A Complex Diffusion Based Modified Fuzzy C- Means Approach for Segmentation of Ultrasound Image in Presence of Speckle Noise for Breast Cancer Detection

In present section, a hybrid model is introduced for segmentation of ultrasound image in presence of speckle noise which follows Rayleigh distribution [14]. Single framework is designed for segmentation and restoration of ultrasound images. At first, the proposed approach is derived for segmentation to extract region of interest of cancerous or suspicious area. Secondly, it removes noise from segmented image during each of the iteration for better segmentation. Thirdly, it avoids segmentation problem of blocky artifacts and preserve the edges.

The present section is organized into four sub-sections, Section 2.1, presents the analysis of original FCM segmentation approach and examines its limitations and applicability for proposed approach in terms of speckle noise. Section 2.2 presents the modified FCM to overcome the limitations in original FCM. Section 2.3 presents the restoration/ de-noising concept of proposed model for restoration of ultrasound image using adaptive complex diffusion based PDE nonlinear filter adapted (ACDPDE) to noise. Section 2.4 presents the proposed model for segmentation of ultrasound images in presence of speckle noise.

2.1 Classical Fuzzy C Means

It is type of clustering technique which is more robust, prone to blurring and depends upon the strength of segmenting picture elements. Segmenting picture element may belong to more than one cluster. FCM implementation is easy but highly sensitive to noise. It works in intensity of image, boundary condition and spatial context [18, 31].

Value is assigned for each picture element in image by a fuzzy membership function, and starting point is chosen.

J represents the performance index or cost function for membership matrix (U) = [u_ij], u_ij degree of a membership function of the data point of i^th cluster is x_i.

$J\left(U, C_{1}, C_{2}, \ldots . . C_{n})=\sum_{i=1}^{c} J_{j}=\sum_{i=1}^{c} \sum_{j=1}^{N} u_{i j}^{m} d_{i j}^{2}\right.$      (1)

where, $d_{i j}=\left\|x_{j}-c_{i}\right\|$  is the Euclidian distance between i^th centroid (c_i) of a cluster and j^th data point x_j. m $\epsilon(1, \infty)$  is a weighting exponent. U have values in the vicinity of 0 and 1 but the summation of degrees of membership of a data point to all partition or segments is constantly equivalent to unity:

$\sum_{i=1}^{c} u_{i j}=1, \forall j=1 \ldots n$       (2)

For getting a minimum dissimilarity function or minimum cost function, Eq. (1) must be satisfied by below mentioned Eqns. (3) and (4).

$c_{i}=\frac{\sum_{j=1}^{n} u_{i j}^{m} x_{j}}{\sum_{j=1}^{n} u_{i j}^{m}}$     (3)

$u_{i j}=\frac{1}{\sum_{k=1}^{c}\left(\frac{d_{i j}}{d_{k j}}\right)^{\frac{2}{m-1}}}$       (4)

Its execution depends on the initial value of membership matrix. The number of clusters must define before execution. The algorithm is kept running for a few iterations, each iteration with various estimations of membership grades of data points to all clusters is always equal to one.

2.2 Modified Fuzzy C Means (MFCM)

The original intensity-based FCM algorithm fails to segment image corrupted by noise, intensity inhomogeneity, imaging artifacts, and outliers [31]. Only considering grey level information, it is efficient for simple texture and background but unable to segment complex texture or corrupted by noise. To improve the segmentation effect, local spatial information is incorporated in cost function [37].

$x=R^{c}(I)$         (5)

where, x is the reconstructed image by R^C morphological operation erosion and dilation and I denotes noisy image.

Modified cost function is given below [37]:

$J\left(U, C_{1}, C_{2}, \ldots C_{n}\right)=\sum_{i=1}^{c} J_{j}=\sum_{i=1}^{c} \sum_{j=1}^{q} \alpha_{j} u_{i j}^{m} d_{i j}^{2}$     (6)

$\sum_{j=1}^{n} \alpha_{j}=N$      (7)

where, α_j is number of grey value for computing histogram of image, q denote the no. of grey levels contained in x which is generally much smaller than N. And, u_ij represent degree of fuzzy membership of grey value q with respect c cluster.

To minimize the cost function or to reach minimum dissimilarity from Eq. (6), the following two conditions must be satisfied.

$c_{i}=\frac{\sum_{j=1}^{q} \alpha_{j} u_{i j}^{m} x_{j}}{\sum_{j=1}^{q} u_{i j}^{m}}$       (8)

$u_{i j}=\frac{1}{\sum_{k=1}^{c}\left(\frac{d_{i j}}{d_{k i}}\right)^{\frac{2}{m-1}}}$        (9)

Hence, FCM and MFCM lack enough robustness to noise and outliers, TVFCM was proposed to make FCM robust with noisy data [38]. TVFCM method was used for segmentation and restoration corrupted by Gaussian noise.

For TVFCM the cost function is given below [38]:

$J\left(u, c_{1}, c_{2} \ldots c_{n}\right)=\sum_{i=1}^{c} J_{i}=\sum_{i=1}^{c} J_{i}+\sum_{k=1}^{c} T V\left(u_{i j}\right)$    (10)

Similarly, AFPDEFCM [31, 39] was used for segmentation, restoration and enhancement corrupted by poison noise for biopsy images.

Here, in this paper we propose to couple a complex diffusion based partial differential equation (CDPDE) non linear filter adapted to speckle noise with MFCM segmentation method. This method is achieved for reducing blocking artifacts with preservation of fine edge details for noisy ultrasound image in cancer detection.

2.3 Adaptive complex diffusion based PDE non filter adapted to Rayleigh noise (ACDPDE)

During the acquisition of ultrasound images, when more than one echoes are travelling toward the probe then constructive and destructive additive summation of ultrasound echoes occurred, this phenomenon is called interference, which fluctuate the ultrasound pixel intensity that causes the speckle noise. Speckle noise is type of multiplicative noise which follows Rayleigh distribution that may affect the over segmentation process. To remove the speckle noise, filtering process is required for denoising.

Speckle is having complex amplitude nature, which as

$Z(i, j)=Z_{R}(i, j)+j Z_{I}(i, j)$      (11)

where, Z_R and Z_I represent as Gaussian independent random variable and having zero mean and variance $\sigma^{2}$  and $(i, j)$  represents pixel coordinates.

The intensity of speckle noise as follows [38]:

$n(i, j)=|(Z(i, j))|^{2}=Z_{R}^{2}+Z_{I}^{2}$       (12)

where, $n(i, j)$  is random variable which follows Rayleigh distribution, Z_R and Z_I representing real and imaginary part of speckle noise.

The general model of observation reads:

$u_{o}(i, j)=u(i, j) * n(i, j)+\eta(i, j)$       (13)

$u_{o}(i, j)=$ Rayleigh $* u(i, j)+\eta(i, j)$       (14)

where, u₀ is observed blurred and noisy ultrasound image; u is true image; $n(i, j)$  is speckle noise; η(i, j) is additive noise. Assuming additive noise is zero.

The probability density function (PDF) of observed or blurred image is [40]:

$p\left(\frac{u}{u_{n}}\right)=\frac{u_{o}}{\sigma^{2}} \exp \left(-\frac{u^{2}}{2 \sigma^{2}}\right)$     (15)

Need to estimate maximum likelihood of u that can be calculated by minimization of negative log likelihood of Rayleigh PDF, is given as follows [40].                                           

$\begin{aligned} U_{M L}=\arg \min \{&\left.-\operatorname{In} p\left(\frac{u}{u_{o}}\right)\right\} =\arg \min \left\{\operatorname{In} \frac{u_{o}}{\sigma^{2}}-\frac{u^{2}}{2 \sigma^{2}}\right\} \end{aligned}$      (16)

Estimated maximum likelihood of u assuming Rayleigh noise is given as:

$U_{M L}=-\frac{u}{\sigma^{2}}$       (17)

The obtained result of $U_{M L}$  can be noise corrupted, so regularization is necessary. By framing the problem in variational framework, the Gibbs prior model, based on energy functional which is obtained in terms of gradient norm of image, gives best estimate of u for additive noise removal [30, 31].

$p(u)=\exp (-\lambda . E(u))$       (18)

$E(u)=\arg \min _{\Omega} \int\left[L\left(p\left(\frac{u}{u_{o}}\right)\right)+\lambda \varphi\left\|\nabla^{2} u\right\|\right] d \Omega$        (19)

where, $E(u)$  is energy functional.

In the case of complex diffusion PDE [22].

$\varphi\left\|\nabla^{2} u\right\|=\nabla^{2}(c(\operatorname{Im}(u)) \nabla u)$       (20)

Putting the value of Eq. (20) in Eq. (19). 

$\begin{aligned} E(u)=\arg \min _{\Omega} \int\left[L\left(p\left(\frac{u}{u_{o}}\right)\right)\right.\left.+\lambda \nabla^{2}(c(\operatorname{Im}(u)) \nabla u)\right] d \Omega \end{aligned}$        (21)

The obtained energy functional $E(u)$ , from Eq. (21) is minimized using Euler-Lagrangian minimization technique with combination of gradient decent approach for minimizing the speckle noise which follows Rayleigh distribution from ultrasound image that leads to non linear complex diffusion based PDE filter [41-44].

$\frac{\partial u}{\partial t}=L^{\prime} p\left(\frac{u}{u_{o}}\right)+\lambda \nabla^{2}(c(\operatorname{Im}(u)) \nabla u)$      (22)

With initial condition.

$u_{t=0}=u_{o}$      (23)

$\frac{\partial u}{\partial t}=-\frac{u}{\sigma^{2}}+\lambda \nabla^{2}(c(\operatorname{Im}(u)) \nabla u)$        (24)

Above derived Eq. (24) can be used as prior or second term in proposed frame work.

$c(\operatorname{Im}(u))=\frac{e^{j \theta}}{1+\left(\frac{\operatorname{Im}(u)}{k \theta}\right)^{2}}$        (25)

And $k=\frac{1}{A v g S N R}, A v g S N R=\frac{\operatorname{mean}(u)}{s t d \cdot \operatorname{dev}(u)}$      (26)

For making proposed framework in adaptive nature, k speckle index value must be calculated based on inverse average SNR. To differentiate among homogeneous area, region of contour and edges, k value must be defined positively and keep on changing throughout iterative process of PDE till its convergence.

2.4 The proposed model for segmentation of ultrasound image in presence of Rayleigh noise

The proposed single framework for segmentation and restoration is defined as follows:

$\mathrm{J}=\mathrm{J} 1+\mathrm{J} 2=\sum_{i=1}^{c} J_{i}+\sum_{k=1}^{c} A C P D E(U)_{i j}$       (27)

The main purpose of Eq. (27) is to minimize cost functional. 

where, J1 is modified FCM cost function in Eq. (27), taken from Eq. (6) as explained in section 2.2.

$J 1=\sum_{l=1}^{c} \sum_{j=1}^{q} \alpha_{j} u_{i j}^{m} d_{i j}^{2}$        (28)

and J2 is adaptive complex diffusion based non linear filter adapted to Rayleigh noise, as explained in section 2.3, which is responsible for reduction of Rayleigh noise from segmented image in each iteration obtained by minimizing modified cost functional of modified FCM algorithm as mentioned above in Eq. (24). The solution of functional J2 obtained after its minimization as explained in section 2.3.

$J 2=-\frac{u_{i j}}{\sigma^{2}}+\lambda \nabla^{2}\left(c\left(\operatorname{Im}\left(u_{i j}\right)\right) \nabla u_{i j}\right)$    (29)

The overall combined process of segmentation and restoration is as follows:

$\begin{aligned} \frac{u_{i j}^{n+1}-u_{i j}^{n}}{\Delta t}=\arg \min &\left[\sum_{i=1}^{c} \sum_{j=1}^{q} \alpha_{j} u_{i j}^{m} d_{i j}^{2}\right]+\left(-\frac{u_{i j}}{\sigma^{2}}+\lambda \nabla^{2}\left(c\left(\operatorname{Im}\left(u_{i j}\right)\right) \nabla u_{i j}\right)\right) \end{aligned}$       (30)

Finite difference scheme is used for digital implementation of Eq. (30), which is as follows:

$\begin{aligned} u_{i j}^{n+1}=u_{i j}^{n}+\Delta t &\left.\sum_{i=1}^{c} \sum_{j=1}^{q} \alpha_{j} u_{i j}^{m} d_{i j}^{2}\right]+\left(-\frac{u_{i j}^{n}}{\sigma^{2}}\right.+\lambda \nabla^{2}\left(c\left(\operatorname{Im}\left(u_{i j}^{n}\right)\right) \nabla u_{i j}^{n}\right) \end{aligned}$       (31)

where, $u_{i j}^{n}$  is n-th iteration of estimated image, $u_{i j}^{n+1}$  is current (n+1)th iteration of obtained image. For stability purpose $\Delta t$  is grid constant set to 0.25. $\lambda$  (Regularization parameter) is used for establishing balance between likelihood (data fidelity) term and the complex diffusion PDE during restoration process as discuss in section 2.3. The model flow chart for the proposed method is shown in Figure 1. The block diagram showing each component and its operations for proposed method is given in Figure 2.

If Rayleigh noise is not present in image, then the functional frame work is changed i.e. only applicable for Gaussian noise.

$u_{i j}^{n+1}=u_{i j}^{n}+\Delta t\left[\arg \min \left[\sum_{i=1}^{c} \sum_{j=1}^{q} \alpha_{j} u_{i j}^{m} d_{i j}^{2}\right]\right.\left.+\lambda\left(\nabla^{2}\left(c\left(\operatorname{Im}\left(u_{i j}^{n}\right)\right) \nabla u_{i j}^{n}\right)\right)\right]$

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Figure 1. Flowchart for the proposed method

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Figure 2. The proposed method for segmentation and restoration of ultrasound images (An iterative process)