Abstract
In complex color images, colors inside a homogeneous region might be contradistinctive and the distribution could not be described by a simple Gaussian distribution as used in traditional image segmentation algorithms. Based on the characteristics that the red, green, and blue color planes are not independent and pixels in the same neighborhood system might stand for the same object, we introduce a Gaussian model containing the interactions between different color planes to strengthen the connections both on a color plane and between color planes in a neighborhood system. Consequently, a Gaussian mixture model with the prior distribution, defined by Markov random field and acting as the weight, is employed to describe the distribution of color measures inside a homogeneous region. With the Gaussian mixture model containing the interactions between color planes, we proposed a fuzzy clustering approach for complex color image segmentation. Experiments on synthetic and real-color images, in which homogeneous regions are complex, show that the proposed algorithm compares favorably with the compared algorithms developed for the same purpose.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China [No. 41271435; No. 41301479].
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Appendix
Appendix
To obtain the derivation of the fuzzy membership u ij , we should minimize the objective function J m in Eq. (12) over it under the constraint
Introducing a Lagrange multiplier to the objective function, we have
Differentiating L with respect to u ij leads to
u ij is solved from Eq. (20) as
Since u ij satisfies Eq. (18), substitute Eq. (21) in to Eq. (18), the expression of u ij is obtained
To obtain the estimates of μ j , we need to conduct the minimization of J m in Eq. (12) over it
For 8-Neighborhood system, we have its expression as follows
Derivate J m over ∑ (−1) j , we can get
The expression of ∑ j is
The third term of Eq. (15) for d ij can be rewritten as
The derivation J m over α j is
Let Eq. (28) be equal to 0, we can solve α j as
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Zhao, X., Li, Y. & Zhao, Q. A Fuzzy Clustering Approach for Complex Color Image Segmentation Based on Gaussian Model with Interactions between Color Planes and Mixture Gaussian Model. Int. J. Fuzzy Syst. 20, 309–317 (2018). https://doi.org/10.1007/s40815-017-0411-1
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DOI: https://doi.org/10.1007/s40815-017-0411-1