Valley-peak modulation in phase space: a law-invariant VPM and its theta-function structure

Abstract

Valley-peak modulation (VPM) was introduced as a metric for quantifying read noise in deep sub-electron read noise (DSERN) CMOS image sensors. In its original amplitude-domain definition VPM depends on both read noise and quanta exposure, yet Starkey and Fossum demonstrated exposure-independent approximations valid in the DSERN regime. Here we identify the invariant object those approximations probe, and find its invariance extends beyond exposure to the electron-number law itself. A phase mapping quotients the sensor model by the integer electron lattice, yielding a wrapped-Gaussian density parameterized only by read noise and admitting lattice-sum and Jacobi theta-function representations. The invariant is the theta ratio $R=\vartheta_4(q)/\vartheta_3(q)$ with nome $q$, of which any VPM is a contrast normalization; the existing approximations are low-order truncations of its lattice sums, and the amplitude-domain metric converges to its phase-space counterpart at large exposure. A closed-form inverse for read noise in terms of VPM follows from elliptic integrals. The identification yields a moment method of characterization: conversion gain and read noise are estimated jointly from the modulus of the empirical characteristic function of the raw gray values, without specifying the number law, and with leading-order precision depending on it only through its variance.