLO mammogram presentation.
In these, the right and left breasts point to the opposite sides in the
mammogram image. Therefore, flipping one of the breasts to the same direction
as the other one ensures that all images pointed in the same direction,
preventing changes in the wavelet transform coefficients due only to the
directionality change between right and left images. The sharp edge between
the tissue and the dark background is a major feature in all images that
affects this change. As shown in Fig. 2, the intensity of right breast images
falls from left to right across this edge, while it rises in left breast
images. This would change the sign of the calculated wavelet coefficient. Fig.
2 shows the result of orientation matching of an example of Medial Lateral
Oblique (MLO) view mammogram. Fig. 2 A and B respectively show the right and
left breast images of a patient with tiny microcalcifications in her breast
tissue. Fig. 2 C shows the reflected image of orientation matching of the
right breast.

Figure 3: A. Mammogram image before background thresholding; B. The
thresholded binary image used to mask the original image.

  2. ii)

Background Thresholding: signal outside the tissue is non-informative and was
removed by binary masking. Concretely, a threshold is set to create binary
images. Pixels with lower intensity value than the threshold are set to zero.
A satisfactory threshold can remove all irrelevant information in the
background pixels, and leave foreground objects unaltered. One of the most
commonly used method to choose the threshold is Otsu’s Method [12], which
assumes that the image to be thresholded contains two classes of pixels or bi-
modal histogram (e.g. foreground and background). The method then calculates
the optimum threshold separating those two classes so that their combined
spread (intra-class variance) is minimal [12]. It also assumes that the
foreground and background intensities are normally distributed, and it chooses
the threshold level which minimizes the segmentation error between the two
regions. The attenuation of x-rays passing through the tissue affects the
intensity in the images, and is influenced by the thickness and density of the
tissue. Therefore, tissue pixels which fall below the conservative threshold
are predominantly from the edges of the tissue region where the breast tissue
is thin and uncompressed. While a few pixel layers may be removed by this
method, it was deemed acceptable as any pathology that exists this close to
the surface of a patient’s skin should be readily detectable by conventional
examination without the aid of mammography. In this work, the binary
thresholding, which sets all pixels below a threshold, was set to an intensity
of zero and all pixels above the threshold to an intensity of one (see Fig.
3). The output image of the process is the pixel-by-pixel product of the
binary mask image and the original image. In this way, all background pixels
of the output image are set to zero of intensity, while all foreground pixels
are unaffected.

Figure 4: Mammogram image before A and after B intensity matching.

  3. iii)

Intensity Matching: intensity matching is the last pre-processing step applied
to the images before they are ready for data transforms. In this step, all
mammograms are linearly scaled to an intensity of 0.0 to 1.0. This intensity
matching process can be defined by

$img\\_out=\frac{img\\_in}{max(img\\_in)}$ (1)

where $img\\_in$ is the input image following the background thresholding
step, and $img\\_out$ is the intensity-matched image whose pixel intensities
range from zero to one. This step ensures the uniformity across all different
mammogram images, because their pixel intensities ranges could differ with
machines settings. It can be seen in Fig. 4 that there is tiny difference
before and after the intensity matching procedure. The broader spread in
intensities would increase the variations in different tissue types and
densities. (the maximum relative intensity prior to normalization was 0.92).

####  III-A2 Data Transforms

once the images are pre-processed to minimize the differences between images
that were not related to differences in the physical composition of the breast
tissue, the wavelet and Fourier transforms were performed on the images. The
images were all sampled to 1024×1024 pixels, which would allow maximum 10
levels of decomposition, since dyadic sampling reduces the dimensions by a
factor of two in each direction after each pass. In this work, only eight
levels of decomposition were used. Because the final two levels would consist
of four-pixel and one-pixel images, respectively, which are basically useless
for mammogram analysis, compared to the size of the entire breast. As a
result, these levels are omitted from the wavelet analysis to speed
calculation.

  1. i)

Choice of Transform Methods: Fig. 5 shows the original mammogram and its four
detail views obtained at the first decomposition level when the Db4 wavelet
basis is used. It is shown that the wavelet maps have a lower resolution than
the original image. Each view is sensitive to different features in the image.
For example, the horizontal detail detects vertical changes in intensity, the
vertical detail detects horizontal changes in intensity, the diagonal detail
responds when the intensity is varying in both directions, and the
approximation image is a low resolution version of the original image used as
an input to the next coarser level of the decomposition. Fig. 6 shows the
Fourier transform view of the original mammogram. Compared with the wavelet
maps, it can be seen that the wavelet transform provides multi-resolution
decomposition, which means the wavelet maps at different levels reflect the
image features of different sizes. Furthermore, spatial information is
partially conserved. The wavelet maps in Fig. 5 show the spatial distribution
of information at particular size scales; in contrast, the Fourier transform
would lose the spatial information and simply produce a map of the relative
contributions of different frequencies over the entire image. This spatial
information is useful for finding localized structures, such as
microcalcifications and masses. These structures remain localized after the
wavelet transform is applied, and they can then be distinguished from a more
homogeneous background.

Figure 5: First level db4 wavelet decomposition: A. Original mammography
image; B. Approximation view; C. Horizontal detail view; D. Vertical detail
view; E. Diagonal view. Figure 6: The Fourier transform view of the mammogram
in Fig. 5.

  2. ii)

Choice of Measurement: in our experiment, four statistical features were
extracted: mean intensity, standard deviation, skewness and kurtosis of the
pixel intensities. Then, the mammogram analysis system uses some of these
features to classify mammogram images as being normal or cancerous.

     * •

Mean: the mean $\mu$ in this paper is obtained by calculating the average
pixel value of the tissue region in the mammogram image. The equation is given
by

$\mu=\frac{1}{N}\sum_{i,j}I(i,j)$ (2)

where $I(i,j)$ the pixel value at point $(i,j)$ of the mammogram image. N is
the number of pixels in the tissue region of the image. The mean feature
measures the average value of each detail views at different decomposition
levels. Microcalcifications are usually tiny and bright. Compared with normal
samples, microcalcifications have a slightly higher intensity in the high
resolution maps. While masses are usually different in sizes and shapes, they
could range from millimetres to several centimetres in width. Therefore,
masses cannot be extracted from the background tissue through single scale or
wavelet basis. However, masses are located in one region of tissue, and they
are usually brighter than normal tissue. As a result, a slightly larger mean
intensity can be measured through a wavelet basis, especially when different
scales are used to detect masses.

     * •

Standard Deviation: the standard deviation $\sigma$, the estimate of the mean
square deviation of grey pixel values, describes the dispersion of a local
region. It is defined as

$\sigma=\sqrt{\frac{1}{N}\sum_{i,j}[I(i,j)-\mu]^{2}}$ (3)

It measures the variability in the brightness of the image over the tissue
region. The value of the standard deviation would increase in the high spatial
resolution levels of the wavelet map images that contain microcalcifications
or masses, because they are brighter than normal parts of mammogram images.

     * •

Skewness: the third statistic feature measured from each wavelet map image is
the skewness of the pixel intensities, which measures the degree of asymmetry.
The skewness of a distribution of values is defined as the third central
moment of the distribution, normalized by the cube of the standard deviation.
It is given by

$S=\frac{1}{N}\sum_{i,j}[\frac{I(i,j)-\mu}{\sigma}]^{3}$ (4)

When a distribution has a larger right tail, then it shows a positive
skewness. Even there is no significantly difference in the mean value or
standard deviation, the skewness still changes because it is sensitive to the
addition of a small number of unusually small or large values on a
distribution.

     * •

Kurtosis: the fourth statistic measured from the wavelet maps is the kurtosis
of the pixel intensities. The kurtosis of a distribution of values is defined
as the fourth central moment of the distribution, normalized by the fourth
power of the standard deviation of the distribution. The kurtosis K is given
by

$K=\frac{1}{N}\sum_{i,j}[\frac{I(i,j)-\mu}{\sigma}]^{4}$ (5)

Kurtosis measures the narrowness of the central peak of a distribution
compared with the size of the distribution’s tails. A distribution with a
narrow peak and tails that drop off slowly has a large kurtosis compared with
a distribution with a relatively wide peak but suppressed tails. The kurtosis
and standard deviation of a distribution may be similar, but kurtosis is more
sensitive to points distant from the mean than the standard deviation. Because
of this, kurtosis is sensitive to the presence of microcalcifications and
masses. It will rise when the number of unusual bright pixels increases in a
wavelet map.

###  III-B Feature Selection

Since a large number of potential classification features are generated from
each mammogram image, a selection process is needed to choose those features
that are most effective at differentiating between normal and cancerous
images. Specifically, there are four parameters measured from each wavelet
map, with four wavelet maps per level and eight levels of decomposition. Thus,
16 features could be generated form each level of decomposition. To eliminate
some of these, it was noted in N. Terki, etc. [16] that peak signal to noise
ratio (PSNR) improved when the level of decomposition increases, and the image
quality was better from third level of decomposition. Therefore, level 3 to
level 8 of decomposition of the proposed three wavelet transform methods were
applied in this work. In this case, 96 features would be generated from each
of three wavelet transforms based on the 6 levels of wavelet decomposition.

Then, the generated 96 features from the wavelet transform were combined with
the 6 features extracted from the Fourier transform. In other words, 3
different feature sets were created, and each of the feature sets contains
features from one wavelet transform and the Fourier transform.

We adopted entropy-based feature selection in our work. The entropy of a
variable X is defined as

$H(X)=-\sum_{i}P(x_{i})\log_{2}(P(x_{i}))$ (6)

and the entropy of X after observing values of another variable Y is defined
as

$H(X|Y)=-\sum_{j}P(y_{j})\sum_{i}P(x_{i}|y_{j})\log_{2}(P(x_{i}|y_{j}))$ (7)

where $P(x_{i})$ the prior probabilities for all values of X, and
$P(x_{i}|y_{j})$ is the posterior probabilities of X given the values of Y.
The amount by which the entropy of X decreases reflects additional information
about X provided by Y, and is called information gain [11], given by

$IG(X|Y)=H(X)-H(X|Y)$ (8)

If we have $IG(X|Y)>IG(Z|Y)$, it means a feature Y is regarded more correlated
to feature X than to feature Z.

The entropy with the feature selection algorithm was implemented by the
following steps:

  * 1)

Order features based on decreasing entropy values using (6), and build a link
list for all features;

  * 2)

Calculate the entropy of each feature in the link list related to the
classification results using (7);

  * 3)

Calculate the information gain of each feature using (8) based on its two
entropies obtained from step 1 and 2;

  * 4)

Compare each feature’s information gain with the next feature, and move the
larger one ahead till the end of the link list;

  * 5)

Select the features with the information gain larger than the threshold set in
the program.

In order to select the most effective features for differentiating between
normal and cancerous mammogram images, less significant features are removed
by entropy-based algorithm. This selection was achieved by sorting and
selecting features with higher information gain values. The experimental
results suggested that the information gain of features from the db4-Fourier
transform was higher than that of features from the bior6.8-Fourier transform,
and the information gain of features from the db2-Fourier transform was the
lowest among the three feature sets. In the features from the db2, db4,
bior6.8, and Fourier transforms, we selected the top 12 features (the optimal
features) with their information gain values higher than 0.74.

###  III-C Image Classification

In the final image classification stage, mammogram images were determined as
either normal or cancerous based on the selected features. In proposed system,
three classifiers (Linear Discriminant Analysis, Back-propagation Network, and
Naive Bayes Classifier) were trained and tested. Moreover, combining the above
mentioned classifiers (LDA, BP, NB), a voting classification scheme is further
proposed for the mammogram analysis system in this research. In the voting
classification scheme, where “1” represents cancerous mammograms from a
classifier, and “0” represents normal mammograms. When classifying a mammogram
image, the voting classification decision is made by taking opinions of the
majority of the three classifiers.

  1. 1)

Linear Discriminant Analysis: the objective of LDA is to make the data points
of different classes as far apart from each other as possible. In addition, it
also aims at making the data points from the same class as close as possible.
It can be implemented as:

     * •

Constructing a matrix of feature vectors: all feature samples were read in as
a matrix $X=\\{x_{1},x_{2},...,x_{n}\\}$. Each feature data $x_{i}$ was
regarded as a node i, and in the same way, another feature data $x_{j}$ was
regarded as a node j. Node i and j were connected with a line if $x_{i}$ and
$x_{j}$ were close, and they belonged to the same class.

     * •

Calculating scatter matrices: in this step, between-class scatter matrix
$S_{b}$ and within-class matrix $S_{w}$ were calculated using (9) (10).

$\displaystyle S_{b}=\frac{1}{n}\sum_{i=1}^{c}(m^{i}-m)(m^{i}-m)^{T}$ (9)
$\displaystyle
S_{w}=\frac{1}{n}\sum_{i=1}^{c}(\sum_{j=1}^{n_{i}}(x_{j}^{i}-m^{i})(x_{j}^{i}-m^{i})^{T})$
(10)

where $m$ is the total sample mean vector, $n_{i}$ is the number of samples in
class $C_{i}$, $m^{i}$ is the average vector associated to $C_{i}$ class,
$x_{j}^{i}$ is the $j-th$ sample vector in the $C_{i}-th$ class. $S_{b}$ and
$S_{w}$ are named between-class scatter matrix and within-class matrix,
respectively.

     * •

LDA projection: data points $x_{i}$ were projected into the LDA subspace so
that the matrix $S_{w}$ was non-singular. The transformation matrix of LDA was
presented here as $W_{LDA}$. After projection, $S_{b}$ and $S_{w}$ became

$\displaystyle\overline{S_{b}}=W_{LDA}^{T}S_{b}W_{LDA},$ (11)
$\displaystyle\overline{S_{w}}=W_{LDA}^{T}S_{w}W_{LDA}$ (12)

     * •

Computing the projection matrices: after adding the Lagrange multiplier and
some derivation steps, the following function was achieved. It is also called
the Fisher Linear Discrimination.

$S_{w}^{-1}S_{b}W=\lambda W$ (13)

It can be seen that $W$ is the eigenvector of matrix $S_{w}^{-1}S_{b}$.

     * •

Linear embedding: with the substitution of eigenvector, $W_{best}$ is easy to
find by the following equation:

$W=S_{w}^{-1}(\mu_{1}-\mu_{2})$ (14)

where $\mu$ is the mean value (central point) of samples in each class.

$\mu_{i}=\frac{1}{n_{i}}\sum_{x\in C}x$ (15)

  2. 2)

Back-propagation Network: the aim of the algorithm was to classify mammograms
into two categories: cancerous or normal. Because the input features are 14
dimensional, and there are two kinds of mammograms to be classified, the
construction of the BP network can be defined as “14-15, 2”. It means that
there are 14 nodes in the input layer, 15 nodes in the hidden layer, and 2
nodes in the output layer. Furthermore, after random sorting of 670
mammograms, 520 of them were randomly selected as the training dataset, the
remaining 150 were chosen to test the classification performance of the BP
network.

     * •

Network initialization: according to the input and desired output values (X
and Y) of the network, we can set n nodes in the input layer, l nodes in the
hidden layer, and m nodes in the output layer. The weight values ($w_{ij}$ and
$w_{jk}$), the threshold value $a$ in the hidden layer, the threshold value
$b$ in the output layer, the learning speed, and the activation functions
should also be initialized.

     * •

Calculation of the hidden layer output: this output $H$ can be achieved
through $X$, $w_{ij}$, and $a$.

$H_{j}=f(\sum_{i=1}^{n}w_{ij}x_{i}-a_{j}),\quad j=1,2,...l$ (16)

here, $l$ is the number of nodes in the hidden layer, $f$ is an activation
function. In this work, the activation function is chosen as

$f(x)=\frac{1}{1+e^{x}}$ (17)

     * •

Calculation of the output layer output: O is determined through $H$,$w_{jk}$,
and $b$.

$O_{k}=\sum_{j=1}^{l}H_{j}w_{jk}-b_{k},\quad k=1,2,...m$ (18)

Calculation of error. Based on the output O and the estimated output Y, we can
obtain the prediction error $e$. $e_{k}=Y_{k}-O_{k}$.

     * •

Weight update: the values can be updated using the following equations:

$\displaystyle w_{ij}=w_{ij}+\eta
H_{j}(1-H_{j})x(l)\sum_{k=1}^{m}w_{jk}e_{k},j=1,2,...l$ (19) $\displaystyle
w_{jk}=w_{jk}+\eta H_{j}e_{k},\quad j=1,2,...,l,k=1,2,...m$ (20)

in which, $\eta$ is the learning speed.

     * •

Threshold update: the threshold value can be updated using the following
equations:

$\displaystyle a_{i}=a_{i}+\eta H_{j}(1-H_{j})\sum_{k=1}^{m}w_{jk}e_{k},\quad
j=1,2,...l$ (21) $\displaystyle b_{k}=b_{k}+e_{k},\quad k=1,2,...m$ (22)

     * •

If the iteration is not over, the algorithm goes back to the second step.

  3. 3)

Naive Bayes Classifier: suppose that there are m classes, $C$ =
$\\{C_{1},C_{2},...,C_{m}\\}$. $A=\\{A_{1},A_{2},...,A_{n}\\}$ are the
features for one dataset. Given an instance, its feature is $X$ =
$\\{X_{1},X_{2},...,X_{n}\\}$, then the posterior probability that instance
belongs to a class $C_{i}$ is $P$ = $P(C_{i})$ = $(X|C_{i})$. The Naive Bayes
Classifier can be represented as $C(X)$ =
$argmax(P(C_{i})P(X|C_{i})),{C_{i}\in C}$. It indicates that the prediction
accuracy reaches the maximum value when instance $X$ has the largest posterior
probability. However, the posterior probability is difficult to calculate.
Therefore, the “Naive Bayes hypothesis”, which assumes all features $A_{i}$
are independent from each other, is introduced to the Naive Bayes Classifier.
Thus, $P(A_{i}|C,A_{j})=P(A_{i}|C)$, $\forall A_{i},A_{j},P(C)>0$.

The Naive Bayes classification algorithm can independently learn either the
conditional probability of each feature $A_{i}$ in the class $C(P(A_{i}|C))$,
or the probability of each feature $A_{i}$. Replaced with a normalization
factor “$a$”, the posterior probability becomes

$\begin{split}P(C=c|A_{1}=a_{1}...A_{n}=a_{n})=&\\\ \alpha
P(C=c)\prod\limits_{i=1}^{n}P(A_{i}|C=c)\end{split}$ (23)

According to (19), the optimal classification ($C=C_{i}$) should satisfy

$\displaystyle
P(C_{i}|<a_{1},...,a_{n}>)=\frac{P(<a_{1},...,a_{n}>C_{i})}{P(<a_{1},...,a_{n}>)}P(C_{i})$
(24) $\displaystyle
P(C_{i}|<a_{1},...,a_{n}>)>P(C_{j}|<a_{1},...,a_{n}>),j\neq i$ (25)

##  IV Experiments

In this section, we conduct experiments to evaluate the performance of our
system on mammogram image dataset. Since we adopted entropy-based feature
selection method, we evaluated and compared the performance of three unique
classifiers (LDA, BP, NB) using optimal and non-optimal features. Also, the
classification performances of the three classifiers were compared with
Receiver Operating Characteristic (ROC) curves. Furthermore, we compared the
performance of voting scheme and unique classifier scheme (LDA, BP, NB).

###  IV-A Dataset

The dataset to be analyzed in this work were a gift from the Eastern Health in
Newfoundland and Labrador of Canada. It consists of 1487 mammogram images,
which is divided into a training set with 1040 mammogram images and a test set
with 447 mammogram images. The images were all anonymous in the format of
DICOM, which is a set of standard protocols in medical image processing,
storage, printing, and transmission. They were authorized for our experiment
by the Health Research Ethics Authority (HREA) in the reference number of
11312. All DICOM mammogram images were sampled to 1024×1024 pixels and
reconfigured to PGM format.

###  IV-B Implementation Details

In this experiment, the programs of the image processing and the
classification were developed in Matlab 2010b, and the program of feature
selection was developed in Eclipse using JAVA. The computer used was based on
Windows 10, Intel(R) Core(TM) CPU i7-8700 CPU @ 3.2GHz, 16GB RAM.

###  IV-C Evaluation Metrics

The performance of a mammography screening system can be measured by two
parameters: sensitivity and specificity. Sensitivity (true positive rate) is
the proportion of the cases deemed abnormal when breast cancer is present. For
example, if 100 women do have breast cancer among 1000 screened patients but
only 90 are detected, then the sensitivity is 90/100 or 90%. Sensitivity may
depend on several factors, such as lesion size, breast tissue density, and
overallS image quality. In cancer screening protocols, sensitivity is deemed
more important than specificity, because failure to diagnose breast cancer may
result in serious health consequences for a patient. Almost fifty percent of
cases in medical malpractice relate to “false-negative mammograms” [21].
Specificity (true negative fraction) is the proportion of cases deemed normal
when breast cancer is absent. For example, if 100 cases of breast cancer are
diagnosed in a set of 1000 patients, and the screening system finds 720 cases
to be normal, the specificity is 720/900 or 80%. Although the consequences of
a false positive (diagnosing a normal patient as having breast cancer) are
less severe than missing a positive diagnosis of cancer, specificity should
also be as high as possible. False positive examinations can result in
unnecessary follow-up examinations and procedures, and may lead to significant
anxiety and concern for the patient.

###  IV-D Results

Table I summarizes the accuracy, sensitivity and specificity performances of
the three classifiers based on the selected features from db2-Fourier,
bior6.8-Fourier, db4-Fourier, and the optimal features. Features including
standard diversion, kurtosis and skewness extracted from the Fourier transform
are obtained by the proposed entropy-based feature selection algorithm. By
ranking the information gain, 12 optimal features with top information gain
value for classification are selected. The Receiver Operating Characteristics
curves were also plotted to facilitate comparison of the three classifiers as
shown in Fig. 7.

TABLE I: Classification performances of three classifiers for the training dataset Classifier | LDA | BP | NB | VOTING
---|---|---|---|---
db2- Fourier Features | Accuracy | 80.69% | 85.05% | 81.03% | 88.07%
Sensitivity | 71.18% | 83.05% | 90.06% | 90.06%
Specificity | 89.03% | 88.06% | 70.08% | 92.03%
bior6.8- Fourier Features | Accuracy | 81.01% | 86.07% | 83.03% | 89.01%
Sensitivity | 72.06% | 84.65% | 91.71% | 91.70%
Specificity | 90.32% | 89.01% | 72.15% | 90.41%
db4- Fourier Features | Accuracy | 84.03% | 89.06% | 86.02% | 89.08%
Sensitivity | 74.24% | 87.55% | 93.20% | 95.01%
Specificity | 93.06% | 92.05% | 74.07% | 95.81%
The optimal Features | Accuracy | 88.02% | 94.14% | 89.83% | 96.06%
Sensitivity | 78.26% | 90.45% | 96.21% | 96.45%
Specificity | 96.88% | 95.03% | 80.60% | 97.45%

Figure 7: ROC curves with the classifiers: A. LDA; B. BP; C. NB; D. Voting
(a), (b), (c), and (d): performances of classifiers based on db2-Fourier,
bior6.8-Fourier, db4-Fourier, and the optimal features respectively.

According to the results in Table I, classifiers achieved their highest
classification performances using the optimal features, followed by the
features selected from the db4 wavelet and Fourier transforms; whereas their
performances are the lowest using features selected from the db2 wavelet and
Fourier transforms. The specificity of the voting classifier with the optimal
features is 1.64% higher than that of the features of the db4-Fourier
transform, 7.04% higher than that of the bior6.8-Fourier transform, and 5.42%
higher than that of the db2-Fourier transform. Moreover, the specificity of
the LDA, BP, NB classifiers with optimal features is better than that of the
three feature methods, i.e., the db4-Fourier, db2-Fourier, and bior6.8-Fourier
transform.

In addition, it is also clear that the voting classification scheme
outperforms each individual base classifier. The reason that optimal features
achieve the highest performance could be the information gain of the optimal
features is the highest among the four different feature sets. In other words,
the features in the optimal feature set are more correlated to the mammogram
class than any of the other features.

What’s more, it can be found from Table I and ROC curves that the voting
scheme achieves the highest accuracy using the default parameters in the
proposed mammogram analysis system. On the other hand, the Naive Bayes
Classifier among the three single classifiers (LDA, BP, NB) achieves the
highest specificity, and the Back Propagation network achieves the highest
accuracy based on all the four feature sets. This result suggests that among
the three classifiers, the NB classifier is more sensitive to classify
cancerous mammograms, the LDA classifier gives better classification in normal
mammograms, and the BP neural network works well in both of normal and
cancerous mammograms. The voting classification scheme performs best.

##  V Conclusions

In this study, a computer-aided automatic mammogram analysis system, which
consists of image processing, feature selection and image classification, is
proposed to improve the detection performances. In conclusion, the
experimental results on mammogram image dataset from the Eastern Health in
Newfoundland and Labrador of Canada demonstrated that the proposed automatic
mammogram analysis system could effectively improve the classification
performances, especially using the voting classification scheme based on the
selected optimal features. In the future, more wavelet bases can be explored
to extract valuable features with high information gain.

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