Object-based feature extraction for hyperspectral data using firefly algorithm

Abstract

Object-based classification methods can improve the accuracy of hyperspectral image classification due to the fact that they incorporate spatial information into the classification procedure by assigning neighboring pixels into the same class. In this paper, a new object-based feature extraction method is proposed which makes use of information theory to reduce the Bayes error. In this way, the proposed method exploits higher order statistics for feature extraction which are very effective for non Gaussian data such as hyperspectral images. The criterion to be minimized is composed of three mutual information terms. The first and second terms, consider the maximal relevance and minimal redundancy, respectively, while the third term takes into account the segmentation map containing disjoint spatial regions. To obtain the segmentation map, we apply the firefly clustering algorithm whose fitness function simultaneously considers the intra-distance between samples and their cluster centroids, and inter-distance between centroids of any two clusters. Our experimental results, performed using a variety of hyperspectral scenes, indicate that the proposed framework gives better classification results than some state-of the-art spectral–spatial feature extraction methods.

Appendix

The gradients of different terms used in the objective function $Q\left( {R,_{{}} W,_{{}} E} \right)$ with respect to the (n₁, n₂)^th entry of the transformation matrix A are:

$$\frac{{\partial H\left( {v_{i} } \right)}}{{\partial {\mathbf{A}}\left( {n_{1} ,n_{2} } \right)}} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {i \ne n_{1} } \hfill \\ {\frac{1}{{N_{p} }}\left( {\frac{1}{{\sigma_{i}^{2} }}\sum\limits_{k = 1}^{{N_{p} }} {\frac{{\sum\nolimits_{e = 1}^{{N_{p} }} {G\left( {v_{i}^{kt} - v_{i}^{et} ;\sigma_{i}^{2} } \right)\left( {v_{i}^{kt} - v_{i}^{et} } \right)\left( {y_{{n_{2} }}^{kt} - y_{{n_{2} }}^{et} } \right)} }}{{\sum\nolimits_{e = 1}^{{N_{p} }} {G\left( {v_{i}^{kt} - v_{i}^{et} ;\sigma_{i}^{2} } \right)} }}} } \right)} \hfill & {i = n_{1} } \hfill \\ \end{array} } \right.$$

(20)

$$\frac{{\partial \hat{H}\left( {v_{i} } \right)}}{{\partial {\mathbf{A}}\left( {n_{1} ,n_{2} } \right)}} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {i \ne n_{1} } \hfill \\ {\frac{1}{{N_{h} }}\left( {\frac{1}{{\hat{\sigma }_{i}^{2} }}\sum\limits_{k = 1}^{{N_{h} }} {\frac{{\sum\nolimits_{e = 1}^{{N_{h} }} {G\left( {v_{i}^{k} - v_{i}^{e} ;\hat{\sigma }_{i}^{2} } \right)\left( {v_{i}^{k} - v_{i}^{e} } \right)\left( {y_{{n_{2} }}^{k} - y_{{n_{2} }}^{e} } \right)} }}{{\sum\nolimits_{e = 1}^{{N_{h} }} {G\left( {v_{i}^{k} - v_{i}^{e} ;\hat{\sigma }_{i}^{2} } \right)} }}} } \right)} \hfill & {i = n_{1} } \hfill \\ \end{array} } \right.$$

(21)

$$\frac{{\partial H\left( {v_{i} |C} \right)}}{{\partial {\mathbf{A}}\left( {n_{1} ,n_{2} } \right)}} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {i \ne n_{1} } \hfill \\ {\frac{1}{{N_{p} }}\sum\limits_{o = 1}^{{N_{c} }} {\sum\limits_{k = 1}^{{N_{o} }} {\frac{{\sum\nolimits_{e = 1}^{{N_{o} }} {G\left( {v_{i}^{o,kt} - v_{i}^{o,et} ;\sigma_{i,o}^{2} } \right)\left( {v_{i}^{o,kt} - v_{i}^{o,et} } \right)\left( {y_{{n_{2} }}^{o,kt} - y_{{n_{2} }}^{o,et} } \right)} }}{{\sigma_{i,o}^{2} \sum\nolimits_{e = 1}^{{N_{p} }} {G\left( {v_{i}^{o,kt} - v_{i}^{o,et} ;\sigma_{i,o}^{2} } \right)} }}} } } \hfill & {i = n_{1} } \hfill \\ \end{array} } \right.$$

(22)

$$\frac{{\partial \hat{H}\left( {v_{i} |C} \right)}}{{\partial {\mathbf{A}}\left( {n_{1} ,n_{2} } \right)}} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {i \ne n_{1} } \hfill \\ {\frac{1}{{N_{h} }}\sum\limits_{\lambda = 1}^{{N_{b} }} {\sum\limits_{k = 1}^{{N_{\lambda } }} {\frac{{\sum\nolimits_{e = 1}^{{N_{\lambda } }} {G\left( {v_{i}^{\lambda ,k} - v_{i}^{\lambda ,e} ;\hat{\sigma }_{i,\lambda }^{2} } \right)\left( {v_{i}^{\lambda ,k} - v_{i}^{\lambda ,e} } \right)\left( {y_{{n_{2} }}^{\lambda ,k} - y_{{n_{2} }}^{\lambda ,e} } \right)} }}{{\hat{\sigma }_{i,\lambda }^{2} \sum\nolimits_{e = 1}^{{N_{\lambda } }} {G\left( {v_{i}^{\lambda ,k} - v_{i}^{\lambda ,e} ;\hat{\sigma }_{i,\lambda }^{2} } \right)} }}} } } \hfill & {i = n_{1} } \hfill \\ \end{array} } \right.$$

(23)

$$\begin{aligned} \frac{{\partial H\left( {v_{i} ,v_{j} } \right)}}{{\partial {\mathbf{A}}\left( {n_{1} ,n_{2} } \right)}} & = \frac{1}{{N_{p} \sigma_{i}^{2} }}\sum\limits_{k = 1}^{{N_{p} }} {\frac{{\sum\nolimits_{e = 1}^{{N_{p} }} {G\left( {v_{i}^{kt} - v_{i}^{et} ;\sigma_{i}^{2} } \right)G\left( {v_{j}^{kt} - v_{j}^{et} ;\sigma_{j}^{2} } \right)\left( {v_{i}^{kt} - v_{i}^{et} } \right)\left( {y_{{n_{2} }}^{kt} - y_{{n_{2} }}^{et} } \right)} }}{{\sum\nolimits_{e = 1}^{{N_{p} }} {G\left( {v_{i}^{kt} - v_{i}^{et} ;\sigma_{i}^{2} } \right)G\left( {v_{j}^{kt} - v_{j}^{et} ;\sigma_{j}^{2} } \right)} }}\delta_{{in_{1} }} } \\ & \quad + \frac{1}{{N_{p} \sigma_{j}^{2} }}\sum\nolimits_{k = 1}^{{N_{p} }} {\frac{{\sum\nolimits_{e = 1}^{{N_{p} }} {G\left( {v_{j}^{kt} - v_{j}^{et} ;\sigma_{j}^{2} } \right)G\left( {v_{i}^{kt} - v_{i}^{et} ;\sigma_{i}^{2} } \right)\left( {v_{j}^{kt} - v_{j}^{et} } \right)\left( {y_{{n_{2} }}^{kt} - y_{{n_{2} }}^{et} } \right)} }}{{\sum\nolimits_{e = 1}^{{N_{p} }} {G\left( {v_{j}^{kt} - v_{j}^{et} ;\sigma_{j}^{2} } \right)G\left( {v_{i}^{kt} - v_{i}^{et} ;\sigma_{i}^{2} } \right)} }}} \delta_{{jn_{1} }} \\ \end{aligned}$$

(24)

where $\delta$ is the Kronecker delta function:

$$\delta_{{q_{1} q_{2} }} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {q_{1} = q_{2} } \hfill \\ 0 \hfill & {q_{1} \ne q_{2} } \hfill \\ \end{array} } \right.$$

(25)

Using Eqs. (20) to (25), one can easily obtain Eq. (19).