Histogram tomography
Histogram tomography
复制标题
直方图断层扫描
DOI：
10.3934/mine.2020004
复制
发表时间：
2018
影响因子：
1
通讯作者：
W. Lionheart
中科院分区：
文献类型：
--
作者：
W. Lionheart
在许多层析成像问题中，数据由沿直线或曲线的积分组成。我们越来越多地遇到“富层析成像”问题，其中成像的数量比每体素的标量高维，包括向量张量和函数。数据也可以是高维的，在许多情况下由每条射线的一维或二维光谱组成。在许多这样的情况下，数据不仅包含沿射线的积分，还包含沿射线的值分布。如果把它离散到箱子里我们可以把它看成一个直方图。本文引入了“直方图断层扫描”的概念。对于直方图数据的标量问题，这可以用更少的射线进行重建。在矢量和张量问题中，它保证了在相关积分变换的零空间中的图像的重建。对于标量直方图断层扫描问题，我们展示了直方图中的bin如何对应于重建函数的水平集，而分布的矩是未知函数的幂的x射线变换。在矢量情况下，我们给出了场的势分量的重建过程。我们演示了如何从Bragg边缘中子谱数据中提取直方图纵向射线变换数据，因此，使用矩，导出了应变张量的非线性偏微分方程系统。在应变的x射线衍射层析中，可以从衍射图中推导出横向射线变换，但不能从全直方图中推导出横向射线变换。我们给出了产生相同衍射图样的应变沿直线分布的明确例子，并描述了相关变换的零空间。
In many tomographic imaging problems the data consist of integrals along lines or curves. Increasingly we encounter "rich tomography" problems where the quantity imaged is higher dimensional than a scalar per voxel, including vectors tensors and functions. The data can also be higher dimensional and in many cases consists of a one or two dimensional spectrum for each ray. In many such cases the data contain not just integrals along rays but the distribution of values along the ray. If this is discretized into bins we can think of this as a histogram. In this paper we introduce the concept of "histogram tomography". For scalar problems with histogram data this holds the possibility of reconstruction with fewer rays. In vector and tensor problems it holds the promise of reconstruction of images that are in the null space of related integral transforms. For scalar histogram tomography problems we show how bins in the histogram correspond to reconstructing level sets of function, while moments of the distribution are the x-ray transform of powers of the unknown function. In the vector case we give a reconstruction procedure for potential components of the field. We demonstrate how the histogram longitudinal ray transform data can be extracted from Bragg edge neutron spectral data and hence, using moments, a non-linear system of partial differential equations derived for the strain tensor. In x-ray diffraction tomography of strain the transverse ray transform can be deduced from the diffraction pattern the full histogram transverse ray transform cannot. We give an explicit example of distributions of strain along a line that produce the same diffraction pattern, and characterize the null space of the relevant transform.
影响因子：
2.1
作者：
Desai, Naeem M.;Lionheart, William R. B.;Schmidt, Soren
通讯作者：
Schmidt, Soren