Finite-Resolution Inverse Readout for Density-Selected Phase Kicks: Pixelized Contrast, Elliptic Reconstruction, and Spectral Identifiability

This preprint develops a finite-resolution mathematical imaging theory for inverse readout after density-selected phase kicks. The paper studies how a continuum inverse-readout identity can be connected to realistic blurred, pixel-averaged, and noisy density images. It introduces a measurement model with a point-spread function, a pixel grid, and a deterministic noise envelope, constructs a discrete weak contrast functional, proves finite-resolution consistency estimates, and derives a weighted elliptic reconstruction theorem with separated error contributions from finite readout time, pixel quadrature, optical blur, noise amplification, and density-coefficient approximation. The manuscript also treats finite-band multi-probe spectral identifiability of the kernel generating the phase profile, including perturbation, aliasing, windowing, and regularization effects. A minimal manufactured sanity-check example and a reproducibility protocol are included. This version corresponds to the manuscript submitted to the Journal of Mathematical Imaging and Vision in May 2026.