Uttam Kumar | S. Kumar Raja | C. Mukhopadhyay | T. V. Ramachandra* |
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) (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)
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*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. 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). |