Physics-Informed Unrolled Learning for ATR-FTIR Spectral Unmixing

Source abstract

Quantitative infrared spectroscopy of bone reads chemistry from ratios of band intensities, but attenuated total reflectance distorts those intensities before they are measured. We treat the distortion as a model to be inverted rather than a nuisance to be filtered. A causal Lorentz parameterisation of the dielectric function, an effective-medium mixture, the Fresnel relations for a finite contact crystal and the instrument line shape are composed into one differentiable operator, and its Jacobian is derived in closed form. Three consequences are quantified. Anomalous dispersion displaces the apparent phosphate maximum by 7 to 36 wavenumbers depending on packing, so band position thresholds do not transfer between laboratories. The displacement varies with composition, so the I1030/I1020 crystallinity ratio responds 1.93 times more strongly to mineral content at fixed crystallinity than to the entire plausible crystallinity range. Absorbance is not affine in composition, which invalidates bilinear unmixing. We prove that closure and structural anticorrelation force the sufficiently scattered condition to fail for bone, and replace it with spectral separability, a checkable condition on band spacing that yields a local identifiability theorem with an explicit lower bound on the smallest singular value. Inference uses a Levenberg–Marquardt iteration unrolled into a network with five learned scalars per layer, for which we prove local linear convergence under a per-layer contraction condition, for constant and for layer-dependent schedules, and a Rademacher bound of order TlogT/n that is conditional on non-expansive layers; on the trained networks the median contraction is below one at every layer, but non-expansiveness does not hold. On a generator whose band ratios all lie inside published ranges, the estimator reaches the Cramér–Rao bound to within 3 to 6 per cent across a sixteenfold range of noise, recovers volume fractions 143 times more accurately than multivariate curve resolution, and outperforms a network with 1.1×105 parameters by three orders of magnitude at twenty-four training spectra. These figures hold when the data follow the operator. Under deliberate mismatch in band positions, line shape, baseline, effective-medium law, contact physics and noise, the advantage over curve resolution persists at factors of 2 to 108, but the loss rises to many times the bound and the nominal intervals cover the truth in as few as 5 per cent of cases, so sampling uncertainty is reported separately from systematic model uncertainty. The same information matrix prices the experiment: contact variability inflates the standard error on composition by 3.6 while leaving the band parameters untouched, and an optimal choice of crystal and geometry recovers 23.6 nats under signal-independent noise, falling to between 0.6 and 17.9 nats once throughput loss and detector-limited noise are included.