Spectrally Regularized Spline Mapping with Recursive Cubic Residual Correction

Abstract

Greg Passmore

This work introduces the Fourier-Space Spline with Recursive Polynomial Residuals (FSS-RPR), a hybrid empirical georegistration and inversion method intended for ISR exploitation when physical sensor models are incomplete, denied, vendor-specific, or operationally impractical. The design fuses two mature mathematical lineages; spectral representations originating in Fourier’s heat-diffusion analysis and spline-based smooth function estimation formalized by Schoenberg and later developed for computation by de Boor; with a residual-stacking estimator path that is structurally analogous to stagewise additive modeling [1-4]. The principal competing approach is the rigorous sensor-model/bundle-adjustment family, which can exceed surrogate accuracy when calibration/metadata are complete, but often fails operationally due to metadata availability, datum ambiguity, or workflow constraints; the competing critique is that any surrogate can overfit and can degrade sharply under extrapolation outside its fitted domain.

The method targets a specific ISR failure mode: georegistration error dominated by smooth, low-frequency spatial bias (platform/attitude bias, timing skew, ephemeris drift, or systematic terrain/height convention mismatch), with a residual component that is structured but not captured by a single global smooth warp. The FSS component enforces controlled smoothness and bandwidth in a way that is auditable and computationally efficient, especially when knots are uniform and spectral penalties are applied using FFT-class methods [5]. The RPR component then absorbs remaining deterministic structure in the residual field using a fixed low-order polynomial basis and recursive residual fitting, preserving the simplicity of evaluation and enabling explicit failure-mode diagnosis through layerwise residual decay.

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