Map Making from Tables

Abstract

This paper presents a geometric approach to the coordinatization of a measured space called the Map Maker's algorithm. The measured space is defined by a distance matrix for sites which are reordered and mapped to points in a two-dimensional Euclidean space. The algorithm is tested on distance matrices created from 2D random point sets and the resulting coordinatizations compared with the original point sets for confirmation. Tolerance levels are set to deal with the cumulative numerical errors in the processing of the algorithm. The final point sets are found to be the same apart from translations, reflections and rotations as expected. The algorithm also serves as a method for projecting higher dimensional data to 2D.

The N-Dimensional Map Maker Algorithm

“…The Map Maker algorithm [1] serves a fundamental purpose. Given geodetic survey data on N sites P i for i = 1 to N in the form of an NxN distance matrix D the algorithm generates 2D coordinates for each point i.e.…”

Source Signal Strength Independent Localization

“…ψ is the RSSI of the source at unit distance from S. It will be assumed that Having found the position (x S ,y S ) of the radio source by equations (2), (3), (4), (5) and (6), its strength at the source is found by:…”

“…To accurately determine the position of a radio source S all other radio sources (at the given frequency) must be turned off and then the strength of the radio signal from S is measured at the 5 antennas as 1 (3) and then x S and y S from equation (6). Then compute r 1 from equation (8) Then the radio source is located at S = (x S ,y S ) with signal intensity 0 ψ .…”

“…If the distances between all nodes in the network can be measured by one of these techniques then a distance matrix can be formed and using the Map Maker algorithm [6] the coordinates of all the nodes are known up to a system translation, rotation or inversion. The hardware of the location detector needs to be small, portable, low cost and unobtrusive and yet also sufficiently accurate in determining positions [3].…”

Multi-Antenna Localization Method

“…The distance between sites is determined by RSSI measurements on the assumption that the signal intensity at the source is a known constant. In another work [2] a method is given for computing the positions of sites when the distance matrix (giving the distances between all sites) is known. Triangulation on the other hand involves the use of directional antennas at points A and B which each point to the radio source at C the third point of a triangle whose position is then computed trigonometrically.…”