02 Nov 2017

We consider a new approach to image segmentation following the Chan-Vese Model. The idea would be to find drawbacks, followed by their removal and to create a faster model by combining it with Split Bregman Method and Globally Convex Image Segmentation method.

Globally Convex Method transforms a non-convex problem into a convex one .Split Bergman method is used to update values of the Level Set Functions in Chan-Vese Model. We are optimizing the time of computation, and enhancing the segmentation of an image in the Chan-Vese model.\\

Keywords: Chan-Vese Model, Real Images, Image Segmentation, Split Bregman.

INTRODUCTION

Overview

Image segmentation is an important part of the Digital Image Processing \cite{RafalC.Gonzalez2009}. Segmentation is the method to split a digital image into many segments. Each segment is the set of pixel and this process is carried until we do not get the results. Most of the algorithm in segmentation are based on the two basic properties Similarity and Discontinuity. Similarity is the same intensity level of that particular part. Discontinuity is the change in intensity in the digital image. This also follows some features that are based on certain boundaries, color, intensity or any other pattern.

The first method is follow two steps, first being feature extraction, which is found out by the different

characteristics of each pixel in the particular region and secondly is clustering which is divide the part in the

meaningful way. Second method is finds the boundary between segments on the basis of some rules. On the basis of pixel

value two pixels can be distinguished in one segment and then split them in to two segments. We can also merge them on

the basis of pixel value. Third method is edge based detection that's follow the two patterns one is edge detection and

another is edge linking. Edge detection is based on the masking method and edge linking also follows this.

Many models have been proposed till today in image segmentation field. Some of them are currently using in many real

problem. Mumford-Shah Model \citep{D.Mumford1989} is one of the famous models in this field and they also proposed the many problems in the image segmentation field. Halo, Damson, Paragons are the some other models. These all models are using image gradient. Image gradient is directional change of the intensity or the color of the image. These methods are very slow and also have the natural vacuum and overlap.

\subsection{New Models in Segmentation Field}

In 2001 Vese and Chan \citep{TonyF.Chan2001} proposed the new model on the basis of Mumford and Shah Model \cite{D.Mumford1989}. This model is much better than the previous model. Chan-Vese model suggested in two cases Piecewise Smooth and Piecewise Constant.

Chan-Vese model for active contour \cite{TonyF.Chan2001} is a perfect and flexible method which is can segment many kinds of images, as well as some of them very though to segment when we use the classical model to segment the image on the basis of threshold value and gradient method. This model is based on the Mumford -Shah Method \cite{D.Mumford1989} for segmentation and it is extensively used in various fields like medical imaging area, such as if we want to get the image means segment of brain from the whole the image. This is very sensitive work in image segmentation field.

Chan-Vese model \citep{TonyF.Chan2001} is grounded on the energy minimization problem, which can be reformulated in the level se formulation, leading to an easier way to solve the problem. This model is compared in three points

\begin{enumerate}[(i)]

\item This model is mechanically remove the problem of vacuum and overlap by creation.

\item It needs fewer level set functions to represent the same number of phases in the piecewise constant case.

\item At last good point is that it can symbolize boundaries with compound topologies, including triple junctions.

\end{enumerate}

In this report first we will explain about the previous models then about the talking about Chan-Vese Model and at last about the new approach. The image given below represent in two parts first is using Chan-Vese Model and second is using gradient based method as Mumford-Shah Model.

\begin{figure}[h]

% Requires \usepackage{graphicx}

\centering{\includegraphics[scale=1]{First}

\caption{Comparison of Chan-Vese model image and Old model's image \citep{TonyF.Chan2001}}}

\end{figure}

\clearpage

\section{LITERATURE REVIEW}

\subsection{Modification of Mumford-Shah Model}

Let $Sample-Image$ represent the specified gray scaling image on a territory $\Omega$ to be segmented. The Chan-Vese method \cite{TonyF.Chan2001} is stimulated through the Mumford-Shah model \cite{D.Mumford1989}. Mumford and Shah estimated the image $Sample-Image$ by a piecewise-smooth function u as the solution of the minimization problem

\begin{equation}

\underset{u,C}{arg min}\; \mu Length(C)+ \lambda \int_{\Omega}{(f(x))-u(x))^{2}}dx + \int_{\Omega\\C} \mod{\triangledown

u(x)}^{2}dx

\end{equation}

Where C is an edge set curve and u is permitted to be discontinuous. The first part of the given above formula confirms the symmetry of C, the second part inspires u to be nearby to $Sample-Image$, and the third part of the formula verifies u is differentiable on $\Omega/C$. The Mumford-Shah approximation advises picking this edge set C as the segmentation border. The following example exemplifies the variance between the Mumford-Shah and Chan-Vese function models. The Mumford-Shah solution was designed using the reduced Ambrosio-Tortorelli method of Vese and Chan.

\begin{figure}[h]

\centerline{\includegraphics[scale=1]{Chen}}

\caption{Comparison of models \citep{TonyF.Chan2001}}

\end{figure}

Even though this is a usual way to pose segmentation, algorithms for answering the general Mumford-Shah model incline to be moderately complicated and computationally expensive. As a explanation, Mumford and Shah also measured a piecewise constant formulation,

\begin{equation}

\underset{u,C}{arg min}\; \mu Length(C) + \int_{\Omega }{f(x))-u(x))^{2}}dx,

\end{equation}

In this formula u is compulsory to be constant on each linked component of $\Omega/C$. In this situation, C is essentially the boundary of a closed set and that C is composed of closed curves. The main problem of minimization is non-convex. Presence of a solution of the piecewise constant model was showed in \citep{D.Mumford1989}. When Chan-Vese model is related to the piecewise constant Mumford-Shah model, the main variances are an additional term penalizing the surrounded area and a further explanation that u is permissible to have only two values,

\begin{equation}

u(x) = \begin{cases}

& \text{ c1 \! where x is inside C,} \\

& \text{ c2 \! where x is outside C,}

\end{cases}

\end{equation}

Where C is the border of a closed set and the inside and outside values of C are c1 and c2. $\varphi(x)$ is define 0 for the boundary case. The Chan-Vese model is to catch among all u of this form and that is the best approximates for the given image,

$\underset{c1,c2,C}{arg min} \mu Length(C) + \nu Arrea(inside(C)) +$

\begin{equation}

\lambda _{1} \int_{inside(C)}{\left | f(x)-c_{1} \right |^{2}}dx + \lambda _{2} \int_{outside(C)}{\left | f(x)-c_{2} \right

|^{2}}dx

\end{equation}

The first part of the formula controls the orderliness by disciplining the length. The second part disciplines the enclosed zone of C to control its size. The third and fourth parts discipline difference between the piecewise constant model u and the input image $Sample-Image$. By conclusion a local minimizer of this problem, a segmentation is gained as the best two-phase piecewise constant approximation u of the image $Sample-Image$.

\subsection{Chan-Vese Model's Level Set Functions}

The minimization problem needs reducing over all fixed boundaries C. This is consummate by put on the level set technique. In its place of manipulating C clearly, it is signified as the zero-crossing of a level set function $\varphi$ by the connection.

\begin{equation}

\emph{C} = {\emph{x}\; \epsilon\; \Omega : \varphi(\emph{x}) = 0}.

\end{equation}

Additionally the inside and outside of C are eminent by the sign of $\varphi$.On condition that $\varphi$ is smooth enough and that $\varphi$ is definitely a distance function (i.e., $|\varphi| = 1$), then the Chan-Vese minimization can be consistently rewritten in terms of the level set function $\varphi$ as

$\underset{c1,c2,\varphi }{arg min} \mu \int_{\Omega }{\delta (\varphi (x))\left| \triangledown \varphi (x) \right |}dx+$

\begin{equation}

\nu \int_{\Omega }{H(\varphi (x))}dx +\\ \lambda_{1} \int_{\Omega }{\left | f(x)- c_{1} \right |^{2}H(\varphi (x))}dx

+ \lambda _{2} \int_{\frac{\Omega}{C}}{\left | f(x)-c_{2} \right |^{2}(1-H(\varphi (x)))}dx

\end{equation}

Where H represents the Heaviside function and $\delta$ the Dirac mass, its distributional imitative,

\begin{equation}

H(t) = \begin{cases}

& \text{1 if } t\geq 0, \\

& \text{0 if } t< 0,

\end{cases}

\delta (t)= \frac{d}{dt}H(t)

\end{equation}

$H(\varphi)$ is the pointer function of the set surrounded by C. Through this linking, the area of $\varphi$ is gained in the second integral as the integral of $H(\varphi)$ in the second part of the minimization. The first integral holding the Dirac mass $\varphi$ is not actually an integral, but a curve integral laterally the boundary C. On condition that $\varphi$ is a distance function, the length of C is gained as the total difference of $H(\varphi)$,

\begin{equation}

Length(C) = \int_{\Omega }{\left | \triangledown H (\varphi (x))\right |}dx = \int_{\Omega }{\delta (\varphi (x))

\left | \triangledown \varphi (x) \right |}dx

\end{equation}

The minimization is explained by alternatingly modernizing c1, c2 and $\varphi$. For fixed value of the $\varphi$,the finest values of c1 and c2 are the region averages

\begin{equation}

c1 = \frac{\int{_\Omega}f(x)H(\varphi(x))dx}{\int{_\Omega}H(\varphi(x))dx},

c2 = \frac{\int{_\Omega}f(x)H(\varphi(x))dx}{\int{_\Omega}H(\varphi(x))dx},

\end{equation}

For the minimization with respect to $\varphi$, H is normalized as

\begin{equation}

H_{\varepsilon} = \frac{1}{2}(1+\frac{2}{\pi}arctan(\frac{t}{\varepsilon}))

\end{equation}

and $\delta_{\varepsilon}$ is its derivative,

\begin{equation}

\delta_{\epsilon}(t) := \frac{d}{dt}H_{\epsilon}(t) = \frac{\epsilon}{\pi(\epsilon^{2}+t^{2}}.

\end{equation}

In the implementation, $\varepsilon = 1$. Tor fixed c1 and c2, gradient descent with respect to $\varphi$ is

\begin{equation}

\begin{cases}

& { \frac{\partial\varphi}{\partial x} = \delta _{\varepsilon}(\varphi ) \left [ \mu div(\frac{\triangledown \varphi }

{\left | \triangledown \varphi \right |})-\nu -\lambda _{1}(f-c_{1})^{2} + \lambda _{2}(f-c_{2})^{2} \right ] } \; in \;

\Omega, \\

& { \frac{\delta _{\epsilon }{\varphi }}{\left |\triangledown \varphi \right |}\frac{\partial \varphi }

{\partial\overrightarrow{n}} = 0 } \; on \; \partial \Omega

\end{cases}

\end{equation}

where $\vec{n}$ is the outward normal on the image boundary.For the initialization, an operative choice is the function

\begin{equation}

\varphi(x) = \sin(\frac{\pi}{5}x)\sin(\frac{\pi}{5}y)

\end{equation}

Which describes the initial segmentation as a checkerboard shape. This initialization has been witnessed to have dissolute convergence. When using this initialization with $\nu = 0$, the meaning of the result's \lq\lq inside\rq\rq ~vs. \lq\lq outside\rq\rq ~are arbitrary.On the other hand, one may specify a contour C for the initialization. A level set representation of C can be created as

\begin{equation}

\varphi(x) = \{\frac{+1 where x is inside C}{-1 where x is outside C}

\end{equation}

\clearpage

\subsection{Chan-Vese Model is Extension to Vector-Valued Data}

\cite{TonyF.Chan2000} model is an extension to segmentation of vector-valued data,

$\underset{c1,c2,C}{arg min} \mu Length(C) + \nu Arrea(inside(C))+$

\begin{equation}

\lambda _{1} \int_{inside(C)}{\left || f(x)-c_{1} \right ||^{2}}dx + \lambda _{2} \int_{outside(C)}{\left || f(x)-c_{2}

\right ||^{2}}dx

\end{equation}

Here, the variety of $Sample-Image$ is d-dimensional, and c1 and c2 are the vectors of size d. As in the scalar part, c1 and c2 are the region averages of $Sample-Image$ inside and outside of C. In the case of color images, c1 and c2 are the average color values of the two segments. The source code involved with this article implements Chan-Sandberg-Vese for any number of variety dimensions. The Chan-Sandberg-Vese \citep{TonyF.Chan2000} problem is the same as the Chan-Vese minimization excluding that the absolute values $|f(x) - c1|$ and $|f(x) - c2|$ are swapped with the Euclidean vector norms $||f(x) - c1||$ and $||f(x) - c2||$. All image channels contribute mutually in driving the segmentation process. The problem can be characterized by a level set and discretized equally, foremost to the semi-implicit update of $\varphi$.

\subsection{Multiphase Chan-Vese Model}

In this section, we show how we can generalize the 2-phase active contour model without edges \cite{TonyF.Chan2001}. We note again that, using only one level set function, we can represent only two phases or segments in the image. Also, other geometrical features, such as triple junctions, cannot be represented using only one level set function. Our goal is to look for a new multiphase level set model with which we can represent more than two segments or phases, triple junctions and other complex topologies, in an efficient way. We will need only $log_{2}n$ level set functions to represent n phases or segments with complex topologies, such as triple junctions \cite{VESE2002}. In addition, our formulation automatically removes the problems of vacuum and overlap, because our partition is a disjoint decomposition and covering of the domain $\Omega$ by definition.

Let us consider m = log n level set functions $\varphi$ : $\Omega$ ??$\rightarrow$ R. The union of the zero-level sets of $\varphi_{i}$ will represent the edges in the segmented image. We also introduce the "vector level set function" $\varphi = (\varphi_{1}, . . . , \varphi_{m})$, and the ??"vector Heaviside function" $H(\varphi) = (H(\varphi_{1}), . . . , H(\varphi_{m}))$ whose components are only 1 or 0.We can now define the segments or phases in the domain$\Omega$, in the following way: two pixels $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in $\Omega$ will belong to the same phase or class, if and only if $H(\varphi(x_{1}, y_{1})) = H(\varphi(x_{2}, y_{2}))$. In other words, the classes or phases are given by the level sets of the function $H(\varphi)$, i.e. one class is formed by the set in which one phase or class contains those pixels (x, y) of $\Omega$ having the same value $H(\Omega(x, y))$. There are up to $n = 2^{m}$ possibilities for the vector values in the image of $H(\varphi)$. In this way, we can define up to $n = 2^{m}$ phases or classes in the domain of definition $\Omega$. The classes defined in this way form a disjoint decomposition and covering of $\Omega$. Therefore, each pixel (x, y) of $\Omega$ will belong to one, and only one class, by definition, and there is no vacuum or overlap among the phases.

We label the classes by $\textit{I}$ , with $1 \leq \textit{I} \leq 2^{m} = n$. Now, let us introduce a constant vector of averages $c = (c_{1}, . . . , c_{n})$, where $c_{\textit{I}} = mean(u_{0})$ in the class $\textit{I}$ , and the characteristic function $\chi_{I}$ for each class $\textit{I}$.

\begin{equation}

F_{n}^{MS}(c,\varphi)= \sum_{1\leq I\leq n=2^{m}}\int _{\Omega} (u_{0}(x,y)-c_{I})^{2}\chi_{I}dx dy + \nu \frac{1}{2}\sum_{1\leq I\leq n=2^{m}}\int _{\Omega}\left | \triangledown \chi_{I} \right |

\end{equation}

In order to simplify the model, we will replace the length term by $\sum _{i} \int _{\Omega}\left | \triangledown H(\varphi_{i}) \right|$ (i.e. the sum of the length of the zero-level sets of $\varphi_{i}$ ). Thus, in some cases, some parts of the curves will count more than once in the total length term, or in other words, some edges will have a different weight in the total length term.We will see that with this slight modification and simplification, we still obtain very satisfactory results (it may have only a very small effect in most of the cases, because the fitting term is dominant).Therefore, the energy that we will minimize is given by:

\begin{equation}

F_{n}(c,\varphi)= \sum_{1\leq I\leq n=2^{m}}\int _{\Omega} (u_{0}-c_{I})^{2}\chi_{I}dx dy + \sum_{1\leq I\leq n=2^{m}}\nu \int _{\Omega}\left | \triangledown H(\varphi_{i}) \right|

\end{equation}

\begin{figure}[h]

\centerline{\includegraphics[scale=1]{Multiphase}}

\caption{2 curves partition the domain into 4 regions}

\end{figure}

Clearly, for n = 2 (and therefore m = 1), we obtain the active contour model without edges. For the purpose of illustration, let us write the above energy for n = 4 phases or classes (and therefore using m = 2 level set functions; see in Figure above): \\

$F_{4}(c,\varphi)=\int_{\Omega}(u_{0}-c_{11})^{2}H(\varphi_{1})H(\varphi_{2})dxdy+\int_{\Omega}(u_{0}-c_{10})^{2}

H(\varphi_{1})(1-H(\varphi_{2}))dxdy+$

$\int_{\Omega}(u_{0}-c_{01})^{2}(1-H(\varphi_{1}))H(\varphi_{2})dxdy+\int_{\Omega}(u_{0}-c_{10})^{2}(1-H(\varphi_{1}))

(1-H(\varphi_{2}))dxdy+$

\begin{equation}

\nu \int _{\Omega}\left | \triangledown H(\varphi _{1}) \right |+\nu \int _{\Omega}\left | \triangledown H(\varphi _{2}) \right |

\end{equation}

where $c = (c_{11}, c_{10}, c_{01}, c_{00})$ is a constant vector, and $\varphi = (\varphi_{1}, \varphi_{2})$.

With these notations, we can express the image function u as:

$u=c_{11}H(\varphi _{1})H(\varphi _{2})+c_{10}H(\varphi _{1})(1-H(\varphi _{2}))+c_{01}(1-H(\varphi _{1}))H(\varphi _{2})+c_{00}(1-H(\varphi _{1}))(1-H(\varphi _{2}))$

The Euler-Lagrange equations obtained by minimizing (18) with respect to c and $\varphi$, embedded in a dynamical scheme, are: given $\varphi_{1}(0, x, y) = \varphi_{1,0}(x, y)$,$\varphi_{2}(0, x, y) = \varphi_{2,0}(x, y)$,

\begin{eqnarray*}

c_{11}(\varphi)=mean(u_{0})in{(x,y):\varphi_{1}(t,x,y)>0,\varphi_{2}(t,x,y)>0}\\

c_{10}(\varphi)=mean(u_{0})in{(x,y):\varphi_{1}(t,x,y)>0,\varphi_{2}(t,x,y)<0}\\

c_{01}(\varphi)=mean(u_{0})in{(x,y):\varphi_{1}(t,x,y)<0,\varphi_{2}(t,x,y)>0}\\

c_{00}(\varphi)=mean(u_{0})in{(x,y):\varphi_{1}(t,x,y)<0,\varphi_{2}(t,x,y)<0}

\end{eqnarray*}

We note that the equations in $\varphi=(\varphi_{1},\varphi_{2})$ are governed by both mean curvature and jump of the data energy terms across the boundary.

It is easy to extend the proposed model to vector valued functions, such as color images \cite{TonyF.Chan2000}. In this case, $u_{0} = (u_{0,1}, . . . , u_{0,N})$ is the initial data, with N channels (N = 3 for color RGB images), and for each channel i =1, . . . , N, we have the constants $c_{I} = (c_{I,1}, . . . , c_{I,N})$. In this case, the model for multichannel segmentation will be:

\begin{equation}

F_{n}(c_{I},\varphi)= \sum_{1\leq I\leq n=2^{m}}\sum_{i=1}^{N}\int _{\Omega} (u_{0,i}-c_{I,i})^{2}\varphi_{I}dx dy +\sum_{1\leq I\leq n=2^{m}} \nu \int _{\Omega}\left | \triangledown H (\varphi_{i}) \right |

\end{equation}

Note that, even if we work with vector-valued images, the level set functions are the same for all channels (i.e.

we do not need additional level set functions for each channel). The associated Euler-Lagrange equations can

easily be deduced.

\clearpage

\section{MOTIVATION}

Image segmentation is the essential part of image processing. It can found everywhere suppose we want to check the one table into the image then we need to apply the image segmentation to get the detail about that table. Suppose we want to check out the every part in the single image then we image segmentation to get the detail. Image segmentation commonly assists as pre-processing before image pattern recognition, image feature extraction and image compression. Researches of it started nearby 1970, find out the new models which make this more perfect field. They used the models in various field. Now this is very important part of the medical field, security field etc.

Chan-Vese Model is very good model. It do not need image gradient that's make this much better. After Mumford-Shah

Model's modification this model becomes best model in this field. Currently many researcher works on this model. Some good points of this model like triple junction, vacuum make this favorite of the researcher. But in the medical image field we need faster model. That's why we need good modification in this model. I will work on the multiphase segmentation model which is primarily focus on the homogeneous as well as inhomogeneity.Every medical machine need faster model today.

\clearpage

\section{OBJECTIVES}

To devise a model which optimally performs image segmentation incurring less time. Our model shall address the drawbacks of Chan-Vese model, and will enhance its efficiency. I will exploit Split Bregman method \cite{T.Goldstein2009} and Globally Convex image segmentation method \cite{Chan2007} to optimize the chan-vese model.

\begin{enumerate}[(1)]

\item We envisage a model performing an efficient segmentation.

\item To compare our model with others to ensure its desired efficiency.

\end{enumerate}

\clearpage

\section{METHODOLOGY}

\subsection{Overview}

I will try to modify the Chan -Vese Model. But there is some drawbacks which is already mentioned.For the improvement \cite{EthanS.BrownMay2012} of this model i will use the two methods

\begin{enumerate}[(i)]

\item Split Bregman Method \citep{T.Goldstein2009}.

\item Globally Convex image segmentation method \citep{Chan2007}.

\end{enumerate}

The Split Bregman method is a technique used for solving image segmentation problems which uses compressed sensing to remove noise from the image. Split Bregman is one of the fastest solvers for image segmentation problems uses compressed sensing. This method removes noise from the image and make this model more fast and optimize.

\begin{figure}[h]

\centerline{\includegraphics[scale=0.8]{Second}}

\caption{Two part Chan-Vese Model : Work Methodology}

\end{figure}

Globally Convex Image Segmentation Method behaves convexly on triples of collinear points that are widely dispersed. Chan-Vese model is best for triple junction problem but not optimized. For that we will use the globally convex image segmentation method to remove the triple junction problem.

Further step taken is:

\begin{enumerate}[(i)]

\item We will take the multiphase chan-vese model in which we combine the Globally Convex Image Segmentation and Split Begman Method.

\end{enumerate}

Proposed algorithm will be implemented in \textbf{}Matlab. Results obtained from the algorithm will be compared and verified with existing solutions. Models which we will be used for the comparison are Chan - Vese model, Gaussian Mixture Model \cite{GuangJianTianMAY2011}, Dimensionality Reduction Model \cite{Mignotte2011}, and Level Set Method for Image Segmentation \cite{ChunmingLiJULY2011}.

\clearpage

\textbf{Effect of Varying $\mu$}:\\

While Chan???Vese has quite a few tuning parameters, the most important is ??. Parameter ?? adjusts the length penalty, which balances between fitting the input image more accurately (smaller $\mu$) vs. producing a smoother boundary (larger $\mu$). In the example below, there are two groups of circles.

Depending on ??, the circles are either segmented individually or as two clusters.

\begin{figure}[h]

\centerline{\includegraphics[scale=0.7]{Mu}}

%\caption{}

\end{figure}

\textbf{Effect of Varying $\nu$}:\\

Parameter $\nu$ sets the penalty (or reward, if $\nu < $0) for area inside C. Note that this parameter is meaningful only when there is a prescribed inside vs. outside of the segmentation boundary. In the figure below, the evolution is shown over 1000 iterations with five different values of $\nu$ where the initialization is a circle. When $\nu$ is too negative, the boundary expands to fill the full domain, and when $\nu$ is too positive, the boundary shrinks until it vanishes.

\begin{figure}[h]

\centerline{\includegraphics[scale=0.7]{Nu}}

%\caption{}

\end{figure}

\textbf{Changes of Iteration}:\\

Increment of iterations tends to produce more accurate results as compare to earlier results. On an average about 800 Iteration are sufficient to segment the image correctly but in case the input too noisy then we may have to increase the iteration up to 3000 but the overhead here is its time taken by algorithm, the more the iteration are the more time it will take.

Time Complexity of the Chan-Vese Model is ascertained to be O($n^2$). \\

\clearpage

\subsection{Implementation of Update Model}

\begin{itemize}

\item We are proposing a new and fast model for multiphase image segmentation based on the piecewise constant case of Chan-Vese Mode. The Globally convex image segmentation method\cite{Chan2007} and the Split Bregman method \cite{T.Goldstein2009}, \cite{Yin2010}, \cite{.Yang2011} are using for the improvement of this model.In this section we are using color images for segmentation. Let $u_{0}:\Omega\rightarrow R^{3}$ be a given color image and $u_{0}$ is the given color image. Let c$_{i}=(c_{1},c_{2},c_{3},c_{4})(i=1,2,3,4)$ be the average vector of the $i_{th}$ region. First we propose a new energy function equation.

\begin{equation}

\underset{\varphi _{1},\varphi _{2}}{min}E(\varphi _{1},\varphi _{2})=\underset{\varphi _{1},\varphi _{2}}{min}(\left | \triangledown H(\varphi _{1}) \right |+\left | \triangledown H(\varphi _{2}) \right |+\sum_{1\leq I\leq 2^{m}}((u_{0}-c_{I})^{2}\chi_{I}))

\end{equation}

Where first term represent Length of the first curve and second term represent Length of second curve and the last term represent difference of piecewise constant value and the pixels values of particular $I^{th}$ region and chi is characteristic function of particular region.

\item Now apply the Globally Convex on the given Image. Globally convex method is used to nonconvex optimization problems that arise in image processing and computer vision can be restated as convex minimization problem. Often a function surface can be very un-smooth, having many sharp local minima, making it hard to find the overall global minimum. In particular it's finding the global minima. Chan-Vese Model is used the nonconvex optimization that's why this is always finding the local minima not the Global minima. Globally convex uses the edge detector to make its edge more smooth and sharp. Through this graph you can understand if the noise if the noise in the image too much then it cannot able to find global minima. So first remove noise from the image with edge detector.

\begin{figure}[h]

\centerline{\includegraphics[scale=0.6]{Global}}

\caption{https://www.mathworks.com/matlabcentral/fileexchange/27631-derivative-based-optimization}

\end{figure}

\clearpage

Where as mentioned by \cite{Chan2007},

\begin{equation}

g(\left | \triangledown u_{0}(x) \right |)=\frac{1}{1 + \beta(\left | \triangledown u_{0}(x) \right |)^{2}}

\end{equation}

is an edge detector function \cite{X.Bresson2009}, \cite{T.Goldstein2010} and $\beta$ is a parameter that determines the detail level of the segmentation.

When $u_{0}$ is a color image, for each pixel x = (x,y), $(\left | \triangledown u_{0}(x) \right |)^{2}$ is defined as

\begin{equation}

\left | \triangledown u_{0}(x) \right |^{2}= \left \{ \left [ \left (\frac{\partial u_{01}}{\partial x} \right )^{2}+\left (\frac{\partial u_{02}}{\partial x} \right )^{2}+\left (\frac{\partial u_{03}}{\partial x} \right )^{2} \right ]+\left [ \left (\frac{\partial u_{01}}{\partial y} \right )^{2}+\left (\frac{\partial u_{02}}{\partial y} \right )^{2}+\left (\frac{\partial u_{03}}{\partial y} \right )^{2} \right ] \right \}

\end{equation}\

Where $u_{0i}$ belongs to image's ith color part. Given $\varphi_{1}(0,x)=\varphi_{1,0}(x),\varphi_{1}(0,x)=\varphi_{1,0}(x)$,Once the minimization problem is solved in the equation(20), the image domain can be partitioned into four region $\Omega_{i}(i=1,2,3,4)$ by thresholding the level set functions as follows:

\begin{eqnarray*}

\Omega _{1}=\left \{ x:\varphi_{1}(x)> \alpha,\varphi_{2}(x)> \alpha \right \},\\

\Omega _{2}=\left \{ x:\varphi_{1}(x)> \alpha,\varphi_{2}(x)<\alpha \right \},\\

\Omega _{3}=\left \{ x:\varphi_{1}(x)< \alpha,\varphi_{2}(x)>\alpha \right \},

\end{eqnarray*}

\begin{equation}

\Omega _{4}=\left \{ x:\varphi_{1}(x)< \alpha,\varphi_{2}(x)<\alpha \right \},

\end{equation}

for some $\alpha$ belongs to (0,1) range, in this project we choose $\alpha$= 0.5. This is different from \cite{VESE2002} where the zero level set is used to identify the boundary. We use the $\alpha$ = 0.5 level set to identify the boundary because we restricted the range (0,1).

Then the average vector $c_{i}=(c_{1},c_{2},c_{3},c_{4})(i=1,2,3,4)$ for each region can be updated by

\begin{eqnarray*}

c_{1i}=mean(u_{0i}) in \Omega{1},\\

c_{2i}=mean(u_{0i}) in \Omega{2},\\

c_{3i}=mean(u_{0i}) in \Omega{3},

\end{eqnarray*}

\begin{equation}

c_{4i}=mean(u_{0i}) in \Omega{4}

\end{equation}

Let u be the fitting image of the given color image $u_{0}$.Minimized the Integration and Heaviside function then fitting image u can be expressed as

\begin{equation}

u_{0}=c_{1}\varphi _{1}\varphi_{2}+c_{2}\varphi_{1}(1-\varphi_{2})+c_{3}(1-\varphi _{1})\varphi_{2}+c_{4}(1-\varphi _{1})(1-\varphi_{2})

\end{equation}

The fitting image can be represent in four segment phases of u as $Phase_{1},Phase_{2}$,$Phase_{3},Phase_{4}$ and these can be represent as

\begin{eqnarray*}

Phase_{1}=c_{1}\varphi_{1}\varphi_{2}\\

Phase_{2}=c_{2}\varphi_{1}(1-\varphi_{2})\\ Phase_{3}=c_{3}(1-\varphi_{1})\varphi_{2}\\

Phase_{4}=c_{4}(1-\varphi_{1})(1-\varphi_{2})

\end{eqnarray*}

In fact u can be obtained by $u=Phase_{1}+Phase_{2}+Phase_{3}+Phase_{4}$

\item Now we apply the Split Bregman Method. Split Bregman method only work on the convex problem that's why initially we apply the globally convex method.To minimize the energy functional E($\varphi_{1},\varphi_{2}$) in (20) with respect to $\varphi_{1}$ and $\varphi_{2}$, one traditional method is using the standard gradient descent method directly. In this section, split Bregman method used to solve our proposed minimization problem more efficiently. Here the rang of value $\varphi_{1} and \varphi_{2}$ is (0,1). First, we introduce two auxiliary variables: $\overrightarrow{d_{1}}\leftarrow\nabla\varphi_{1}$ and $\overrightarrow{d_{2}}\leftarrow\nabla\varphi_{2}$. Initially we assign the value of $\overrightarrow{d_{1}}$ and $\overrightarrow{d_{2}}$ is 0.5. Then, we add two quadratic penalty functions to weakly enforce the resulting equality constraints and get the unconstrained problem as follows:

$(\varphi_{1}^{*},\varphi_{2}^{*},\overrightarrow{d_{1}}^{*},\overrightarrow{d_{2}}^{*})$

\begin{equation}

=arg \underset{0\leq\varphi_{1},\varphi_{2}\leq 1,\overrightarrow{d_{1}},\overrightarrow{d_{2}}}{min}(\left | \overrightarrow{d_{1}} \right |+\left | \overrightarrow{d_{2}} \right |+\sum_{1\leq I\leq 2^{m}}((u_{0}-c_{I})^{2}\chi_{I} + \frac{\lambda}{2}\left \| \overrightarrow{d_{1}}-\nabla\varphi_{1} \right \|+\frac{\lambda}{2}\left \| \overrightarrow{d_{2}}-\nabla\varphi_{2} \right \|)

\end{equation}

Where $\lambda$ is a positive constant. We then strictly enforce the constraints by applying the Bregman method, and this results in the following optimization problem:

$(\varphi_{1}^{k+1},\varphi_{2}^{k+1},\overrightarrow{d_{1}}^{k+1},\overrightarrow{d_{2}}^{k+1})=$

\begin{equation}

arg \underset{0\leq\varphi_{1},\varphi_{2}\leq 1,\overrightarrow{d_{1}},\overrightarrow{d_{2}}}{min}(\left | \overrightarrow{d_{1}} \right |+\left | \overrightarrow{d_{2}} \right |+\sum_{1\leq I\leq 2^{m}}((u_{0}-c_{I})^{2}\chi_{I} + \frac{\lambda}{2}\left \| \overrightarrow{d_{1}}^{k}-\nabla\varphi_{1}^{k}\right \|+\frac{\lambda}{2}\left \| \overrightarrow{d_{2}}^{k}-\nabla\varphi_{2}^{k}\right \|)

\end{equation}

With the help of these equation we update the value of $\varphi_{1}$ and $\varphi_{2}$ and then we apply these values in chan-vese model's equation. Through these two method we optimize the time and improve the segmentation. Through this method we get the exact value of level set which is very useful for the energy minimization. After one iteration of above equation then $\varphi_{1}^{k+1}-\varphi_{1}^{k}<\epsilon$. If the this condition satisfied then terminate the process. Where $\epsilon$ is $\frac{55}{{255}^{2}}$ for the range of (0,1). The calculation of $\varphi_{1},\varphi{2},\overrightarrow{d_{1}},\overrightarrow{d_{2}}$ done with the help of above equation.

Keep $\varphi{2},\overrightarrow{d_{1}},\overrightarrow{d_{2}}$ fixed, the Euler-Lagrange equation of the optimization problem(27) with respect to $\varphi_{1}$ is $\triangle\varphi_{1}=\frac{\mu}{\lambda}+\nabla.\overrightarrow{d_{1}}$. Similarly calculate the $\overrightarrow{d_{1}}$,$\overrightarrow{d_{2}}$ and $\varphi_{2}$. This can be understand by the example suppose chan-vese model using the 100 iteration to get the segmentation(perfect curve) for the image. But if we know the exact value of level set function. Then it can work with 50 iteration. So through this you can understand its minimum energy as compare to previos. After the level set process at last we apply the standard chan-vese equation to the optimized result for getting better segmentation.

\end{itemize}

\clearpage

\subsection{Algorithm}

\begin{enumerate}

\item Input Image $(u_{0})$,

\item Compute $g(|\nabla_{0}(x)|)$ by (21)

\item Input $\varphi_{1}^{0},\varphi_{2}^{0}$

\item Compute Initial $\Omega_{i}$ by (23)

\item Compute $c_{i}$ by (24)

\item Initialize $\overrightarrow{d_{1}},\overrightarrow{d_{2}}$

\item While $\left|\varphi_{1}^{k+1}-\varphi_{1}^{k}\right| > \epsilon$ or $\left|\varphi_{2}^{k+1}-\varphi_{2}^{k}\right| > \epsilon $ do

\item Update $\varphi_{1}^{k+1},\varphi_{2}^{k+1}$ and $\overrightarrow{d_{1}}^{k+1},\overrightarrow{d_{2}}^{k+1}$ by (26) and (27)

\item end while

\item update $\Omega_{i}$ by (23)

\item update $c_{i}$ by (24)

\item Compute $F_{n}(c_{I},\varphi)$ by (19)

\end{enumerate}

\subsection{Future Work}

\begin{enumerate}

\item To overhaul the model in order to make it function efficiently.

\item Compare the model with other similar ones to ascertain it's effectiveness.

\end{enumerate}

\clearpage

\section{Results and Discussion}

\subsection{Implementation of Chan-Vese Model}

\begin{figure}[h]

\centerline{\includegraphics[scale=0.65]{Brain}}

\caption{Input Image and Output Image(Segmented)- 1}

\end{figure}

\begin{figure}[h]

\centerline{\includegraphics[scale=0.6]{flower1}}

\caption{Input Image and Output Image(Segmented)- 2}

\end{figure}

\begin{figure}[h]

\centerline{\includegraphics[scale=0.6]{flower2}}

\caption{Input Image and Output Image(Segmented)- 3}

\end{figure}

\begin{figure}[h]

\centerline{\includegraphics[scale=0.6]{flower3}}

\caption{Input Image and Output Image(Segmented)- 4}

\end{figure}

\clearpage

\subsection{Execution Time of Chan-Vese Model}

\begin{figure}[h]

\centerline{\includegraphics[scale=0.6]{Time}}

\caption{Execution Time of Test Cases}

\end{figure}

\subsection{Output of Update Model}

\begin{figure}[!h]

\centerline{\includegraphics[scale=0.6]{Output1}}

\caption{Update Result, Real Image and Base Result(Segmented)}

\end{figure}

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