Objective

2

Multiphase image segmentation

an extension of two-phase image segmentation

is designed to partition images automatically into different regions according to different image features. It is a basic problem in image processing

image analysis

and computer vision. Variational image segmentation Vese-Chan model is a basic model of multiphase image segmentation that can construct characteristic functions for different phases or regions using fewer label functions

thus producing small-scale solutions. Graph cut (GC) algorithm can transform the optimization problem of energy function into the min-cut/max-flow problem

which greatly improves computational efficiency. In the spatially discrete setting

the computational results of the min-cut method are influenced by the discrete grid

resulting in measurement errors. In recent years

the continuous max-flow (CMF) method was proposed. As a continuous expression of the classical GC algorithm

CMF can keep the high efficiency of the GC algorithm and overcome measurement errors caused by the discretization of the classical GC algorithm. On the basis of the framework of variational theory

the CMF method for multiphase image segmentation Potts model and two-phase image segmentation Chan-Vese model was proposed and studied. However

a CMF method for the Vese-Chan model has not been studied. Therefore

we propose a CMF method for multiphase image segmentation Vese-Chan model based on convex relaxation and study its computational effectiveness and efficiency.

Method

2

In this study

binary label functions are used to construct different characteristic functions for different phases according to the relationship between a natural number and a binary representation of partitioned regions. The characteristic functions are divided into two parts according to the value of binary expression

i.e.

0 or 1. The characteristic functions are different. The date term of the segmentation model is also divided into two parts. Therefore

multiphase image segmentation can be transformed into two-phase image segmentation. This model can be expressed as a symmetric form of label functions

which are beneficial to interpret and realize. We introduce three dual variables

namely

source flow

sink flow

and spatial flow field. Next

we rewrite the optimization problem of energy function for the Vese-Chan model

including the above three dual variables. Flow conservation conditions are obtained by solving minimum problems of energy function. The Vese-Chan model can be transformed into a continuous max-flow problem corresponding to the min-cut problem. To improve computational efficiency in the experiments

we likewise design the alternating direction method of multipliers (ADMM) by introducing Lagrange multipliers and penalty parameters for the proposed model. The main idea of alternating optimization is to solve the optimization problem of one variable by fixing other variables. Therefore

the optimization problem of energy function can be transformed into three simple sub-problems of optimization

which can be achieved and solved more easily. For example

to solve one sub-problem on the source flow variable

the sink flow variable and spatial flow field variable should be fixed. Three dual variables must be calculated and Lagrange multipliers should be updated at each step. All of these variable need projection to satisfy the range of values. When the energy error formula is satisfied

the computational iteration stops. To represent the boundary of images after segmenting

it is necessary to threshold convex relaxed label functions into binary label functions. Finally

we can obtain segmentation results according to the binary label functions.

Result

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Numerical experiments are performed on gray and colored images. According to the area numbers of images

numerical experiments are divided into three parts: experiments using two binary label functions

three binary label functions

and four binary label functions. Segmentation results are represented by curves of different colors. In particular

to accurately represent the segmentation effects for complicated images

we obtain the approximate segmentation results. For the segmentation effectiveness

experimental results prove that the ADMM method and our proposed method have the same segmentation effectiveness for simple synthetic images. However

compared with ADMM

our proposed method is more accurate for complex images

such as medical and remote sensing images. Moreover

our method can achieve better separation for segmented objects and background. For the computational efficiency

we use two binary label functions to compare the efficiency of four gray images

including synthetic images

a natural image

and a medical image. The acceleration ratios of computational time for our proposed method are 6.35%

10.75%

12.39%

and 7.83%

respectively. For the experiment of three binary label functions

three gray images are compared

including synthetic images and a remote sensing image are compared. The computational times of our proposed method improve by 12.32%

15.45%

and 14.04% for each image. For the experiment of four binary label functions

we compare two color images

including a natural image and a synthetic image. The computational times of our proposed method improve by 16.69% and 20.07%. In the experiments

our proposed method reduces the number of iterations and improves the convergence speed. By comparing the acceleration ratio of computational time with the increase of region phases or the complexity of the image

the advantages of the computational efficiency for our method are more evident.

Conclusion

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Continuous max-flow method is used to solve the multiphase image segmentation Vese-Chan model. Numerical experiments are performed to demonstrate the superiority of our method in terms of computational effectiveness and efficiency for medical images

remote sensing images

and color images. Our method can be applied to multiphase segmentation for three-dimensional reconstruction of medical images in the future.