具有多爆炸衰减假设的发射层析成像
Emission tomography with a multi-bang assumption on attenuation
摘要：
本文研究了二维发射层析成像中衰减$a$和源密度$f$的联合重建问题。这有时被称为单光子发射计算机断层扫描（SPECT）识别问题，或被称为SPECT中的衰减校正。假设$a$只取100多个值，而$f \in C_c^1(\mathbb{R}^2)$，我们能够消除衰减拉东变换$R_a f$中出现的奇异性，该变换对发射断层扫描数据进行建模。利用这一特性，我们证明了在某些情况下可以确定$a$和$f$。当$a$只从一个已知的有限列表中取值时，我们还提出了一个基于弱凸正则化子的从$R_af$联合计算$a$和$f$的数值算法，并在一些合成示例上表明该算法性能良好。
英文摘要：
We consider the problem of joint reconstruction of both attenuation $a$ and source density $f$ in emission tomography in two dimensions. This is sometimes called the Single Photon Emission Computed Tomography (SPECT) identification problem, or referred to as attenuation correction in SPECT. Assuming that $a$ takes only finitely many values and $f \in C_c^1(\mathbb{R}^2)$ we are able to characterise singularities appearing in the Attenuated Radon Transform $R_a f$, which models emission tomography data. Using this characterisation we prove that both $a$ and $f$ can be determined in some circumstances. We also propose a numerical algorithm to jointly compute $a$ and $f$ from $R_af$ based on a weakly convex regularizer when $a$ only takes values from a known finite list, and show that this algorithm performs well on some synthetic examples.
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