Robust spectral unmixing using the Hough transform.

Let us re-write the equation of the linear model for spectral mixing for the case of 3 pure classes:

w_j = a x_j + b y_j + (1-a-b) z_j                               (14)

where a and b are the unknown proportions and x_j, y_j, z_j and w_j are the reflectances of the pure and the mixed class in band j. For given values of x_j, y_j, z_j and w_j, equation (14) represents a straight line in a coordinate system (a,b) where we measure the values of proportions a and b along the two axes. If we consider the equations for two bands, we can solve for the two unknowns a and b. This way, a quadruple of pixels (i1, i2, i3, i4) with values (x_i1j, y_i2j, z_i3j, w_i4j) in band j and values (x_i1k, y_i2k, z_i3k, w_i4k) in band k defines a point in parameter space (a,b). This process is called the Hough transform.

Imagine now that we use a grid in the parameter space (a,b), i.e. imagine that we discretise the values of parameters a and b and create discrete cells each of which corresponds to a particular set of ranges of values of a and b. Every time we find a pair (a,b) that is within the range of a cell, the value of the cell will be incremented by 1.

Imagine then that we pick at random quadruples of pixels, one from each pure class and one from the mixed class. For every quadruple of pixels and for every pair of bands we identify a point (a,b). We accumulate the value of the corresponding cell by 1. We repeat the process for all quadruples of pixels and all pairs of bands. The cell in the (a,b) space that will end up having the maximum value will indicate the correct values of the mixing proportions a and b.

This approach was proposed by Bosdogianni, Petrou and Kittler (Mixed Pixel Classification with Robust Statistics, IEEE Transactions on Geoscience and Remote Sensing, vol 35,pp 551-559, 1997 ). Figure 11 shows an example of an accumulator array with the peak clearly identified.

Figure 11

Advantages of robust spectral unmixing 

Disadvantages of robust spectral unmixing 

What if data representing the pure classes are not available?