http://www.iisc.ernet.in/
HYBRID BAYESIAN CLASSIFIER FOR IMPROVED CLASSIFICATION ACCURACY
http://wgbis.ces.iisc.ernet.in/energy/

Uttam Kumar
[uttam@ces.iisc.ernet.in]

S. Kumar Raja
[s.kumar.raja@yahoo.com]

C. Mukhopadhyay
[cm@mgmt.iisc.ernet.in]

T. V. Ramachandra*
[cestvr@ces.iisc.ernet.in]

Citation: Uttam Kumar, S. Kumar Raja, C. Mukhopadhyay and T. V. Ramachandra , 2011, Hybrid Bayesian Classifier for Improved Classification Accuracy. IEEE Geoscience and Remote Sensing letters, Vol. 8, No. 3, pp. 473 – 476.

REVIEW OF METHODS

Linear Unmixing: With K spectral bands and M classes (C1,..,CM), 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)

 

*T. V. Ramachandra, Senior Member, IEEE, is with the Centre for Ecological Sciences, Centre for Sustainable Technologies and Centre for Infrastructure, Sustainable Transport and Urban Planning, Indian Institute of Science (IISc), Bangalore, India.
(Corresponding author phone: 91-80-22933099; fax: 91-80-23601428; e-mail: cestvr@ces.iisc.ernet.in).

Uttam Kumar, Student Member, IEEE, is with the Department of Management Studies and Centre for Sustainable Technologies, Indian Institute of Science, India.  (e-mail: uttam@ces.iisc.ernet.in).

Chiranjit Mukhopadhyay is with the Department of Management Studies, Indian Institute of Science, Bangalore, India (e-mail: cm@mgmt.iisc.ernet.in).

S. Kumar Raja is with the VISTA Group, IRISA, Rennes, France and Thomson R&D France, SNC Cesson - Sévigné, France (email: s.kumar.raja@yahoo.com).

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