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Characterization of Simple Closed Surfaces in ℤ3: A New Proposition with a Graph-Theoretical Approach [chapter]

Rémy Malgouyres, Jean-Luc Toutant
2009 Lecture Notes in Computer Science  
In the present paper, we propose a topological characterization of digital surfaces. We introduce simple local conditions on the neighborhood of a voxel.  ...  If each voxel of a 26-connected digital set satisfies them, we prove a Jordan theorem and ensure that this set is strong 6-separating in Z 3 . Thus, we consider it as a digital surface.  ...  preservation of the parity of the number of intersections between a path and a digital surface in the case of the second condition of Definition 7.  ... 
doi:10.1007/978-3-642-04397-0_25 fatcat:juatdm7rsvc3tbqinstpzqfw2u

Comparison of intraoral and extraoral scanners on the accuracy of digital model articulation

Jason L. Porter, Caroline K. Carrico, Steven J. Lindauer, Eser Tüfekçi
2018 Journal of orthodontics  
Orthodontists increasingly rely on digital models in clinical practice. The ability of modern scanners to articulate digital models must be scientifically evaluated.  ...  An acceptable range of 0.5mm more or less than the gold standard was used for evaluation.  ...  The 3M™ True Definition scanner and Ortho Insight 3D® laser surface scanner with Regisil® bite registration were the next most accurate scanning method of articulating digital models.  ... 
doi:10.1080/14653125.2018.1500773 pmid:30024348 fatcat:h5uqsfra3rdsjhkovukv6kgvse

Curves, Hypersurfaces, and Good Pairs of Adjacency Relations [chapter]

Valentin E. Brimkov, Reinhard Klette
2004 Lecture Notes in Computer Science  
In this paper we propose several equivalent definitions of digital curves and hypersurfaces in arbitrary dimension.  ...  Thus this work appears to be the first one on digital manifolds where the definitions involve the notion of dimension.  ...  Acknowledgements The authors thank the three anonymous referees for their useful remarks and suggestions.  ... 
doi:10.1007/978-3-540-30503-3_21 fatcat:ubcltcvcfrgj5p4dxp6yfj6cye

Coding cells of digital spaces: a framework to write generic digital topology algorithms

Lachaud Jacques-Olivier
2003 Electronic Notes in Discrete Mathematics  
Discrete subsets of multidimensional images (e.g. regions, digital surfaces, cubical cell complexes) have then a common and compact representation.  ...  This paper proposes a concise coding of the cells of n-dimensional finite regular grids.  ...  The paper is organized as follows: (i) coding of cells and implementation of low-level digital topology definitions, (ii) definition of data structures for subsets of C n (e.g., digital surfaces, complexes  ... 
doi:10.1016/s1571-0653(04)00497-4 fatcat:wcnleshl2jhplhq67jra6cjyre

Digital planarity—A review

Valentin Brimkov, David Coeurjolly, Reinhard Klette
2007 Discrete Applied Mathematics  
Digital planarity is defined by digitizing Euclidean planes in the three-dimensional digital space of voxels; voxels are given either in the grid-point or the grid-cube model.  ...  The paper summarizes results (also including most of the proofs) about different aspects of digital planarity, such as supporting or separating Euclidean planes, characterizations in arithmetic geometry  ...  The authors dedicate it to this outstanding researcher who established the field of digital geometry with numerous fundamental contributions.  ... 
doi:10.1016/j.dam.2006.08.004 fatcat:3xosyl4ofjfeparnocewylur74

Graph Theoretical Models of Discrete Spaces with Locally Non-spherical Topology

Alexander V. Evako
2017 Applied Mathematics and Physics  
We define and investigate properties of digital 3-and 5-dimensional closed surfaces with a local toroidal and projective plane topology.  ...  This paper defines and studies properties of a new class of digital n-dimensional spaces with a locally non-spherical topology. We prove that such spaces have the dimension n≥3.  ...  Definition 3 The digital 0-dimensional surface S 0 (a, b) is a disconnected graph with just two points a and b.  ... 
doi:10.12691/amp-5-2-3 fatcat:7iduyoqtzbdvfnaxvgxrm7ctuy

Three Dimensional Digital Sieving of Asphalt Mixture Based on X-ray Computed Tomography

Chichun Hu, Jiexian Ma, M. Kutay
2017 Applied Sciences  
In order to perform three-dimensional digital sieving based on X-ray computed tomography images, the definition of digital sieve size (DSS) was proposed, which was defined as the minimum length of the  ...  It was suggested that concave surface of aggregate was the possible reason for the disparity between DSS and mechanical sieve size.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/app7070734 fatcat:maz5fygoavdrra2v75qrwh457i

Intersection Number of Paths Lying on a Digital Surface and a New Jordan Theorem [chapter]

Sébastien Fourey, Rémy Malgouyres
1999 Lecture Notes in Computer Science  
The purpose of this paper is to define the notion of "real" intersection between paths drawn on the 3d digital boundary of a connected object.  ...  We consider two kinds of paths for different adjacencies, and define the algebraic number of oriented intersections between these two paths.  ...  Given Y ⊂ X two subsets of a digital surface, a relationship is established, between the property of X to be reducible to Y by sequential deletion of simple surfels, and the fundamental groups ( [3] )  ... 
doi:10.1007/3-540-49126-0_9 fatcat:qjkj5l6danajhitov7ryqehksq

COMMUTATIVE MONOID OF THE SET OF k-ISOMORPHISM CLASSES OF SIMPLE CLOSED k-SURFACES IN Z3

Sang-Eon Han
2010 Honam Mathematical Journal  
Besides, with some hypothesis the set of k-homotopy equivalence classes of closed k-surfaces in Z 3 is also proved to be a commutative monoid with the above operation, k ∈ {18, 26}.  ...  In this paper we prove that with some hypothesis the set of k-isomorphism classes of simple closed k-surfaces in Z 3 forms a commutative monoid with an operation derived from a digital connected sum, k  ...  This kinds of two monoids of the sets of k-isomorphism classes of simple closed k-surfaces in Z 3 and k-homotopy equivalence classes of closed k-surfaces can be used in classifying simple closed k-surfaces  ... 
doi:10.5831/hmj.2010.32.1.141 fatcat:wcwlxonm5re27mjodm4miz2o3y

Border and SurfaceTracing - Theoretical Foundations

V.E. Brimkov, R. Klette
2008 IEEE Transactions on Pattern Analysis and Machine Intelligence  
This work appears to be the first one on digital manifolds based on a graph-theoretical definition of dimension.  ...  We also discuss possible applications of the presented definitions and results.  ...  We are indebted to the three anonymous referees for their valuable remarks and suggestions.  ... 
doi:10.1109/tpami.2007.70725 pmid:18276965 fatcat:hd4jmado5vd3flzakiv4apr4ai

Topology preserving representations of compact 2D manifolds by digital 2-surfaces. Compressed digital models and digital weights of compact 2D manifolds. Classification of closed surfaces by digital tools [article]

Alexander V. Evako
2014 arXiv   pre-print
Using digital topology approach, we construct digital models of closed surfaces as the intersection graphs of LCL covers of the surfaces.  ...  It is proved that digital models of closed surfaces are digital 2-dimensional surfaces preserving the geometry and topology of their continuous counterparts.  ...  Intersection graphs of LCL coves of curves and closed surfaces and digital models of closed surfaces In this section, we investigate a connection between LCL covers of closed surfaces and intersection  ... 
arXiv:1412.0143v1 fatcat:wdyeistrozgyjmeyili4ouhhai

Presentation of the Fundamental Group in Digital Surfaces [chapter]

Rémy Malgouyres
1999 Lecture Notes in Computer Science  
A classical mathematical way to encode a discrete group is to find a presentation of this group. In this paper, we define a presentation for the fundamental group of any subset of a digital surface.  ...  As its analogue in the continuous framework, the digital fundamental group represents a major information on the topology of discrete objects.  ...  In [9] , we introduce two complementary notions of adjacency between surfels on a digital surface, which can be used to study the topology of such subsets X of a digital surface, and we study the problem  ... 
doi:10.1007/3-540-49126-0_11 fatcat:z4r3qfqzpjbmngtmt53osqxdc4

An isomorphism for digital images

Christian Ronse
1985 Journal of combinatorial theory. Series A  
In this paper, an isomorphism between two digital images as a relation, not a map, which satisfies several requirements related to the equivalence of the two digital structures is defined.  ...  These properties, together with the correspondence with the Euclidean topology in the.case of the rectangular grid, validate our definition of the digital isomorphism.  ...  The property of surrounding is defined from the frame FV of V, not from the image 9. We will thus assume in this subsection that V and v' have a frame or are infinite.  ... 
doi:10.1016/0097-3165(85)90034-2 fatcat:jx3l5xqyhvb45dfkgwl4mmz3ty

Digital Geometry, a Survey [article]

Li Chen, David Coeurjolly
2018 arXiv   pre-print
It also covers recent developments in the applications of digital geometry and topology including image processing, computer vision, and data science.  ...  This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements.  ...  The method was a nice bridge between digital space and continuous space. Definition 3.  ... 
arXiv:1807.02222v1 fatcat:a5q5mhmly5hwfbjgliywabeosi

A Tutorial on Well-Composedness

Nicolas Boutry, Thierry Géraud, Laurent Najman
2017 Journal of Mathematical Imaging and Vision  
Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld.  ...  Believing in its prime importance in digital topology, we then propose this state-of-the-art of well-composedness, summarizing its different flavours, the different methods existing to produce well-composed  ...  Now let us recall the definition of (discrete) n-surfaces [55] .  ... 
doi:10.1007/s10851-017-0769-6 fatcat:pwvvx5rvlndblkexm5tj6y4y7i
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