AI 中文总结
研究利用变形映射诱导非平稳依赖，通过线性化近似邻域协方差行为，给出切向协方差及局部谱，据此构建模拟方案，还应用局部雅可比矩阵摘要探索图像变形，展示了心肌变形局部几何在诊断组间的差异。
AI 中文摘要
空间变形通过扭曲平稳随机场的坐标，为非平稳依赖提供了一条灵活途径。虽然精确的诱导协方差完全取决于变形映射，但我们表明其在邻域内的行为可通过线性化精确近似。这产生了一个切向协方差，我们明确界定了它与真实协方差的差异，其傅里叶变换产生了封闭形式的局部谱。基于此谱描述，我们引入了一种模拟方案，该方案在考虑局部谱的邻域内生成变形高斯场，使得模拟场通过构造再现有限维切向协方差。对于在许多参考点上的重复采样，截断奇异值分解将空间和频率权重压缩成可重复使用的形式。我们进一步将基于局部雅可比矩阵的摘要应用于从图像估计的变形的探索工具，使用来自自动心脏诊断挑战的心脏磁共振数据以及光流。由此产生的局部几何通过心肌变形的方向和各向异性特征，在诊断组之间表现出差异，这些特征超越了简单的局部扩张或压缩测量。
英文摘要
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.
