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Abstract: In this paper we provide an analytical description of various classes of digital circles, spheres and in some cases hyperspheres, defined in a morphological framework. The topological properties of these objects, especially the separation of the digital space, are discussed according to the shape of the structuring element. The proposed framework is generic enough so that it encompasses most of the digital circle definitions that appear in the literature and extends them to dimension and sometimes dimension . [Copyright &y& Elsevier]

Antheridiogen systems, whereby notch-bearing, archegoniate gametophytes induce maleness in ameristic neighbors, have been detected in many core leptosporangiate ferns. Previous studies have failed to detect an antheridiogen system in Osmundales, which is sister to all other extant leptosporangiates; hence, antheridiogen systems are thought to have evolved after their divergence. Detailed studies of morphological development and patterns of gender expression in Osmundales and other basal leptosporangiate clades are needed to clarify how antheridiogen systems evolved. Here, we tracked the development and gender expression of gametophytes of Osmundastrum cinnamomeum grown in isolation and multispore populations exposed to basal media (control), gibberellic acid (GA3), and an older generation of gametophytes. A notch-producing apical meristem invariably preceded antheridia and archegonia production in O. cinnamomeum. Exposure to either GA3 or an older generation of gametophytes delayed growth rates and prolonged asexual and male status, whereas the multispore control possessed significantly greater proportions of females relative to isolates. Our observations con?rm the existence of a putative antheridiogen system in Osmundales and a mechanism by which male-first expression is bypassed by a subset of the population. The evolution of fully developed antheridiogen systems in core leptosporangiate families may have involved the decoupling of the formation of a notch meristem from the production of antheridia in an ancestor with an Osmundales-type pattern of gender expression. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the relationship between Euclidean and discrete space. We study discrete operations based on Euclidean functions: discrete smooth scaling and discrete-continuous rotation. Conversely, we study Euclidean operations based on discrete functions: the discrete based simplification, the Euclidean-discrete union and the Euclidean-discrete co-refinement. These operations operate partly in discrete, and partly in continuous space. Especially for the discrete smooth scaling operation, we provide error bounds when such different operations are chained. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to show that the supercover of an -flat (i.e. a Euclidean affine subspace of dimension ) in Euclidean -space is a discrete analytical object. The supercover of a Euclidean object is a discrete object consisting of all the voxels that intersect . A discrete analytical object is a set of discrete points that is defined by a finite set of inequalities. A method to determine the inequalities defining the supercover of an -flat is provided. [Copyright &y& Elsevier]

This paper presents a method for fitting digital hyperspheres to a given set of n D points in an image in the presence of noise by maximizing the number of inliers, namely the consensus set. The digital Hyperspheres are defined using the k -Flake Digitization models [25] . We present an algorithm, that provides optimal fitting solutions for digital k -Flake hyperspheres within a time complexity O ( ( ( n k ) 2 n − k ) n N n + 1 log ⁡ N ) for dimension n , N being the number of points. We have implemented this algorithm for the particular case of 3D 2-Flake spheres that corresponds to the classical so called Naive digital spheres. [ABSTRACT FROM AUTHOR]

In this paper, we propose an implementation of the 3-D Ridgelet transform: the 3-D discrete analytical Ridgelet transform (3-D DART). This transform uses the Fourier strategy for the computation of the associated 3-D discrete Radon transform. The innovative step is the definition of a discrete 3-D transform with the discrete analytical geometry theory by the construction of 3-D discrete analytical lines in the Fourier domain. We propose two types of 3-D discrete lines: 3-D discrete radial lines going through the origin defined from their orthogonal projections and 3-D planes covered with 2-D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3-D DART adapted to a specific application. Indeed, the 3-D DART representation is not orthogonal, It is associated with a flexible redundancy factor. The 3-D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3-D DART and its extension to the Local-DART (with smooth windowing) to the denoising of 3-D image and color video. These experimental results show that the simple thresholding of the 3-D DART coefficients is efficient. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we describe the theoretical foundations of the -arithmetization. This method provides a multi-scale discretization of a continuous function that is a solution of a differential equation. This discretization process is based on the Harthong–Reeb line . The Harthong–Reeb line is a linear space that is both discrete and continuous. This strange line stems from a nonstandard point of view on arithmetic based, in this paper, on the concept of -numbers introduced by Laugwitz and Schmieden. After a full description of this nonstandard background and of the first properties of , we introduce the -arithmetization and we apply it to some significant examples. An important point is that the constructive properties of our approach leads to algorithms which can be exactly translated into functional computer programs without uncontrolled numerical error. Afterwards, we investigate to what extent fits Bridges’s axioms of the constructive continuum. Finally, we give an overview of a formalization of the Harthong–Reeb line with the Coq proof assistant. [Copyright &y& Elsevier]