The single-scatter approximation is fundamental in many tomographic imaging problems including x-ray scatter imaging and optical scatter imaging for certain media. In all cases, noisy measurements are affected by both local scatter events and nonlocal attenuation. Prior works focus on reconstructing one of two images: scatter density or total attenuation. However, both images are media specific and useful for object identification. Nonlocal effects of the attenuation image on the data are summarized by the broken ray transform (BRT). While analytic inversion formulas exist, poor conditioning of the inverse problem is only exacerbated by noisy measurements and sampling errors. This has motivated interest in the related star transforms incorporating BRT measurements from multiple source-detector pairs. However, all analytic methods operate on the log of the data. For media comprising regions with no scatter a new approach is required. We are the first to present a joint estimation algorithm based on Poisson data models for a single-scatter measurement geometry. Monotonic reduction of the log-likelihood function is guaranteed for our iterative algorithm while alternating image updates. We also present a fast algorithm for computing the discrete BRT forward operator. Our generalized approach can incorporate both transmission and scatter measurements from multiple source-detector pairs. Transmission measurements resolve low-frequency ambiguity in the joint image estimation problem, while multiple scatter measurements resolve the attenuation image. The benefits of joint estimation, over single-image estimation, vary with problem scaling. Our results quantify these benefits and should inform design of future acquisition systems.

中文翻译：

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折断射线变换数据的联合散射和衰减估计的迭代算法

单散射近似是许多层析成像问题的基础，包括某些介质的X射线散射成像和光学散射成像。在所有情况下，噪声测量都会受到本地散射事件和非本地衰减的影响。先前的工作着重于重建两个图像之一：散射密度或总衰减。但是，这两个图像都是特定于媒体的，并且对于对象识别很有用。衰减图像对数据的非局部影响通过断射线变换（BRT）进行了总结。尽管存在解析反演公式，但反响问题的不良条件只会因嘈杂的测量结果和采样误差而加剧。这引起了对相关星形转换的兴趣，该星形转换合并了来自多个源-探测器对的BRT测量。然而，所有分析方法都对数据的日志进行操作。对于包含没有散射的区域的媒体，需要一种新的方法。我们是第一个提出基于Poisson数据模型的单点测量几何联合估计算法的公司。我们的迭代算法可保证对数似然函数的单调减少，同时交替更新图像。我们还提出了一种用于计算离散BRT前向运算符的快速算法。我们的通用方法可以合并来自多个源-探测器对的透射和散射测量。透射测量解决了联合图像估计问题中的低频模糊问题，而多次散射测量则解决了衰减图像。与单幅图像估计相比，联合估计的好处随问题缩放而变化。