equation is solvable as
f(p) = X4 v=1
ckelvp (7.4.10)
where lk=√
ω eπi(v −1)/
2 i=√
−1. The boundary conditions require
X4 v=1
cv = 0, X4
v=1
cvl2v = 1 (7.4.11)
X4 v=1
cvelv = 0, X4
v=1
cvl2velv = 0 (7.4.12) Solving these equations the particular solution can be determined. The particular solution sat- isfies the initial conditions u1(0, t) = f(p) = P∞
v=−∞
cveiπvp and ∂u∂t1 (0, t) = 0, where f(p) is expandable in a complex Fourier series as an odd valued function such that
f(p) =f(p+ 2) =−f(2−p) 0< p <1 (7.4.13) This implies thatf(p)is represented as a sine series,
f(p) = X∞
v=1
avsin (vπp) (7.4.14)
withav =−2 [imag(cv)]. The homogeneous solution is represented as u2(p, t) =−
X∞ v=1
avcos¡ π2v2t¢
sin (vπp) (7.4.15)
so thatu1 andu2 combine to satisfy the desired initial conditions.
7.5 Derivative of Gaussian (DroG) weighting function
The Gaussian weighting function is defined as wGauss(x) =
√1 2πσ2e
³
−2σx22´
, x∈ROI
0, elsewhere, (7.5.1)
whereσ is the usual standard deviation. The derivative of the Gaussian function (Gauss(x)) is obtained [116] by,
∂n
∂xnGauss(x) = (−1)n 1
¡σ√ 2¢nHn
µ x σ√
2
¶
Gauss(x) (7.5.2)
7.5. DERIVATIVE OF GAUSSIAN (DROG) WEIGHTING FUNCTION 115 wherenis the order of the derivative andHn(x)is the Hermite polynomial. In our case, we take n = 1. Simplifying the above function we get,
wDroG(x) = −x 2√
2πσ3e−2σx22 (7.5.3)
Here, we consider the absolute value of the above weighting function. The characteristic function due to Gaussian weighting function is
EvGauss= Z2t
0
v(x)wGauss(x)dx. (7.5.4)
and for DroG is
EvDroG = Zt
0
x 2√
2πσ3e−2σx22e−(x−t)22σ2s dx+ Zt
t
x 2√
2πσ3e−2σx22dx (7.5.5) The difference between two characteristic functions is expressed as
EvDroG−EvG =
1 2√
2πσ3
"
Exp Ã
1 + t(lnt−1)−
µ
1 2σ2+ 1
2σ2 s
¶
t3 3+ t2
4σ2 s
t3Exp µ
− µ
1 2σ2+2σ12
s
¶¶
t2
!
−0.25σ2
³ Γ
³ 2,2tσ22
´
−Γ
³ 2,2σt22
´´i
−√12σ
µ 0.5
1 2σ2+2σ12
s
¶3
2
µ³ t 2σ2s
´2
−
³ 1 2σs2+2σ2
´ ³ t2 2σ2s
´¶ n
erf³q
1
2σ2 + 2σ12 st−
t 2σ2s
q 1
2σ2+ 1
2σs2
¶
−erf
µ −t
2σ2s
q 1
2σ2+ 1
2σ2s
¶¾ +√σ2
³ erf
³q 1 2σ22t
´
erf
³q 1 2σ2t
´´i
whereerf is the error function andΓ (a, x) = R∞
z
e−tta−1dt. Usually, for edge detection, the first derivative of the image function convolved with a Gaussian is equivalent to the image function convolved with the first derivative of a Gaussian. Therefore, it is possible to combine the smooth- ing and detection stages into a single convolution in one dimension, either convolving with the first derivative of the Gaussian and looking for peaks, or with the second derivative and look- ing for zero crossings. Meanwhile, in context with our objective, DroG is used as a weighting function.
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Chapter 8 Publications
8.1 J OURNAL P APERS (P UBLISHED /A CCEPTED )
1. S. P. Dakua and J. S. Sahambi, “Modified Active contour Model and Random Walk Ap- proach for Left Ventricular Cardiac MR Image Segmentation”, International Journal for Numerical Methods in Biomedical Engineering, Wiley, (2011) 27: 1350-1361, DOI: 10.1002/
cnm.1430.
2. S. P. Dakua and J. S. Sahambi, “Detection of Left Ventricular Myocardial Contours from Is- chemic Cardiac MR Images”, IETE Journal of Research, (2011) 57:372-384, DOI: 10.4103/
0377-2063.86338.
3. S. P. Dakua and J. S. Sahambi, “A Strategic Approach for Left Ventricular Cardiac MR Im- age Segmentation, Cardiovascular Engineering, Springer, (2010) 10:163-168, DOI 10.1007/
s10558-010-9102-3, 2010.
4. S. P. Dakua and J. S. Sahambi, “Automatic Contour Extraction of Multi-labeled Left Ven- tricle from CMR Images Using CB and Random Walk Approach”, Cardiovascular Engi- neering, Springer, (2010) 10:3043, DOI 10.1007/s10558-009-9091-2, 2010.
5. S. P. Dakua and J. S. Sahambi, “LV Contour Extraction from Cardiac MR Images Using Random Walk Approach,” International Journal of Recent Trends in Engineering, Academy Publisher, vol. 1, no. 3, pp. 101 - 105, 2009.