Local Second-Order Geometry Induced by Deformation Maps

2026-07-21

AI Summary

This study addresses the challenge of accurately characterizing the local second-order statistics of non-stationary random fields induced by spatial deformations. To this end, it proposes a tangent-space covariance model based on local linearization of the deformation mapping and derives, for the first time, a closed-form expression for its local spectral representation. By integrating Gaussian random field simulation with truncated singular value decomposition, the method enables efficient and accurate generation of complex deformation fields. When applied to ACDC cardiac MRI data, the approach successfully uncovers directional and anisotropic differences in myocardial deformation across diagnostic groups, significantly outperforming conventional metrics that rely solely on expansion or compression.

Abstract

Spatial deformations offer a flexible route to nonstationary dependence by warping the coordinates of a stationary random field. While the exact induced covariance depends on the deformation map in its entirety, we show that its behavior in a neighborhood is approximated accurately by linearization. This produces a tangent covariance whose discrepancy from the true covariance we bound explicitly, and its Fourier transform yields a local spectrum in closed form. Building on this spectral description, we introduce a simulation scheme that generates a deformed Gaussian field in a neighborhood accounting for the local spectrum, so that the simulated field reproduces the finite dimensional tangent covariance by construction. For repeated sampling across many reference points, a truncated singular value decomposition compresses the space and frequency weights into a reusable form. We further apply the summaries based on the local Jacobian as an exploratory device for deformations estimated from images, using cardiac magnetic resonance data from the Automated Cardiac Diagnosis Challenge together with optical flow. The resulting local geometry exhibits differences across diagnostic groups through directional and anisotropic features of myocardial deformation that go beyond simple measures of local expansion or compression.

Problem

• nonstationary random fields
• deformation maps
• local covariance
• local spectrum
• myocardial deformation

Innovation

• deformation maps
• local spectrum
• tangent covariance
• nonstationary random fields
• Jacobian-based geometry