A Fuzzy C-Means Clustering Algorithm Based on Spatial Context Model for Image Segmentation

Abstract

An improved Fuzzy C-Means (FCM) algorithm, which is called Reliability-based Spatial context Fuzzy C-Means (RSFCM), is proposed for image segmentation in this paper. Aiming to improve the robustness and accuracy of the clustering algorithm, RSFCM integrates neighborhood correlation model with the reliability measurement to describe the spatial relationship of the target. It can make up for the shortcomings of the known FCM algorithm which is sensitive to noise. Furthermore, RSFCM algorithm preserves details of the image by balancing the insensitivity of noise and the reduction of edge blur using a new fuzzy measure indicator. Experimental data consisting of a synthetic image, a brain Magnetic Resonance (MR) image, a remote sensing image, and a traffic sign image are used to test the algorithm’s performance. Compared with the traditional fuzzy C-means algorithm, RSFCM algorithm can effectively reduce noise interference, and has better robustness. In comparison with state-of-the-art fuzzy C-means algorithm, RSFCM algorithm could improve pixel separability, suppress heterogeneity of intra-class objects effectively, and it is more suitable for image segmentation.

Appendices

Appendix 1

See Table 7.

Table 7 List of abbreviations

Appendix 2

The objective function of RSFCM is defined as (10), and the constraint conditions are as (18).

$$ \mathop \sum \limits_{k = 1}^{c} u_{ki} = 1,i = 1,2, \ldots ,N. $$

(18)

Using the Lagrangian multiplier method, combined with constraint condition (18), adding the coefficient $\lambda$, and expanding all k of (10), then the objective function J is obtained, as shown in (19).

$$ J(U,V) = \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{k = 1}^{c} u_{ki}^{m} \times \left\{ {x_{i} - v_{k}^{2} + \frac{1}{{N_{R} }}\mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} (k)} \right]x_{j} - v_{k}^{2} } \right\} + \lambda_{1} \left( {\mathop \sum \limits_{k = 1}^{c} u_{k1} - 1} \right) + \cdots + \lambda_{i} \left( {\mathop \sum \limits_{k = 1}^{c} u_{ki} - 1} \right) \cdots + \lambda_{N} \left( {\mathop \sum \limits_{k = 1}^{c} u_{kN} - 1} \right). $$

(19)

Further simplify (20).

$$ J\left( {\varvec{U},\varvec{V}} \right) = \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{k = 1}^{c} u_{ki}^{m} \times \left\{ {x_{i} - v_{k}^{2} + \frac{1}{{N_{R} }}\mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} (k)} \right]x_{j} - v_{k}^{2} } \right\} + \mathop \sum \limits_{i = 1}^{N} \left( {\mathop \sum \limits_{k = 1}^{c} \lambda_{i} u_{ki} - \lambda_{i} } \right). $$

(20)

Analyze (20), finding the extreme value of the objective function. Then the derivatives $u_{ki}$ and $v_{k}$ are, respectively, derived, and first $u_{ki}$ is derived. Analyze (17), unfold the summation, and derive the two-level summation $u_{ki}$ of the two parts of the equation, respectively, to obtain (21).

$$ \frac{{\partial J\left( {U,V} \right)}}{{\partial u_{ki} }} = mu_{ki}^{m - 1} \times \left\{ {x_{i} - v_{k}^{2} + \frac{1}{{N_{R} }}\mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} (k)} \right]x_{j} - v_{k}^{2} } \right\} + \lambda_{i} . $$

(21)

Using the Euclidean norm, let (21) equal zero, and simplify the equation to (22) and (23).

$$ u_{ki}^{(m - 1)} = \frac{{( - \lambda_{i} )}}{{m\left\{ {||x_{i} - v_{k} ||^{2} + \frac{1}{{N_{R} }}\sum\nolimits_{\begin{subarray}{l} j \in N_{i} \\ j \ne i \end{subarray} } {[1 - SC_{ij} (k)]} ||x_{j} - v_{k} ||^{2} } \right\}}}, $$

(22)

$$ u_{ki} = \left[ {\frac{{( - \lambda_{i} )}}{{m\left\{ {||x_{i} - v_{k} ||^{2} + \frac{1}{{N_{R} }}\sum\nolimits_{\begin{subarray}{l} j \in N_{i} \\ j \ne i \end{subarray} } {[1 - SC_{ij} (k)]} ||x_{j} - v_{k} ||^{2} } \right\}}}} \right]^{{\frac{1}{m - 1}}} . $$

(23)

Combining formula (18) gets (24), where m > 1.

$$ \left( {\frac{{ - \lambda_{i} }}{m}} \right)^{{\frac{1}{m - 1}}} = \frac{1}{{\mathop \sum \nolimits_{s = 1}^{c} \frac{1}{{\left\{ {x_{i} - v_{s}^{2} + \frac{1}{{N_{R} }}\mathop \sum \nolimits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} (s)} \right]x_{j} - v_{s}^{2} } \right\}^{{\frac{1}{m - 1}}} }}}}. $$

(24)

Bring (24) into (23) to get the final iteration (12) of $u_{ki}$.

It can be seen from (19) that only the first term is related to $v_{k }$. The second-order sum is expanded, and $v_{k }$ is derived to (25).

$$ \frac{{\partial J\left( {U,V} \right)}}{{\partial v_{k} }} = \mathop \sum \limits_{i = 1}^{N} \left( { - 2} \right)u_{ki}^{m} x_{i} - v_{k} - 2\mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} \left\{ {\frac{1}{{N_{R} }}\mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} \left( k \right)} \right]x_{j} - v_{k} } \right\}. $$

(25)

Using the Euclidean norm, let (25) be equal to 0 to get the clustering center $v_{k }$. The detailed process is as (26), (27), and (28), the formula of $v_{k }$ is as shown in (11).

$$ \left( { - 2} \right)\mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} x_{i} - v_{k} - \frac{2}{{N_{R} }}\mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} \left\{ {\mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} \left( k \right)} \right]x_{j} - v_{k} } \right\} = 0, $$

(26)

$$ - \mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} x_{i} + \mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} v_{k} - \frac{1}{{N_{R} }}\mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} \left\{ {\mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} \left( k \right)} \right]x_{j} - \mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} \left( k \right)} \right]v_{k} } \right\} = 0, $$

(27)

$$ \left\{ {\mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} + \frac{1}{{N_{R} }}\mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} \mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} \left( k \right)} \right]} \right\}v_{k} = \mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} x_{i} + \frac{1}{{N_{R} }}\mathop \sum \limits_{i = 1}^{N} u_{ki}^{m} \mathop \sum \limits_{{\begin{array}{*{20}c} {j \in N_{i} } \\ {j \ne i} \\ \end{array} }} \left[ {1 - SC_{ij} \left( k \right)} \right]x_{j} . $$

(28)