## Registration

Registration denotes the proces of aligning geometric objects, e.g. images, points, lines, surfaces, and signals. The summer school on registration aims to enable the participants to work with registration in computer graphics, image and signal analysis, and be able to

- understand 2D and 3D image registration in the above areas,
- understand numerical schemes for representation and optimization in image registration,
- use several similarity measures for image co-registrration,
- use several linear and non-rigid image transformations for image co-registration,
- use relevant optimization schemes for image co-registration,
- use image registration in their research and choose relevant image registration methods to solve their problem,
- have and overview of the use of image registration methods within medical image analysis.

The general setup in registration covers many different types of geometric object but, in order to be concrete, we describe the particular case of image registration below.

### Image Registration:

In image registration, a moving image I_{m} is transformed in order to match a fixed
image I_{f}. If the domain of both images I_{m} and
I_{f} is denoted Ω, the registration finds a suitable map
φ:Ω → Ω so that I_{m}⚪φ is
close to I_{f}. Registration requires a *deformation model* to describe
and represent the map φ and a *similarity measure* to measure the closeness
between the transformed moving image I_{m}⚪φ and I_{f}. In addition, the
registration problem is most often ill-posed in the sense that multiple φ
exist that can bring I_{m} suitably close to I_{f}, and a
choice between these candidate φ's must be made. This is performed by a
regularization scheme.

### Applications:

Image registration plays a fundamental role in a range of applications. Examples include alignment of medical images and image based morphometry, alignment of satellite images in geographical information systems, and alignment of astronomical images.

### Variational Formulation:

The combination of regularization and similarity often leads to the registration problem
being formulated as an energy minimizing tasks: If we let R(φ) denote the reqularization measure,
let S(I_{m}⚪φ,I_{f}) be the similarity measure, and let λ be a weight,
the optimal φ can be found by minimizing

E(φ) = R(φ) + λS(I_{m}⚪φ,I_{f}) .

### Degrees of Freedom:

There exists different choices for the allowed range of deformations φ. Most
often an *affine* or *rigid* registration is used to remove the
effect of rotation, translation, and, in the affine case, scaling. A closer
match can then be obtained by extending the set of admissble φ and hence allowing
more freedom in the transformation. These *non-rigid* transformations have
more degrees of freedom, from many to infinite, and the require in particular
a computational representation.

### Deformation Models and Computational Representations:

Various ways of representing φ exist, each with pros and cons in terms of smoothing properties, computational complexity, and numerical stability. Among the most widely used are B-spline basis functions that interpolates between a finite number of control points distributed over the image domain Ω. The smoothness of B-splines and the finite number of control points provide implicit regularization, and B-splines are computationally practical.

A more mathically oriented deformation model is the *large deformation
diffeomorphic metric mapping* framework (LDDMM) that models φ as the
solution of a differential equation. This provides the
registration setup with a Riemannian manifold structure and a solid
mathematical foundation.

A range of other methods are in use, to many to be listed here: various flavours of the Demons algorithm, fluid registration methods, etc.

### Image Similarities:

Perhaps the simplest and most widely used measure for similarity between images is the
sum of squared errors:

S(I_{m}⚪φ,I_{f})
=
Σ_{Ω}|I_{m}⚪φ(x)-I_{f}(x)|^{2}dx .

SSE can be problematic in several ways, for example if the images are illuminated differently.
Statistical measures such as cross correlation, mutual information, and normalized mutual information
seek to avoid some of the deficiencies of SSE.