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VOI is extracted, the autocorrelation matrix may be computed and used as a charac-
teristic vector. In the second stage, the characteristic vectors of the training tumors
will be fed into the input layer of the neural network and the learning process starts.
Also, a relatively simple feed-forward, error back-propagation artificial neural net-
work (ANN) with one hidden layer was adopted. The output of each ANN will be a
number between 0 and 1.
The coefficient matrix of each 2-D image within a 3-D US dataset will be calcu-
lated and added to an autocorrelation coefficient matrix. The pixel relation charac-
teristic of tissues may be lost by such an accumulative operation and result in a false
diagnosis. To improve the diagnostic accuracy, another pixel relation feature
extraction scheme was proposed by Chen et al. [14]. This scheme of multiple
autocorrelation matrixes (MULTI-US) explores more than one autocorrelation
matrix for keeping the coefficients of each 2-D image in the3-D volume. Because the
number of frames in a 3-D volume dataset is large, it is impossible to use all of the
coefficient matrixes. Hence, only five matrixes were used to reduce the computation
time. Also, the diagnostic model needs to be modified to fit multiple coefficient
matrixes.
In the first stage, the VOI extraction method is just like the previous scheme;
however, the autocorrelation matrix is replaced by using five autocorrelation
matrixes. Next, these five autocorrelation matrixes are fed into the neural network.
In the second stage, the diagnosis method is modified. Each matrix is fed into the
neural network and five neural network outputs will be obtained. These diagnosis
outputs need to be combined into a final diagnostic output. In their study, if more
than two diagnostic outputs were malignant, then the final diagnosis result was
malignant.
The third scheme is a combination of the preceding two schemes. That is, the
five matrixes of the second scheme are added into an accumulative matrix. The
method can not only reduce the computation time of the first scheme, but can also
avoid the problem of too many ROIs influencing the feature extraction result,
because some ROIs may contain incomplete pixel relation information on the
tumor. The hybrid scheme is called an autocorrelation matrix with five frames
(ACC5-US).
Including the predefined threshold, the multilayer feed-forward neural network
in their work [14] included 25 input nodes, 10 hidden nodes, and a single output
node. The degree of freedom (i.e., the number of connections) was 260. The training
process was stopped when the improvement of error distortion was smaller than
0.001 or when the number of training iterations was more than 20,000. The error is
computed by the absolute difference between the desired output and the actual out-
put of the neural network. Due to the small number of test cases, the cross-valida-
tion method was adopted. Because each set was used as the test set once, five sets of
parameters for neural network were used.
The accuracy of this proposed ACC5-US scheme was 90.68% (146 of 161), the
sensitivity 90.74% (49 of 54), the specificity 90.65% (97 of 107), the PPV 83.05%
(49 of 59), and the NPV 95.1% (97 of 102).
The spiculation feature is used primarily to diagnose the breast tumors using
3-D US. Ina study by Rotten et al. [28], the coronal section in a 3-D US image allows
precise demonstration of the tissue surrounding the central lesion. A converging pat
-
194 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
tern of the peripheral tissue is highly suspicious of malignancy. The converging pat-
tern is a spiculation in which alternating hypoechoic and hyperechoic lines radiate
in multiple directions from the mass into the surrounding tissue. This feature is eas-
ily seen on mammograms, and malignancy cannot be excluded based on this find-
ing. In their experiments, 91% (53 of 58) of malignant tumors and 6% (8 of 128) of
benign tumors have spiculations. Although several successful spiculation detection
techniques are available for mammograms [29–32], automatic spiculation detec-
tion on US images is not easily implemented due to the speckle noise in US images.
Moreover, a mature segmentation algorithm is needed to segment the tumor. There-
fore, Chen et al. [14] provide another way to use 3-D US. Because the multilayer
feed-forward neural network can extract higher-order statistics by adding one or
more hidden layers in their model, it has become extremely popular in terms of clas-
sification and prediction. Hence, the multilayer feed-forward neural network was
adopted for classification. The computation time is rather critical for clinical appli-
cations. This proposed algorithm uses only the middle five consecutive frames in a
3-D US dataset to quickly obtain the pixel relation information. That is, their
proposed method can be used for real-time clinical applications.
Although all of the pixel relation information of a tumor was used in the first
scheme using an autocorrelation matrix, the first scheme’s performance is worse
than that of the third scheme, in which only the pixel relation information of five
consecutive frames was adopted. This does not mean, however, that the whole pixel
relation information of a tumor is useless.
8.2.2 Solid Breast Nodules with Neural Networks
Chen et al. [27] have developed a CAD algorithm with setting-independent features
and ANNs to differentiate benign from malignant breast lesions. In their work, two
sets of breast sonograms were evaluated. The first set, containing 160 lesions, was
stored directly on the magnetic optic disks from the US system. Four different
boundaries were delineated by four persons for each lesion in the first set. The sec-
ond set comprised 111 lesions that were extracted from the hardcopy images. Seven
morphologic features were used, five of which were newly developed. A multilayer
feed-forward neural network was used as the classifier, and the reliability,
extendibility, and robustness of the proposed CAD algorithm were evaluated. The
results of the proposed algorithm were compared with those from two previous
CAD algorithms. All performance comparisons were based on paired-samples
t-tests. The proposed CAD algorithm could effectively and reliably differentiate
between benign and malignant lesions. The proposed morphologic features were
nearly setting independent and could tolerate reasonable variation in boundary
delineation.
Chen et al.’s proposed CAD algorithm was composed of three essential compo-
nents: feature extraction, feature selection, and classification. To relax the con-
straints on the system settings, the morphologic features rather than the regional
features were adopted. The potential dependence of the morphologic features on
the contour extraction process was minimized by capturing important topologic
properties of the lesions, which may not vary drastically with the delineated con
-
tour. Feature selection was necessary to alleviate dimensionality [33]. A set of essen
-
8.2 Related Works 195
tial morphologic features that yields the curve of the best performance was selected
on the basis of stepwise logistic regression [34]. Classification was accomplished
with a multilayer feed-forward neural network (MFNN) [35] on the basis of the
essential morphologic features. The advantage of the MFNN is that arbitrarily
complex convex separation surfaces can be approximated.
Seven morphologic features were extracted from each lesion to account for such
sonographic features as shape, contour, and size. Five of these morphologic fea-
tures were developed by including the number of substantial protuberances and
depressions (NSPD), lobulation index (LI), elliptic-normalized circumference
(ENC), elliptic-normalized skeleton (ENS), and long axis–to–short axis (L:S) ratio.
The other two features were clinically useful indicators [36]: depth-to-width (D:W)
ratio and size of the lesion.
•
NSPD: The spiculation [37] and irregular shape and contour [38] of a lesion
are two important sonographic features that characterize a malignant breast
lesion. The NSPD is an effective descriptor in a lesion to quantify these two
sonographic features. Consider thisgeographic analogy: A protuberance and a
depression are like a peninsula and a bay, respectively. Because protuberances
and depressions may easily result from a wobbly delineation process, only the
substantial protuberances and depressions defined by the representative con-
vex and concave points, respectively, were used to characterize a breast lesion.
•
LI: This feature was devised to characterize the size distribution of the lobes in
a lesion. A lobe is defined as the gray region enclosed by the lesion contour and
the dashed line connected by two adjacent representative concave points. The
size of the lobe is the area of the gray region. The LI can correctly characterize
a benign lesion with multiple large lobes of similar sizes. This type of benign
lesion may be misclassified as a malignant lesion with the NSPD.
•
ENC: Anfractuosity is a common morphologic characteristic of malignant
lesion boundaries that provides at least two visually appreciable geometric
features. One feature is the multiple protuberances and depressions that may
be well described with the NSPD. The other feature is the lengthened circum-
ference due to the circuitous boundaries that define the protuberances and
depressions. Because the boundary of a smaller lesion would appear to be
more winding than that of a larger lesion with the same circumference, the cir-
cumference itself is not a good descriptor with which to characterize the
anfractuosity of the lesion boundary. Alternatively, a more reasonable
approach is quantification of the anfractuosity with the percentage of circum-
ference increment relative to a lesion-dependent baseline. An ideal baseline
would be a smooth curve such that the lesion boundary would look like oscil-
lating around the curve. To quantify the anfractuosity of a lesion contour, the
circumference ratio ofthe lesionand itsequivalent ellipse is proposed, which is
termed the ENC. The equivalent ellipse of a lesion [39, 40] is an ellipse with
the same area and center of mass as those of the lesion when the interiors of the
lesion and its equivalent ellipse are both set to the same constant gray level.
Perceptually, one can see that the equivalent ellipse roughly captures the shape
of the lesion, and the lesion boundary.
196 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
•
ENS: A skeleton is an effective representation of a region [40] that is used fre-
quently in such areas as computer vision and pattern recognition. The skele-
ton is sensitive to the anfractuous property of the lesion boundary. The more
protuberances and depressions contained in the lesion boundary, the more
complex the skeleton is. Therefore, it seems to be reasonable to quantify the
shape complexity by the number of points in the skeleton. Nevertheless, the
number of skeleton points is also a function of the lesion size. Just as for the
ENC, to eliminate the size effect, it is suggested that the number of skeleton
points be normalized by the circumference of the equivalent ellipse of the
lesion, which gives the ENS.
In addition to the NSPD, LI, ENC, and ENS descriptors, which capture the con-
tour and shape characteristics, three more mathematic features are considered to
incorporate two clinically useful indicators. The first feature is the D:W ratio of the
lesion. The depth and the width of a lesion are the horizontal and vertical edge
lengths, respectively, of the minimal circumscribed rectangle of the lesion. The
larger the D:W ratio, the more likely the lesion is malignant.
Because the D:W ratio may vary with the scanning angle and the compressing
pressure, it is suggested that another quantity be used to describe the shape of the
lesion, namely, the L:S ratio. The L:S ratio is the length ratio of the major (long) axis
to the minor (short) axis of the equivalent ellipse of the lesion. Clearly, the L:S ratio
is independent of the scanning angle but may be affected by the compressing pres-
sure. The last feature is the size of the lesion (i.e., the area within the lesion bound-
ary). Clinically, the larger the breast lesion, the more likely the lesion is malignant.
Feature selection was performed in two stages for each training dataset. In the
first stage, the best NSPD value was selected from five candidate NSPD values.
Then, in the second stage, the selected NSPD value along with the other six features
were used to select the essential features that yield the best classification accuracy
for the underlying training dataset. The classifier used in the present study is an
MFNN. Once the essential features were selected by means of the logistic discrimi-
nation function for a set of training data, the training data were used to train the
MFNN to divide the training data into benign and malignant categories. The
MFNN used in this study was a two-layer feed-forward neural network with one
hidden layer. The number of inputs for the MFNN was set to be the same as the
number of essential features, and the number of neurons in the output layer was set
to 1 for the underlying two-class classification. The number of neurons in the hid-
den layer was determined through exhaustive experiments to be 2 to 10 neurons. As
a result, the number of neurons in the hidden layer was set to 2 because this number
gave the best performance for almost all cases evaluated.
The training algorithm used to train the MFNN was the widely used error
back-propagation training algorithm [35]. The training data with the essential fea-
tures were fed into the MFNN in a cyclic manner. For each datum, the estimated
output was computed on the basis of the synaptic weights determined in the previ-
ous iterations and the sigmoidal activation function. The discrepancy between the
desired output and the estimated output was back propagated to modify the synap-
tic weights until the discrepancy was within the acceptable range. Because the final
output was a number between 0 and 1, a threshold neural network (TNN) was
8.2 Related Works 197
required to assign the datum to the benign or the malignant category. In that work,
the TNN was determined on the basis of the value thatresulted in a dichotomization
with the bestclassification accuracy for the training data, or TNN varied from 0 to 1
to generate the ROC curve.
The area under ROC curve (A
z
) was 0.952 ± 0.014 for the first set, 0.982 ±
0.004 for the first set as the training set and the second set as the prediction set,
0.954 ± 0.016 for the second set as the training set and the first set as the prediction
set, and 0.950 ± 0.005 for all 271 lesions. At the 5% significance level, the perfor-
mance of the proposed CAD algorithm was shown to be extendible from one set of
US images to the other set and robust for both small and large sample sizes. More-
over, the proposed CAD algorithm was shown to outperform the two previous CAD
algorithms in terms of the A
z
value.
However, as pointed out before, the texture-based CAD is highly machine
dependent and utilizes these machine settings all the time.
As outlined earlier, this chapter proposes a five-step novel and automatic meth-
odology for breast lesion classification in 3-D US images.
8.3 Theoretical Background
8.3.1 Tsallis Entropy
Entropy is an idea born in the heart of classic thermodynamics, not as something
fundamentally intuitive, but as some fundamentally quantitative. It was defined by
an equation and has been known as Boltzman-Gibbs (BG) entropy. Later, Shannon
redefined the concept of BG entropy (now called BGS entropy) as an uncertainty
measure associated with the content of system information. This traditional form of
entropy is well known by:
()
Spp
ii
i
=−
∑
ln
(8.1)
Generically speaking, systems having statistics of the BGS type are called exten-
sive systems and have an additive property, defined as follows. Let P and Q be two
random variables, with probability densities functions P = (p
1
, ..., p
n
) and Q = (q
1
,...,
q
m
), respectively, and S be the entropy associated with P or Q.IfP and Q are inde-
pendent, under the context of the probability theory, the entropy of the composed
distribution
1
verifies the so-called additivity rule:
()()()
SPQ SP SQ∗= +
(8.2)
This traditional form of entropy is well known and for years has achieved rela-
tive success in explaining several phenomena if the effective microscopic interac-
tions are short ranged (i.e., close spatial connections) and the effective spatial
microscopic memory is short-ranged (i.e., close time connections) and the boundary
conditions are non(multi)fractal. Roughly speaking, the standard formalisms are
applicable whenever (and probably only whenever) the relevant space–time is
198 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
1. We define the composeddistribution, also calleddirect product of P =(p
1
, ..., p
n
) and Q = (q
1
,..., q
m
), as P * Q
= {p
i
q
j
}
i,j
, with 1 ≤ i ≤ n and 1 ≤ j ≤ m.
non(multi)fractal. If this is not the case, some kind of extension appears to become
necessary. We can make a complete analogy with Newtonian mechanics, when it
becomes only anapproximation (anincreasingly bad one) when the involved veloci-
ties approach that of light or the masses are as small like, for instance, the electron
mass; the standard statistical mechanics do not apply when the above requirements
[short-range microscopic interactions, short-ranged microscopic memory and
(multi)fractal boundary conditions] are not the case.
Recent developments, based on the concept of nonextensive entropy, also called
Tsallis entropy, have generated new interest in the study of Shannon entropy for
information theory [41, 42]. Tsallis entropy (or q-entropy) is a new proposal for the
generalization of BGtraditional entropyapplied to nonextensivephysical systems.
The nonextensive characteristic of Tsallis entropy has been applied through the
inclusion of a parameter q, which generates several mathematical properties. The
general equation is as follows:
()
Sp p
p
q
q
k
i
q
i
k
1
1
1
1
,,K =
−
−
=
∑
(8.3)
where k is the total number of possibilities of the whole system, and the real number
q is the entropic index that characterizes the degree of nonextensiveness. In the limit
q → 1, (8.3) meets the traditional BGS entropy defined by (8.1). These characteris-
tics give to q-entropy flexibility in explaining several physical systems. In addition,
this new kind of entropy does not fail to explain traditional physical systems since it
is a generalization.
Furthermore, a generalization of some theory may suppose the violation of one
of its postulates. In the case of the generalized entropy proposed by Tsallis, the addi-
tive property described by (8.2) is violated in the form of (8.4), which applies if the
system has a nonextensive characteristic. In this case, the Tsallis statistic is useful
and the q-additivity better describes the composed system. In our case, the experi-
mental results (Section 8.5) show that it is better to consider our systems as having
nonextensive behavior:
( ) () ( ) ( ) () ( )
SPQ SP SQ qSPSQ
qqq qq
∗= + +−1
(8.4)
In this equation, the term (1 − q) stands for the degree of nonextensiveness.
Considering S
q
≥ 0 in the pseudoadditive formalism of (8.4), the following clas-
sification for entropic systems is defined:
•
Subextensive entropy (q > 1):
()()()
SPQ SP SQ
qqq
∗> +
•
Extensive entropy (q = 1):
() ()()
SPQ SP SQ
qqq
∗= +
•
Superextensive entropy (q < 1):
8.3 Theoretical Background 199
()()()
SPQ SP SQ
qqq
∗< +
Taking into account the similarities between the formalisms of Shannon and BG
entropy, it is interesting to investigate the possibility of the generalization of Shan-
non entropy to the case of information theory, as has been shown by Yamano [43].
This generalization may be extended to image segmentation tasks by applying
Tsallis entropy, which has nonadditive information content.
Albuquerque et al. [21] proposed an algorithm that uses the concept of
q-entropy to segment US images. Because this concept may be naturally applied over
any statistical distribution, in this chapter we propose a natural extension of the
algorithm proposed by Albuquerque et al. [21]. Our proposal is a recursive proce-
dure of [21] in that, for each distribution P and Q, we applied the concept of
q-entropy. We named our extended algorithm NESRA and also proposed to apply it
in an initial segmentation of ourproposed five-step methodology for CADsystems.
The motivations to use the q-entropy are: (1) managing only a simple parameter
q yields a more controllable system; (2) as suggested in [21], mammographic images
and possibly several other natural images have a nonextensive profile; and (3) it is
simple and makes the implementation easy because it has a low computational
overload.
In the following section, we fully describe the NESRA proposal.
8.3.2 The Nonextensive Segmentation Approaches
Applying the concept of entropy in order to segment a digital image is a common
practice since Pun [15] showed that by maximizing the foreground and background
Shannon entropy of a gray-level image good results were generated concerning the
prominent regions. Then, other works following the same line were proposed. For
example, Kapur et al. [16] maximized an upper bound of the total a posteriori
entropy in order to obtain the threshold level. Abutaleb [17] extended the method
using two-dimensional entropies. Li and Lee [18] and Pal [19] used the directed
divergence of Kullback for the selection of the threshold, and Sahoo et al. [20] used
the Reiny entropy model for image thresholding.
In 2004, Albuquerque et al. [21] presented the concept of nonextensive entropy
applied to mmamographic gray-scale images. They assume a probability distribu-
tion, one for background and the other for foreground, and take the threshold that
maximizes the nonadditivity characteristic given by (8.4). However, as with several
methods designed to produce binary images, this approach does not work for
multiregion segmentation. So we proposed an extension of the method presented in
[21] by applying recursively the maximization of (8.4) over the background and the
foreground in order to achieve multiple regions of homogenous gray-level or color
distribution.
It can be argued that any parametric or nonparametric method with the objec-
tive of finding an ideal threshold to produce a binary image can be recursively used
in the foreground and in the background in order to achieve multiple regions. How-
ever, our approach outperforms several well-known methods under the same condi-
tions. In this section, we formalize the NESRA algorithm. First, we review the
nonextensive procedure for image segmentation proposed in [21].
200 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
8.3.2.1 Nonextensive Segmentation Algorithm for Image Binarization
Suppose we have an image with k gray levels. Let the probability distribution of
these levels be P ={p
i
= p
1
; p
2
; ... ; p
k
}. Then, weconsider twoprobability distribution
from P, one for the foreground (P
A
) and another for the background (P
B
). We can
make a partition at luminance level t between the pixels from P into A and B.To
maintain the constraints 0 ≤ P
A
≤ 1 and 0 ≤ P
B
≤ 1 we must to renormalize both dis-
tribution as:
P
p
p
p
p
p
p
A
AA
t
A
:,,,
12
K
(8.5)
P
p
p
p
p
p
p
B
t
B
t
B
k
B
:, ,,
++12
K
(8.6)
where
pp
Ai
i
t
=
=
∑
1
and
pp
Bi
it
k
=
=+
∑
1
.
Now, following (8.3), we calculate the a priori Tsallis entropy for each
distribution:
S
p
p
q
A
i
A
q
i
t
=
−
⎛
⎝
⎜
⎞
⎠
⎟
−
=
∑
1
1
1
(8.7)
S
p
p
q
B
i
B
q
it
k
=
−
⎛
⎝
⎜
⎞
⎠
⎟
−
=+
∑
1
1
1
(8.8)
Observe that the Tsallis entropy represented by (8.3), (8.7), and (8.8) depends
directly on the parameter t for the foreground and background, and it is formulated
as the sum of each entropy, allowing the pseudoadditive property, given by (8.4),
for statistically independent systems, defined by the following equation:
()
St
p
p
q
p
p
q
AB
i
A
q
i
t
i
B
q
it
k
+
==+
=
−
⎛
⎝
⎜
⎞
⎠
⎟
−
+
−
⎛
⎝
⎜
⎞
⎠
⎟
−
∑∑
1
1
1
11
()
1
1
1
1
1
1
11
+−
−
⎛
⎝
⎜
⎞
⎠
⎟
−
−
⎛
⎝
⎜
⎞
⎠
⎟
−
==+
∑∑
q
p
p
q
p
p
q
i
A
q
i
t
i
B
q
it
k
(8.9)
To accomplish the segmentation task, in [21] the information measure between
the two classes (foreground and background) is maximized. In this case, the lumi-
nance level t is considered to be the optimum threshold value (t
opt
), which can be
achieved with a cheap computational effort of
() () ( ) () ()
[]
t St St qStSt
opt
AB AB
=++−arg max 1
(8.10)
8.3 Theoretical Background 201
Note that the value of t that maximizes (8.10) depends primarily on the parameter
q. This is an advantage due to its simplicity. Furthermore, the value of q = 0.0001,
which generates t
opt
, is not explicitly calculated and, in this work, it was empirically
defined as that which generated the best CAD system performance. This value, q <
1, then suggests a system nonextensivity. An example of application of q ≠ 1
(nonextensive system) against q = 1.0 (traditional extensive system) is shown in
Figure 8.1.
Among the results presented in Figure 8.1, only parts (c) and (d) seem to be use-
ful in a postprocessing module. With part (c) we can extract the central region and
used its boundary curve to initialize some deformable model framework (level set,
snake, dual snake, and so forth) in order to better determine the tumor’s boundary
location. The same can be thinking about part (d), but in this case we can initialize
the deformable model from inside to outside, such as in [44, 45]. However, both
initializations do not guarantee that the narrow boundary information (as explained
in Section 8.1) will be included in the tumor region. Then, one more level of segmen-
tation is needed, yielding to a multisegmentation process instead of the simple
binarization. For this reason we create an extension of the extensive algorithm that
is able to obtain the same result as in Figure 8.1 but with more levels of segmenta-
tion. This is a multisegmentation algorithm. Its advantages are that: (1) it does not
oversegment the image; (2) it is as fast as the image binarization algorithm; (3) the
number of regions is well controlled through the number of recursions; and (4) it
preserves the original result of the binarization algorithm, adding a narrowband
region, which means a transition between the foreground and background that can
later be analyzed and included (or refuted) in the tumor region. This algorithm is
formalized in the next section.
202 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
(d)(c)
(b)(a)
Figure 8.1 (a) Original ultrasound benign image and the nonextensive entropy segmentation
results for different q values: (b) 1.0, (c) 6.0, and (d) 10.0, respectively.
8.3.2.2 The Nonextensive Segmentation Recursive Algorithm
Following the definitions and formulations from Section 8.3.2.1, we can take each
distribution P
A
and P
B
and subdivide them into two news distribution, P
A1
, P
A2
, P
B1
,
and P
B2
, as following:
P
p
p
p
p
p
p
A
AA
t
A
1
1
1
2
11
:,,,K
(8.11)
P
p
p
p
p
p
p
A
t
A
t
AA
2
1
2
2
22
:, ,,
++
K
∂
(8.12)
P
p
p
p
p
p
p
B
BB
v
B
1
1
1
2
11
:, ,,
∂∂++
K
(8.13)
P
p
p
p
p
p
p
B
v
B
v
B
k
B
2
1
2
2
22
:, ,,
++
K
(8.14)
having the constraints
ppp pp pp
Ai
i
t
AiBiB
v
t
1
1
21 2
11
== =
=++
∑∑∑
,
,,.and
∂
∂
For each one of the four preceding distributions, we can compute its respective
nonextensive entropy as follows:
S
p
p
q
A
i
A
q
i
t
1
1
1
1
1
=
−
⎛
⎝
⎜
⎞
⎠
⎟
−
=
∑
(8.15)
S
p
p
q
A
i
A
q
it
2
2
1
1
1
=
−
⎛
⎝
⎜
⎞
⎠
⎟
−
=+
∑
∂
(8.16)
S
p
p
q
B
i
B
q
i
v
1
1
1
1
1
=
−
⎛
⎝
⎜
⎞
⎠
⎟
−
=+
∑
∂
(8.17)
S
p
p
q
B
i
B
q
iv
k
2
2
1
1
1
=
−
⎛
⎝
⎜
⎞
⎠
⎟
−
=+
∑
(8.18)
Using (8.4), (8.15) and (8.16) to compute S(A) = S(A1 + A2) and, similarly
(8.17) and (8.18) to compute S(B) = S(B1 + B2), we have:
8.3 Theoretical Background 203