International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
  
2. SPECTRAL SIMILARITY MEASURES 
In spectral matching issue, the algorithms need the definition of 
some criteria for-measuring the similarity and closeness of 
pixels. 
As mentioned, regarding deterministic measure, there are tree 
major types of measures. They are distance based measures, 
angle based and correlation based measures. 
In this paper we have used a classical notation, which may be 
finding in the literature. If we consider a set of hyperspectral 
images as a cube, then, each pixel can be consider as an 
observation vector (see Figure 1). 
  
  
  
  
  
Figure 1. Each pixel corresponds to a vector of observations 
This consideration corresponds to spectral space beside of the 
image and feature spaces, defined in (Landgrebe 1999). With 
this definition, the spectra are variations within pixels as a 
function of wavelength. From here, we define a pixel vector as 
b and a target vector as / . The number of hyperspectral 
channels is n. In other word, n is the dimensionality of sample 
data. 
2.1 Spectral Distance Similarity (SDS) 
In statistical analysis and signal processing, the metric of 
distance is frequently used for measuring the separation or the 
closeness of data samples. Based on the distance, the use of 
different norms generates different metrics. For example, City 
block distance, Euclidian distance and Tchebyshev distance are 
some measures corresponding to £,, 6, and £, -norms (Keränen 
2002). We use here the Euclidian distance defined as: 
Y -py 
For a logical comparison, we prefer to scale the distances 
between 0 and 1: 
Fd=(Ed - m)/(M —m) (2) 
where m and M are the minimum and maximum of Ed; 
values respectively. 
Ed, = (1) 
orig 
   
orig 
2.2 Spectral Correlation Similarity (SCS) 
The Pearson statistical correlation, p can be used as a similarity 
measure. It shows how two vectors are correlated. We define it 
here as: 
  
  
; i - u Xp; - us) 3) 
p 
n-1 gc 
where p and 6 are the mean and. standard deviation of target 
vector and pixel vector, respectively. To have the value 
between 0 and 1 as the previous measure, negative values are 
disregarded. 
50 
2.3 Spectral Similarity Value (SSV) 
The spectral similarity value is a combined measure of the 
correlation similarity and the Euclidian distance. It can be 
formulated as: 
SSV = JEd* +(1-p)’ (4) 
Identical vectors have identical magnitudes and directions. For 
a spectrum considered as a vector, the magnitude corresponds 
to the average spectral reflectance (brightness) and the direction 
corresponds to the spectral shape (including all the absorptions 
and emissions due to physical processes). Both dimensions of 
vector identity must be quantified when determining the 
similarity, or ‘closeness’ between two spectra. Euclidean 
distance primarily measures the brightness difference between 
two vectors. Correlation compares the shapes of two spectra. By 
definition, the SSV combines brightness and shape similarity. It 
has a minimum of zero and a maximum of the square root of 
two. In other word, smaller SSV indicates spectra that are more 
similar (Granahan 2001). 
2.4 Modified Spectral Angle Similarity (MSAS) 
Given two vectors as the target and pixel spectra, a spectral 
angle between this pair of vectors can be defined (Yuhas 92). In 
the case of a hyperspectral image, the "hyper-angle" is 
calculated with: 
Shp, 
izl 
a = arecos 
  
The smaller angle means more similarity between the pixel and 
target spectra. Here, we prefer to use a modified spectral angle 
presented by (Schwarz 2001). In above equation a is between 0 
and z/2, so we can easily obtain: 
MEAS mal (6) 
Z. 
by this rescaling the values of measure convert to [0, I]. It can 
be helpful for comparison with other measures. 
2.5 Constrained Energy Minimizing (CEM) 
The CEM technique (Harseany 93) has become quite popular in 
recent years as a mean for constructing a linear operator to 
perform matched filtering of hyperspectral images. The CEM 
algorithm tries to maximize the response of the target spectral 
signature while suppressing the response of the unknown 
background signatures. In other words, in this technique we try 
to find a linear operator or filter such as e, which could reduce 
all bands of hyperspectral images to one image. It emphasizes 
on target spectrum and minimizes the background energy 
(Farrand 97), as: 
yeu an (7) 
I 
Where r, is the set of total image pixels. If t is our target 
spectrum of interest, then this operator must grant: 
te =l (8) 
In other words the operator @ minimizes the filter output energy 
subject to the constraint (8). With this consideration we obtain: 
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