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 ( α , β , 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: the Legendre polynomial case ( α = β =0) minimizes reconstruction root-mean-square error (RMSE) under low-noise conditions ( σ <0.1) with up to nine filters. In higher-noise regimes or under tighter filter-count constraints, alternative Jacobi parameters are required to minimize RMSE of reconstruction, demonstrating the importance of exploring the full polynomial design space. Through the application of PCA in the filter design process, we find a method of reconstruction that retrieves the principal scores of the measured reflectance spectra, thus likening the measurement through these designed transmission filters to an optical computation. These retrieved principal scores can then be used in spectral reconstruction, demonstrating significantly (around an order-of-magnitude) lower RMSE than when the traditional matrix-inversion method is applied for the same filter.