Analytical Description of Digital Circles

Andrès, Éric; Roussillon, Tristan

doi:10.1007/978-3-642-19867-0_20

Cited by 19 publications

(19 citation statements)

References 27 publications

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“…There are two parameters to work with: the distance d and a thickness parameter r. Let us note that the parameter r can also be dened as a function. See [31,26,33] for examples of digital objects dened with a non-constant thickness. Considering the points that are close to the original continuous object seems reasonable if we want the digital object to look like the original.…”

Section: Morphological Digitizationsmentioning

confidence: 99%

“…It is interesting to note that, for these digitizations, the structuring element is a polytope and therefore all the linear objects, at least, can be described analytically as linear digital objects (with linear inequalities There is one now [26,33]. An interesting property of such Andres hyperspheres is that concentric Andres hyperspheres pave digital space.…”

Section: Direct Dened Analytical Digital Objectsmentioning

confidence: 99%

“…An immediate example is the Bresenham Straight line Segment [22] that has been shown to be a Reveilles straight line segment [20]. In the same way, in [26], most notions of digital circles that have been introduced have been analytically characterized [23,7]. An extension to higher dimensions has been proposed in [33] with an explicit mention of Morphological…”

Section: Digitized Analytical Objectsmentioning

confidence: 99%

See 2 more Smart Citations

Digital Analytical Geometry: How Do I Define a Digital Analytical Object?

Andrès

2015

Lecture Notes in Computer Science

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Abstract. This paper is meant as a short survey on analytically dened digital geometric objects. We will start by giving some elements on digitizations and its relations to continuous geometry. We will then explain how, from simple assumptions about properties a digital object should have, one can build mathematical sound digital objects. We will end with open problems and challenges for the future.

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Section: Morphological Digitizationsmentioning

confidence: 99%

Section: Direct Dened Analytical Digital Objectsmentioning

confidence: 99%

Section: Digitized Analytical Objectsmentioning

confidence: 99%

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Digital Analytical Geometry: How Do I Define a Digital Analytical Object?

Andrès

2015

Lecture Notes in Computer Science

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“…Representing a discretized object (such as curve or surface) by the set of integer solutions of a system of inequalities, called the analytical representation, has been studied [1-6, 12, 21, 27, 28]. For some curves and surfaces, the morphological discretization (with some speci c structuring elements) is known to have its analytical representation [2,5,6,27,28]. With the analytical representation, the morphological discretization is straightforwardly computed, just by evaluating inequalities for each integer point.…”

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confidence: 99%

On properties of analytical approximation for discretizing 2D curves and 3D surfaces

Sekiya

Sugimoto

2017

Mathematical Morphology - Theory and Applications

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Abstract:The morphological discretization is most commonly used for curve and surface discretization, which has been well studied and known to have some important properties, such as preservation of topological properties (e.g., connectivity) of an original curve or surface. To reduce its high computational cost, on the other hand, an approximation of the morphological discretization, called the analytical approximation, was introduced. In this paper, we study the properties of the analytical approximation focusing on discretization of 2D curves and 3D surfaces in the form of y = f (x) (x, y ∈ R) and z = f (x, y) (x, y, z ∈ R). We employ as a structuring element for the morphological discretization, the adjacency norm ball and use only its vertices for the analytical approximation. We show that the discretization of any curve/surface by the analytical approximation can be seen as the morphological discretization of a piecewise linear approximation of the curve/surface. The analytical approximation therefore inherits the properties of the morphological discretization even when it is not equal to the morphological discretization.

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“…It can also be extended to digital spheres or hyperspheres. This is a step forward compared to the results previously presented in [19].…”

Section: Introductionmentioning

confidence: 99%

Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties

Toutant

Andrès

Roussillon

2013

Discrete Applied Mathematics

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In this paper we provide an analytical description of various classes of digital circles, spheres and in some cases hyperspheres, defined in a morphological framework. The topological properties of these objects, especially the separation of the digital space, are discussed according to the shape of the structuring element. The proposed framework is generic enough so that it encompasses most of the digital circle definitions that appear in the literature and extends them to dimension 3 and sometimes dimension n.

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Analytical Description of Digital Circles

Abstract: Abstract. In this paper we propose an analytical description of different kinds of digital circles that appear in the literature and especially in digital circle recognition algorithms.

Cited by 19 publications

References 27 publications

Digital Analytical Geometry: How Do I Define a Digital Analytical Object?

Digital Analytical Geometry: How Do I Define a Digital Analytical Object?

On properties of analytical approximation for discretizing 2D curves and 3D surfaces

Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties

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