SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 134-147, March 2024.
Abstract. The elastic potential is a valuable modeling tool for many applications, including medical imaging. One reason for this is that the energy and its Gâteaux derivative, the elastic operator, have strong coupling properties. Although these properties are desirable from a modeling perspective, they are not advantageous from a computational or operator decomposition perspective. In this paper, we show that the elastic operator can be spectrally decomposed despite its coupling property when equipped with sliding boundary conditions. Moreover, we present a discretization that is fully compatible with this spectral decomposition. In particular, for image registration problems, this decomposition opens new possibilities for multispectral solution techniques and fine-tuned operator-based regularization.

中文翻译：

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滑动边界条件下连续和离散线弹性算子的谱分解

《SIAM 矩阵分析与应用杂志》，第 45 卷，第 1 期，第 134-147 页，2024 年 3 月。
摘要。弹性势对于许多应用（包括医学成像）来说是一种有价值的建模工具。原因之一是能量及其 Gâteaux 导数（弹性算子）具有很强的耦合特性。尽管从建模的角度来看这些属性是理想的，但从计算或算子分解的角度来看它们并不有利。在本文中，我们表明，当配备滑动边界条件时，尽管弹性算子具有耦合特性，但它可以进行谱分解。此外，我们提出了与这种谱分解完全兼容的离散化。特别是，对于图像配准问题，这种分解为多光谱求解技术和基于算子的微调正则化开辟了新的可能性。