REVIEW OF METHODS

Linear Unmixing: With K spectral bands and M classes (C₁,..,C_M), each pixel has an associated K-dimensional pixel vector  whose components are the gray values corresponding to the K spectral bands. Let , where, for ,  is a  column vector representing the endmember spectral signature of the mth target material. For a pixel, let  denote the fraction of the mth target material signature, and  denote the M-dimensional abundance column vector. The linear mixture model for  is given by

                                                                                  (1)

where, , are i.i.d.  [11]. Equation (1) represents a standard signal detection model where  is a desired signal vector to be detected. Since, OSP detects one signal (target) at a time, we divide a set of M targets into desired  and undesired  targets. A logical approach is to eliminate the effects of undesired targets that are considered as “interferers” to  before the detection of  takes place. Now, in order to find , the desired target material is . The term  in (1) can be rewritten to separate the desired spectral signature :

                                                                   (2)                                                                                                                                                                                                                                                                                                                                               
where  and . Thus the interfering signatures in  can be removed by the operator,

                                                           (3)

which is used to project  into a space orthogonal to the space spanned by the interfering spectral signatures, where  is the  identity matrix [12]. Operating on  with , and noting that ,

.                                                                   (4)

After maximizing the SNR, an optimal estimate of  is

                                                                        (5)

Note that  and thus  can be taken as proportional probabilities of the M classes i.e. .

Bayesian classifier: Associated with any pixel, there is an observation . With M classes , Bayesian classifier calculates the posterior probability of each class conditioned on  [13]:

                                                           (6)

In (6), since  is constant for all classes, only  is considered.  is computed assuming class conditional independence, so,  is given by

                                                             (7)