Topic
Adjacency list
About: Adjacency list is a research topic. Over the lifetime, 4419 publications have been published within this topic receiving 78449 citations.
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TL;DR: A supervised dimensionality method DAG-DNE is proposed for classification tasks, which keeps the local structure of data and finds a good projection matrix for them and compares it with the state-of-the-art dimensionality reduction techniques such as DNE and MFA on several publicly available datasets.
38 citations
TL;DR: This chapter presents the advanced concept of image analysis: a generalization of the subgraph isomorphism problem based on the notion of a cell list, which represents a segmented image as a cell complex.
Abstract: Publisher Summary Problems of image analysis and image understanding are closely related to geometrical notions, such as shape or size, and also to some topological notions, such as boundaries and connectivity of subsets The boundary surface of a three-dimensional region is defined as a set of faces: space elements separating two adjacent voxels (volume elements) from each other Consistent topological concept for digital images must include space elements of various kinds; this chapter verifies this concept It is shown that the resolution of these problems consists in considering the digital plane as a finite topological space in full accordance with topological axioms It is also shown that the most suitable for practical purposes is the particular case of finite topological spaces, known as “abstract cell complexes” The chapter introduces important notions of Cartesian complexes and coordinates In the chapter, a definition of boundaries of subcomplexes is given and the advantages of this concept as compared to boundaries in adjacency graphs are discussed The chapter also discusses the applications of finite topology to image analysis—tracking and filling of boundaries and thinning of regions—and a new topologically founded data structure—the cell list, which represents a segmented image as a cell complex The chapter also presents the advanced concept of image analysis: a generalization of the subgraph isomorphism problem based on the notion of a cell list
38 citations
31 Jul 2005
TL;DR: This chapter provides an implementation of a DEC-friendly tetrahedral mesh data structure in C++ and documents the ideas behind the implementation.
Abstract: The methods of Discrete Exterior Calculus (DEC) have given birth to many new algorithms applicable to areas such as fluid simulation, deformable body simulation, and others. Despite the (possibly intimidating) mathematical theory that went into deriving these algorithms, in the end they lead to simple, elegant, and straightforward implementations. However, readers interested in implementing them should note that the algorithms presume the existence of a suitable simplicial complex data structure. Such a data structure needs to support local traversal of elements, adjacency information for all dimensions of simplices, a notion of a dual mesh, and all simplices must be oriented. Unfortunately, most publicly available tetrahedral mesh libraries provide only unoriented representations with little more than vertex-tet adjacency information (while we need vertex-edge, edge-triangle, edge-tet, etc.). For those eager to implement and build on the algorithms presented in this course without having to worry about these details, we provide an implementation of a DEC-friendly tetrahedral mesh data structure in C++. This chapter documents the ideas behind the implementation.
37 citations
DOI•
07 Dec 2017
TL;DR: This document provides the theoretical basis that hides behind the modules of MAJA processor, which is applicable to time series of Sentinel-2, Landsat, Venµs, and Formosat satellites.
Abstract: This document provides the theoretical basis that hides behind the modules of MAJA processor. MAJA stands
for MACCS-ATCOR Joint Algorithm, where MACCS was the Multi-Temporal Atmospheric Correction and
Cloud Screening software, developed by CNES and CESBIO, and ATCOR is the Atmospheric Correction soft-
ware developed by DLR. MAJA is based on MACCS architecture and includes modules that come from AT-
COR.
The MAJA processor is applicable to time series of Sentinel-2, Landsat, Venµs, and Formosat satellites.
This Algorithmic Theoretical Basis Document (ATBD) provides a scientific description of the methods used
within MAJA and some justifications of the choices made, as well as basic validation results.
37 citations
01 Jan 1992
TL;DR: In this article, the first eigenvalue of the Laplacian on a connected regular graph is compared to the size of the kernel of the universal covering map, and the main results have been proven in (C, G, P).
Abstract: Let & be a ?-regular graph and T the covering tree of S. We define a cogrowth constant of & in T and express it in terms of the first eigenvalue of the Laplacian on S. As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Lapla- cian on 9 is zero. Grigorchuk's criterion for amenability of finitely generated groups follows. In this note, we shall relate the first eigenvalue of the Laplacian on a connected regular graph to the size of the kernel of the universal covering map. The main results have been proven in (C, G, P). The proof presented here appears simpler; it depends on the explicit formula for minimal positive solutions of AF + eF = -I. Let f be a connected simple graph with constant vertex degree d > 3, F be the universal covering tree of "§, and 6 the covering map (i.e., 6 is a vertex surjection of T on ^ that preserves adjacency and vertex degree). We let T and & denote the vertex sets of the corresponding graphs. Note that F has constant vertex degree d. Since F is connected, F may be considered a metric space with the usual graph metric a (S(x, y) is the length of the shortest path connecting x and y). For x £ T and zz > 0, let (x) = 6~x(6(x)) and Sn(x) = {y: S(x, y) = n} . For x, y £ T, note that
37 citations