Spectral Unmixing (SMA – Spectral Mixture Analysis)

Spectral Unmixing is a technique used to decompose the mixed spectral signal of a pixel into fractional abundances of a limited set of pure spectral components called endmembers. This method helps to quantify the proportion of distinct materials or substances within a pixel, which is particularly useful when spatial resolution does not allow pure pixels for each material.

How it works #

Our unmixing process uses a linear mixing model expressed mathematically as:

$\mathbf{R}=\sum _{i=1}^n{f}_i{\mathbf{E}}_i+\epsilon$

where:

• $\mathbf{R}$ is the observed reflectance vector of the pixel (across spectral bands),
• $n=3$ is the number of selected endmembers,
• $f_i$ is the fractional abundance of the i-th endmember,
• ${\mathbf{E}}_i$ is the spectral signature vector of the i-th endmember,
• $\epsilon$ represents residual error or noise.

The algorithm solves this system for the abundance fractions $f_i$ with the following physical constraints:

$f_i\ge 0\forall i,\text{and}\sum _{i=1}^n{f}_i=1$

These constraints guarantee that the abundances are physically meaningful as proportions that sum to one within each pixel.

Step-by-step process:

• Endmember Selection: The user selects exactly three pure spectral signatures (endmembers) from the available spectral library. These correspond to materials of interest in the scene.
• Band Alignment: The system extracts the pixel reflectance values $\mathbf{R}$ and aligns spectral bands to match the wavelengths and bandwidths of the endmembers ${\mathbf{E}}_i$.
• Abundance Estimation: Using a constrained least squares approach, the algorithm estimates the vector $\mathbf{f}=[f_1,f_2,f_3]$ that best reconstructs $\mathbf{R}$ as a linear combination of ${\mathbf{E}}_i$, respecting the sum-to-one and non-negativity constraints.
• Result Output: The fractional abundances are stored as three new image bands, conceptually representing Red, Green, and Blue channels, which can be visualized or used for further geospatial analysis.

Why it matters #

Spectral unmixing provides detailed material composition information at sub-pixel scales, which is critical in complex landscapes such as mining areas where multiple minerals or vegetation types coexist within the same pixel footprint.

By enforcing the sum-to-one constraint, the results represent reliable fractional estimates of each material, enabling accurate mapping and quantitative interpretation.

Applications #

• Quantitative mineral mapping in heterogeneous mining zones.
• Vegetation species or health differentiation in mixed land cover areas.
• Soil and surface composition analysis in environmental monitoring.