Spinal Canal and Spinal Marrow Segmentation by Means of the Hough Transform of Special Classes of Curves

Perasso, Annalisa; Campi, Cristina; Massone, Anna Maria; Beltrametti, Mauro C.

doi:10.1007/978-3-319-23231-7_53

Cited by 11 publications

(20 citation statements)

References 12 publications

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“…In this section we provide the examples we come back on in next Section 4. Such examples belong to classes of curves of interest in astronomical and medical imaging, and widely used in recent literature to best approximate bone profiles and typical solar structures such as coronal loops (for instance, see [2,12,13]). These families of curves mainly come from atlas of plane curves as [17], as well as from knowledge of classical tools in algebraic geometry.…”

Section: Examples Of Interestmentioning

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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Section: Examples Of Interestmentioning

confidence: 99%

Geometry of the Hough Transforms with Applications to Synthetic Data

et al. 2020

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“…Curves selected for the recognition of the bone profiles highlighted in red in the images of the first row. From left to right: curve with three convexities, 5, 13 Wassenaar curve, 5 Lamet curve, 14, 15 ellipse 13,15. …”

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confidence: 99%

“…The results of the HT algorithm for the recognition of ellipses13,15 in a slice containing a right femur.…”

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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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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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“…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. This paper formally investigates a known relationship between RT and HT which has been so far discussed just from a heuristic perspective.…”

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“…where p is the HT kernel (12) used in the rescaled Hough counter of Definition 3. According to (12) and (2), the thesis (15) can be recast as…”

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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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Spinal Canal and Spinal Marrow Segmentation by Means of the Hough Transform of Special Classes of Curves

Cited by 11 publications

References 12 publications

Geometry of the Hough Transforms with Applications to Synthetic Data

Geometry of the Hough Transforms with Applications to Synthetic Data

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

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

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