Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Finite topology as applied to image analysis
Computer Vision, Graphics, and Image Processing
Digital topology: introduction and survey
Computer Vision, Graphics, and Image Processing
A Jordan surface theorem for three-dimensional digital spaces
Discrete & Computational Geometry
Discrete multidimensional Jordan surfaces
CVGIP: Graphical Models and Image Processing
Multidimensional digital boundaries
CVGIP: Graphical Models and Image Processing
Dimensional properties of graphs and digital spaces
Journal of Mathematical Imaging and Vision - Special issue on topology and geometry in computer vision
Graphical Models and Image Processing
Strong thinning and polyhedric approximation of the surface of a voxel object
Discrete Applied Mathematics
Discretization in 2D and 3D orders
Graphical Models - Special issue: Discrete topology and geometry for image and object representation
Derived neighborhoods and frontier orders
Discrete Applied Mathematics - Special issue: Advances in discrete geometry and topology (DGCI 2003)
Derived neighborhoods and frontier orders
Discrete Applied Mathematics - Special issue: Advances in discrete geometry and topology (DGCI 2003)
Equivalence between Closed Connected n-G-Maps without Multi-Incidence and n-Surfaces
Journal of Mathematical Imaging and Vision
Derived neighborhoods and frontier orders
Discrete Applied Mathematics - Special issue: Advances in discrete geometry and topology (DGCI 2003)
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Many applications require the extraction of an object boundary from a discrete image. In most cases, the result of such a process is expected to be, topologically, a surface, and this property might be required in subsequent operations. However, only through careful design can such a guarantee be provided. In the present article we will focus on partially ordered sets and the notion of n-surfaces introduced by Evako et al. to deal with this issue. Partially ordered sets are topological spaces that can represent the topology of a wide range of discrete spaces, including abstract simplicial complexes and regular grids. It will be proved in this article that (in the framework of simplicial complexes) any n-surface is an n-pseudomanifold, and that any n-dimensional combinatorial manifold is an n-surface. Moreover, given a subset of an n-surface (an object), we show how to build a partially ordered set called frontier order, which represents the boundary of this object. Similarly to the continuous case, where the boundary of an n-manifold, if not empty, is an (n驴1)-manifold, we prove that the frontier order associated to an object is a union of disjoint (n驴1)-surfaces. Thanks to this property, we show how topologically consistent Marching Cubes-like algorithms can be designed using the framework of partially ordered sets.