# 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

#### 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

## SEARCH

### 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