Data Fusion Mathematics: Theory and Practice, 1st Edition (Paperback) book cover

Data Fusion Mathematics

Theory and Practice, 1st Edition

By Jitendra R. Raol

CRC Press

504 pages | 70 B/W Illus.

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Fills the Existing Gap of Mathematics for Data Fusion

Data fusion (DF) combines large amounts of information from a variety of sources and fuses this data algorithmically, logically and, if required intelligently, using artificial intelligence (AI). Also, known as sensor data fusion (SDF), the DF fusion system is an important component for use in various applications that include the monitoring of vehicles, aerospace systems, large-scale structures, and large industrial automation plants. Data Fusion Mathematics: Theory and Practice offers a comprehensive overview of data fusion, and provides a proper and adequate understanding of the basic mathematics directly related to DF. The material covered can be used for evaluation of the performances of any designed and developed DF systems. It tries to answer whether unified data fusion mathematics can evolve from various disparate mathematical concepts, and highlights mathematics that can add credibility to the data fusion process.

Focuses on Mathematical Tools That Use Data Fusion

This text explores the use of statistical/probabilistic signal/image processing, filtering, component analysis, image algebra, decision making, and neuro-FL–GA paradigms in studying, developing and validating data fusion processes (DFP). It covers major mathematical expressions, and formulae and equations as well as, where feasible, their derivations. It also discusses SDF concepts, DF models and architectures, aspects and methods of type 1 and 2 fuzzy logics, and related practical applications. In addition, the author covers soft computing paradigms that are finding increasing applications in multisensory DF approaches and applications.

This book:

  • Explores the use of interval type 2 fuzzy logic and ANFIS in DF
  • Covers the mathematical treatment of many types of filtering algorithms, target-tracking methods, and kinematic DF methods
  • Presents single and multi-sensor tracking and fusion mathematics
  • Considers specific DF architectures in the context of decentralized systems
  • Discusses information filtering, Bayesian approaches, several DF rules, image algebra and image fusion, decision fusion, and wireless sensor network (WSN) multimodality fusion

Data Fusion Mathematics: Theory and Practice incorporates concepts, processes, methods, and approaches in data fusion that can help you with integrating DF mathematics and achieving higher levels of fusion activity, and clarity of performance. This text is geared toward researchers, scientists, teachers and practicing engineers interested and working in the multisensor data fusion area.


"An application's guide to sensor fusion - Raol's comprehensive yet succinct handling of the mathematical fundamentals of sensor fusion make this a reference source for every practitioner."

—Ajith K. Gopal, The Council for Scientific and Industrial Research in South Africa

"… comprehensively presents tools for data fusions. Initial two chapters cover basic of data fusion and state estimations, especially Bayesian framework. The rest of chapters deal with advance topics that include fuzzy-logic based design, centralized and decentralized strategies, and image fusion. I feel the content of the book will useful both academia and industry."

—Dr. Mangal Kothari, Indian Institute of Technology Kanpur

Table of Contents

Introduction to Data Fusion Process

Data Fusion Aspects

Data Fusion Models

Sensor Data Fusion Configurations

Sensor Data Fusion Architectures

Data Fusion Process


Statistics, Probability Models and Reliability: Towards Probabilistic Data Fusion



Probability Models

Probabilistic Methods for DF

Reliability in DF

Information Methods

Probability Concepts for Expert System and DF

Probabilistic Methods for DF: Theoretical Examples

Bayesian Formula and Sensor/DF: Illustrative Example


Fuzzy Logic and Possibility Theory-Based Fusion


Fuzzy Logic Type I

Adaptive Neuro-fuzzy Inference System

Fuzzy Logic Type

Fuzzy Intelligent Sensor Fusion

FL-based Procedure for Generating the Weights for a DF Rule

FL-ANFIS for Parameter Estimation and Generation of DF Weights: Illustrative Examples

Possibility Theory

Fusion of Long-Wave IR and EOT Images Using Type 1 and Type 2 Fuzzy Logics: Illustrative Examples

DF Using Dempster-Shafer and Possibility Theory: Illustrative Example

A: Type 1 - Triangular MF-MATLAB Code

B: Type 2 - Gaussian MF-MATLAB Code

Fuzzy Inference Calculations - MATLAB Code


Filtering, Target Tracking and Kinematic Data Fusion


The Kalman Filter

The Multi-Sensor Data Fusion and Kalman Filter

Non-linear Data Fusion Methods

Data Association in MS Systems

Information Filtering

HI Filtering-Based DF

Optimal Filtering for Data Fusion with Missing Measurements

Factorisation Filtering and Sensor DF: Illustrative Example


Decentralised Data Fusion Systems


Data Fusion Architectures

Decentralised Estimation and Fusion

Decentralised Multi-Target Tracking

Millman's Formulae in Sensor Data Fusion

SRIF for Data Fusion in Decentralised Network with Four Sensor Nodes: Illustrative Example


Component Analysis and Data Fusion


Independent Component Analysis

An Approach to Image Fusion Using ICA Bases

Principal Component Analysis

Discrete-Cosine Transform

WT: A Brief Theory

An Approach to Image Fusion Using ICA and Wavelets

Non-Linear ICA and PCA

Image Fusion Using MR Singular Value Decomposition


Image Algebra and Image Fusion

S. Sethu Selvi


Image Algebra

Pixels and Features of an Image

Inverse Image

Red, Green and Blue, Grey Images and Histograms

Image Segmentation

Noise Processes in an Observed/Acquired Image

Image Feature Extraction Methods

Image Transformation and Filtering Approaches

Image Fusion Mathematics

Image Fusion Algorithms

Performance Evaluation

Multimodal Biometric Systems and Fusion: Illustrative Examples


Decision Theory and Fusion


Loss and Utility Functions

Bayesian DT

Decision Making with Multiple Information Sources

Fuzzy Modelling Approach for Decision Analysis/Fusion

Fuzzy-Evolutive Integral Approach

Decision Making Based on Voting

DeF Using FL for Aviation Scenarios

DeF Strategies

SA with FL and DeF for Aviation Scenarios: Illustrative Examples


Wireless Sensor Networks and Multimodal Data Fusion


Communication Networks and Their Topologies in WSNs

Sensor/Wireless Sensor Networks

Wireless Sensor Networks and Architectures

Sensor Data Fusion in WSN

Multimodality Sensor Fusion

Decision Fusion Rules in WSN

Data Aggregation in WSN

Hybrid Data and Decision Fusion in WSN

Optimal Decision Fusion in WSN


Soft Computing Approaches to Data Fusion


Artificial Neural Networks

Radial Basis Function Neural Network

Recurrent Neural Networks

FL and Systems as SC Paradigm

FL in Kalman Filter for Image-Centroid Tracking: A Type of Fusion

Genetic Algorithms

SDF Approaches Using SC Methods: Illustrative Examples

Machine Learning

Neural-Fuzzy-Genetic Algorithm Fusion

Image Analysis Using ANFIS: Illustrative Example



A: Some Algorithms and/or Their Derivations

B: Other Methods of DF and Fusion Performance Evaluation Metrics

C:Automatic Data Fusion

D: Notes and Information on Data Fusion Software Tools

E: Definitions of Sensor DF in Literature

F: Some Current Research Topics in DF

About the Author

Jitendra R. Raol received a BE and ME in electrical engineering from the MS University of Baroda, Vadodara in 1971 and 1973, respectively, and a PhD (in electrical and computer engineering) from McMaster University, Hamilton, Canada in 1986. He taught for two years at the MS University of Baroda before joining the National Aeronautical Laboratory in 1975. He retired in 2007 as Scientist G and head, flight mechanics and control division at CSIR-NAL. His main research interests are DF, system identification, state/parameter estimation, flight mechanics–flight data analysis, H-infinity filtering, ANNs, fuzzy systems, genetic algorithms, and soft technologies for robotics.

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