SIAM Journal on Imaging Sciences, Volume 14, Issue 2, Page 506-529, January 2021.
Although characterized by two different mathematical definitions, both the Radon and the Hough transforms ultimately take an image as input and provide, as output, functions defined on a preassigned parameter space, i.e., the so-called Radon and Hough sinograms, respectively. The parameters in these two spaces describe a family of curves, which represent either the integration domains considered in the Radon transform, or the kind of curves to be detected by the Hough transform. It is heuristically known that the Hough sinogram converges to the corresponding Radon sinogram when the discretization step in the parameter space tends to zero. However, as far as we know, no formal result has been proven so far about such convergence. Therefore, by considering generalized functions in a multidimensional setting, in this paper we give an analytical proof of this heuristic rationale when the input digital image is described as a set of grayscale points, that is, as a sum of weighted Dirac delta functions. On these grounds, we also show that this asymptotic equivalence may lead to a visualization process relying on the interpretation of the Radon sinogram as a Hough sinogram.

中文翻译：

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Images与数字图像的霍夫变换之间的渐近等价

SIAM影像科学杂志，第14卷，第2期，第506-529页，2021年1月。
尽管Radon变换和Hough变换具有两个不同的数学定义，但最终都将图像作为输入，并提供在预先指定的参数空间上定义的函数作为输出，即分别称为Radon和Hough正弦图。这两个空间中的参数描述了一系列曲线，这些曲线代表Radon变换中考虑的积分域或Hough变换要检测的曲线的种类。启发式地知道，当参数空间中的离散步长趋于零时，霍夫正弦图收敛到相应的Radon正弦图。但是，据我们所知，到目前为止，关于这种融合的正式结果尚未得到证实。因此，通过考虑多维设置中的广义函数，在本文中，当将输入数字图像描述为一组灰度点（即加权Dirac delta函数之和）时，我们给出了这种启发式原理的分析证明。基于这些理由，我们还表明，这种渐近等价可能导致依赖于将Radon正弦图解释为Hough正弦图的可视化过程。