放射变换与霍夫变换：一种统一的视角
The Radon transform and the Hough transform: a unifying perspective
AI总结：
本文从统一视角探讨放射变换与霍夫变换，证明在参数空间离散化步长趋近于零时，输入图像的霍夫正弦图收敛于对应的放射正弦图，并探讨其对医学影像重建的影响。
摘要：
放射变换是一种线性积分变换，模拟医学成像模态如X射线计算机断层扫描和正电子发射断层扫描的数据形成过程。霍夫变换是一种模式识别技术，主要用于检测数字图像中的直线，并最近扩展到自动识别代数平面曲线。尽管定义方式截然不同，在数值应用中，两种变换最终都将图像作为输入，并提供在参数空间上定义的函数作为输出。此空间中的参数描述了一族曲线，这些曲线代表了放射变换（广义）中考虑的积分域或霍夫变换所检测的曲线。在两种情况下，输出函数强度值的2D图称为所谓的（放射或霍夫）正弦图。霍夫正弦图由一种需要将参数空间离散化为单元格的算法生成，而放射正弦图在数学上定义在连续参数空间上，这可能仅出于物理或数值原因需要离散化。在本文中，通过考虑更一般且n维的设置，我们证明，无论输入图像被描述为一组点（可能具有不同强度值）还是分段常数函数，其（重新缩放）霍夫正弦图在参数空间离散化步长趋近于零时收敛于对应的放射正弦图。我们还表明，这一结果可能对通过医学成像扫描仪记录的放射正弦图的图像重建问题产生显著影响，并且在反问题正则化理论框架内描述霍夫变换问题值得深入研究。
英文摘要：
The Radon transform is a linear integral transform that mimics the data formation process in medical imaging modalities like X-ray Computerized Tomography and Positron Emission Tomography. The Hough transform is a pattern recognition technique, which is mainly used to detect straight lines in digital images and which has been recently extended to the automatic recognition of algebraic plane curves. Although defined in very different ways, in numerical applications both transforms ultimately take an image as an input and provide, as an output, a function defined on a parameter space. The parameters in this space describe a family of curves, which represent either the integration domains considered in the (generalized) Radon transform, or the curves to be detected by means of the Hough transform. In both cases, the 2D plot of the intensity values of the output function is the so-called (Radon or Hough) sinogram. While the Hough sinogram is produced by an algorithm whose implementation requires that the parameter space be discretized in cells, the Radon sinogram is mathematically defined on a continuous parameter space, which in turn may need to be discretized just for physical or numerical reasons. In this paper, by considering a more general and n-dimensional setting, we prove that, whether the input image is described as a set of points (possibly with different intensity values) or as a piecewise constant function, its (rescaled) Hough sinogram converges to the corresponding Radon sinogram as the discretization step in the parameter space tends to zero. We also show that this result may have a notable impact on the image reconstruction problem of inverting the Radon sinogram recorded by a medical imaging scanner, and that the description of the Hough transform problem within the framework of regularization theory for inverse problems is worth investigating.
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