Multi-Energy Computed Tomography (ME-CT) is a medical imaging modality aiming to reconstruct the spatial density of materials from the attenuation properties of probing x-rays. For each line in two- or three-dimensional space, ME-CT measurements may be written as a nonlinear mapping from the integrals of the unknown densities of a finite number of materials along said line to an equal or larger number of energy-weighted integrals corresponding to different x-ray source energy spectra. ME-CT reconstructions may thus be decomposed as a two-step process: (i) reconstruct line integrals of the material densities from the available energy measurements; and (ii) reconstruct densities from their line integrals. Step (ii) is the standard linear x-ray CT problem whose invertibility is well-known, so this paper focuses on step (i). We show that ME-CT admits stable, global inversion provided that (a well-chosen linear transform of) the differential of the transform in step (i) satisfies appropriate orientation constraints that makes it a P-matrix. We introduce a notion of quantitative P-function that allows us to derive global stability results for ME-CT in the determined as well as over-determined (with more source energy spectra than the number of materials) cases. Numerical simulations based on standard material properties in imaging applications (of bone, water, contrast agents) and well accepted models of source energy spectra show that ME-CT is often (always in our simulations) either (i) non-globally injective because it is non-injective locally (differential not of full rank), or (ii) globally injective as soon as it is locally injective (differentials satisfy our proposed constraints).

中文翻译：

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多能 CT 中的唯一性标准

多能量计算机断层扫描 (ME-CT) 是一种医学成像方式，旨在根据探测 X 射线的衰减特性重建材料的空间密度。对于二维或三维空间中的每条线，ME-CT 测量可以写成非线性映射，从沿所述线的有限数量材料的未知密度的积分到相等或更多数量的能量加权积分对应不同的X射线源能谱。ME-CT 重建因此可以分解为两步过程：（i）根据可用能量测量重建材料密度的线积分；(ii) 从它们的线积分重建密度。步骤(ii)是标准的线性X射线CT问题，其可逆性是众所周知的，因此本文重点讨论步骤(i)。我们表明 ME-CT 允许稳定的全局反演，前提是步骤 (i) 中的变换的微分（精心选择的线性变换）满足适当的方向约束，使其成为 P 矩阵。我们引入了定量 P 函数的概念，它使我们能够在确定和超确定（源能谱多于材料数量）情况下推导出 ME-CT 的全局稳定性结果。基于成像应用（骨骼、水、造影剂）中标准材料特性的数值模拟和广为接受的源能谱模型表明 ME-CT 通常（总是在我们的模拟中）要么（i）非全局注入，因为它局部非单射（微分非满秩），