A Modified Active Contour Model for Cardiac MR Image Segmentation Based on Entropy Information

In this article, a Modified Geodesic Active Contour Model has been presented. Normally, Active Contourbased methods are categorized as either edge-based models or region-based models. Active Contour Models not only uses data on Gradient, but also can use global features of image for evolutionary curve to move and segment target boundaries which leads to detection of their weak ones. This type of function, introduce a window to calculate the average surrounding data on any pixel, and fashion a Signed Pressure Force based on entropy to make the contour reach the boundaries. In order to optimizing the efficiency and stability of the algorithm, this article has provided two sorts of level set methods for segmentation to be done. Simulation results demonstrate that due process can provide precise readings even in cases of inhomogeneity or not well-defined edges.


Introduction
Active Contour Models [1] are used in image processing, mainly in image segmentation, which was first introduce by Kass et al.Generally, Active Contour Models are categorized as Edge-Based [1], [2] and Region-Based Models [3].Edge-Based models can perfectly segment target boundaries but are not that precise at noisy images.Region-Based models can easily overcome noisy or weak borders, using the mean or variance of the gray levels.However, these models are highly sensitive to intensity inhomogeneous; therefore, cannot segment bias-fielded images.Lee et al. [4] proposed Local Binary Fitting model which uses image's local information, and they managed to segment images with high intensity inhomogeneity to a certain point.By using local mean data, LBF model can easily overcome inhomogeneity of images, although cannot be relied on in heavily-noised or weak boundaries.A Geodesic Active Contour model [5] based on area was talked about by Zhang et al. which could categorize noisy image with weak borders perfectly.This model forms a function of Signed Pressure Force which will be replacing the function of Stopping Edge Model in GAC.This model assumes that the gray levels of image is homogenous, therefore, mentioned method cannot segment images with bias field accurately.Usually in medical images, gray levels has been distributed with Gaussian Function which can be describe by statistical distributions.This paper presents a novel method to segment medical images with bias field, noisy and weak boundaries using Signed Pressure Force based on image local entropy and statistics.The results show that the proposed method provide a better class of segmentation which uses the whole gray levels information.

Geodesic Active contour
Geodesic Active contour model was first introduced by Casseles et al. in 1997 [2].Energy function of this method is defined as followed: In above equation L(C) present the length of contour C, I  image gradient, g(.) stopping edge function and s, the curve length of image.Level set Evolution equation of (2.1) is as followed: In which div(.) is divergence operator,  a constant, ) ( I g  stopping edge function, which gets reduced by growth of image gradient I  .
Generally, medical images suffer from fuzzy, noisy and weak boundaries.Classic GAC models only use the gradient data, which can benefit from image global features for evolutionary curve movement to segment target boundaries, resulting in evolution of contours even in weak borders.

Image Entropy
, entropy information for each pixel presents Gaussian distribution function in local region, ) ( y x w  window function which is centered by pixel x.Entropy information is maximized in cases that gray level distribution is uniform.Image information entropy [6] is practiced in numerous image processing fields, specially, in image segmentation.As an instance, region based active contour model by local entropy, which was proposed by http://www.ispacs.com/journals/cacsa/2017/cacsa-00077/International Scientific Publications and Consulting Services Hee et al. [7]; and also Chatterjee et al. [8] who used the maximum fuzzy entropy for brain CT image segmentation.

Signed pressure force based on local entropy information
Function g(.) in geodesic active contour model was replaced by a signed pressure force function in a region-based GAC by Zhang et al. [5] spf is defined as followed:

 
Since RGAC models is still related to intensity inhomogeneity and medical images usually include bias field, hence cannot segment boundaries accurately.Gray levels of these images in local regions are illustrated as Gaussian distribution [9].This paper propose w window function for image gray level distribution; moreover, entropy of image data in spf to form a force activation function as followed:     This method can segment images with bias field and weak boundaries precisely.In order to obtain better results or having a comparison for accuracy and performance of segmentation in each model, 20 sets of 110×110 left ventricle MR images are used.Jaccard similarity measure [10] is used for quantitative analysis.Results show that proposed segmentation algorithm accuracy is better in comparison with other methods.Chan-Vese and RGAC models cannot accurately segment images with bias field, and GAC model pass the borders easily.LBF performance is lower, according to number of convolutions.The results obtained in less than 200 iterations with an appropriate calculation time.

Conclusion
This paper presents a modified GAC model for noisy, bias-filed images with weak borders, which uses the window function for analyzing local intensity information.This method defines it by acquiring the local information entropy.In a comparison with GAC model, our method is significantly more accurate.Moreover, comparing with CV and RGAC, this method could segment the bias-fielded images precisely.Finally, this method contains more Kernel parameters when compared to LBF method.Proposed method RGAC LBF

4 )
In which  (|.|) demonstrates maximum difference of gray levels. 1 c and 2 c are the gray level means in equation (4.5).
the image background and foreground area, simultaneously.Ω is image domain, function of image background and foreground area, which calculate the prior probability of them[9]: f are the gray level means, b  and m  are the gray level variance of background and foreground area (Equatuin 8, and 9).

2 2( 4 . 9 )
In local regions, gray levels changes follow Gaussian distribution.Consider a given point x is in contour and y  is a local region which is controlled by window function w, centered by x. b  and m  are background and foreground area and y  is divided into interior and exterior sections.If b x   , then Ep gets increased and Em gets decreased; Therefore, 0  lespf and evolution curve gets compressed and moves toward object boundary and vice versa.
Figure 2 shows the results of left ventricle MR image segmentation for sigma = [3, 9, 15, and 21].Input images include bias field, noise, and weak boundaries.

Figure 2 (
c) is the result of RGAC model with 100 iterations and different sigma's which can overcome noises and weak boundaries.

Figures 4 and 5 Figure 3 :Figure 4 :
Figures 4 and 5 show the accuracy and performance of segmentation corresponding to different sigma's.Obviously, in LBF model, these criterions decline as sigma is increased; therefore, regarding the results, proposed method is clearly better than RGAC and LBF.

Figure 5 :
Figure 5: Segmentation performance of proposed method in sigma parameter.
International Scientific Publications and Consulting ServicesAs it was mentioned, gray level data in local regions follow Gaussian distribution, hence maximum prior probability of interior and exterior sections is equal to one when contour moves toward boundaries.This movement continues till 0  lespf which leads in stopping the evolution curve.Equation (4.6), demonstrates that lespf function changes between [-1,1].Equation 11 illustrates the Gradient descent relations of RGAC model with lespf instead of spf function.