磁感应断层成像中支撑恢复的因子分解方法
A Factorization Method for Support Recovery in Magnetic Induction Tomography
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种磁感应断层成像的因子分解框架，通过谱指示器非迭代地恢复电导率支撑，无需正演求解，并能准确分离夹杂物、处理非凸形状。
AI中文摘要：
我们针对涡流状态下的磁感应断层成像发展了一种因子分解框架。感应电流密度由一个无散度的电流矢量势表示，外部激励则由观测球面上的磁偶极子密度表示。这一选择揭示了近场算子的一种自然物理因子分解，将其分解为数据算子、内部边界响应算子及其伴随算子。通过在通量空间上引入Riesz映射，我们在希尔伯特空间框架下重新表述了这一因子分解，从而得到一个具有强制性中间因子的半正定算子。Douglas值域定理随后将这一耗散算子的平方根的值域与数据算子的值域等同起来。对于具有规则边界的夹杂物，这一值域恒等式导出了电导率支撑的唯一性定理。这一表征导致了一种非迭代的谱指示器，该指示器基于从近场数据矩阵的厄米虚部计算的正则化Picard商。重建阶段不需要任何正演求解。数值实验证实，该指示器能够准确定位并分离夹杂物，在噪声增加时性能优雅地退化，并且无需任何凸性先验即可捕获非凸的支撑几何形状。
英文摘要：
We develop a factorization framework for magnetic induction tomography in the eddy-current regime. The induced current density is represented by a divergence-free current vector potential, and the external excitation by magnetic dipole densities on an observation sphere. This choice reveals a natural physical factorization of the near-field operator into a data operator, an interior boundary response operator, and an adjoint counterpart. By introducing the Riesz map on the flux space, we recast this factorization in a Hilbert-space setting, which yields a positive-semidefinite operator with a coercive middle factor. Douglas' range theorem then identifies the range of the square root of this dissipative operator with the range of the data operator. For inclusions with a regular boundary, this range identity yields a uniqueness theorem for the support of the conductivity. This characterization leads to a non-iterative spectral indicator based on a regularized Picard quotient computed from the Hermitian imaginary part of the near-field data matrix. The reconstruction stage requires no forward solves. Numerical experiments confirm that the indicator localizes and separates the inclusions accurately, degrades gracefully under increasing noise, and captures non-convex support geometry without any convexity prior.
