Multiset Combinatorial Gray Codes with Application to Proximity Sensor Networks

TL;DR

This paper introduces multiset combinatorial Gray codes for source coding in proximity sensor networks, focusing on efficient object tracking by leveraging grid-based codes and color mappings.

Contribution

It proposes a novel multiset Gray code framework, including product codes for high-dimensional grids, and demonstrates their application in object tracking over 2D sensor networks.

Findings

Efficient coding schemes for proximity sensor networks.

Construction of color multiset codes on 1D and 2D grids.

Numerical results showing improved tracking performance.

Abstract

We investigate coding schemes that map source symbols into multisets of an alphabet set. Such a formulation of source coding is an alternative approach to the traditional framework and is inspired by an object tracking problem over proximity sensor networks. We define a \textit{multiset combinatorial Gray code} as a mulitset code with fixed multiset cardinality that possesses combinatorial Gray code characteristic. For source codes that are organized as a grid, namely an integer lattice, we propose a solution by first constructing a mapping from the grid to the alphabet set, the codes are then defined as the images of rectangular blocks in the grid of fixed dimensions. We refer to the mapping as a \textit{color mapping} and the code as a \textit{color multiset code}. We propose the idea of product multiset code that enables us to construct codes for high dimensional grids based on 1-dimensional (1D) grids. We provide a detailed analysis of color multiset codes on 1D grids, focusing on codes that require the minimal number of colors. To illustrate the application of such a coding scheme, we consider an object tracking problem on 2D grids and show its efficiency, which comes from exploiting transmission parallelism. Some numerical results are presented to conclude the paper.