Information Geometry of Imaging Operators

TL;DR

This paper develops a geometric framework based on the Fisher--Rao metric for analyzing the spectral properties of imaging operators, providing a new invariant and mathematically tractable way to understand information structure in imaging systems.

Contribution

It introduces a novel Riemannian geometric structure on the space of normalized singular spectra of imaging operators, invariant under unitary transformations and rescaling.

Findings

Closed-form expressions for distances and geodesics in the spectral space

The geometry has constant positive curvature

Spectral distances are preserved only in uniform cases

Abstract

Imaging systems are represented as linear operators, and their singular value spectra describe the structure recoverable at the operator level. Building on an operator-based information-theoretic framework, this paper introduces a minimal geometric structure induced by the normalised singular spectra of imaging operators. By identifying spectral equivalence classes with points on a probability simplex, and equipping this space with the Fisher--Rao information metric, a well-defined Riemannian geometry can be obtained that is invariant under unitary transformations and global rescaling. The resulting geometry admits closed-form expressions for distances and geodesics, and has constant positive curvature. Under explicit restrictions, composition enforces boundary faces through rank constraints and, in an aligned model with stated idealisations, induces a non-linear re-weighting of spectral states. Fisher--Rao distances are preserved only in the spectrally uniform case. The construction is abstract and operator-level, introducing no optimisation principles, stochastic models, or modality-specific assumptions. It is intended to provide a fixed geometric background for subsequent analysis of information flow and constraints in imaging pipelines.