Continuous Analogs of Digital Boundaries: A Topological Approach to Iso-Surfaces

Lachaud, Jacques-Olivier; Montanvert, Annick

doi:10.1006/gmod.2000.0522

Cited by 79 publications

(55 citation statements)

References 42 publications

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“…In the literature the adjacency relation of lattice points (or pixels) is usually characterised by a neighbourhood graph and the complementarity of adjacencies is defined via the Jordan surface theorem (Jordan-Brouwer theorem), see, e.g., Lachaud and Montanvert (2000). Here we follow Ohser et al (2003); Schladitz et al (2006) where the definition of adjacency is based on a consistency relation for the Euler number.…”

Section: Adjacency and Euler Numbermentioning

confidence: 99%

Miles Formulae for Boolean Models Observed on Lattices

Ohser¹,

Nagel²,

Schladitz³

2011

Image Anal Stereol

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The densities of the intrinsic volumes -in 3D the volume density, surface density, the density of the integral of the mean curvature and the density of the Euler number -are a very useful collection of geometric characteristics of random sets. Combining integral and digital geometry we develop a method for efficient and simultaneous calculation of the intrinsic volumes of random sets observed in binary images in arbitrary dimensions. We consider isotropic and reflection invariant Boolean models sampled on homogeneous lattices and compute the expectations of the estimators of the intrinsic volumes. It turns out that the estimator for the surface density is proved to be asymptotically unbiased and thus multigrid convergent for Boolean models with convex grains. The asymptotic bias of the estimators for the densities of the integral of the mean curvature and of the Euler number is assessed for Boolean models of balls of random diameters. Miles formulae with corresponding correction terms are derived for the 3D case.

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Section: Adjacency and Euler Numbermentioning

confidence: 99%

Miles Formulae for Boolean Models Observed on Lattices

Ohser¹,

Nagel²,

Schladitz³

2011

Image Anal Stereol

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“…Fig. 67: n-D approach of Lachaud [98], based on the convex hull. reason, we are not able to make any well-composed interpolation or to use any topological repairing method, the use of the Modified Marching Cubes algorithm [170], shortly MMC, is a good choice, but it assumes that the digitized object is r-regular and that the sampling grid has a sufficient resolution.…”

Section: Marching Cubesmentioning

confidence: 99%

A Tutorial on Well-Composedness

2017

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Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called wellcomposed images, with no topological issues. Furthermore, well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well-defined, and so on. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. 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 signals, and the various topics that are related to well-composedness.

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“…To have properties on the topology of the reconstruction, we need a process that disambiguates the configurations according to the topology of the input discrete surface. The configurations presented in the figure 1 correspond to a (18, 6)−surface [12,13]. Hence, if the binary object is 6-connected, the triangulated surface is a combinatorial 2-manifold, i.e.…”

Section: The Marching-cubes Algorithmmentioning

confidence: 99%

“…Hence, if the binary object is 6-connected, the triangulated surface is a combinatorial 2-manifold, i.e. closed, oriented and without self crossing [12,13]. In the following, we consider the Object Boundary Quantization (OBQ for short) scheme, also called Gauss digitization [10]: given an Euclidean region P in R 3 , the OBQ digitization of P is the set of voxels P ∩ Z 3 .…”

Section: The Marching-cubes Algorithmmentioning

confidence: 99%

“…To prove the topology, since the MC surface is a combinatorial 2-manifold [12,13] and we can locally prove that treatments on both H triangle and N H triangle do not change the topology: no holes are created, no self-crossings are introduced since we remain on the MC cell, and both the orientation and the combinatorial aspects of the surface are maintained. Hence, the final overall surface is still a combinatorial 2-manifold (see [4] for details on the H triangle treatment).…”

Section: Overall Algorithmmentioning

confidence: 99%

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