Medical Image Segmentation using Level set ... - Semantic Scholar

Medical Image Segmentation using Level set Method without reinitialization Manish Khare

Rajneesh Kumar Srivastava

Department of Electronics and Communication University of Allahabad, India [email protected]

Department of Electronics and Communication University of Allahabad, India [email protected]

____________________________________________________________________________ ABSTRACT In this paper we have proposed a segmentation method based on level set without re-initialization approach, applied with certain specific shape based model, for medical images. Level set method without re-initialization, with certain specific shape based model has advantages over level set method with reinitialization. Large time steps are possible with the proposed method which speeds up the process of curve evolution. We have applied the proposed approach on several medical images. Results on six different kidney images are being presented in this paper. We have also compared performance of the proposed method with other existing methods. The proposed method is found to be better as compared to the other existing methods.

Keywords: Active Contour Model, Level set Methods, Image Segmentation, Medical Images.

____________________________________________________________________________ I. INTRODUCTION Image Segmentation subdivides an image into its constituent regions or objects [1,2]. The level of this subdivision depends on the problems which are being solved. Medical image segmentation is an important task for identification and location of tumors, diagnosis, and computer guided surgery etc. [3]. Researches have developed the methods for image segmentation like watershed segmentation [4], region based segmentation [5], clustering based segmentation [6], histogram based segmentation [7] and level set method [8, 9,10], but all of these methods deal with specific category of images. In the present work we are proposing a level set method, without reinitialization, with certain specific shape model for image segmentation. Level set methods have their own importance in segmentation due to their accuracy and fast speed. In this paper, some specific shape based models are applied on level a set method which does not uses any reinitialization function for updation of energy function. Rest of the paper is organized as follows: Section 2 describes basics of active contour model. Section 3 describes level set method. The proposed segmentation algorithm (Level set method without reinitialization with some specific shape based model) is described in section 4. Achieved experimental results and comparison from other existing methods are given in section 5, and finally conclusions of the proposed segmentation algorithm are given in section 6.

II.

ACTIVE CONTOUR MODEL

Active Contour model is a useful framework for marking an object outlines in an image. Minimization of energy term, associated with active contour is a sum of two energy functions - internal energy function and external energy function. Active Contour provides a methodology for different applications [11] such as – detection of edges, motion tracking, stereo matching etc. Active Contour model is also called snake [12]. External energy will be minimum when the snake is at the object boundary position, and internal energy will be minimum when the snake has a shape which is relevant to consider the shape of object. A snake is initialized near the target objects and then iteratively attracted towards the salient contour. A snake in an image can be represented as a set of n points as: Ci = (xi, yi) where i=0, 1, 2, -----, n-1 and energy function for this is given as:

(2.1)

1

Esnake( contour )   Eint ernal (C ( s))ds  Eexternal (C ( s))

(2.2)

0

External energy term is combination of two other energies as external constraint force applied by user (E con) and image force acting on spline (Eimage) [12]. i.e. Eexternal = Eimage + Econ Thus energy function for contour is converted into:

(2.3)

___________________________________________________________________________________________ International Conference on Signal, Image and Video Processing (ICSIVP) 2012

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Medical Image Segmentation using Level set Method without Reinitialization ___________________________________________________________________________________________ 1

Esnake ( contour )   Eint ernal (C ( s))ds  Eimage (C (s ))  Econ (C (s ))ds

(2.4)

0

The evolution equation for level set function ф for the equation 3.1 can be written in general form as a nonlinear Partial Differential Equation:

Internal energy is combination of energy of snake contour (Econt) and energy of spline curvature (Ecurv) [12]. i.e. Einternal = Econt + Ecurv (2.5) This internal energy is given by spline energy curve: dC d C  2 2   ( s ) || ds  s  ||  ( s ) || ds 2  s  ||     2

Eint eral

(2.6)

2

Here large value of η(s) increases the internal energy of the snake as it stretches more and more and small value of η(s) make energy function insensitive to amount of stretch curve, Large value of ψ(s) increase the internal energy of the snake as it develop more curves and small value of ψ(s) make energy function insensitive to amount to curves in snake. Zero value of ψ(s) make the snake to become second order discontinuous and develop a corner.

III. Level Set METHOD Level Set method introduced by Osher and Sethian [13, 14], is a way to denote active contours and it has been widely used in image segmentation, motion segmentation, object tracking etc. Level Set methods depend on position, time, the geometry of interface and some energy function [15]. For any given image I0 we can create a level set function ф (x, y) to describe the contour. The contour is defined as the zero level set function ф [16]: C = {(x, y) | ф (x, y) = 0} (3.1) Inside region and the outside region of the curve is given by Lipshitz continuous function ф, with following properties: inside the contour    ( x, y )  0 (3.2)   at the contour   ( x, y )  0    ( x, y )  0 outside the contour   By changing the value of ф, some regions will turn to positive, that were originally negative and some regions turn to negative that were originally positive. Hence contour will change position according to update value of level set function, as shown in figure 1:

  V |  | 0 t

(3.3)

This equation is called level set equation. Here V is given velocity field called speed function for Image segmentation part. Function V depends on only image data and level set function ф. Now one question arises, how to evaluate contour with level set equation? We try to explain this by using figure 2:

FIGURE 2: Illustration of Contour evolution (a)symbol of each area, (b) – The sign of update value in each region, (c) – final evolution result In figure 2 the white circle is initial contour, and a, b, c, d are four regions, here ф value of a and b takes positive value and ф value of c and d takes negative value at time of initial contour. Here (gray level intensities in b and c) > (average gray level intensities in a and b) > (average gray level intensities in c and d) > (gray level intensities in a and d). Using level set computation, after some iteration, ф value of b and c will become positive and ф value of a and c will become negative and contour stop at the desired place. IV.

THE P ROPOSED SEGMENTATION METHOD

Level set method with reinitialization has disadvantage as, this method is not able to work with signed distance function, and Level set method without reinitialization also has disadvantage as, this method is not able to work with some shapes. Therefore we are proposing the level set method without reinitialization with some specific shape model. In following subsection 4.1 and 4.2, we have described, how to establish the shape model and how to take level set equation for level set method without reinitialization. IV.1 Establishment of Shape Model Once we get the initial shape model we get the location and size of the segmented region for shape model. We have used Chan-Vese model [16] for getting information of segmented region, using following equation:

FIGURE 1: Representation of Level set function with contour changes positions


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Medical Image Segmentation using Level set Method without Reinitialization ___________________________________________________________________________________________       ( )  .div  1 (u0  c1 )2  2 (u0  c2 ) 2    |   |  

(4.1)

where u0 is the original image, ф is the level set function, μ, λ1, λ2, ʋ are parameters which adjust the weights, c1 and c2 are average gray level intensity in ф > 0 region and ф < 0 region respectively, δ is the Dirac delta function and calculated as:

The Dirac function in equation (4.5) is used for smoothing and is given as in equation (4.8): 0 | x |    (4.8)  ( x)   1   x   1  cos( ) | x |   2       where ϵ = 2.5 is fixed for the proposed method.

(4.2)

In the proposed method, the time step ʋ is chosen as 10. Range of detection of time step with some restrictions has been described by Chunming et al [18].

then we can use this location and the size of the inside region of the shape model, and after that this location can be used as zero level set function for level set equation.

Further, we will minimize total energy function given in equation (4.7) for evolving curve.

IV.2 Level set method without reinitialization

IV.3 Complete Algorithm for the Proposed method

Reinitialization is the process of periodically reinitializing the level set function during the evolution for maintaining stable curve evolution. According to Gomes and Faugeras [17], re-initializing the level set function is a disagreement between the theory of level set method and its implementation. Chunming et al. [18] proposed level set method without reinitialization, but this approach fails with square and diagonal shape objects, therefore we have proposed a method based on level set without reinitialization with specific shape, and this approach works well for both circular and square shape objects.

Steps of the proposed algorithm are as follows: 1) Apply Gaussian Convolution function for smoothness of image. 2) Apply Dirac function into smooth image. 3) Establish the initial shape model 4) Find the region of segmentation from established shape by using equation (4.1) and equation (4.2). 5) Use value got from step 4 for determining the contour. 6) Initialize the level set function in image region R. (Negative value indicate that contour is inside the region, positive value indicate that contour is outside the region and zero value indicate that contour is at the region). 7) Iterate following steps for the user defined number of iterations: 7.1. Call level set function with updated value. 7.2. Update contour with update value.

 ( ) 

.   2  2 1

We define external energy which will move the zero level curves towards the boundaries of object. Let I be an image, and g be edge indicator function which is defined as-

g

1 1 | G * I |2

(4.3)

where, Gσ is the Gaussian kernel with some standard deviation σ, then external energy function for a level set function ф (x, y) is defined as:

 g , , ( )   Lg ( )   Ag ( )

(4.4)

where λ > 0, ʋ is time step function, and Lg (ф) and Ag (ф) are given as:



(4.5)



(4.6)

Lg ( )  g ( ) |  | dxdy 

Ag ( )  g H ( ) dxdy 

where δ is dirac delta function and H is Heaviside function. Further we define the total energy of contour as: E ( )   ( )  g , , ( ) (4.7) The external energy ψg, λ, ʋ derives the zero level set towards the boundary of object, and the internal energy term μ∂(ф) penalize the deviation of ф from a signed distance function during its evolution.

V. EXPERIMENT AND RESULTS In this section, we have shown the experimental results of the proposed algorithm. We have implemented segmentation method described in section 4 and tested on several images including noisy images and blurred images. In which we are showing following categorized results. Results of the proposed algorithm are shown for Kidney image. We have tested on six sets of images corresponding to kidney image. First set is original image, second one is noisy image with 5% Uniform noise, third one is noisy image with 5% Gaussian noise, fourth one is blurred image with shape blur of 10 pixel value, fifth one is motion blurred image with 25 pixel value and sixth one is corrupted with combination of noise (5% uniform noise) and blur (motion blur with 25 pixel value). Figure 3 shows results for kidney image, in which six sets of images are given as figure 3(a), 3(b), 3(c), 3(d), 3(d), 3(e) and 3(f).


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Medical Image Segmentation using Level set Method without Reinitialization ___________________________________________________________________________________________

Original Image

Initial Contour

1000 iteration

2500 iteration

FIGURE 3(c)

1000 iteration

2500 iteration FIGURE 3(a)

Original Image

Initial Contour

Original Image

Initial Contour

1000 iteration

2500 iteration FIGURE 3(d)

1000 iteration

3000 iteration FIGURE 3(b)

Original Image

Original Image

Initial Contour

Initial Contour

1000 iteration

3000 iteration FIGURE 3(e)


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Original Image

Initial Contour

1000 iteration

2500 iteration FIGURE 3(f)

Fig(a)

Fig (c)

Fig (b)

Fig (d)

FIGURE 3: Segmentation Results for Kidney image in different case (A - Original Kidney image, B - Noisy kidney image with 5% Uniform noise, C - Noisy kidney image with 5% Gaussian noise, D - Blurred kidney image with shape blur of 10 pixel value, E – Blurred kidney image with motion blur of 25 pixel value, F – Kidney image with mixture of noise (5% uniform noise) and blur (motion blur with 25 pixel value)

We have compared the proposed method with Global Region Merging method [19] and Fast Global Minimization method [20] for above discussed set of kidney image. Comparison results are shown in figure 4. By observing figure 4, one can conclude that performance of the proposed method is better than others.

Fig(a)

Fig (c)

Fig(a)

Fig (b)

Fig (c)

Fig (d)

Fig(a)

Fig (b)

Fig (b)

Fig (d)


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VI. CONCLUSION

Fig (c)

Fig (d)

In this paper, we proposed a segmentation algorithm using level set method without reinitialization with some specific shape model. Here, we have chosen the shape model as square. We have used large time step to speed up the curve evolution. The experimental results show that the proposed method work well with specific shape model used for medical images, in comparison with global region merging method and fast global minimization method.

VII. ACKNOWLEDGEMENT

Fig(a)

Fig (b)

The authors are thankful to University Grants Commission (UGC), New Delhi, India for providing research grant vide its grant no. 36-246/2008(SR) for major research project.

REFERENCES

Fig (c)

Fig (d)

Fig(a)

Fig (b)

Fig (c)

Fig (d)

FIGURE 4: Segmentation Results for kidney image in different case (Fig (a) - Original image, Fig (b) – Result of proposed method, Fig (c) – Result of Global Region Merging Method, Fig (d) – Result of Fast Global Minimization)

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