next up previous
Next: Transformation Space Up: No Title Previous: Introduction

   
Mathematical Formulation

The standard registration problem is to find a transformation that best aligns a reference image Ir to another (floating) image If. In this context, an image defines the intensity for each spatial location. That is, it is a mapping of positions to intensity values; $I : {\mathcal R}^3 \rightarrow {\mathcal R}$. Given this definition, registration is formulated as a mathematical problem by taking a cost function, C(I1,I2), that quantifies the quality of the registration and then finding the transformation T* which gives the minimum cost:

\begin{displaymath}T^* = \arg\min_{T \in S_T} C(I^r,I^f \circ T)
\end{displaymath} (1)

where ST is the space of allowable transformations.



 

Mark Jenkinson
2000-05-10