Polyhedral Representation and Adjacency Graph in n-dimensional Digital Images

A new model of an adaptive adjacency graph (AAG) for representing a 2-D image or a 2-D view of a 3-D scene is introduced. The model makes use of image representation similar in form to a region adjacency graph. Adaptive adjacency graph, as opposed to region adjacency graph, is an active representation of the image. The AAG can adapt to the image or track features and maintain the topology of the graph. Adaptability of the AAG is achieved by incorporating active contours (`snakes') in the graph. Various methods for creating the AAGs are discussed. Results obtained for dynamic tracking of features in sequence of images and for registration of retinal images are presented.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

SUMMARY An automatic planning system for robotic assembly must have knowledge of the workspace. In particular it must be capable of reasoning about the shape of objects within the workspace. This requires a knowledge of object features and their behaviour in assembly situations. In this paper, a system is presented which extracts features relevant to assembly and machining from a boundary model derived from a Constructive Solid Geometry representation of a complex shape. In particular, finding concavities plays an important role in identifying possible mating features for assembly. These relevant features can then be used for assembly planning based on finding compatible sets of mating features. In this work the CAD data is extracted from a boundary model represented in an ASCII file and converted into the data-structures of the AI language, Prolog.

In digital geometry and topology, there are two popular kinds of digital spaces: point spaces and raster spaces. In point-spaces, a digital object is presented by a set of elements. In raster spaces as defined in this note, a digital object is a subset of a 'relation' on the space. In an Euclidean space, given a set S of points which are called sites, we can get the Voronoi diagram of S and its Delaunay triangulation. The Voronoi diagram is just a raster space as well as Delaunay simple decomposition is a point space. Thus, a point space is a dual space of a raster space. This note reviews some research results in point spaces and raster spaces and present the author's opinions on the following problems: how to define digital curves, surfaces, and manifolds in point spaces or raster spaces. What are the difference and relationship between them. What are the advantages and/or disadvantages to use point spaces or raster spaces in practical computation. The purpose of the note is to show a global consideration and to unify some basic concepts in digital geometry and topology.

The primary focus of this paper is to explore the concept of an initial condition for a noninstantaneous impulsive fractional integro-differential equation of order 0<ϑ<1 in an n-dimensional Euclidean space. Using the Laplace transform method, we derive the solution representation of the given dynamical system and investigate the existence and uniqueness of the mild solution through the topological degree method and Gronwall’s inequality. Furthermore, we present a filter model featuring a finite impulsive response, which serves as a practical demonstration of the proposed system because it effectively captures the memory effects inherent in fractional-order systems and enhances system reliability with minimal input. Finally, the numerical computations with graphical illustrations provide concrete examples that validate the theoretical results, showcasing how the behavior of the system is impacted by changes in fractional order and emphasizing the versatility of our proposed method.

Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner stability, asymptotic stability, and BIBO (Bounded Input, Bounded Output) stability, are well understood for classical linear control systems in Euclidean spaces. This paper aims to thoroughly address the stability problem for a class of linear control systems defined on matrix Lie groups. This approach generalizes classical models corresponding to the latter when the group is Abelian and non-compact. It is important to note that this generalization leads to a very difficult control system, due to the complexity of the state space and the special dynamics resulting from the drift and control vectors. Several mathematical concepts help us understand and characterize stability in the classical case. We first show how to extend these algebraic, topological, and dynamical concepts from Euclidean space to a connected Lie group of matrices. Building on classical results, we identify a pathway that enables us to formulate conjectures about stability in this broader context. This problem is closely linked to the controllability and observability properties of the system. Fortunately, these properties are well established for both classes of linear systems, whether in Euclidean spaces or on Lie groups. We are confident that these conjectures can be proved in future work, initially for the class of nilpotent and solvable groups, and later for semi-simple groups. This will provide valuable insights that will facilitate, through Jouan’s Equivalence Theorem, the analysis of an important class of nonlinear control systems on manifolds beyond Lie groups. We provide an example involving a three-dimensional solvable Lie group of rigid motions in a plane to illustrate these conjectures.

Topological concepts in Euclidean space

Using topological methods, we study the structure of the set of forced oscillations of a class of parametric, implicit ordinary differential equations with a generalized [Formula: see text]-Laplacian-type term. We work in the Carathéodory setting. Under suitable assumptions, involving merely the Brouwer degree in Euclidean spaces, we obtain global bifurcation results. In some illustrative examples, we provide a visual representation of the bifurcating set.

Prediction of production indicators of fractured-vuggy reservoirs based on improved Graph Attention Network

We consider the problem of image representation and clustering. Traditionally, an n1 x n2 image is represented by a vector in the Euclidean space R n1 x n2. Some learning algorithms are then applied to these vectors in such a high dimensional space for dimensionality reduction, classification, and clustering. However, an image is intrinsically a matrix, or the second order tensor. The vector representation of the images ignores the spatial relationships between the pixels in an image. In this paper, we introduce a tensor framework for image analysis. We represent the images as points in the tensor space Rn1 mathcal Rn2 which is a tensor product of two vector spaces. Based on the tensor representation, we propose a novel image representation and clustering algorithm which explicitly considers the manifold structure of the tensor space. By preserving the local structure of the data manifold, we can obtain a tensor subspace which is optimal for data representation in the sense of local isometry. We call it TensorImage approach. Traditional clustering algorithm such as k-means is then applied in the tensor subspace. Our algorithm shares many of the data representation and clustering properties of other techniques such as Locality Preserving Projections, Laplacian Eigenmaps, and spectral clustering, yet our algorithm is much more computationally efficient. Experimental results show the efficiency and effectiveness of our algorithm.

An inverse design in electromagnetic method based on the topological derivative approach is presented. Topological derivative method is used to measure the sensitivity of a given functional with respect to an infinitesimal perturbation in a domain. The topological derivative concept has been successfully applied in many relevant fields such as geophysics, multi-scale material design and inverse problems. In this work, to design the electromagnetic devices an objective function dependent on the scattering parameters is considered. Finally, numerical results are presented to illustrate the performance of the optimisation approach.

A number of strategies have been used to include spatial and topological properties in the image segmentation stage. It is generally accepted that grouping of nearby pixels by modelling neighbourhood relationships as (a, b) connected graphs may lead to meaningful image objects. In such approach, however, topological concepts may suffer from ambiguity since image elements (pixels) are two dimensional entities. This paper evaluates whether an alternative representation of digital images based both on Cartesian complexes and oriented matroids may improve multispectral image segmentation by enforcing topological and geometric properties and then be used in the classification stage. A conceptual model is defined, using Cartesian complexes, in order to link combinatorial properties of axiomatic locally finite spaces and their associated oriented matroids for involving topological properties. The proposed approach uses a layered architecture going from a physical level, going next through logical geospatial abstraction level and then through the Cartesian complex logical level. Additionally, there is a layer of oriented matroids composed by conceptual elements in terms of combinatorics for encoding relevant features to multispectral image segmentation. First, it is conducted an edge detection task, next an probability contour map using a Cartesian complex space rather than the conventional image space and finally, an image classification using random forest method. A computational solution including several components was developed using a framework for parallel computing. The performance of this solution was assessed using a small subset of GEOBIA2016 benchmark dataset. It is shown that the usage of a partial implementation of Cartesian complexes and associated oriented matroids is computationally but does not increase classification accuracy.

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

Spatial Temporal Graph Convolution Networks (STGCNs) have been proposed to embed spatio-temporal graphs. However, these networks used the Euclidean space as the embedding space which does not exploit the structure of the embedded graphs. Euclidean space has been shown not to be the ideal space for embedding graphs especially with tree-like structures. In this work, we make use of hyperbolic geometry and introduce a compact tangent space-free Lorentz ST-GCN and call it LSTGCN that perform the network operations directly on the manifold without resorting to the tangent space. The network uses spatial and temporal modules to propagate features between adjacent nodes in both, the spatial domain and the temporal domain, respectively. In addition, we introduce an attention module which can automatically determine the similarity of nodes without the need for the graph adjacency matrix. Experiments have been conducted on traffic flow forecasting tasks to show the effectiveness of the proposed compact Lorentz model.

This study aims to analyze the representation of millennial students in digital literacy on Instagram using Norman Fairclough's critical discourse analysis. This study describes how millennial students represent themselves in the context of digital literacy through popular social media platforms such as Instagram. A critical discourse analysis approach is used to reveal the power relations involved in this representation. This research method involves analyzing visual and text content uploaded by millennial students on their Instagram accounts. This study's results indicate that millennial students' representation of digital literacy on Instagram is strongly influenced by their cultural, religious, and social context factors. They use Instagram to express their religious identity, share religious knowledge, and promote Islamic values relevant to the digital era. However, there are also constructions of representation that face conflict with the dominant narrative. Millennial students face challenges in navigating the complex digital literacy space, often facing negative stereotypes or discrimination. This research identifies the efforts of millennial students to challenge negative narratives and build strong communities on Instagram. Thus, this research provides a deeper understanding of the representation of millennial students in digital literacy on Instagram. The implications of this research can be used to strengthen the role of millennial Santri in influencing public narratives and promoting a better understanding of Islam in the digital era.

The Support Vector Machine (SVM) is a powerful tool for classification. We generalize SVM to work with data objects that are naturally understood to be lying on curved manifolds, and not in the usual d-dimensional Euclidean space. Such data arise from medial representations (m-reps) in medical images, Diffusion Tensor-MRI (DT-MRI), diffeomorphisms, etc. Considering such data objects to be embedded in higher dimensional Euclidean space results in invalid projections (on the separating direction) while Kernel Embedding does not provide a natural separating direction. We use geodesic distances, defined on the manifold to formulate our methodology. This approach addresses the important issue of analyzing the change that accompanies the difference between groups by implicitly defining the notions of separating surface and separating direction on the manifold. The methods are applied in shape analysis with target data being m-reps of 3 dimensional medical images.