Hough Transform of Special Classes of Curves

Beltrametti, Mauro C.; Massone, Anna Maria; Piana, Michele

doi:10.1137/120863794

Cited by 40 publications

(74 citation statements)

References 12 publications

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“…Finally, we look for the cell corresponding to the maximum of the accumulator function: the parameter set associated to that cell provides the curve of the family which best approximates the profile of interest in the image. 4,5,7 Several curves and their properties have been presented in Refs. 4, 5, and 13: the Lamet curve, the curve with three convexities, and the conchoid of Slüse have been used to approximate the sternum, the spinal canal and the vertebra profiles, respectively (see Refs.…”

Section: Hough Transform For Algebraic Curvesmentioning

confidence: 99%

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HT-BONE: a graphical user interface for the identification of bone profiles in CT images via extended Hough transform

et al. 2016

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It has been recently proved that the computational analysis of X-ray Computed Tomography (CT) images allows clinicians to assess the alteration of compact bone asset due to hematological diseases. HT-BONE implements a new method, based on an extension of the Hough transform (HT) to a wide class of algebraic curves, for accurately measuring global and regional geometric properties of trabecular and compact bone districts. In the case of CT/PET analysis, the segmentation of the CT images provides masks for Positron Emission Tomography (PET) data, extracting the metabolic activity in the region surrounded by compact bone tissue. HT-BONE offers an intuitive, user-friendly, Matlab-based Graphical User Interface (GUI) for all input/output procedures and the automatic managing of the segmentation process also from non-expert users: the CT/PET data can be loaded and browsed easily and the only pre-preprocessing required from the user is the drawing of Regions Of Interest (ROIs) around the bone districts under consideration. For each bone district, specific families of curves, whose reliability has been already tested in previous works, is automatically selected for the recognition task via HT. As output, the software returns masks of the segmented compact bone regions, images of the Standard Uptake Values (SUV) in the masked regions of PET slices, and the values of the parameters in the curve equations utilized in the HT procedure. This information can be used for all pathologies and clinical conditions for which the alteration of the compact bone asset or bone marrow distribution plays a crucial role.

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Section: Hough Transform For Algebraic Curvesmentioning

confidence: 99%

“…On the other hand, let us stress how a good choice of the family F makes the HT-based procedure extremely robust both in presence of very noisy framework and against random perturbation of point positions 4. …”

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confidence: 99%

HT-BONE: a graphical user interface for the identification of bone profiles in CT images via extended Hough transform

et al. 2016

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“…Thanks to algebraic geometrical arguments, the Hough transform definition has been extended to include special classes of curves [2,12]. In [2], a characterization of families of irreducible algebraic plane curves of the same degree for which is defined a Hough-type correspondence is provided. In fact, given a family F of algebraic curves, a general point p in the image plane corresponds to an algebraic locus, Γ p (F), in the parameter space.…”

Section: Introductionmentioning

confidence: 99%

Geometry of the Hough Transforms with Applications to Synthetic Data

et al. 2020

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In the framework of the Hough transform technique to detect curves in images, we provide a bound for the number of Hough transforms to be considered for a successful optimization of the accumulator function in the recognition algorithm. Such a bound is consequence of geometrical arguments. We also show the robustness of the results when applied to synthetic datasets strongly perturbed by noise. An algebraic approach, discussed in the appendix, leads to a better bound of theoretical interest in the exact case.

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“…Improved algorithms based on HT voting schemes and enabling very rapid and efficient recognition of specific geometric features in large images or data sets, have been introduced in [8,9]. From a more theoretical point of view, a recent research [10] provided a rigorous mathematical foundation, based on algebraic-geometry arguments, for the case of algebraic plane curves of whatever degree, together with a key lemma stating equivalent conditions under which the existence and uniqueness of the intersection point of the HTs is guaranteed. This framework was then applied in [11,12,13,14] to provide an atlas of algebraic curves used to recognize profiles in real astronomical and biomedical images.…”

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confidence: 99%

“…, t, a sampling distance d k with respect to the component λ k . Then, set (10) λ k,n k := λ * k + n k d k , k = 1, . .…”

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On the asymptotic equivalence between the radon and the hough transforms of digital images

Aramini

Delbary

Beltrametti

et al. 2019

Preprint

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Although characterized by 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 either Radon or Hough sinograms. 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. By considering generalized functions in multi-dimensional setting, in this paper we give an analytical proof of this heuristic rationale when the input grayscale 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 have a valuable impact on the image reconstruction problem of inverting the Radon sinogram recorded by a medical imaging scanner.1. Introduction. The Radon transform (RT) [1] is an important tool in harmonic analysis and inverse problems theory with significant impacts on both group theory and applied mathematics. The classical definition of this transform considers integrals over hyperplanes with specific orientation and distance from a reference hyperplane. In this case, many functional properties of the RT have been investigated, including the characterization of its kernel and range, the ill-posedness of the inverse problem, as well as several inversion formulas and algorithms. The RT has also been extended to integration on manifolds [2], although here important functional and computational problems are still open. In biomedical imaging, RT is the well-established mathematical model for data formation in X-ray Computerized Tomography (CT) and in Positron Emission Tomography (PET) [3,4] and all software tools for image visualization implemented in current industrial CT and PET scanners must realize, at same stage, the numerical inversion of RT.While RT plays a crucial role in image reconstruction, the Hough transform (HT) [5,6,7] provides an important computational technique in pattern recognition. Indeed, HT is widely used in image-processing to detect specific algebraic plane

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Hough Transform of Special Classes of Curves

Cited by 40 publications

References 12 publications

HT-BONE: a graphical user interface for the identification of bone profiles in CT images via extended Hough transform

HT-BONE: a graphical user interface for the identification of bone profiles in CT images via extended Hough transform

Geometry of the Hough Transforms with Applications to Synthetic Data

On the asymptotic equivalence between the radon and the hough transforms of digital images

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