147 research outputs found
Discrete euclidean skeletons in increased resolution
Orientadores: Roberto de Alencar Lotufo, Michel CouprieTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: A extração de esqueletos Euclidianos é uma tema de grande importância na área de processamento de imagens e tem sido discutido pela comunidade cientÃfica já há mais de 20 anos. Hoje é consenso que os esqueletos Euclidianos devem ter as seguintes caracterÃsticas: ï¬?nos, centrados, homotópicos e reversÃveis, i.e., suficientes para a reconstrução do objeto original. Neste trabalho, introduzimos o Eixo Mediano Euclidiano Exato em Resolução Aumentada -HMA, com o objetivo de obter um eixo mediano mais ï¬?no do que o obtido pela definição clássica. Combinando o HMA com um eï¬?ciente algoritmo de afinamento paralelo homotópico, propomos um esqueleto Euclidiano que é centrado, homotópico, reversÃvel e mais ï¬?no que os já existentes na literatura. O esqueleto proposto tem a particularidade adicional de ser único e independente de decisões arbitrárias. São dados algoritmos e provas, assim como exemplos de aplicações dos esqueletos propostos em imagens reais, mostrando as vantagens da proposta. O texto inclui também uma revisão bibliográfica sobre algoritmos de transformada de distância, eixo mediano e esqueletos homotópicosAbstract: The extraction of Euclidean skeletons is a subject of great importance in the domain of image processing and it has been discussed by the scientiï¬?c community since more than 20 years.Today it is a consensus that Euclidean skeletons should present the following characteristics: thin, centered, homotopic and reversible, i.e., sufï¬?cient for the reconstruction of the original object. In this work, we introduce the Exact Euclidean Medial Axis in Higher Resolution -HMA, with the objective of obtaining a medial axis which is thinner than the one obtained by the classical medial axis deï¬?nition. By combining the HMA with an efï¬?cient parallel homotopic thinning algorithm we propose an Euclidean skeleton which is centered, homotopic, reversible and thinner than the existing similars in the literature. The proposed skeleton has the additional particularity of being unique and independent of arbitrary choices. Algorithms and proofs are given, as well as applicative examples of the proposed skeletons in real images, showing the advantages of the proposal. The text also includes an overview on algorithms for the Euclidean distance transform algorithms, the medial axis extraction, as well as homotopic skeletonsDoutoradoEngenharia de ComputaçãoDoutor em Engenharia Elétric
Minimal simple pairs in the cubic grid
International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. This paper constitutes an introduction to the study of non-trivial simple sets in the framework of cubical 3-D complexes. A simple set has the property that the homotopy type of the object in which it lies is not changed when the set is removed. The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs
Distance, granulometry, skeleton
In this chapter, we present a series of concepts and operators based on the notion of distance. As often with mathematical morphology, there exists more than one way to present ideas, that are simultaneously equivalent and complementary. Here, our problem is to present methods to characterize sets of points based on metric, geometry and topology considerations. An important concept is that of the skeleton, which is of fundamental importance in pattern recognition, and has many practical application
Cellular Skeletons: A New Approach to Topological Skeletons with Geometric Features
This paper introduces a new kind of skeleton for binary volumes called the cellular skeleton. This skeleton is not a subset of voxels of a volume nor a subcomplex of a cubical complex: it is a chain complex together with a reduction from the original complex.
Starting from the binary volume we build a cubical complex which represents it regarding 6 or 26-connectivity. Then the complex is thinned using the proposed method based on elementary collapses, which preserves significant geometric features. The final step reduces the number of cells using Discrete Morse Theory. The resulting skeleton is a reduction which preserves the homology of the original complex and the geometrical information of the output of the previous step.
The result of this method, besides its skeletonization content, can be used for computing the homology of the original complex, which usually provides well shaped homology generators
Nonparametric ridge estimation
We study the problem of estimating the ridges of a density function. Ridge
estimation is an extension of mode finding and is useful for understanding the
structure of a density. It can also be used to find hidden structure in point
cloud data. We show that, under mild regularity conditions, the ridges of the
kernel density estimator consistently estimate the ridges of the true density.
When the data are noisy measurements of a manifold, we show that the ridges are
close and topologically similar to the hidden manifold. To find the estimated
ridges in practice, we adapt the modified mean-shift algorithm proposed by
Ozertem and Erdogmus [J. Mach. Learn. Res. 12 (2011) 1249-1286]. Some numerical
experiments verify that the algorithm is accurate.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1218 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Topological properties of thinning in 2-D pseudomanifolds
International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z^2) such procedures are usually based on the notion of simple point. In contrast to the situation in Z^n , n>=3, it was proved in the 80s that the exclusive use of simple points in Z^2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to cubical complexes in 2-D pseudomanifolds
Exact Lagrangian immersions with one double point revisited
We study exact Lagrangian immersions with one double point of a closed
orientable manifold K into n-complex-dimensional Euclidean space. Our main
result is that if the Maslov grading of the double point does not equal 1 then
K is homotopy equivalent to the sphere, and if, in addition, the Lagrangian
Gauss map of the immersion is stably homotopic to that of the Whitney
immersion, then K bounds a parallelizable (n+1)-manifold. The hypothesis on the
Gauss map always holds when n=2k or when n=8k-1. The argument studies a filling
of K obtained from solutions to perturbed Cauchy-Riemann equations with
boundary on the image f(K) of the immersion. This leads to a new and simplified
proof of some of the main results of arXiv:1111.5932, which treated Lagrangian
immersions in the case n=2k by applying similar techniques to a Lagrange
surgery of the immersion, as well as to an extension of these results to the
odd-dimensional case.Comment: 39 pages, 2 figures. Version 2: A lengthy appendix now contains a
detailed and largely self-contained proof of the existence of a C1-smooth
structure on a certain compactified moduli space of Floer disks. The rest of
the text is accordingly somewhat re-organised. Version 3: minor further
change
Singularities and stable homotopy groups of spheres II
We establish a connection between Morin singularities and stable homotopy
groups of spheres. This connection allows us to describe how the images of
singularity strata behave around the image of a more complicated stratum.Comment: 31 pages, submitted to Journal of Singularitie
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