Spatial indices

As the abovementioned indicators work based on comparing pixel values of the fused and reference images with no care about the pixel neighbors therefore some probable drawbacks are sensible. These problems return to the fact that, it is probable for two compared images with the same statistical parameters (like mean and variance) to have different spatial arraignments of pixels. For example, assume a 5*5 window from an image (Figure 2-4(a)); after fusion process, several possible outputs could be appeared while in all cases statistical properties are the same.

Figure 2-4. 5*5 window from an image: (a) original image and (b-f) different probable results from a pixel level data fusion process.

With reference to figure (2-4(a)) after an assumptive pixel level fusion about five possible spatial arrangements of pixels and consequently different spatial

1. First (i.e. the best) is the case which pixel values and their spatial arraignments are the same (b);

2. Second (i.e. the worst) is the case where no pixel values and no spatial arraignments are the same (c);

3. Third is the case when one of the data properties i.e. spatial arraignment (d) or spectral pixel values (e) is exactly the same while another property is different.

4. Fourth case could be any possible situation in which the pixel values and their spatial arraignments could partly be the same (f).

With reference to the mentioned possibilities the weakness of spectral based parameters is more sensible. On the other hand the spatial based calculations lonely will also mislead the evaluation process. For example in the case (d) the spatial indicators will show the same value while it is definitely different from the reference data (a). Thus we do believe two spectral and spatial properties must be evaluated and used in a complementary fashion for image quality assessments. Based on the number four property (added property to Wald’s properties) which considers the spatial similarity of fused dataset to compare with Pan image the next two indices are evaluated as two considerable spatial indicators.

2.2.3.1 Normalized Difference of Entropies (NDE)

Entropy, as a measure of texture, has been applied in several domains of image processing. The texture-based entropy filter in the form of co-occurrence measure uses a gray-tone spatial dependence matrix to calculate texture values. This is a matrix of relative frequencies with which pixel values occur in two neighboring processing windows separated by a specified distance and direction. It shows the number of occurrences of the relationship between a pixel and its specified neighbor [Anys, 1994]. Thus the normalized differences of the entropy will show the amount of spatial similarity of the fused image in compare to its reference image.

∑= + the reference band that here is referred to as Pan image; N is the number of bands of the fused dataset. The NDE as a normalized measurement can get any value between 0 and 1. Therefore NDE=0 is the maximum spatial similarity between fused and Pan image. And with the increment of this value the amount of similarity will get lower.

As another possibility the difference of entropies can be measured in absolute difference value.

2.2.3.2 Normalized Difference of Autocorrelations (NDA)

Another robust indicator that can offer some spatial information about an image is autocorrelation. This measurement looks for an overall pattern between contiguous pixels and their similarity in an image. This measurement can be calculated globally across the whole image or locally throughout local filters. Consequently it provides a single global value that describes the spatial dependency of dataset as a whole or can offer an image in which every pixel has a value which shows its spatial dependency in relation to its neighbors. Irrespective of the global or locality of the measurement there are several indicators that calculate autocorrelation e.g. Moran’s I, Geary’s C and semivariance [Curran, 1988 and Woodcock and Strahler, 1987]. The same as NDE, the amount of NDA is calculated based on the normalized differences of autocorrelation of fused image and reference.

∑= +

Where A_Fi denotes the autocorrelation of a single fused band from dataset (F) and R is the reference band that here is referred to Pan image; N is the number of bands of the fused dataset. The same as NDE this indicator can be calculated as the absolute differences of two datasets.

Here the local Moran’s I and Geary’s C of Anselin’s LISAs [Anselin, 1995] [adapted from Yee Leung et al. 2002] are briefly explained and for more dialed discussions readers are referred to related references.

1. Local Moran’s I

For illustration purposes let xi (x1, x2… xn)^T be the vector of observations on random variable X at n locations and W=(wij)n*n be a symmetric spatial link matrix which is defined by the underlying spatial structure of the geographical units where the observations are made. Based on Moran’s I for pixel i, the local Moran’s I_i in its standardized form is [Anselin, 1995].

∑ large positive value of I_i indicates spatial clustering of similar values (either high or low) around location i and a large negative value indicates a clustering of dissimilar values that is a location with a high value which surrounded by neighbors with low values and vice versa.

2. Local Geary’s C

The local Geary’s C_i at a reference location i is defined by Anselin [1995]

With this assumption that, without loss of generality, W_ii =0. A small value of C_i suggests a positive spatial association (similarity) of observation i with its surrounding observations, whereas a large value of Ci suggests a negative association (dissimilarity) of observation i with its surrounding observations.

In this work the two mentioned spatial evaluations are calculated and the absolute value of differences has been adapted and used for image evaluation purposes.