Chapter 3. Statistical Description of Images

  1. Maria Petrou1 and
  2. Panagiota Bosdogianni2

Published Online: 17 OCT 2001

DOI: 10.1002/0470841907.ch3

Image Processing: The Fundamentals

Image Processing: The Fundamentals

How to Cite

Petrou, M. and Bosdogianni, P. (2001) Statistical Description of Images, in Image Processing: The Fundamentals, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/0470841907.ch3

Author Information

  1. 1

    University of Surrey, Guildford, UK

  2. 2

    Technical University of Crete, Chania, Greece

Publication History

  1. Published Online: 17 OCT 2001

ISBN Information

Print ISBN: 9780471998839

Online ISBN: 9780470841907



  • ergodicity;
  • random field;
  • correlation matrix;
  • Karhunen–Loeve transform;
  • least mean square error approximation


This chapter gives the basic theory behind the description of an image as a random field. All fundamental concepts necessary to define a random field are explained, starting from the concept of a random variable. It includes a detailed analysis on the assumption of ergodicity, what it means and when it is valid. It defines Karhunen–Loeve transform and gives worked out examples with step by step calculation of the correlation or covariance matrix of an image, its Karhunen–Loeve transform and the basis images in terms of which Karhunen–Loeve transform expands it. Everything is demonstrated with a help of small 3 by 3 or 4 by 4 images where all steps of the analysis are shown. Results of truncating the Karhunen–Loeve transform are compared with those of chapter 2 using the same 8 by 8 grey image.