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In this paper, we propose a new soft multi-phase segmentation model where it is assumed that the pixel intensities are distributed as a Gaussian mixture. The model is formulated as a minimization problem through the use of the maximum likelihood estimator and phase-transition theory. The mixture coefficients, which are estimated using a spatially varying mean and variance procedure, are used for image segmentation. The experimental results indicate the effectiveness of the method.

Image segmentation is one of the most extensively studied problems in image processing and computer vision. Many different approaches have been proposed for the partitioning of images based on a variety of criteria including brightness (intensity), color, or texture. In general the partitioning of an image or detection of edges is under the assumption that an image consists of several patterns, and each point on the image domain belongs exclusively to only one pattern. Finding boundaries separating the different patterns in this sense is called a hard segmentation. Different to hard segmentation, soft segmentation assumes that each point may belong to more than one pattern. The goal of soft segmentation is to find all the probabilities that each pixel can belong to each pattern. This probability is also called membership (or ownership) in the literature.

One of the most extensively studied approaches for hard segmentation is the variational method. Many effective variational models have been developed, for instance, the Mumford-Shah model [

To overcome the non-convexity problem mentioned above, one approach is to replace the composition of the heaviside function with the level set function in level set formulation by use of a weight/membership function. Through the implementation of this modification the energy is convex with respect to the membership function. For example, Chan et al. [

Soft segmentation is also motivated by its applications to real world problems. In medical imaging, due to limited spacial resolution of the equipment, not all the voxels in a segmented region contain the same tissue type, especially near the boundary of two subregions. A typical example is the partial volume effect in MRI brain image segmentation. Instead of labeling each image voxel with a unique tissue type [

There have been many soft segmentation methods. Mory and Ardon extended the original region competition model [

partitions a data set

where

cluster centers. The original FCM method is very sensitive to noise. An adaptive fuzzy c-means (AFCM) was proposed by Pham et al. [

where

where

ter

Another class of soft segmentation is based on stochastic approaches. It considers that pixel intensities are independent samples from one or several distributions. The likelihood functions have been widely used in soft segmentation. In [

In [

The phase transition theory in material sciences and fluid mechanics have inspired people to borrow ideas from contemporary material sciences, e.g., the diffuse interface model of Cahn-Hilliard [

where

converges to 0 or 1 as

In paper [

viewed as independent random variables indexed by

Denote by

belonging to the i-th pattern. Then the pdf of the image

Under the assumption that all random variables are independent, we have the following joint pdf

The regularization is made using a double well potential borrowed from the phase-transition theory. By assuming that all patterns are Gaussian distributions with mean fields

where

defines the Gaussian probability density function,

with constraints

where

In this paper, we propose a new multiphase soft segmentation model that integrates phase-transition theory into a mixture of Gaussian model for image intensities. The proposed model is an extension of the paper [

The difference between this work and [

This paper is organized as follows: Section 2 addresses the proposed model development. Section 3 presents the implementation details and experimental results. Finally, the conclusion is given in section 4.

In this section, we develop a soft multiphase segmentation model under the assumption that the intensity of the image is distributed as a mixture of Gaussians.

We assume the intensity

is to estimate the optimal vectorial pair of ownerships

Through the Bayesian formula, the posterior given

assuming that the mixture patterns

By taking the logarithmic likelihood

As assumed, all the samples

where

For energies

Finally, for energy

convergence theory.

where

Now, in combination of (15), (16), (18) and (19), the final proposed segmentation model is the minimization of:

where

Since the energy functional contains three group parameters, in order to minimize the energy, we use the alternating iteration scheme:

Each group parameter can be iterated with its Euler-Lagrange equation. The Euler-Lagrange equation for patterns

where

Considering that

and since

We can solve the equations (21)-(23) using the flow from their associated Euler-Lagrange equations. The flow equation for

The flow equations for

The numerical solution was obtained using finite differences to discretize the flow equations. For the numerical implementation it is supposed that the images are represented by

We denote

Following the same procedure for

where

and

To start the iteration process, we need to choose the initial values for the ownership functions

adopted procedures is: given

In

In

In the following experiments, we tested our model on real images. In

Finally, we take the experiment on a color image, as shown in

In this paper, we extended the idea in paper [

C.A.Z. Barcelos is partially supported by CNPq-Conselho Nacional de Desenvolvimento Científico e Tec- nológico; Y. Chen is partially supported by NSF grants IIP-1237814 and DMS-1319050.