A New and Fast Multiphase Image Segmentation Model for Color Images

This paper presents a new and fast multiphase image segmentation model for color images. We propose our model by incorporating the globally convex image segmentation method and the split Bregman method into the piecewise constant multiphase Vese-Chan model for color images. We have applied our model to many synthetic and real color images. Numerical results show that our model can segment color images with multiple regions and represent boundaries with complex topologies, including triple junctions. Comparison with the Vese-Chan model demonstrates the efficiency of our model. Besides, our model does not require a priori denoising step and is robust with respect to noise.


Introduction
Image segmentation 1-8 is an important technique for detecting objects and analyzing images in computer vision and image processing.Image segmentation is concerned with the task of partitioning a given image into different classes or regions corresponding to different objects and the background in the image.
Mumford and Shah 9 formulated the image segmentation problem and proposed the famous Mumford-Shah model.For a special case of the Mumford-Shah model, Chan and Vese 1 proposed the famous Chan-Vese model without using the image gradient.Chan and Vese gave the two-phase level set formulation for gray images in 1 .Then, the authors extended their scalar Chan-Vese algorithm to the vector-valued case in 2 .There are many other image segmentation models for color images, and we mention the works in 10, 11 .Either the scalar Chan-Vese algorithm in 1 or the Chan-Vese model for vector-valued images in 2 is mainly for images with two phases.To deal with images with multiple regions, Vese and Chan extended their original two-phase Chan-Vese model 1, 2 to a multiphase model by using a multiphase level set formulation in 3 .In their multiphase Vese-Chan model 3 , multiple regions can be represented by multiple level set functions.Vese and

Our Model for Gray Images
In our previous work 19 , we have proposed an improved active contour model for multiphase image segmentation based on the piecewise constant multiphase Vese-Chan model 3 , the globally convex image segmentation method 20 and the split Bregman method 15-17 .The model we proposed in 19 is mainly for gray images.For simplicity, we only review the four-phase model in this section.
Let Ω ⊂ 2 be the image domain and u 0 : Ω → be a given gray level image.Our proposed model is given in the following level set formulation: where φ 1 and φ 2 are two level set functions, and μ is a positive constant.|∇φ i | g and φ i , r i are defined as is an edge detector function 17, 18, 21 , and β is a parameter that determines the detail level of the segmentation.r 1 and r 2 are defined as where c c 11 , c 10 , c 01 , c 00 is a constant vector of averages in the four phases.Given φ 1 0, x φ 1,0 x , φ 2 0, x φ 2,0 x , once the minimization problem 2.1 is solved, the image domain Ω can be partitioned into four regions Ω i i 1, 2, 3, 4 by thresholding the level set functions as follows: for some α ∈ 0, 1 , we choose α 0.5 in this paper.This is different from 3 where the zero level set is used to identify the boundary.We use the α 0.5 level set to identify the boundary because in our paper we have restricted 0 ≤ φ i ≤ 1.The average vector c c 11 , c 10 , c 01 , c 00 can be obtained by

Our Model for Color Images
In this section, we extend our model in Section 2.1 from gray images to color images.Let u 0 : Ω → 3 be a given color image and u 0i i 1, 2, 3 the three channels of u 0 .Let c i c 1i , c 2i , c 3i , c 4i i 1, 2, 3 be the average vector of the ith channel.
The extension of our model to color images is still in the same formulation as 2.1 :

2.7
The difference between 2.7 and 2.1 lies in the definition of r 1 and r 2 .In our model for color images 2.7 , r 1 and r 2 are defined differently from 2.4 :

2.8
Besides, another difference is the computing of the edge detector function g |∇u 0 x | .According to the definition of g in 2.3 , we need to compute |∇u 0 x | 2 when computing g.
When u 0 is a gray image, for each pixel x x, y , |∇u 0 x | 2 is defined as When u 0 is a color image, |∇u 0 x | 2 is defined as x .

2.10
Similarly to the case of gray images, once the minimization problem 2.7 is solved, the image domain Ω can also be partitioned into four regions Ω i i 1, 2, 3, 4 by 2.5 .Then, the average vector c i c 1i , c 2i , c 3i , c 4i i 1, 2, 3 for each channel can be updated by

2.11
Let u be the fitting image of the given color image u 0 and u i i 1, 2, 3 the three channels of u.Then, each channel u i of the fitting image u can be expressed as 12 Thus, we can obtain the fitting image u by u :, :, 1 u 1 , u :, :, 2 u 2 , u :, :, 3 u 3 .

2.13
We denote the four segmented phases of u as phase 1 , phase 2 , phase 3 , and phase 4 , then the four phases can be given as
In fact, u can also be obtained by u phase 1 phase 2 phase 3 phase 4 .

2.15
Remark 2.1.The multiphase image segmentation model we proposed is different from the piecewise constant multiphase Vese-Chan model in 3 for the following reasons.First, our energy functional E φ 1 , φ 2 in 2.1 or 2.7 is different from 3 .Second, we incorporate information from the edge into the energy functional by using a nonnegative edge detector function g to detect boundaries more easily.Third, we apply the split Bregman method, and our model is much more efficient than the model in 3 .Last, we do not need the procedure to reinitialize φ i to the signed distance function to its zero-level curve as 1-3 , we simply initialize φ i as a binary step function which takes 1 inside a region and 0 outside.We have restricted φ i to 0, 1 , thus we do not need the Heaviside function H or its smooth versions H ε as 3 .

Application of the Split Bregman Method to Update φ 1 and φ 2
To minimize the energy functional E φ 1 , φ 2 in 2.1 or 2.7 with respect to φ 1 and φ 2 , one traditional method is using the standard gradient descent method directly 1-3 .In this section, we apply the split Bregman method to solve our proposed minimization problem 2.7 more efficiently.First, we introduce two auxiliary variables: Then, we add two quadratic penalty functions to weakly enforce the resulting equality constraints and get the unconstrained problem as follows: where λ is a positive constant.

Mathematical Problems in Engineering
We then strictly enforce the constraints by applying the Bregman iteration, and this results in the following optimization problem: and φ 2 fixed, the Euler-Lagrange equation of the optimization problem 2.17 with respect to φ 1 is

2.18
Note here that from the definition of r 2 in 2.8 , r 2 is a function of φ 1 .Here, when we minimize 2.17 with respect to φ 1 , we just consider r 2 as a constant.From the numerical results in the next section, we can see that this consideration does not affect the performance our model.Similarly, when we minimize 2.17 with respect to φ 2 for fixed − → d 1 , − → d 2 , and φ 1 , we also consider r 1 as a constant and get the following Euler-Lagrange equation: For 2.18 and 2.19 , the central difference and the backward difference are used for the Laplace operator and the divergence operator, respectively.Then the resulting numerical scheme is where l 1 or 2 denotes the two schemes for 2.18 and 2.19 .

The Algorithm
The split Bregman algorithm for the proposed minimization problem 2.7 in Section 2.3 can be summarized as in Algorithm 1.

Numerical Results
Our model has been applied to synthetic and real color images with multiple regions in this section.When working with the level set function φ i , we simply initialize the level set function φ i as a binary step function which takes 1 inside a region and 0 outside.The advantage of using binary step functions as the initial level set functions is that new contours can emerge easily and the curve evolution is significantly faster than the evolution from initial functions as signed distance maps.In this paper, λ 1 and β 0 are used for all by 2.20 and 2.21 images.Unless otherwise specified, we choose μ 50/255 2 for all synthetic color images and μ 30/255 2 for all real images.The size of each image is specified in each figure .Figure 1 shows the application of our model to a synthetic color image with a triple junction.Most of the models 12, 13 need three level set functions to represent the triple junction.Here, we need only two level set functions.The α 0.5 level sets of the final φ 1 and φ 2 are shown in d and h , which have to overlap on a segment of the triple junction.
We then apply our model to another synthetic image in Figures 2 and 3.In Figure 2, we show the results for the clean image with three different initial conditions.When we seed with small initials in a and b , our model can work well, both the final contours and the final fitting images are correct.However, when we use two disjoint circles as the initial curves, our model will be trapped in a local minimum, and the pink object will be missed as shown in f and i .Then we add random noises with standard deviation 30.0 to the clean image.Figure 3 shows the results for the noisy version.We can observe that our model works well even if the noise level is high.In the following experiments, the noises we added are all random noise with standard deviation 30.0.
The results for another synthetic color image without or with noise are shown in Figures 4 and 5 Figures 6, 7, and 8 show the applications of our model to two synthetic images and their noisy versions.Both the two images have three objects with different shapes, which may be difficult to detect.However, our model can identify the objects correctly for both clean images and noisy versions.
We have declared that the standard deviation of the random noise added to the above synthetic images is 30.0.Now, we increase the noise level and give the results with our model in Figures 9 and 10.For the above five synthetic images, when we increase the noise level to 50.0, our model can give the correct final contours and final fittings, which can be clearly seen from Figure 9.Even the noise is increased to a much higher level 80.0, our model can still detect the object boundaries of the synthetic images and give good fitting images.The corresponding initial contours, final contours and final fitting images are shown in Figure 10.Here, we only need to choose μ 1/255 2 to avoid detecting the high-level noise for the noisy versions in Figures 9 and 10.This demonstrates the robustness of our model to noise.
We then apply our model to three real color images, and the results are shown in Figures 11, 12, 13, 14, and 15.We seed with small initial curves for all of these real images.Figures 11 and 12 show the results for an image of flowers, while Figures 13 and 14 give the results for a 4-colors image.We can observe that our model can segment these color images well.In Figure 15 we apply our model to an image with contours without gradient cognitive contours .From the final fitting image shown in d , it can be seen that our model can also detect "contours without edges".Figures 16 and 17 show the results of our model for two real color images from the Berkeley image set.In Figures 16 and 17, we give the curve evolution process in Row 1 and the corresponding fitting images in Row 2, while the final four phases are shown in Row 3. It can be observed that our model can handle these two real color images very well.
Our model can be applied to segment real medical MRI images.We apply our model to two real brain MRI images in Figures 18 and 19.Similarly, the curve evolution process, corresponding fitting images and final four phases are shown in Row 1, Row 2 and Row 3, respectively.We can see that our model can identify quite well the gray matter, the white matter, and so forth.
To demonstrate the efficiency of our model, we compare the computation CPU time for the above five synthetic images with the Vese-Chan model from 3 and our model in Tables 1 and 2. In Table 1, we test the CPU time for all the clean synthetic images used in this paper.Except the image from Figure 1, we use the noisy versions for other four synthetic images for Table 2. From Tables 1 and 2, we can see that our model is much more efficient than

Conclusion
In this paper, we propose a new and fast multiphase image segmentation model for color images.Our model is an extension of the model for gray images we proposed in 19 .We have tested our model with many synthetic and real color images.Numerical results show that our model has all the benefits of the Vese-Chan model 3 , including robustness even with noisy data and automatic detection of interior contours.However, our model is much more efficient by applying the split Bregman method.Of course, our model has its limitation.Our model is a piecewise constant multiphase segmentation model, and it mainly focuses on homogeneous multiphase images.It only considers the global information of the given image, thus it cannot deal with images with inhomogeneity.In future work, we will extend our model to handle inhomogeneous multiphase images by taking both local and global information into consideration.

Figure 1 :
Figure 1: Results for a synthetic image with our model.a -c : The active contour evolving process from the initial contour to the final contour; e -g : The corresponding fitting images u at different iterations; d and h : The α level sets of the final φ 1 and φ 2 ; i -l : The final four phases; size 256 × 256.

Figure 2 :
Figure 2: Results for a synthetic image with different initial conditions; a -c : The original image with three different initial contours; d -f : The corresponding final contours; g -i : The corresponding final fitting images; size 256 × 256.

Figure 3 : 1 :φ 0 2 2:
Figure 3: Application of our model to the noisy version of the image from Figure 2. a -c : The curve evolution process; e -g : The corresponding fitting images u; d and h : The α level sets of the final φ 1 and φ 2 ; i -l : The final four phases.

Figure 4 :
Figure 4: Results for a synthetic image with two different initial conditions.a and g : The two different initial contours; b and h : The corresponding final contours; c and i : The corresponding final fitting images; d and j : The α level sets of the final φ 1 and φ 2 with the initial condition a ; i -l : The final four phases with the initial condition a ; size 240 × 110.

Figure 5 :
Figure 5: Application of our model to the noisy version of the image from Figure 4. a -d : The curve evolution process; e -h : The corresponding fitting images u.

Figure 6 :
Figure 6: Results for a synthetic image without noise and with noise.a -c and e -g : The initial contour, final contour and final fitting image for the clean and noisy images; d , h and i -l : The α level sets of the final φ 1 , φ 2 and the final four phases for the noisy version; size 200 × 200.

Figure 7 :Figure 8 :Figure 9 :
Figure 7: Results for a clean synthetic image with our model.a -c : The curve evolution process; e -g : The corresponding fitting images u; d and h : The α level sets of the final φ 1 and φ 2 ; i -l : The final four phases; size 256 × 256.

Figure 19 :
Figure 19: Results for a brain MRI image with our model.a -d : The curve evolution process; e -h : The corresponding fitting images u; i -l : The final four phases; size 125 × 160.

2 7 :
Update Ω i and c by 2.5 and 2.11 8