We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calder\'on's inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be shrunk to obtain the exact crack locations upon verifying certain operator inequalities for differences of the local Neumann-to-Dirichlet maps. This method can simultaneously handle perfectly insulating and perfectly conducting cracks, and it appears to be the first rigorous reconstruction method capable of this. Our second method assumes that only perfectly insulating cracks or only perfectly conducting cracks are present. Once more using operator inequalities, this method generates approximate cracks that are guaranteed to be subsets of the unknown cracks that are being reconstructed.

翻译：我们针对Calderón逆电导率问题中由Lipschitz超曲面并集构成的裂缝，推导了精确重构方法。第一种方法通过验证局部Neumann-to-Dirichlet映射差分的特定算子不等式，获得未知裂缝的上界，该上界可逐步收缩至裂缝精确位置。该方法能同时处理完美绝缘裂缝与完美导电裂缝，是目前首个具备此能力的严格重构方法。第二种方法假设仅存在完美绝缘裂缝或完美导电裂缝，再次利用算子不等式生成近似裂缝，这些裂缝被保证是待重构未知裂缝的子集。