Conceptual framework for Variable Endmember Constrained Least Square (VECLS)

Intra-class variability of remotely sensed data is often too high or too low for unmixing a single pixel (Petrou & Foschi, 1999). In such situations, the endmember of a particular class are not strictly unique vectors and there is always a variation within that class. Therefore, we modify Eq. (1) to represent variable endmembers as random vectors, and introduce the following statistical LMM

                                                         (7)

where,  is the observed set of M reflectance values and  is the random vector representing the variable endmember of class n. The endmembers of each class n are drawn from a probability distribution defined by the probability density function (p.d.f.). It is reasonable to make an assumption that endmemberp.d.f.s of distinct LC categories are mutually independent. For example, the spectral signature distribution of builtup and vegetation can be treated independently. As a consequence of the above assumption,  are also mutually independent. If we denote as ,  Eq. (7) can be rewritten as

                                                      (8)

andp.d.f. of  can be shown to be . Since {} are mutually independent, p.d.f. of  is given by

,                (9)

where, * denotes a convolution operator.Note that p.d.f. of  implicitly depends on the abundance values . Given the observation vector y, we define the likelihood function L as

.       (10)

Here, we estimate abundance values by seeking  which maximises the likelihood function subject to the constraint that sum of all abundance values add to 1, which is the value of  that maximises the likelihood function. This means, given ,  is the value for which  has the maximum chance to occur. Also note that abundance values are deterministic quantities.

We first assume a Gaussian case and then subsequently move on to a generalised case, where the form of p.d.f. is not assumed. Let  and  be mean vector and covariance of nth target material, respectively. It is clear from LMM in Eq. (7) that the distribution of observation () is also Gaussian whose mean and covariance are given by

.                                          (11)

In our framework, in order to find the ML (Maximum Likelihood) abundances, we maximise the log-likelihood

   (12)

The above approach is based on the assumption that the underlying distribution of the spectral signatures are normal.

Next, in what follows we show a framework where no assumption on the form of p.d.f. is made. Let Z be a matrix whose column are the means of the spectral signatures i.e.,  Suppose that the actual observation is o, then we seek to minimise the deviation between o and  subject to constraints given in Eq. (3). We minimise,

    (13)

where, E{.} is the expectation operator and λ is the Lagrangian multiplier for enforcing the constraint on the abundance. Using (11), it is easy to simplify the expectation term in expression (13) as follows:

                (14)

                (15)

.                               (16)

Note that in Eq. (16) no cross-correlation terms appear. Denoting V as the diagonal matrix whose nth diagonal element is the trace of  (covariance of nth target material), the objective function F(α) can be written in a more compact form using Z and V as

                     (17)

where 1 is a vector whose components are all 1. In order to find the minima, we first take the first derivative of F(α) with respect to , λ and equate it to 0.

                                 (18)  

                                                       (19) 

                                                     (20)

.                                    (21)

Now we derive an explicit expression for λ. Denoting , Eq. (20) can be written as

                                       (22)
Multiplying 1T on both sides, we get

.                                (23)

Note that 1Tα = 1. Therefore the final expression for λ is

                                          (24)

                                               (25)

In conventional approach, where the endmembers are assumed to be constant, Z represents the endmember matrix (V=0), and the endmembers are obtained by minimising

                                                (26)

subject to the constraint that . However, in this approach, we have assumed the endmembers to be variable, and as a consequence the endmember variability term namely V has been accounted.

The spectral mixture analysis (SMA) techniques for addressing endmember variability have been broadly categorised (Somers et al., 2011) based on five basic principles: (i) iterative mixture analysis cycle, (ii) spectral feature selection, (iii) spectral weighting, (iv) spectral transformations, and (v) spectral modelling. Table 1 compares VECLS with the existing techniques (Somers et al 2011) and it is evident that VECLS is conceptually different as VECLS is not iterative, does not involve spectral feature selection, spectral weighting, spectral transformation or even spectral modelling. Hence VECLS has been placed as a separate category in the table for comparison. The underlying genesis behind the methods, advantages and disadvantages are also highlighted.

Table 1: Comparative analysis of the existing endmember variability spectral mixture analysis techniques with VECLS.