Comparison of the classiﬁcation methods for the images modeled by Gaussian Random Fields

. In image classiﬁcation often occur such situations, when images in some level are corrupted by additive noise. Such noise in image classiﬁcation can be modeled by Gaussian random ﬁelds (GRF). In image classiﬁcation supervised and unsupervised methods are used. In this paper we compare our proposed supervised classiﬁcation methods based on plug-in Bayes discriminant functions (PBDF) (see [ 6] and [11]) with unsupervised classiﬁcation method based on grey level co-occurrence matrix (GLCM) (see e.g. [8] and [1]). The remotely sensed image is used for classiﬁcation (USGS Earth Explorer). Also GRF with diﬀerent spatial correlation range are generated and added to the original remotely sensed image. Such situation can naturally occur during forest ﬁre, when smoke covers some territory. These images are used for classiﬁcation accuracy examination.


Introduction
Image classification is a problem of dividing an observed image into several homogeneous regions by labeling pixels based on feature information and information about spatial adjacency relationships with training sample. Switzer [13] was the first to treat classification of spatial data. For features based on Gausian Markov RF model, the influence of texture rotation to image classification is considered by [5]. Spatial contextual classification problems arising in geospatial domain is considered by [10]. Atkinson and Lewis [3] reviewed geostatistical techniques for classification of remotely sensed images.
In the present paper for image classification we use method which is retracting the popular requirement of conditional independence in Bayes classification rules (see [6,12]). Also the observation to be classified is assumed to be dependent on the training sample. In the series of papers (see e.g., [9,2,4]) the incorporation of geostatistical information of features into plug-in versions of classifiers is based on the marginal distribution of the observation to be classified. Thus we investigated the geostatistical Bayes classifiers based on conditional feature distribution of the observation to be classified. The importance and effectiveness of proposed techniques was examined in the examples on images with strictly separated classes corrupted by spatial Gaussian noise (see [12]). In this paper we use proposed techniques for classification of remotely sensed images which classes for classification are not strictly separated before GRF noise is added. The stationary GRF model for features and MRF model for class labels are considered. For the model mentioned above and in case of known population parameters the error rates associated with PBDF is investigated (see [7,12]).
In the present paper the comparison of proposed supervised PBDF methods and unsupervised GLCM based classification method is made. The performance is evaluated numerically and visually. For the numerical comparison of the tested classification procedures the empirical errors of misclassification are used.

The main concepts and definitions
Suppose that the feature is modeled by Gaussian random field {Z(s): s ∈ D}, D ⊂ R 2 . In the context of image analysis index s means pixel. The marginal model of observation Z(s) in class Ω l is where µ l is the constant mean, and the error term is generated by zero-mean stationary Gaussian random field {ε(s): s ∈ D} with covariance function defined by model for where r(s − u) is the spatial correlation function and σ 2 is the variance.
Assume that the model of Z for given Y = y is where X y is a design matrix, µ ′ = (µ 1 , µ 2 ) and E is the n-vector of random errors that has multivariate Gaussian distribution N n (0, σ 2 R). Denote by r 0 and by R the vector of spatial correlations between Z 0 and Z n and the matrix of spatial correlations among components of Z n , respectively. Since Z 0 is correlated with training sample, we have to deal with conditional Gaussian distribution of Z 0 given T = t (Z = z, Y = y) with means µ 0 lt and variance σ 2 0t .
Denote the three component vector of parameter estimates byΨ ′ = (μ ′ ,σ 2 ). Then PBDF associated with BDF specified in (1) is , and σ 2 0t =σ 2 R 0n . Denote it by PBDFD. If Z0 is assumed to be independent to T , then PBDF has the following form whereμ andσ 2 are the estimates of µ and σ 2 , based on T = t. Denote it by PBDFI.

Numerical example
We have tested our approach on a real world image, namely on areal remotely sensed image obtained with the Landsat7 satellite. The image shows the area from Lithuania territory. We use grey version of the RGB image for the experiment. The crop of the image containing forest and grassland is used. The cropped image dimensions are 500 × 500 pixels. Here we compare proposed supervised classification methods with unsupervised classification method based on GLCM with relative distance d = 1, 32 grey levels, 7 × 7 window size and relative orientation quantified in four directions (0 • , 45 • , 90 • , 135 • ) is calculated. GLCM mean is implemented as the texture feature. The sub image of size 100 × 100 pixels is extracted from the cropped image for classification. Suppose that original cropped image is corrupted by the noise modelled by GRF with zero mean and Gaussian spatial correlation function given by r(h) = exp{−|h| 2 /α} and also belonging to the Matern class. Here α is a spatial correlation range parameter. Such noise can naturally occur from smoke or fog. For supervised classification methods the training sample with n 1 = n 2 = 30 is selected and 8 nearest neighbour scheme was used. GLCM are calculated from sub images of size 30 × 30 pixels using three sub images from forest territory and three images from grassland territory. These GLCM are used as texture examples for classification. The original and corrupted sub images are classified and results using proposed supervised classification methods and unsupervised GLCM method are shown in (Fig. 1) The empirical errors of misclassification are presented in Table 1. As we can see from the Table 1 the proposed supervised methods perform better then GLCM, especially when GRF is applied. We can see from the Table 1 and from pictures ( Fig. 1) that with very small correlation range parameter classification accuracy is not high, but when the range parameter grows bigger accuracy becomes very similar to original image without additive noise.

Conclusions
The proposed methods can be used to increase classification accuracy of remotely sensed images when the smoke of fire, fog or other natural situations cover the territory and the correlation range is large.  The results show us the advantage of these methods against unsupervised classification method based on GLCM. Of course we must have in mind that these methods require training sample from the same territory.
The results of performed calculations give us the strong argument to encourage the users do not ignore the spatial correlation and locational information about training sample in image classification.