A Preferential Interval-Valued Fuzzy C-Means Algorithm for Remotely Sensed Imagery Classification

Abstract

Remotely sensed imagery classification have a large amount of uncertainty related to the intraclass heterogeneity and the interclass ambiguity of objects. Fuzzy set theory can address the uncertainty effectively, while interval-valued model can improve the separability of samples. Therefore, we propose a novel interval-valued fuzzy c-means algorithm, which integrates the interval-valued model and preferential adaptive method. It preferentially adjusts the interval width according to MSE (mean-square-error) and boundary factor for determining the optimal set of features for the data. In this paper, it is proved that the method can make the intraclass MSE and boundary factor always proportional to the separability of objects, so that it can dynamically adjust the interval-valued separability by controlling the interval width. Experimental data consisting of SPOT5 (10-m spatial resolution) satellite data for three case study areas in China are used to test this algorithm. Compared with other state-of-the-art fuzzy classification methods, our algorithm demonstrates the markedly improved overall accuracy and Kappa coefficients.

Appendices

Appendix 1: The Relationship Between Interval Adaptive Factor ${\mathbf{\alpha}}$ and the Separability of the Samples

Suppose the interval sample ${\tilde{\mathbf{A}}} = \left( {\tilde{A}_{1} , \tilde{A}_{2} , \ldots , \tilde{A}_{p} } \right)$ modeled by Eq. (1) method, and the current interval cluster center ${\tilde{\mathbf{V}}} = \left( {\tilde{V}_{1} , \tilde{V}_{2} , \ldots , \tilde{V}_{p} } \right),$ and the separability is expressed as Euclidean distance between ${\tilde{\mathbf{A}}}$ and ${\tilde{\mathbf{V}}}$ is

$$\begin{aligned} S\left( {{\tilde{\mathbf{A}}}, {\tilde{\mathbf{V}}}} \right) & = \sqrt {\mathop \sum \limits_{l = 1}^{p} \left( {\tilde{A}_{l} - \tilde{V}_{l} } \right)^{2} } \\ & = \sqrt {\mathop \sum \limits_{l = 1}^{p} \left[ {\left( {\tilde{A}_{l}^{ - } - \tilde{V}_{l}^{ - } } \right)^{2} + \left( {\tilde{A}_{l}^{ + } - \tilde{V}_{l}^{ + } } \right)^{2} } \right]} \\ & = \sqrt 2 *\sqrt {\mathop \sum \limits_{l = 1}^{p} \left[ {\left( {m_{{A_{l} }} - m_{{V_{l} }} } \right)^{2} + \left( {\alpha_{Al} *\sigma_{A} - d_{l} } \right)^{2} } \right]} \\ \end{aligned}$$

(12)

Since

$$\alpha_{Al} \in \left[ {0, 1} \right]\;{\text{and }}\left( {m_{{A_{l} }} - m_{{V_{l} }} } \right)^{2} \ge 0$$

(13)

Let $f = \left( {\alpha_{Al} *\sigma_{A} - d_{l} } \right)^{2}$ represent $S\left( {{\tilde{\mathbf{A}}},{\tilde{\mathbf{V}}}} \right)$ . Then differentiating $f$ with respect to $\alpha_{Al}$ leads to

$$\frac{\partial f}{{\partial \alpha_{Al} }} = 2\sigma_{A} \left( {\alpha_{Al} *\sigma_{A} - d_{l} } \right) = 0$$

(14)

We thus obtain

$$\sigma_{Al} = 0 \;{\text{or }}\alpha_{Al} = \frac{{d_{l} }}{{\sigma_{A} }}, \;\sigma_{A} \ne 0$$

(15)

Since $f \ge 0$, let $\sqrt f$ be $f$.

As shown in Fig. 5, we can get the detailed relationship rule between $\sqrt f$ and $\alpha_{Al}$ in interval [0, 1]:

Fig. 5

The detailed relationship rule between $\sqrt f$ and $\alpha_{Al}$ in interval [0, 1]

Case 1

Direct proportion if $\alpha_{Al} \in \left[ {0, 1} \right]$ and $\frac{{d_{l} }}{{\sigma_{A} }} = 0$.

Case 2

Direct proportion if $\alpha_{Al} \in \left[ {0, 1} \right]$ and $\frac{{d_{l} }}{{\sigma_{A} }} > 1.$

Case 3

Inverse proportion if $\alpha_{Al} \in \left[ {0, 1} \right]$ and $\frac{{d_{l} }}{{\sigma_{A} }} = 1.$

Case 4

Inverse proportion if $\alpha_{Al} \in \left[ {0, \frac{{d_{l} }}{{\sigma_{A} }}} \right)$ and $\frac{{d_{l} }}{{\sigma_{A} }} \ge 0.5;$

direct proportion if $\alpha_{Al} \in \left[ {\frac{{d_{l} }}{{\sigma_{A} }}, 1} \right]$ and $\frac{{d_{l} }}{{\sigma_{A} }} \ge 0.5.$

Case 5

Inverse proportion if $\alpha_{Al} \in \left[ {0, \frac{{d_{l} }}{{\sigma_{A} }}} \right)$ and $\frac{{d_{l} }}{{\sigma_{A} }} < 0.5;$

direct proportion if $\alpha_{Al} \in \left[ {\frac{{d_{l} }}{{\sigma_{A} }}, 1} \right]$ and $\frac{{d_{l} }}{{\sigma_{A} }} < 0.5.$

Case 6

None if $\alpha_{Al} \in \left[ {0, 1} \right]$ and $\sigma_{A} = 0.$

In summary, the relationship between the interval width adjustment factor ${\mathbf{\alpha}}$ and the degree of separation is as follows:

$$\begin{aligned} & {\mathbf{\theta}}\left( {\alpha_{Al} , S\left( {{\tilde{\mathbf{A}}}, {\tilde{\mathbf{C}}}_{k} } \right)} \right) \\ & = \left\{ \begin{array}{ll} {\text{inverse}} \;{\text{proportion}}, & \quad \alpha_{Al} \in \left[ 0,\text{min} \left( \frac{{d_{l} }}{{\sigma_{A} }}, 1 \right) \right) \ {\text{and}} \ \frac{{d_{l} }}{{\sigma_{A} }} \in \left( {0, + \infty } \right) \\ {\text{direct}} \ {\text{proportion}}, & \quad \alpha_{Al} \in \left[ {\frac{{d_{l} }}{{\sigma_{A} }}, 1} \right] \ {\text{and}} \ \frac{{d_{l} }}{{\sigma_{A} }} \in \left[ {0, 1} \right)\\ {\text{none}}, &\quad \sigma_{A} = 0 \hfill \end{array} \right. .\end{aligned}$$

(16)

Appendix 2: Preferential Interval Width Adaptive Factor ${\mathbf{\alpha}}$

The purpose of ${\mathbf{\alpha}}$ is to increase the separability of the samples when the intraclass MSE ${\mathbf{e}}$ and boundary factor ${\mathbf{\gamma}}$ increase.

Case 1

Under the condition of Fig. 4a, the separability and ${\mathbf{\alpha}} \in \left[ {0, 1} \right]$ are direct proportion relations, so ${\mathbf{\alpha}}$ needs to be proportional to ${\mathbf{e}}$ and ${\mathbf{\gamma}}$, and is calculated as

$${\mathbf{\alpha}} = 1 - w*\exp \left( { - \lambda *\left( {{\mathbf{e}}*{\mathbf{\gamma}}} \right)^{2} } \right)$$

(17)

Case 2

Under the condition of Fig. 4b, the separability and ${\mathbf{\alpha}} \in \left[ {0, 1} \right]$ are inverse proportion relations, so ${\mathbf{\alpha}}$ needs to be inversely proportional to ${\mathbf{e}}$ and ${\mathbf{\gamma}},$ and is calculated as

$${\mathbf{\alpha}} = 1 - w*\exp \left( { - \lambda *\left( {{\mathbf{e}}*{\mathbf{\gamma}} - {\text{H}}} \right)^{2} } \right),$$

(18)

where ${\text{H}}$ is the maximum value of the normalization of ${\mathbf{e}}*{\mathbf{\gamma}}.$

Case 3

Under the condition of Fig. 4c, the separability and ${\mathbf{\alpha}} \in \left[ {0, 1} \right]$ are inverse proportion relations, so ${\mathbf{\alpha}}$ needs to be inversely proportional to ${\mathbf{e}}$ and ${\mathbf{\gamma}}$, and the calculation method is the same as Case 2.

Case 4

Under the condition of Fig. 4d, the separability is proportional to ${\mathbf{\alpha}} \in \left[ {0, \frac{{\mathbf{d}}}{{\mathbf{\sigma}}}} \right)$ and inversely proportional to ${\mathbf{\alpha}} \in \left[ {\frac{{\mathbf{d}}}{{\mathbf{\sigma}}},1} \right]$. In this study, we choose to adjust in a lager interval, which is ${\mathbf{\alpha}} \in \left[ {0, \frac{{\mathbf{d}}}{{\mathbf{\sigma}}}} \right),$ so ${\mathbf{\alpha}}$ needs to be inversely proportional to ${\mathbf{e}}$ and ${\mathbf{\gamma}}$, and is calculated as

$${\mathbf{\alpha}} = \frac{{\mathbf{d}}}{{\mathbf{\sigma}}} *\left( {1 - w*\exp \left( { - \lambda *\left( {{\mathbf{e}}*{\mathbf{\gamma}} - {\text{H}}} \right)^{2} } \right)} \right)$$

(19)

Case 5

Under the condition of Fig. 4e, the separability is proportional to ${\mathbf{\alpha}} \in \left[ {0, \frac{{\mathbf{d}}}{{\mathbf{\sigma}}}} \right)$ and inversely proportional to ${\mathbf{\alpha}} \in \left[ {\frac{{\mathbf{d}}}{{\mathbf{\sigma}}},1} \right]$. Similarly, we choose to adjust in a lager interval, which is ${\mathbf{\alpha}} \in \left[ {\frac{{\mathbf{d}}}{{\mathbf{\sigma}}},1} \right],$ so ${\mathbf{\alpha}}$ needs to be proportional to ${\mathbf{e}}$ and ${\mathbf{\gamma}},$ and is calculated as

$${\mathbf{\alpha}} = \left( {1 - \frac{{\mathbf{d}}}{{\mathbf{\sigma}}}} \right)*\left( {1 - w*\exp \left( { - \lambda *\left( {{\mathbf{e}}*{\mathbf{\gamma}}} \right)^{2} } \right)} \right) + \frac{{\mathbf{d}}}{{\mathbf{\sigma}}}$$

(20)

Case 6

Under the condition of Fig. 4f, the separability is not related to ${\mathbf{\alpha}}.$

In summary, preferential interval width adaptive factor ${\mathbf{\alpha}}$ is calculated as

$$\begin{aligned} & {\mathbf{\alpha}} \\ & = \left\{ {\begin{array}{ll} {{\mathbf{}}\varvec{ } \frac{{\mathbf{d}}}{{\mathbf{\sigma}}} *\left( {1 - w*\exp \left( { - \lambda *\left( {{\mathbf{e}}*{\mathbf{\gamma}} - {\text{H}}} \right)^{2} } \right)} \right),} &{{\mathbf{\alpha}} \in \left[ {0, { \text{min} }\left( {\frac{{\mathbf{d}}}{{\mathbf{\sigma}}}, 1} \right)} \right)\varvec{ }and\varvec{ }\frac{{\mathbf{d}}}{{\mathbf{\sigma}}} \in \left( {0, + \infty } \right)} \\ \begin{array}{ll} {\mathbf{}} \left( {1 - \frac{{\mathbf{d}}}{{\mathbf{\sigma}}}} \right)*\left( {1 - w*\exp \left( { - \lambda *\left( {{\mathbf{e}}*{\mathbf{\gamma}}} \right)^{2} } \right)} \right) + \frac{{\mathbf{d}}}{{\mathbf{\sigma}}}, \hfill \\ 0, \hfill \\ \end{array} & \begin{array}{ll} {\mathbf{\alpha}} \in \left[ {\frac{{\mathbf{d}}}{{\mathbf{\sigma}}}, 1} \right]\varvec{ }and\varvec{ }\frac{{\mathbf{d}}}{{\mathbf{\sigma}}} \in \left[ {0, 1} \right) \hfill \\ \varvec{ }{\mathbf{\sigma}} = 0 \hfill \\ \end{array} \\ \end{array} } \right.. \\ \end{aligned}$$