发表机构
Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多色CT非线性模型中的信号恢复，确定可注入性阈值并证明多材料重建在一般测量下为NP难，与单材料可多项式求解形成对比。
AI 中文摘要
我们考虑一个由多色计算机断层扫描（CT）驱动的非线性模型。在此模型中，$W$ 个不同的 $d$ 维信号 $x^*_1, \ldots, x^*_W \in \mathbb{R}^d$ 需要从 $n$ 个测量值 $(a_i, y_i)_{i = 1}^n$ 中恢复，这些测量值遵循非线性模型 $\mathbb{E}[y_i|a_i] = h(\langle a_i, x^*_1 \rangle, \ldots, \langle a_i, x^*_W \rangle)$，其中 $h: \mathbb{R}^W \to \mathbb{R}$ 是一个已知的非线性函数，用于建模某种类型的指数衰减定律。即使在测量无噪声的情况下，样本量 $n$（对于任何测量集合 $\{a_i\}_{i = 1}^n$）也必须超过未知数数量 $Wd$，才能保证潜在信号是可识别的。我们构造了一个测量集合，在 $n \geq Wd + W - 1$ 的条件下，几乎必然确保完美的信号恢复，从而将可注入性阈值隔离到附加因子 $W - 1$ 的范围内。我们还研究了在具有一般测量集合 $\{a_i\}$ 的该模型中信号恢复的计算复杂度。我们证明，如果 $W \geq 2$，则存在一个测量集合，使得判定是否存在与测量一致的信号是 NP 难的。特别地，这意味着多色、多材料 CT 重建在一般情况下是 NP 难的。这一发现与单材料设置形成鲜明对比，在单材料设置中，对于任何测量集合，多项式时间算法都可以可证明地执行信号恢复。
英文摘要
We consider a nonlinear model motivated by polychromatic computed tomography (CT). Here, $W$ distinct $d$-dimensional signals $x^*_1, \ldots, x^*_W \in \mathbb{R}^d$ must be recovered from $n$ measurements $(a_i, y_i)_{i = 1}^n$ that obey the nonlinear model $\mathbb{E}[y_i|a_i] = h(\langle a_i, x^*_1 \rangle, \ldots, \langle a_i, x^*_W \rangle)$, where $h: \mathbb{R}^W \to \mathbb{R}$ is a known nonlinearity that models a certain type of exponential attenuation law. Even when there is no noise in the measurements, the sample size $n$ (for any measurement ensemble $\{a_i\}_{i = 1}^n$) must exceed the number of unknowns $Wd$ to guarantee that the underlying signals are identifiable. We construct a measurement ensemble that ensures perfect signal recovery almost surely provided $n \geq Wd + W - 1$, thereby isolating the injectivity threshold up to an additive factor $W - 1$. We also study computational complexity of signal recovery in this model with a general measurement ensemble $\{a_i\}$. We show that if $W \geq 2$, then there is a measurement ensemble for which deciding whether there exist signals consistent with the measurements is NP-hard. In particular, this implies that polychromatic, multimaterial CT reconstruction is NP-hard in general. This finding stands in sharp contrast to the single-material setting, for which a polynomial-time algorithm can provably perform signal recovery for any measurement ensemble.
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