7. Geometry

  1. Chris Solomon1 and
  2. Toby Breckon2

Published Online: 5 JAN 2011

DOI: 10.1002/9780470689776.ch7

Fundamentals of Digital Image Processing: A Practical Approach with Examples in Matlab

Fundamentals of Digital Image Processing: A Practical Approach with Examples in Matlab

How to Cite

Solomon, C. and Breckon, T. (2010) Geometry, in Fundamentals of Digital Image Processing: A Practical Approach with Examples in Matlab, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/9780470689776.ch7

Author Information

  1. 1

    School of Physical Sciences, University of Kent, Canterbury, UK

  2. 2

    School of Engineering, Cranfield University, Bedfordshire, UK

Publication History

  1. Published Online: 5 JAN 2011
  2. Published Print: 22 DEC 2010

ISBN Information

Print ISBN: 9780470844724

Online ISBN: 9780470689776

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Keywords:

  • geometry - need for geometric manipulation of images, in image processing applications;
  • concept of shape - implying existence of some boundary;
  • translating an object from one place to another - rotating it or scaling it, operations changing object's shape vector coordinates but not changing its shape;
  • shape transformation and homogeneous coordinates – shape, preserved under linear operations of scaling, translation and rotation;
  • point distribution matrix (PDM);
  • affine transformation in 2-D space - with six free parameters;
  • affine transformation - in homogeneous coordinates;
  • Procrustes transformation matrix - a special case of general affine transformation;
  • projective transform, how one arbitrary quadrilateral in object plane - maps into another quadrilateral in image plane;
  • mapping of intensity values - from one spatial distribution into another spatial distribution, known as image warping

Summary

This chapter contains sections titled:

  • The description of shape

  • Shape-preserving transformations

  • Shape transformation and homogeneous coordinates

  • The general 2-D affine transformation

  • Affine transformation in homogeneous coordinates

  • The Procrustes transformation

  • Procrustes alignment

  • The projective transform

  • Nonlinear transformations

  • Warping: the spatial transformation of an image

  • Overdetermined spatial transformations

  • The piecewise warp

  • The piecewise affine warp

  • Warping: forward and reverse mapping