非线性锥变换的微局部分析及其在康普顿相机成像中的应用
Microlocal analysis of a non-linear cone transform and applications to Compton camera imaging
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中文总结 AI 辅助
本文针对康普顿相机成像，提出结合非线性锥变换微局部分析的方法，可唯一稳定恢复源强度与衰减系数，模拟重建验证了其有效性。
中文摘要 AI 辅助
我们提出一种恢复康普顿相机成像中源强度$f: \mathbb{R}^n \to \mathbb{R}$和衰减系数$\mu: \mathbb{R}^n \to \mathbb{R}$的新方法。我们应用考虑射线衰减的非线性模型，证明数据$h$可建模为$h = \mathcal{R}(f,\mu) = R(fg)$，其中$g = \exp(-G\mu)$用于建模衰减，$G$是线性发散束变换，$R$是定义$fg$在锥上积分的线性算子。文献中通常令$\mu=0$，此时数据$h = Rf$为线性，我们则处理$\mu \neq 0$且变换为非线性的情况。为简化分析，我们首先将数据转换为加权线积分$\tilde{h} = \mathcal{D}_k(f,\mu) = D_k(fg)$，其中$D_k$是加权射线变换。假设满足实际可行的几何条件，我们证明$\tilde{h} = \exp(-X_{w_1}\mu)X_{w_2}f$，其中$X_w$是加权X射线变换，$w_i$为光滑权重。之后我们利用余法分布理论描述$\tilde{h}$的奇点，证明重建中存在伪影并使用Sobolev空间量化其强度，结合该理论与几何论据恢复$f$的边缘，最终证明$f$和$\mu$由$h$唯一确定。值得注意的是，$f$的恢复稳定性显著高于$\mu$，我们也对此进行了讨论。为验证理论，我们给出了所提方法对$f$和$\mu$的模拟重建结果。
英文摘要
We present a novel method to recover the source intensity, $f : \mathbb{R}^n \to \mathbb{R}$, and attenuation coefficient, $μ: \mathbb{R}^n \to \mathbb{R}$, in Compton camera imaging. We apply a non-linear model, which accounts for ray attenuation. We show that the data, $h$, can be modeled $h = \mathcal{R}(f,μ) = R(fg)$, where $g = \exp(-Gμ)$ models attenuation, $G$ is a (linear) divergent beam transform, and $R$ is a linear operator which defines the integrals of $fg$ over cones. Commonly in the literature, $μ$ is set to zero, and the data $h = Rf$ is linear. We address the case when $μ\neq 0$ and the transform is non-linear. To simplify the analysis, we first transform the data into weighted line integrals, $\tilde{h} = \mathcal{D}_k(f,μ) = D_k(fg)$, where $D_k$ is a weighted ray transform. Assuming practically reasonable geometric conditions, we show that $\tilde{h} = \exp(-X_{w_1}μ)X_{w_2}f$, where $X_w$ is a weighted X-ray transform, and the $w_i$ are smooth weights. After which, we use the theory of conormal distributions to describe the singularities of $\tilde{h}$. We show that there are artifacts in the reconstruction, and we quantify their strength using Sobolev spaces. We combine this theory with a geometric argument to recover the edges of $f$ and ultimately prove that $f$ and $μ$ are unique to $h$. The recovery of $f$ is notably more stable than that of $μ$, which we also discuss. To validate our theory, we present simulated reconstructions of $f$ and $μ$ using the proposed method.
