EMPIRICALLY MINIMAX AFFINE MINERALOGY ESTIMATES FROM FOURIER-TRANSFORM INFRARED SPECTROMETRY USING A DECIMATED WAVELET BASIS

The Fourier transform infrared (FT-IR) spectrum of a rock contains inf ormation about its constituent minerals. Using the wavelet transform, we roughly separate the mineralogical information in the FT-IR spectru m from the noise, using an extensive set of training data for which th e true mineralogy is known. We ignore wavelet coefficients that vary t oo much among repeated measurements on rocks with the same mineralogy, since these are likely to reflect analytical noise. We also ignore th ose that vary too little across the entire training set, since they do not help to discriminate among minerals. We use the remaining wavelet coefficients as the data for the problem of estimating mineralogy fro m FT-IR data. For each mineral of interest, we construct an affine est imator x of the mass fraction x of the mineral of the form x = a over arrow pointing right . w over arrow pointing right + b, where a over a rrow pointing right is a vector, w over arrow pointing right is the ve ctor of retained wavelet coefficients, and b is a scalar. We find a ov er arrow pointing right and b by minimizing the maximum error of the e stimator over the training set. When applying the estimator, we ''trun cate'' to keep the estimated mineralogy between 0 and 1. The estimator s typically perform better than weighted nonnegative least-squares.