On parallel thinning algorithms: minimal non-simple sets, P-simple points and critical kernels

International audienceCritical kernels constitute a general framework in the category of abstract complexes for the study of parallel homotopic thinning in any dimension. In this article, we present new results linking critical kernels to minimal non-simple sets (MNS) and P-simple points, which are notions conceived to study parallel thinning in discrete grids. We show that these two previously introduced notions can be retrieved, better understood and enriched in the framework of critical kernels. In particular, we propose new characterizations which hold in dimensions 2, 3 and 4, and which lead to efficient algorithms for detecting P-simple points and minimal non-simple sets

Non-representable relation algebras from vector spaces

Extending a construction of Andreka, Givant, and Nemeti (2019), we construct some finite vector spaces and use them to build finite non-representable relation algebras. They are simple, measurable, and persistently finite, and they validate arbitrary finite sets of equations that are valid in the variety RRA of representable relation algebras. It follows that there is no finitely axiomatisable class of relation algebras that contains RRA and validates every equation that is both valid in RRA and preserved by completions of relation algebras. Consequently, the variety generated by the completions of representable relation algebras is not finitely axiomatisable. This answers a question of Maddux (2018)

Non-representable relation algebras from vector spaces

Extending a construction of Andreka, Givant, and Nemeti (2019), we construct some finite vector spaces and use them to build finite non-representable relation algebras. They are simple, measurable, and persistently finite, and they validate arbitrary finite sets of equations that are valid in the variety RRA of representable relation algebras. It follows that there is no finitely axiomatisable class of relation algebras that contains RRA and validates every equation that is both valid in RRA and preserved by completions of relation algebras. Consequently, the variety generated by the completions of representable relation algebras is not finitely axiomatisable. This answers a question of Maddux (2018)

On four-dimensional 2-handlebodies and three-manifolds

We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, which provides a complete solution to the long-standing Fox-Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S^3 branched over embedded graphs. Then, we factor the functor above through an equivalence functor from H^r to Chb^{3+1}, where H^r is a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category Chb^{3+1}. From this we derive an analogous description of the category Cob^{2+1} of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.Comment: 213 pages, 272 figures, 15 table

Automatic Extraction of Closed Contours Bounding Salient Objects: New Algorithms and Evaluation Methods

The problem under consideration in this dissertation is achieving salient object segmentation of natural images by means of probabilistic contour grouping. The goal is to extract the simple closed contour bounding the salient object in a given image. The method proposed here falls in the Contour Grouping category, searching for the optimal grouping of boundary entities to form an object contour. Our first contribution is to provide both a ground truth dataset and a performance measure for empirical evaluation of salient object segmentation methods. Our Salient Object Dataset (SOD) provides ground truth boundaries of salient objects perceived by humans in natural images. We also psychophysically evaluated 5 distinct performance measures that have been used in the literature and showed that a measure based upon minimal contour mappings is most sensitive to shape irregularities and most consistent with human judgements. In fact, the Contour Mapping measure is as predictive of human judgements as human subjects are of each other. Contour grouping methods often rely on Gestalt cues locally defined on pairs of oriented features. Accurate integration of these local cues with global cues is a challenge. A second major contribution of this dissertation is a novel, effective method for combining local and global cues. A third major contribution in this dissertation is a novel method based on Principal Component Analysis for promoting diversity among contour hypotheses, leading to substantial improvements in grouping performance. To further improve the performance, a multiscale implementation of this method has been studied. A fourth contribution in this dissertation is studying the effect of the multiscale prior on the performance and analysing the method for combining the results obtained in different resolutions. Our final contribution is comparing the performance of univariate distribution models for local cues used by our method with the use of a multivariate mixture model for their joint distribution. We obtain slight improvement by the mixture models. The proposed method has been evaluated and compared with four other state-of-the-art grouping methods, showing considerably better performance on the SOD ground truth dataset

Invariants in Low-Dimensional Topology

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