Chapter 2. Image Transformations

  1. Maria Petrou1 and
  2. Panagiota Bosdogianni2

Published Online: 17 OCT 2001

DOI: 10.1002/0470841907.ch2

Image Processing: The Fundamentals

Image Processing: The Fundamentals

How to Cite

Petrou, M. and Bosdogianni, P. (2001) Image Transformations, in Image Processing: The Fundamentals, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/0470841907.ch2

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



  • singular value decomposition;
  • Walsh transform;
  • Haar transform;
  • Hadamard transform;
  • Fourier transform;
  • basis images;
  • least square error;
  • approximation


This chapter defines in detail the various linear transformations one may apply to an image: singular value decomposition, Walsh, Haar, Hadamard and Fourier. It explains their relationship with the corresponding sets of orthonormal 1D basis functions and with the orthogonal set of basis images in terms of which they expand an image. The application of each transform is formulated in terms of matrix multiplications. All transforms are demonstrated using small images consisting of 3 by 3 or 4 by 4 pixels so that the reader can go through the operations applied in detail. All transforms and the image reconstruction resulting from the truncation of their coefficients are compared and demonstrated by using the same 8 by 8 grey image.