  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.
