PCA-based polynomial transmission filter design for spectral imaging

We introduce a flexible, analytically grounded framework for designing continuous transmission filters for spectral imaging, based on principal component analysis (PCA) on the coefficients of material reflectance spectra when represented by orthogonal polynomial bases. This design strategy yields a closed-form solution that remains continuous, essential for the physical realization of optical transmission filters. This method exposes several tunable parameters (\((\alpha,\beta,N_{terms})\)) that jointly control filter smoothness, endpoint weighting, and spatial frequency content. This enables filter sets to be optimized not only for spectral reconstruction accuracy but also for manufacturability. A coarse parameter search using the USGS mineral database reveals an interesting result: that the Legendre polynomial case \((\alpha=\beta=0)\) minimizes reconstruction root-mean-square error (RMSE) under low-noise conditions (\(\sigma