Valley-Peak Modulation in Phase Space: an Exposure-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 sensors. In the original amplitude-domain definition, VPM depends on both read noise and quanta exposure, yet Starkey & Fossum demonstrated exposure-independent approximations that hold in the DSERN regime. In this note we identify the exposure-invariant object those approximations probe. Starting from the standard Poisson-Gaussian model, we apply a phase mapping that quotients out the integer electron count, yielding a wrapped-Gaussian density parameterized only by read noise and admitting both lattice-sum and Jacobi theta-function representations. The fundamental exposure-invariant quantity is shown to be the theta ratio $R(\sigma)=\vartheta_4(q)/\vartheta_3(q)$, of which any VPM is a contrast normalization; the existing exposure-independent approximations are then recovered as low-order truncations of the lattice-sum representation of $R$. A closed-form inverse expressing read noise in terms of VPM is obtained using elliptic integrals, and a short simulation example illustrates practical estimation of read noise from the VPM in phase space.