What is the LS-Fit model in ENVI, and why would I use it?
Begining with ENVI 3.0, a spectral tool called LS-Fit was added which predicts the values of a selected image band from the other image bands. LS-Fit can be used to identify pixels containing clay minerals in the presence of vegetation, among other applications. This tip discusses the origin, algorithm and application of this tool.
The LS-Fit model was first developed by Andy Green and Maurice Craig at CSIRO in Australia to "defoliate" Landsat TM images and to find pixels which contain absorption features characteristic of clay minerals in TM Band 7. Hydrothermal alteration makes many different clay minerals that absorb at 2.2 microns (which is TM Band 7). When looking for areas of hydrothermal alteration using remotely sensed data, one generally looks for pixels containing clay minerals. It is not accurate to just look for pixels where values in TM Band 7 are low, because there are many reasons that reflectance in Band 7 might be low besides the presence of clay minerals. For example, the pixel may have a lot of shade or shadow, or it may contain a low reflectance material like water, or green vegetation. In particular the CSIRO investigators wanted to eliminate false positives from green vegetation. They realized that they needed to find pixels where Band 7 was lower than expected given the values in the other bands. If the rest of the spectrum looks like green vegetation, the model should predict a low Band 7 value. If the pixels also contain clay minerals, then these vegetation pixels must also have even lower Band 7 values than expected for a vegetation spectrum. So the investigators needed a way to predict the Band 7 reflectance, given the reflectance in the other five TM bands (Band 6 is excluded).
As in principal components analysis and minimum noise fraction transforms, LS-Fit uses second-order statistics: variance and covariance. It is a very simple model that says, "given a bunch of pixels we can calculate the band vs. band covariance matrix, or equivalently the sums of squares matrix." From this we can directly create a linear equation that says Band 7 is a linear combination of some subset of the other bands plus an offset:
b7=a*b1 + b*b2 + c*b3 +d*b4 + e*b5 + f
It is easy to invert this equation and get a least squares prediction of Band 7 value as a function of the values in other bands and an offset. The resulting model of Band 7 depends only on the covariance matrix and the values in the other bands. ENVI calls the modeled Band 7 the LS-Fit Model image. The quantity that actually gives information about the presence of clay minerals is the difference - the actual Band 7 minus the modeled Band 7. ENVI calls the difference image the LS-Fit Residual image. The Residual has zero mean and shows the deviation of Band 7 values from predicted values. Hopefully, the pixels which contain clay minerals will have low values (Band 7 lower than we'd expect given the other bands' responses at that pixel and the scene statistics).
Many investigators are trying to take this idea further. How the analysis applies to hyperspectral data and which bands to use for prediction are currently being explored.
The ENVI developers feel that the LS-Fit model is a more limited way of approaching a question that can be answered more accurately using Matched Filtering. They feel that LS-Fit has been replaced by Matched Filtering, especially when the target spectrum is known. LS-Fit can also be thought of as somewhat like unmixing stood on its side. Instead of saying that each pixel spectrum is a linear combination of endmember spectra, LS-Fit says that a band is a linear combination of the other bands, and solves for the weights which will best model that band as a weighted average of the predictor bands.