Linear model

Linear unmixing assumes that a satellite signal is modeled as a weighted sum of a limited number of basic signals, where each of these signals is characteristic of one of a number of LC types contributing to the signal. If there are M spectral bands and N classes, then associated with each pixel is a M-dimensional vector y whose components are the gray values corresponding to the M bands. Let E = [e₁, …,e_n-1, e_n, e_n+1, . . , e_N] be a M × N matrix, where {en} is a column vector representing the spectral signature (endmember) of the n^th target material. For a given pixel, the abundance or fraction of the nth target material present in a pixel is denoted by αn, and these values are the components of the N-dimensional abundance vector α. Assuming LMM (Shimabukuro and Smith, 1991), the observation vector  is related to E by

                                                                (1)

where accounts for the measurement noise. We further assume that the components of the noise vector  are zero-mean random variables that are i.i.d. (independent and identically distributed). Therefore, the covariance matrix of the noise vector is σ²I, where σ² is the variance, and I is M × M identity matrix. Two constraints imposed on the abundances in Eq. (1) are the non-negativity and sum-to-one given as

                                                 (2)

and

                                                           (3)

This allows proportions of each pixel to be partitioned between classes.The conventional approach (Nielsen, 2001) to extract the abundance values is to minimise , and the Unconstrained Least Squares estimate for the abundance is

.                                                       (4)

Imposing the unity constraint on the abundance values while minimising, gives the Constrained Least Squares (CLS) estimate of the abundance as,

                                                     (5)

where

.                                                   (6)