Linear spectral unmixing

The basic assumption on which linear spectral unmixing is based is that the spectrum of a mixed pixel has been created by the linear superposition of the spectra of the pure components, and the coefficients of the linear superposition are equal to the fractional coverages of the field of view of the pixel by the corresponding pure components. Consider for example the case where we know that we have 3 pure classes in a scene:

• bare soil

• grass

• tree seedlings

Consider that the sensor used has K bands, i.e. the spectrum of each pixel consists of K numbers, one for each band. Let us make the assumptions that: 

If a pixel corresponds to a patch of bare soil, its spectrum is given by the K-dimensional vector x.

If a pixel corresponds to a patch covered by grass only, its spectrum is given by the K-dimensional vector y.

If a pixel corresponds to a patch fully covered by seedlings, its spectrum is given by the K-dimensional vector z.

Let  w_i be the spectrum of mixed pixel i, and a_i, b_i and c_i the proportions of bare soil, grass and seedlings in it respectively. Then we can write:

 w_i = a_i x + b_i y + c_i z                         (1)

In this equation  w_i, x, y, and z are considered known and a_i, b_i and c_i are the unknowns. This equation, therefore, represents a system of K linear equations (one for each band) with 3 unknowns.

• Identify the mixing proportions of individual pixels.
• Identify the mixing proportions of sets of mixed pixels.

Click here to learn how to deal with single pixels

Click here to learn how to deal with sets of mixed pixels