Additive Noise, Shift Recovery, and Signed Signals in the Cumulative Distribution Transform

2026-06-09

AI Summary

This work investigates the propagation of perturbations induced by additive noise in one-dimensional translated density signals within the cumulative distribution transform (CDT) domain, along with the associated displacement recovery problem. Leveraging the linearization property of CDT with respect to translations, the authors derive for the first time a first-order expansion of noise-induced perturbations in the CDT domain, revealing an amplification effect in low-density regions and establishing an explicit covariance structure. Building on this analysis, they propose a unified framework for displacement recovery applicable to both density and signed signals, which jointly estimates the template and displacements via projection or alternating alignment strategies. Theoretical analysis and numerical experiments demonstrate that the proposed method achieves accurate and stable displacement recovery even in the presence of noise.

Abstract

The cumulative distribution transform (CDT) is a quantile-based transport representation that exactly linearizes one-dimensional translations of positive densities. We study how this structure behaves under additive perturbations and how it can be exploited for shift recovery. Under a local nondegeneracy condition, we derive a first-order expansion showing that additive noise in physical space induces a nonlocal perturbation in CDT space through the primitive of the noise, weighted by the reciprocal density. This yields an explicit description of transform-domain sensitivity and shows, in particular, that perturbations are amplified in low-density regions. When the physical-space perturbation is modeled as a centered Gaussian random field, the induced first-order CDT perturbation is again Gaussian, with an explicit covariance kernel. We then use this structure to study recovery in CDT coordinates. In the known-template setting, the transport shift is obtained by projection onto the constant mode, giving an explicit estimator together with exactness in the noiseless case and a stability bound under perturbations. In the unknown-template setting, multiple observations permit joint recovery of the shifts and a common template up to the natural constant-mode gauge, leading to a simple de-shift--and--average procedure. We also consider a signed-signal analogue based on the signed cumulative distribution transform (SCDT), where shifts are estimated numerically by feature matching and unknown templates are recovered by alternating alignment and averaging. Numerical experiments validate the perturbation analysis and illustrate effective recovery for both density-valued and signed signals.

Problem

• Additive Noise
• Shift Recovery
• Cumulative Distribution Transform
• Signed Signals
• Transport Representation

Innovation

• Cumulative Distribution Transform
• Shift Recovery
• Additive Noise
• Signed Signals
• Transport Representation