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Robust semi-supervised spatial picture fuzzy clustering with local membership and KL-divergence for image segmentation

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Abstract

Aiming at existing symmetric regularized picture fuzzy clustering with weak robustness, and it is difficult to meet the need for image segmentation in the presence of high noise. Hence, a robust dynamic semi-supervised symmetric regularized picture fuzzy clustering with KL-divergence and spatial information constraints is presented in this paper. Firstly, a weighted squared Euclidean distance from current pixel value, its neighborhood mean and median to clustering center is firstly proposed, and it is embedded into the objective function of symmetric regularized picture fuzzy clustering to obtain spatial picture fuzzy clustering. Secondly, the idea of maximum entropy fuzzy clustering is introduced into picture fuzzy clustering, and an entropy-based picture fuzzy clustering with clear physical meaning is constructed to avoid the problem of selecting weighted factors. Subsequently, the prior information of the current pixel is obtained by means of weighted local membership of neighborhood pixels, and it is embedded into the objective function of maximum entropy picture fuzzy clustering with multiple complementary spatial information constraints through KL-divergence, a robust dynamic semi-supervised picture fuzzy clustering optimization model and its iterative algorithm are given. In the end, this proposed algorithm is strictly proved to be convergent by Zangwill theorem. The experiments on various images and standard datasets illustrate how our proposed algorithm works. This proposed algorithm has excellent segmentation performance and anti-noise robustness, and outperforms eight state-of-the-art fuzzy or picture fuzzy clustering-related algorithms in the presence of high noise.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (61671377, 51709228) and the Shaanxi Natural Science Foundation of China (2016JM8034, 2017JM6107). The authors would like to thank the anonymous reviewers for their constructive suggestions to improve the overall quality of the paper. Besides, the authors would like to thank the School of Electronic Engineering, Xi’an University of Posts & Telecommunications, Xi’an, China, for financial support.

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  1. School of Electronic Engineering, Xi’an University of Posts and Telecommunications, Xi’an, 710121, People’s Republic of China

    Chengmao Wu & Jiajia Zhang

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  1. Chengmao Wu
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  2. Jiajia Zhang
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Contributions

CW: conceptualization, data curation, funding acquisition, methodology, project administration, resources, supervision, validation, visualization. JZ: formal analysis, investigation, software, writing—original draft, writing—review and editing.

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Correspondence to Jiajia Zhang.

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Appendix A

Appendix A

Proof: The specific derivation process of Eqs. (19)–(22) of the proposed algorithm is as follows. The optimization model of robust semi-supervised spatial picture fuzzy clustering with local membership and KL-divergence is given as follows.

$$ \begin{aligned} \min {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} J\left( {U,V,\eta ,\xi } \right) = & \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{c} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }}d_{ij}^{2} + \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{c} {\left( {\eta_{ij}^{2} \xi_{ij} + \eta_{ij} \xi_{ij}^{2} } \right)} } } } \\ & + \gamma \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{c} {\left( {\left( {{{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right)\ln \frac{{\left( {{{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right)}}{{\tilde{u}_{ij} }} - \frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }} + \tilde{u}_{ij} } \right)} } \\ \end{aligned} $$
(45)

s.t.

$$ (1)\;0 \le u_{ij} ,\eta_{ij} ,\xi_{ij} \le 1,0 \le u_{ij} + \eta_{ij} + \zeta_{ij} \le 1,i = 1,2, \cdots ,n,j = 1,2, \cdots ,c; $$
(46)
$$ (2)\;\sum\limits_{j = 1}^{c} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }} = 1,i = 1,2, \cdots ,n;} $$
(47)
$$ (3)\;\sum\limits_{j = 1}^{c} {\left( {\eta_{ij} + \frac{1}{c}\xi_{ij} } \right) = 1,i = 1,2, \cdots ,n} . $$
(48)

The objective function of minimization of the optimization model Eq. (45) obtained by Lagrange multiplier method, we construct an unconstrained optimization objective function as:

$$ \begin{aligned} L\left( u \right) = & \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{c} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }}d_{ij}^{2} + \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{c} {\left( {\eta_{ij}^{2} \xi_{ij} + \eta_{ij} \xi_{ij}^{2} } \right)} } } } \\ & { + }\gamma \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{c} {\left( {\left( {{{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right) * \ln \frac{{\left( {{{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right)}}{{\tilde{u}_{ij} }} - \frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }} + \tilde{u}_{ij} } \right)} } \\ & + \sum\limits_{i = 1}^{n} {\alpha_{i} \left( {1 - \sum\limits_{j = 1}^{c} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }}} } \right) + \sum\limits_{i = 1}^{n} {\beta_{i} \left( {1 - \sum\limits_{j = 1}^{c} {\left( {\eta_{ij} + \frac{1}{c}\xi_{ij} } \right)} } \right)} } \\ \end{aligned} $$
(49)

where \(d_{ij}^{2} = w_{1} ||x_{i} - v_{j} ||^{2} + w_{2} ||\overline{x}_{i} - v_{j} ||^{2} + w_{3} ||\hat{x}_{i} - v_{j} ||^{2}\), \(\alpha_{i}\), \(\beta_{i}\) and \(\gamma\) are Lagrange multipliers. Taking the partial derivative of Eq. (49) and set the partial derivative of \(v_{j}\), \(u_{ij}\), \(\eta_{ij}\) and \(\xi_{ij}\) to zero, then

$$ \frac{\partial L}{{\partial v_{j} }} = \sum\limits_{j = 1}^{n} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }}\left( { - 2w_{1} ||x_{i} - v_{j} || - 2w_{2} ||\overline{x}_{i} - v_{j} || - 2w_{3} ||\hat{x}_{i} - v_{j} ||} \right)} = 0 $$
(50)

We simplify Eq. (50) and obtain

$$ \sum\limits_{j = 1}^{n} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }}} \left( {w_{1} + w_{2} + w_{3} } \right)v_{j} = \sum\limits_{j = 1}^{n} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }}} \left( {w_{1} x_{i} + w_{2} \overline{x}_{i} + w_{3} \hat{x}_{i} } \right) $$
(51)

Further, obtain

$$ v_{j} = \frac{{\sum\limits_{i = 1}^{n} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }}\left( {w_{1} x_{i} + w_{2} \overline{x}_{i} + w_{3} \hat{x}_{i} } \right)} }}{{\sum\limits_{i = 1}^{n} {\frac{{u_{ij} }}{{1 - \eta_{ij} - \xi_{ij} }}\left( {w_{1} + w_{2} + w_{3} } \right)} }} $$
(52)
$$ \frac{\partial L}{{\partial u_{ij} }} = \frac{1}{{1 - \eta_{ij} - \xi_{ij} }}\left( {d_{ij}^{2} + \gamma \ln \frac{{{{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}}}{{\tilde{u}_{ij} }} - \alpha_{i} } \right) = 0 $$
(53)

We simplify Eq. (53) and obtain

$$ d_{ij}^{2} + \gamma \ln \frac{{{{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}}}{{\tilde{u}_{ij} }} - \alpha_{i} = 0 $$
(54)

Further, obtain

$$ u_{ij} = \left( {1 - \eta_{ij} - \xi_{ij} } \right)\tilde{u}_{ij} \exp \left( {\frac{{\alpha_{i} - d_{ij}^{2} }}{\gamma }} \right) $$
(55)

By substituting Eq. (55) into Eq. (47), we obtain:

$$ \exp \left( {\frac{{\alpha_{i} }}{\gamma }} \right) = \frac{1}{{\sum\limits_{j = 1}^{c} {\exp \left( { - \frac{{d_{ij}^{2} }}{\gamma }} \right)\tilde{u}_{ij} } }} $$
(56)

Combining Eq. (55) and Eq. (56), we can calculate \(u_{ij}\) as follows.

$$ u_{ij} = \left( {1 - \eta_{ij} - \xi_{ij} } \right)\frac{{\tilde{u}_{ij} \exp \left( { - {{d_{ij}^{2} } \mathord{\left/ {\vphantom {{d_{ij}^{2} } \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \right)}}{{\sum\limits_{k = 1}^{c} {\tilde{u}_{ik} \exp \left( { - {{d_{ik}^{2} } \mathord{\left/ {\vphantom {{d_{ik}^{2} } \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \right)} }} $$
(57)

So that

$$ u_{ij} = \left( {1 - \eta_{ij} - \xi_{ij} } \right)\frac{{\tilde{u}_{ij} \exp \left( { - (w_{1} {{||x_{i} - v_{j} ||^{2} + w_{2} ||\overline{x}_{i} - v_{j} ||^{2} + w_{3} ||\hat{x}_{i} - v_{j} ||^{2} )} \mathord{\left/ {\vphantom {{||x_{i} - v_{j} ||^{2} + w_{2} ||\overline{x}_{i} - v_{j} ||^{2} + w_{3} ||\hat{x}_{i} - v_{j} ||^{2} )} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \right)}}{{\sum\limits_{k = 1}^{c} {\tilde{u}_{ik} } \exp \left( { - (w_{1} {{||x_{i} - v_{k} ||^{2} + w_{2} ||\overline{x}_{i} - v_{k} ||^{2} + w_{3} ||\hat{x}_{i} - v_{k} ||^{2} )} \mathord{\left/ {\vphantom {{||x_{i} - v_{k} ||^{2} + w_{2} ||\overline{x}_{i} - v_{k} ||^{2} + w_{3} ||\hat{x}_{i} - v_{k} ||^{2} )} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \right)}} $$
(58)
$$ \frac{\partial L}{{\partial \eta_{ij} }} = \frac{{u_{ij} }}{{\left( {1 - \eta_{ij} - \xi_{ij} } \right)^{2} }}\left( {d_{ij}^{2} + \gamma \ln \frac{{{{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}}}{{\tilde{u}_{ij} }} - \alpha_{i} } \right) + \xi_{ij}^{2} + 2\eta_{ij} \xi_{ij} - \beta_{i} = 0 $$
(59)

Combining Eq. (53) with Eq. (59), we obtain:

$$ {2}\eta_{ij} \xi_{ij} + \xi_{ij}^{2} = \beta_{i} $$
(60)
$$ \frac{\partial L}{{\partial \xi_{ij} }} = \frac{{u_{ij} }}{{\left( {1 - \eta_{ij} - \xi_{ij} } \right)^{2} }}\left( {d_{ij}^{2} + \gamma \ln \frac{{{{u_{ij} } \mathord{\left/ {\vphantom {{u_{ij} } {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \eta_{ij} - \xi_{ij} } \right)}}}}{{\tilde{u}_{ij} }} - \alpha_{i} } \right) + \eta_{ij}^{2} + 2\eta_{ij} \xi_{ij} - \frac{{\beta_{i} }}{c} = 0 $$
(61)

Combining Eq. (53) with Eq. (61), we obtain:

$$ \eta_{ij}^{2} + 2\eta_{ij} \xi_{ij} = c^{ - 1} \beta_{i} $$
(62)

We combine Eq. (62) with Eq. (60) and obtain the following equation.

$$ \eta_{ij}^{2} + 2\eta_{ij} \xi_{ij} = \frac{{2\eta_{ij} \xi_{ij} + \xi_{ij}^{2} }}{c} $$
(63)

According to Eq. (63), we obtain the expression of \(\eta_{ij}\)

$$ \eta_{ij} = \frac{{\xi_{ij} \left( {2\eta_{ij} + \xi_{ij} } \right)}}{{c\left( {\eta_{ij} + 2\xi_{ij} } \right)}} $$
(64)

Lastly, by using famous Yager’s fuzzy complement operator \(N_{\alpha } {(}x){ = }(1 - x^{\alpha } )^{{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }}}\), We can obtain the expression of refusal degree \(\xi_{ij}\) as follows.

$$ \xi_{ij} = 1 - \left( {u_{ij} + \eta_{ij} } \right) - \left( {1 - \left( {u_{ij} + \eta_{ij} } \right)^{\alpha } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }}} $$
(65)

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Wu, C., Zhang, J. Robust semi-supervised spatial picture fuzzy clustering with local membership and KL-divergence for image segmentation. Int. J. Mach. Learn. & Cyber. 13, 963–987 (2022). https://doi.org/10.1007/s13042-021-01429-y

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