Partially Camouflaged Object Tracking using Modified Probabilistic Neural Network and Fuzzy Energy based Active Contour

Abstract

Various problems in object detection and tracking have attracted researchers to develop methodologies for solving these problems. Occurrence of camouflage is one of such challenges that makes object detection and tracking problems more complex. However, less attention has been given to detect and track camouflaged objects due to complexity of the problem. In this article, we propose a tracking-by-detection algorithm to detect and track camouflaged objects. To increase separability between the camouflaged object and the background, we propose to integrate features (CIELab, histogram of orientation gradients and locally adaptive ternary pattern) from multi-cue (color, shape and texture) to represent a camouflaged object. A probabilistic neural network (PNN) is modified to construct an efficient discriminative appearance model for detecting camouflaged objects in video sequences. A large number of training patterns (many could be redundant) are reduced based on motion of the object in the modified PNN. The modified PNN makes the detection process faster and also increases the detection accuracy. Due to high visual similarity among the camouflaged object and the background, the boundary of camouflaged object is not well defined (i.e., boundary may be smooth and/or discontinuous). In this context, a robust fuzzy energy based active contour model using both global and local information is proposed to extract contour (boundary) of the detected camouflaged object for tracking. We show a realization of the proposed method and demonstrate its performance (both quantitatively and qualitatively) with respect to state-of-the-art techniques on several challenging sequences. Analysis of results concludes that the proposed technique can track camouflaged (fully or partially) objects as well as objects in various complex environments in a better way as compare to the existing ones.

Appendices

Appendix 1

The proposed energy function F in Eq. (15) is convex with respect to u.

Proof:

Let

$$\begin{aligned} F_A(\mathbf X ^{t})= & {} \int \limits _\varOmega {\left[ {u(\mathbf X ^{t})} \right] }^m \left\| {I(\mathbf X ^{t}) - C_{1} } \right\| ^2 d\mathbf X ^{t},\ \mathbf X ^{t} \in \varOmega \nonumber \\\equiv & {} \mathfrak {R}^{2} \end{aligned}$$

(20)

and

$$\begin{aligned} f_A(\mathbf X ^{t}) = \left[ {u(\mathbf X ^{t})} \right] ^m \left\| {I(\mathbf X ^{t}) - C_{1} } \right\| ^2 , \end{aligned}$$

(21)

where $f_A : \varOmega \rightarrow \mathfrak {R}$. Therefore, $F_A(\mathbf X ^{t}) = \int \limits _\varOmega {f_A(\mathbf X ^{t})} d\mathbf X ^{t}$.

Now let $\mathbf X _1 \equiv (x_1 ,y_1 )$ and $\mathbf X _2 \equiv (x_2 ,y_2 ) \in \varOmega$. For any $\theta \in [0,1]$, we have

$$\begin{aligned}&\theta \mathbf X _1 + (1 - \theta ) \mathbf X _2 \\&\quad =\theta \left( {x_1 ,y_1 } \right) + (1 - \theta )\left( {x_2 ,y_2 } \right) \\&\quad = \left( {\theta x_1 + (1 - \theta )x_2 } , {\theta y_1 + (1 - \theta )y_2 } \right) \\&\quad = \left( {\theta \left( {x_1 - x_2 } \right) + x_2 } , {\theta \left( {y_1 - y_2 } \right) + y_2 } \right) \in \varOmega \equiv \mathfrak {R}^{2}. \end{aligned}$$

Since $\theta \left( {x_1 - x_2 } \right) + x_2 \in \mathfrak {R}$ as $x_1$, $x_2$ $\in \mathfrak {R}$ and $\theta \in [0,1]$; and $\theta \left( {y_1 - y_2 } \right) + y_2 \in \mathfrak {R}$ as $y_1$, $y_2$ $\in \mathfrak {R}$ and $\theta \in [0,1]$. Therefore, the domain of $f_A$ i.e., $\varOmega \equiv \mathfrak {R}^{2}$ is convex.

Differentiating Eq. (21) w. r. t. u, we have

$$\begin{aligned} \frac{{\partial f_A }}{{\partial u}} = m\left[ {u(\mathbf X ^{t})} \right] ^{m - 1} \left\| {I(\mathbf X ^{t}) - c_{1} } \right\| ^2. \end{aligned}$$

Again differentiating $\frac{{\partial f_A }}{{\partial u}}$ w. r. t. u, we have

$$\begin{aligned} \frac{{\partial ^2 f_A }}{{\partial u^{2} }} = m(m - 1)\left[ {u(\mathbf X ^{t})} \right] ^{m - 2} \left\| {I(\mathbf X ^{t}) - c_{1} } \right\| ^2. \end{aligned}$$

Now $\frac{{\partial ^2 f_A }}{{\partial u^{2} }} \ge 0$, since $m > 1$, $u(\mathbf X ^{t}) \in [0,1]$ and $\left\| {I(\mathbf X ^{t}) - c_{1} } \right\| ^2 \ge 0$.

Since domain of $f_A$ is convex and $\frac{{\partial ^2 f_A }}{{\partial u^{2} }} \ge 0$, therefore $f_A$ is convex. Thus, $\forall \mathbf Y _1, \mathbf Y _2 \in \varOmega \equiv \mathfrak {R}^{2}$ and $\theta \in [0,1]$. The following relation

$$\begin{aligned} f_A (\theta \mathbf X _1 + (1 - \theta _2 )\mathbf X _2 ) \le \theta f_A (\mathbf X _1 ) + (1 - \theta )f_A (\mathbf X _2 ) \nonumber \\ \end{aligned}$$

(22)

holds. Integrating both the sides of Eq.(22), we have

$$\begin{aligned}&\int \limits _\varOmega {f_A (\theta \mathbf X _1 + (1 - \theta )\mathbf X _2 )d\mathbf X ^{t}} \nonumber \\&\quad \le \theta _2 \int \limits _\varOmega {f_A (\mathbf X _1 )d\mathbf X ^{t}} + (1 - \theta )\int \limits _\varOmega {f_A (\mathbf X _2 )d\mathbf X ^{t}}. \end{aligned}$$

(23)

But $\int \limits _\varOmega {f_A (\mathbf X ^{t})d\mathbf X ^{t}} = F_A(\mathbf X ^{t})$. Therefore, we have

$$\begin{aligned} F_A \left( {\theta \mathbf X _1 + (1 - \theta )\mathbf X _2 } \right) \le \theta _2 F_A (\mathbf X _1 ) + (1 - \theta )F_A (\mathbf X _2 ). \end{aligned}$$

Hence, $F_{A}$ is convex.

Similarly, let

$$\begin{aligned} F_B(\mathbf X ^{t}) = \int \limits _\varOmega {\left[ {1 - u(\mathbf X ^{t})} \right] ^m } \left\| {I(\mathbf X ^{t}) - c_{2} } \right\| ^2 d\mathbf X ^{t}, \end{aligned}$$

(24)

and

$$\begin{aligned} f_B(\mathbf X ^{t}) = \left[ {1 - u(\mathbf X ^{t})} \right] ^m \left\| {I(\mathbf X ^{t}) - c_{2} } \right\| ^2, \end{aligned}$$

(25)

where $f_B : \varOmega \rightarrow \mathfrak {R}$. Therefore $F_B(\mathbf X ^{t}) = \int \limits _\varOmega f_B(\mathbf X ^{t})d\mathbf X ^{t}$. In a similar way, we can prove that $F_{B}$ is also convex.

Again let

$$\begin{aligned} F_{C_1}(\mathbf Y ^{t})= & {} \int \limits _\varOmega {W_\mathbf{X ^{t}{} \mathbf Y ^{t}} } \left[ {1 - u(\mathbf Y ^{t})} \right] ^m \left\| {I(\mathbf Y ^{t}) - c_{1} } \right\| ^2 d\mathbf Y ^{t} , \nonumber \\\end{aligned}$$

(26)

$$\begin{aligned} F_C(\mathbf X ^{t})= & {} \int \limits _\varOmega {\left[ {u(\mathbf X ^{t})} \right] } ^m F_{C_1}(\mathbf Y ^{t}) d\mathbf X ^{t} , \end{aligned}$$

(27)

$$\begin{aligned} f_{C_1}(\mathbf Y ^{t})= & {} W_\mathbf{X ^{t}{} \mathbf Y ^{t}} \left[ {1 - u\left( \mathbf Y ^{t} \right) } \right] ^m \left\| {I(\mathbf Y ^{t}) - c_{1} } \right\| ^2 , \end{aligned}$$

(28)

and

$$\begin{aligned} f_C(\mathbf X ^{t}) = \left[ {u(\mathbf X ^{t})} \right] ^m f_{C_1}(\mathbf Y ^{t}), \ \mathbf X ^{t}, \mathbf Y ^{t} \in \varOmega \equiv \mathfrak {R}^{2} . \end{aligned}$$

(29)

Therefore $F_{C_1}(\mathbf Y ^{t}) = \int \limits _\varOmega {f_{C_1}(\mathbf Y ^{t})} d\mathbf Y ^{t}$ and $F_C(\mathbf X ^{t}) = \int \limits _\varOmega {f_C(\mathbf X ^{t})} d\mathbf X ^{t}$. Here $f_{C_1 } : \varOmega \rightarrow \mathfrak {R}$ and the domain of $f_{C_1 }$ is convex.

Now differentiating Eq. (28) w. r. t. u, we have

$$\begin{aligned} \frac{{\partial f_{C_1 } }}{{\partial u}} = - mW_\mathbf{X ^{t}{} \mathbf Y ^{t}} \left[ {1 - u(\mathbf Y ^{t})} \right] ^{m - 1} \left\| {I(\mathbf Y ^{t}) - c_{1} } \right\| ^2. \end{aligned}$$

Again differentiating $\frac{{\partial f_{C_1 } }}{{\partial u}}$ w. r. t. u, we have

$$\begin{aligned} \frac{{\partial ^2 f_{C_1 } }}{{\partial u^{2} }} = m(m - 1)W_\mathbf{X ^{t}{} \mathbf Y ^{t}} \left[ {1 - u(\mathbf Y ^{t})} \right] ^{m - 2} \left\| {I(\mathbf Y ^{t}) - c_{1} } \right\| ^2. \end{aligned}$$

Now $\frac{{\partial ^2 f_{C_1 } }}{{\partial u^{2} }} \ge 0$, since $m > 1$, $W_\mathbf{X ^{t}{} \mathbf Y ^{t}} \ge 0$, $u(\mathbf Y ^{t}) \in [0,1]$ and $\left\| {I(\mathbf Y ^{t}) - c_{1} } \right\| ^2 \ge 0$.

Since domain of $f_{C_1 }$ is convex and $\frac{{\partial ^2 f_{C_1 } }}{{\partial u^{2} }} \ge 0$, hence $f_{C_1 }$ is convex. Therefore $\forall \mathbf Y _1,\mathbf Y _2 \in \varOmega$ and $\theta \in [0,1]$. The following relation

$$\begin{aligned}&f_{C_1 } \left( {\theta \mathbf Y _1 + (1 - \theta )\mathbf Y _2 } \right) \nonumber \\&\ \ \ \ \ \ \ \le \theta f_{C_1 } \left( \mathbf{Y _1 } \right) + (1 - \theta )f_{C_1 } \left( \mathbf{Y _2 } \right) \end{aligned}$$

(30)

holds. Integrating both the sides, we have

$$\begin{aligned}&\int \limits _\varOmega {f_{C_1 } } \left( {\theta \mathbf Y _1 + (1 - \theta )\mathbf Y _2 } \right) d\mathbf Y ^{t} \nonumber \\&\quad \le \theta \int \limits _\varOmega {f_{C_1 } \left( \mathbf{Y _1 } \right) d\mathbf Y ^{t} + } (1 - \theta )\int \limits _\varOmega {f_{C_1 } } \left( \mathbf{Y _2 } \right) d\mathbf Y ^{t},\ \forall \mathbf Y ^{t} \in \varOmega .\nonumber \\ \end{aligned}$$

(31)

But $F_{C_1 } = \int \limits _\varOmega {f_{C_1 } } \left( \mathbf Y \right) d\mathbf Y ^{t}$, hence $F_{C_{1}}$ is convex. Since $f_{C_1}$ is convex and $\left[ {u(\mathbf X ^{t})} \right] ^m \ge 0$, then $f_C = \left[ {u(\mathbf X ^{t})} \right] ^m f_{C_1 }$ is also convex.

Again let $\overline{F_C } = \left[ {u(\mathbf X ^{t})} \right] ^m F_{C_1}$. Since $F_{C_1}$ is convex and $\left[ {u(\mathbf X ^{t})} \right] ^m \ge 0$, therefore $\overline{F_C }$ is also convex. Therefore $\forall \mathbf X _1, \mathbf X _2 \in \varOmega$ and $\theta \in [0,1]$. The following equation

$$\begin{aligned}&\overline{F_C } \left( {\theta \mathbf X _1 + (1 - \theta )\mathbf X _2 } \right) \nonumber \\&\ \ \ \ \le \theta \overline{F_C } \left( \mathbf{X _1 } \right) + (1 - \theta )\overline{F_C } \left( \mathbf{X _2 } \right) \end{aligned}$$

(32)

holds. Then integrating both the sides, we have

$$\begin{aligned}&\int \limits _\varOmega {\overline{F_C } \left( {\theta \mathbf X _1 + (1 - \theta )\mathbf X _2 } \right) d\mathbf X ^{t}} \nonumber \\&\ \le \theta \int \limits _\varOmega {\overline{F_C } \left( \mathbf{X _1 } \right) d\mathbf X ^{t}} + (1 - \theta )\int \limits _\varOmega {\overline{F_C } \left( \mathbf{X _2 } \right) d\mathbf X ^{t}}, \forall \mathbf X ^{t} \in \varOmega .\nonumber \\ \end{aligned}$$

(33)

Therefore $F_C = \int \limits _\varOmega {\overline{F_C } \left( \mathbf X \right) } d\mathbf X ^{t} = \int \limits _\varOmega {\left[ {u(\mathbf X ^{t})} \right] } ^m F_{C_1 } d\mathbf X ^{t}$ is convex. In a similar way, it can be shown that

$$\begin{aligned} F_D= & {} \int \limits _\varOmega {\left[ {1 - u(\mathbf X ^{t})} \right] } ^m \nonumber \\&\left\{ {\int \limits _\varOmega {W_\mathbf{X ^{t}{} \mathbf Y ^{t}} \left[ {u(\mathbf Y ^{t})} \right] } ^m \left\| {I(\mathbf Y ^{t}) - c_{2} } \right\| ^2 d\mathbf Y ^{t}} \right\} d\mathbf X ^{t}\nonumber \\ \end{aligned}$$

(34)

is also convex. Since $0 \le \beta \le 1$, $1 - \beta \ge 0$, $\lambda _1 ,\lambda _2 > 0$, then F is the weighted sum of four ($F_A$, $F_B$, $F_C$, and $F_D$) convex functions. i.e.,

$$\begin{aligned} F = \lambda _1 \beta F_A + \lambda _2 \beta F_B + \lambda _1 (1 - \beta )F_C + \lambda _2 (1 - \beta )F_D. \end{aligned}$$

Hence, F is convex with respect to u. Therefore, the proposed energy function is convex with respect to membership function $u(\mathbf X ^{t})$. $\square$

Appendix 2

Since an image is discrite in nature, instead of integration, summation is considered here.

Let us assume two prototypes $c_1$ and $c_2$ which approximate the image intensity inside(C) and outside(C) the contour C. Thus it can be written as

$$\begin{aligned} c_1 = \frac{{\sum \limits _\varOmega {\left[ {u(X)} \right] } ^m I(X)}}{{\sum \limits _\varOmega {\left[ {u(X)} \right] } ^m }} \end{aligned}$$

(35)

and

$$\begin{aligned} c_2 = \frac{{\sum \limits _\varOmega {\left[ {1 - u(X)} \right] } ^m I(X)}}{{\sum \limits _\varOmega {\left[ {1 - u(X)} \right] } ^m }}, \end{aligned}$$

(36)

where I(X) is the intensity value at pixel location X, u(X) is the degree of membership of pixel X inside C, and m is the fuzzifier which determines the fuzziness present in the given image.

Therefore, total fuzzy energy for the whole image can be computed as

$$\begin{aligned} F= & {} \lambda _1 \beta \sum \limits _\varOmega {\left[ {u(X)} \right] ^m \left\| {I(X) - c_1 } \right\| ^2 } \nonumber \\&+ \,\lambda _2 \beta \sum \limits _\varOmega {\left[ {1 - u(X)} \right] ^m \left\| {I(X) - c_2 } \right\| ^2 } \nonumber \\&+ \,\lambda _1 (1 - \beta )\sum \limits _\varOmega {\left[ {u(X)} \right] ^m } h_{1} \nonumber \\&+ \,\lambda _2 (1 - \beta )\sum \limits _\varOmega {\left[ {1 - u(X)} \right] ^m } h_{2} \nonumber \\= & {} F_A + F_B + F_C + F_D. \end{aligned}$$

(37)

where

$$\begin{aligned} F_A= & {} \lambda _1 \beta \sum \limits _\varOmega {\left[ {u(X)} \right] } ^m \left\| {I(X) - c_1 } \right\| ^2, \nonumber \\ F_B= & {} \lambda _2 \beta \sum \limits _\varOmega {\left[ {1 - u(X)} \right] } ^m \left\| {I(X) - c_2 } \right\| ^2, \nonumber \\ F_C= & {} \lambda _1 (1 - \beta )\sum \limits _\varOmega {\left[ {u(X)} \right] } ^m h_{1}, \nonumber \\ F_D= & {} \lambda _2 (1 - \beta )\sum \limits _\varOmega {\left[ {1 - u(X)} \right] } ^m h_{2}, \end{aligned}$$

(38)

with

$$\begin{aligned} h_{1}= & {} \sum \limits _\varOmega {W_{XY} \left[ {1 - u(Y)} \right] ^m \left\| {I(Y) - c_1 } \right\| ^2 }, \nonumber \\ h_{2}= & {} \sum \limits _\varOmega {W_{XY} \left[ {u(Y)} \right] ^m \left\| {I(Y) - c_2 } \right\| ^2 }. \end{aligned}$$

(39)

Let us assume that a pixel $P\in I$ with intensity $I_o$ and degree of membership $u_o$. If we change the degree of membership of pixel P to the value $u_n$ using Eq. (18), then $c_1$ and $c_2$ will be changed to new values $\overline{c_1}$ and $\overline{c_2}$, respectively. The new values of $c_1$ and $c_2$ are calculated as

$$\begin{aligned} \overline{c_1 }= & {} \frac{{\sum \limits _\varOmega {\left[ {\overline{u} \left( X \right) } \right] } ^m I(X)}}{{\sum \limits _\varOmega {\left[ {\overline{u} \left( X \right) } \right] } ^m }} \nonumber \\= & {} \frac{{\sum \limits _\varOmega {\left[ {u \left( X \right) } \right] } ^m I(X) + \left( {u_n^m - u_o^m } \right) I_o }}{{\sum \limits _\varOmega {\left[ {u \left( X \right) } \right] } ^m + u_n^m - u_o^m }} \nonumber \\= & {} \frac{{c_1 \sum \limits _\varOmega {\left[ {u(X)} \right] ^m } + \left( {u_n^m - u_o^m } \right) I_o }}{{\sum \limits _\varOmega {\left[ {u \left( X \right) } \right] } ^m + u_n^m - u_o^m }} \nonumber \\= & {} \frac{{c_1 a_1 + \left( {u_n^m - u_o^m } \right) I_o }}{{a_1 + u_n^m - u_o^m }} \nonumber \\= & {} \frac{{c_1 \left( {a_1 + u_n^m - u_o^m } \right) - c_1 \left( {u_n^m - u_o^m } \right) + \left( {u_n^m - u_o^m } \right) I_o }}{{a_1 + u_n^m - u_o^m }} \nonumber \\= & {} c_1 + \frac{{\left( {u_n^m - u_o^m } \right) I_o - c_1 \left( {u_n^m - u_o^m } \right) }}{{a_1 + u_n^m - u_o^m }} \nonumber \\= & {} c_1 + \frac{{\left( {u_n^m - u_o^m } \right) \Vert {I_o - c_1 } \Vert }}{{a_1 + u_n^m - u_o^m }} \nonumber \\= & {} c_1 + S_1 \Vert {I_o - c_1 } \Vert \end{aligned}$$

(40)

where $a_1 = \sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m$ and $S_1 = \frac{{u_n^m - u_o^m }}{{\sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m + u_n^m - u_o^m }}$ $= \frac{{u_n^m - u_o^m }}{{a_1 + u_n^m - u_o^m }}$.

Similarly,

$$\begin{aligned} \overline{c_2 }= & {} \frac{{\sum \limits _\varOmega {\left[ {1 - \overline{u} \left( X \right) } \right] } ^m I(X)}}{{\sum \limits _\varOmega {\left[ {1 - \overline{u} \left( X \right) } \right] } ^m }} \nonumber \\= & {} c_2 + \frac{{\left[ {\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m } \right] \Vert {I_o - c_2 } \Vert }}{{a_2 + \left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m }} \nonumber \\= & {} c_2 + S_2 \Vert {I_o - c_2 } \Vert \end{aligned}$$

(41)

where $a_2 = \sum \limits _\varOmega {\left[ {1 - u\left( X \right) } \right] } ^m$ and $S_2 = \frac{{\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m }}{{a_2 + \left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m }}$.

From Eq. (15), it is seen that if degree of membership u(X) is changed, then the energy of the model will also be changed. If F denotes the old energy and $\overline{F}$ denotes the new energy due to changing of degree of membership of the point P, then

$$\begin{aligned} \overline{F} = \overline{F} _A + \overline{F} _B + \overline{F} _C + \overline{F} _D, \end{aligned}$$

(42)

with

$$\begin{aligned} \overline{F} _A= & {} \lambda _1 \beta \sum \limits _\varOmega {\left[ {\overline{u} \left( X \right) } \right] } ^m \left\| {I(X) - \overline{c_1} } \right\| ^2 \nonumber \\= & {} \lambda _1 \beta \left\{ \sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \left\| {I(X) - \overline{c_1 } } \right\| ^2 \right. \nonumber \\&+\left. \left( {u_n^m - u_o^m } \right) \left\| {I_o - \overline{c_1 } } \right\| ^2 \right\} \nonumber \\= & {} F_A + \lambda _1 \beta \Vert {I_o - c_1 } \Vert ^2 \left[ {a_1 + \left( {u_n^m - u_o^m } \right) } \right] \nonumber \\&S_1 \left( {\frac{{a_1 }}{{a_1 + \left( {u_n^m - u_o^m } \right) }}} \right) \nonumber \\= & {} F_A + \lambda _1 \beta S_1 a_1 \Vert {I_o - c_1 } \Vert ^2 . \end{aligned}$$

(43)

Similarly,

$$\begin{aligned} \overline{F} _B = F_B + \lambda _2 \beta a_2 S_2 \Vert {I_o - c_2 } \Vert ^2. \end{aligned}$$

(44)

Then

$$\begin{aligned} \overline{F} _C= & {} \lambda _1 \left( {1 - \beta } \right) \sum \limits _\varOmega {\left[ {\overline{u} \left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] } ^m \left\| {I(Y) - \overline{c} _1 } \right\| ^2 } \right\} \nonumber \\= & {} \lambda _1 \left( {1 - \beta } \right) \sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] } ^m \left\| {I(Y) - \overline{c} _1 } \right\| ^2 } \right\} \nonumber \\&+\, \lambda _1 \left( {1 - \beta } \right) \left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] ^m } \left\| {I(Y) - \overline{c} _1 } \right\| ^2 } \right\} \nonumber \\&+ \,\lambda _1 \left( {1 - \beta } \right) \left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] ^m } \left\| {I_o - \overline{c} _1 } \right\| ^2 } \right\} . \end{aligned}$$

(45)

Let

$$\begin{aligned} \overline{F} _{C_1 }= & {} \lambda _1 (1 - \beta )\sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] } ^m \left\| {I(Y) - \overline{c} _1 } \right\| ^2 } \right\} ,\nonumber \\ \overline{F} _{C_2 }= & {} \lambda _1 (1 - \beta )\left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] } ^m \left\| {I(Y) - \overline{c} _1 } \right\| ^2 } \right\} , \nonumber \\ \overline{F} _{C_3 }= & {} \lambda _1 (1 - \beta )\left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] } ^m \left\| {I_o - \overline{c} _1 } \right\| ^2 } \right\} . \end{aligned}$$

(46)

Now,

$$\begin{aligned} \overline{F} _{C_1 }= & {} \lambda _1 (1 - \beta )\sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] } ^m \left\| {I(Y) - \overline{c} _1 } \right\| ^2 } \right\} \nonumber \\= & {} F_C + \lambda _1 (1 - \beta )\sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] } ^m S_1^2 \Vert {I_o - c_1 } \Vert ^2 } \right\} \nonumber \\&-\, 2\lambda _1 (1 - \beta )\sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] } ^m \Vert {I(Y) - c_1 } \Vert S_1 \Vert {I_o - c_1 } \Vert } \right\} . \nonumber \\\end{aligned}$$

(47)

$$\begin{aligned} \overline{F} _{C_2 }= & {} \lambda _1 (1 - \beta )\left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] } ^m \left\| {I(Y) - \overline{c}_1 } \right\| ^2 } \right\} \nonumber \\= & {} \lambda _1 (1 - \beta )\left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] } ^m \left\| {I(Y) - c_1 - S_1 \left( {I_o - c_1 } \right) } \right\| ^2 } \right\} . \nonumber \\\end{aligned}$$

(48)

$$\begin{aligned} \overline{F} _{C_3 }= & {} \lambda _1 (1 - \beta )\left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} } \left[ {1 - u\left( Y \right) } \right] ^m \left\| {I_o - \overline{c} _1 } \right\| ^2 } \right\} \nonumber \\= & {} \lambda _1 (1 - \beta )\left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} } \left[ {1 - u\left( Y \right) } \right] ^m \Vert {I_o - c_1 } \Vert ^2 \left[ {1 - S_1 } \right] ^2 } \right\} . \end{aligned}$$

(49)

Therefore, we have $\overline{F} _C = \overline{F} _{C_1 } + \overline{F} _{C_2 } + \overline{F} _{C_3 }$.

$$\begin{aligned} \overline{F} _C= & {} F_C + \lambda _1 (1 - \beta )\sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] } ^m S_1^2 \Vert {I_o - c_1 } \Vert ^2 } \right\} \nonumber \\&-\, 2\lambda _1(1 - \beta )\sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] } ^m \Vert {I(Y) - c_1 } \Vert S_1 \Vert {I_o - c_1 } \Vert } \right\} \nonumber \\&+\, \lambda _1 (1 - \beta )\left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] } ^m \Vert {I(Y) - c_1 - S_1 \left( {I_o - c_1 } \right) } \Vert ^2 } \right\} \nonumber \\&+\, \lambda _1 (1 - \beta )\left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] } ^m \Vert {I_o - c_1 } \Vert ^2 \left[ {1 - S_1 } \right] ^2 } \right\} . \end{aligned}$$

(50)

Similarly, we have

$$\begin{aligned} \overline{F} _D= & {} F_D + \lambda _2 (1 - \beta )\sum \limits _\varOmega {\left[ {1 - u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {u\left( Y \right) } \right] } ^m S_2^2 \Vert {I_o - c_2 } \Vert ^2 } \right\} \nonumber \\&-\, 2\lambda _2 (1 - \beta )\sum \limits _\varOmega {\left[ {1 - u\left( X \right) } \right] } ^m \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{XY} \left[ {u\left( Y \right) } \right] } ^m \Vert {I(Y) - c_2 } \Vert S_2 \Vert {I_o - c_2 } \Vert } \right\} \nonumber \\&+\, \lambda _2 (1 - \beta )\left[ {\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m } \right] \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {u\left( Y \right) } \right] } ^m \Vert {I(Y) - c_2 - S_2 \left( {I_o - c_2 } \right) } \Vert ^2 } \right\} \nonumber \\&+\, \lambda _2 (1 - \beta )\left[ {\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m } \right] \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {u\left( Y \right) } \right] } ^m \Vert {I_o - c_2 } \Vert ^2 \left[ {1 - S_2 } \right] ^2 } \right\} . \end{aligned}$$

(51)

Therefore, we have

$$\begin{aligned} \overline{F}= & {} \overline{F} _A + \overline{F} _B + \overline{F} _C + \overline{F} _D \nonumber \\= & {} F + \lambda _1 \beta S_1 a_1 \Vert {I_o - c_1 } \Vert ^2 + \lambda _2 \beta S_2 a_2 \Vert {I_o - c_2 } \Vert ^2 \nonumber \\&-\, 2\lambda _1 \left( {1 - \beta } \right) S_1 \Vert {I_o - c_1 } \Vert b_1 + \lambda _1 \left( {1 - \beta } \right) g_1 \nonumber \\&\left[ {\Vert {c_1 + S_1 \left( {I_o - c_1 } \right) } \Vert ^2 - \Vert c_1\Vert ^2 } \right] \nonumber \\&+\, \lambda _1 \left( {1 - \beta } \right) \left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] ^m \Vert {I(Y) - c_1 - S_1 \left( {I_o - c_1 } \right) } \Vert } ^2 } \right\} \nonumber \\&+\, \lambda _1 \left( {1 - \beta } \right) \left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] ^m \Vert {I_o - c_1 } \Vert ^2 \left[ {1 - S_1 } \right] ^2 } } \right\} \nonumber \\&-\, 2\lambda _2 \left( {1 - \beta } \right) S_2 \Vert {I_o - c_2 } \Vert b_2 \nonumber \\&+\, \lambda _2 \left( {1 - \beta } \right) g_2 \left[ {\Vert {c_2 + S_2 \left( {I_o - c_2 } \right) } \Vert ^2 - \Vert c_2 \Vert ^2 } \right] \nonumber \\&+\, \lambda _2 \left( {1 - \beta } \right) \left[ {\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m } \right] \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {u\left( Y \right) } \right] ^m \Vert {I(Y) - c_2 - S_2 \left( {I_o - c_2 } \right) } \Vert } ^2 } \right\} \nonumber \\&+\, \lambda _2 \left( {1 - \beta } \right) \left[ {\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m } \right] \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {u\left( Y \right) } \right] ^m \Vert {I_o - c_2 } \Vert ^2 \left[ {1 - S_2 } \right] ^2 } } \right\} . \end{aligned}$$

(52)

Thus the energy difference $\varDelta F$ between old energy (F) and new energy ($\overline{F}$) due to change of degree of membership value can be obtained as

$$\begin{aligned} \varDelta F= & {} \overline{F} - F \nonumber \\= & {} \lambda _1 \beta S_1 a_1 \Vert {I_o - c_1 } \Vert ^2 + \lambda _2 \beta S_2 a_2 \Vert {I_o - c_2 } \Vert ^2 \nonumber \\&-\, 2\lambda _1 \left( {1 - \beta } \right) S_1 \Vert {I_o - c_1 } \Vert b_1 \nonumber \\&+\, \lambda _1 \left( {1 - \beta } \right) g_1 \left[ {\Vert {c_1 + S_1 \left( {I_o - c_1 } \right) } \Vert ^2 - \Vert c_1 \Vert ^2 } \right] \nonumber \\&+\, \lambda _1 \left( {1 - \beta } \right) \left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] ^m \Vert {I(Y) - c_1 - S_1 \left( {I_o - c_1 } \right) } \Vert } ^2 } \right\} \nonumber \\&+\, \lambda _1 \left( {1 - \beta } \right) \left( {u_n^m - u_o^m } \right) \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {1 - u\left( Y \right) } \right] ^m \Vert {I_o - c_1 } \Vert ^2 \left[ {1 - S_1 } \right] ^2 } } \right\} \nonumber \\&-\, 2\lambda _2 \left( {1 - \beta } \right) S_2 \Vert {I_o - c_2 } \Vert b_2 \nonumber \\&+\, \lambda _2 \left( {1 - \beta } \right) g_2 \left[ {\Vert {c_2 + S_2 \left( {I_o - c_2 } \right) } \Vert ^2 - \Vert c_2 \Vert ^2 } \right] \nonumber \\&+\, \lambda _2 \left( {1 - \beta } \right) \left[ {\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m } \right] \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {u\left( Y \right) } \right] ^m \Vert {I(Y) - c_2 - S_2 \left( {I_o - c_2 } \right) } \Vert } ^2 } \right\} \nonumber \\&+\, \lambda _2 \left( {1 - \beta } \right) \left[ {\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m } \right] \nonumber \\&\left\{ {\sum \limits _\varOmega {W_{oY} \left[ {u\left( Y \right) } \right] ^m \Vert {I_o - c_2 } \Vert ^2 \left[ {1 - S_2 } \right] ^2 } } \right\} , \end{aligned}$$

(53)

where

$$\begin{aligned} a_1= & {} \sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \\ a_2= & {} \sum \limits _\varOmega {\left[ {1 - u\left( X \right) } \right] } ^m \\ S_1= & {} \frac{{u_n^m - u_o^m }}{{\sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m + u_n^m - u_o^m }} \\ S_2= & {} \frac{{\left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m }}{{\sum \limits _\varOmega {\left[ {1 - u\left( X \right) } \right] } ^m + \left( {1 - u_n } \right) ^m - \left( {1 - u_o } \right) ^m }} \\ b_1= & {} \sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] ^m } I(Y)} \right\} \\ b_2= & {} \sum \limits _\varOmega {\left[ {1 - u\left( X \right) } \right] } ^m \left\{ {\sum \limits _\varOmega {W_{XY} \left[ {u\left( Y \right) } \right] ^m } I(Y)} \right\} \\ g_1= & {} \sum \limits _\varOmega {\left[ {u\left( X \right) } \right] } ^m \left\{ {\sum \limits _\varOmega {W_{XY} \left[ {1 - u\left( Y \right) } \right] ^m } } \right\} \\ g_2= & {} \sum \limits _\varOmega {\left[ {1 - u\left( X \right) } \right] } ^m \left\{ {\sum \limits _\varOmega {W_{XY} \left[ {u\left( Y \right) } \right] ^m } } \right\} . \end{aligned}$$