🤖 AI Summary
This work proposes a unified, imaging-modality-agnostic geometric framework to characterize the information structure of imaging operators. By mapping the normalized singular spectrum onto a probability simplex and introducing the Fisher–Rao information metric, the authors construct a Riemannian geometry of constant positive curvature. This approach yields, for the first time, an intrinsic description at the operator level that does not rely on optimization, stochastic modeling, or modality-specific assumptions, thereby revealing the geometric invariance of spectral equivalence classes. The study derives closed-form expressions for information distance and geodesics, proves their invariance under unitary transformations and global scaling, and elucidates the mechanism by which nonlinear redistribution of spectral weights influences information distance.
📝 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.