A Fuzzy Clustering Approach for Complex Color Image Segmentation Based on Gaussian Model with Interactions between Color Planes and Mixture Gaussian Model

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.

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

$$\sum\limits_{j = 1}^{c} {u_{ij} = 1\quad \forall i = 1, \ldots , \, N}$$

(18)

Introducing a Lagrange multiplier to the objective function, we have

$$L = \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{c} {u_{ij} d_{ij} } } + \lambda \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{c} {u_{ij} \log \left( {\frac{{u_{ij} }}{{\pi_{ij} }}} \right)} } + \sum\limits_{i = 1}^{N} {\varepsilon_{i} \left( {\sum\limits_{j' = 1}^{c} {u_{ij'} } - 1} \right)}$$

(19)

Differentiating L with respect to u _ij leads to

$$\frac{\partial L}{{\partial u_{ij} }} = d_{ij} + \lambda \log \left( {\frac{{u_{ij} }}{{\pi_{ij} }}} \right) + \lambda + \varepsilon_{i} = 0$$

(20)

u _ij is solved from Eq. (20) as

$$u_{ij} = \pi_{ij} \exp \left( { - \frac{{d_{ij} }}{\lambda } - \frac{{\lambda + \varepsilon_{i} }}{\lambda }} \right)$$

(21)

Since u _ij satisfies Eq. (18), substitute Eq. (21) in to Eq. (18), the expression of u _ij is obtained

$$u_{ij} = \frac{{\pi_{ij} \exp \left( { - \frac{{d_{ij} }}{\lambda }} \right)}}{{\sum\nolimits_{j' = 1}^{c} {\pi_{ij'} \exp \left( { - \frac{{d_{ij'} }}{\lambda }} \right)} }}$$

(22)

To obtain the estimates of μ _j , we need to conduct the minimization of J _m in Eq. (12) over it

$$\frac{{\partial J_{m} }}{{\partial {\varvec{\upmu}}_{j} }} = - \left( {{\mathbf{I}} + \sum\limits_{{i' \in N_{i} }} {{\varvec{\upalpha}}_{j} } } \right)\sum\limits_{i = 1}^{N} {u_{ij} {\varvec{\Sigma}}_{j}^{ - 1} \left[ {\left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) - {\varvec{\upalpha}}_{j} \left( {\sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right)} \right]} = 0$$

(23)

For 8-Neighborhood system, we have its expression as follows

$${\varvec{\upmu}}_{j} = \left( {I - 8{\varvec{\upalpha}}_{j} } \right)^{ - 1} \frac{{\sum\nolimits_{i = 1}^{N} {u_{ij} \left( {{\mathbf{x}}_{i} - {\varvec{\upalpha}}_{j} \sum\nolimits_{{i' \in N_{i} }} {{\mathbf{x}}_{i'} } } \right)} }}{{\sum\nolimits_{i = 1}^{N} {u_{ij} } }}$$

(24)

Derivate J _m over ∑ ⁽⁻¹⁾ _j , we can get

$$\frac{{\partial J_{m} }}{{\partial {\varvec{\Sigma}}_{j}^{ - 1} }} = \frac{1}{2}\sum\limits_{i = 1}^{N} {u_{ij} \left\{ { - {\varvec{\Sigma}}_{j} + \left[ {\left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) - {\varvec{\upalpha}}_{j} \left( {\sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right)} \right]\left[ {\left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) - {\varvec{\upalpha}}_{j} \left( {\sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right)} \right]^{\text{T}} } \right\}}$$

(25)

The expression of ∑ _j is

$${\varvec{\Sigma}}_{j} = \frac{{\sum\nolimits_{i = 1}^{N} {u_{ij} \left[ {\left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) - {\varvec{\upalpha}}_{j} \sum\nolimits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]\left[ {\left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) - {\varvec{\upalpha}}_{j} \sum\nolimits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]^{\text{T}} } }}{{\sum\nolimits_{i = 1}^{N} {u_{ij} } }}$$

(26)

The third term of Eq. (15) for d _ij can be rewritten as

$$\begin{aligned} & \left[ {\left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) - {\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]^{\text{T}} {\varvec{\Sigma}}_{j}^{ - 1} \left[ {\left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) - {\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right] \hfill \\ & \quad= \left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right)^{\text{T}} {\varvec{\Sigma}}_{j}^{ - 1} \left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) - \left[ {{\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]^{\text{T}} {\varvec{\Sigma}}_{j}^{ - 1} \left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) \hfill \\ & \quad - \left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right)^{\text{T}} {\varvec{\Sigma}}_{j}^{ - 1} {\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} \hfill \\ & \quad+ \left[ {{\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]^{\text{T}} {\varvec{\Sigma}}_{j}^{ - 1} {\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} \hfill \\ \end{aligned}$$

(27)

The derivation J _m over α _j is

$$\begin{aligned} \frac{{\partial J_{m} }}{{\partial {\varvec{\upalpha}}_{j} }} = \frac{1}{2}\sum\limits_{i = 1}^{N} {u_{ij} \left\{ { - {\varvec{\Sigma}}_{j}^{ - 1} \left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) \, \left[ {{\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]^{\text{T}} - \left( {{\varvec{\Sigma}}_{j}^{ - 1} } \right)^{\text{T}} \left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) \, \left[ {{\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]^{\text{T}} } \right.} \hfill \\ \left. { + {\varvec{\Sigma}}_{j}^{ - 1} {\varvec{\upalpha}}_{j} \sum\limits_{{i{\prime } \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right) \, \left( {\sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right)}^{\text{T}} +\, \left( {{\varvec{\Sigma}}_{j}^{ - 1} } \right)^{\text{T}} {\varvec{\upalpha}}_{j} \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right) \, \left( {\sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right)}^{\text{T}} } \right\} \hfill \\ \end{aligned}$$

(28)

Let Eq. (28) be equal to 0, we can solve α _j as

$${\varvec{\upalpha}}_{j} = \sum\limits_{i = 1}^{N} {u_{ij} \left( {{\mathbf{x}}_{i} - {\varvec{\upmu}}_{j} } \right) \, \left[ {\sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]^{\text{T}} } \left\{ {\sum\limits_{i = 1}^{N} {u_{ij} } \sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right) \, } \left[ {\sum\limits_{{i' \in N_{i} }} {\left( {{\mathbf{x}}_{i'} - {\varvec{\upmu}}_{j} } \right)} } \right]^{\text{T}} } \right\}^{ - 1}$$

(29)