Image registration ordinarily involves the manipulation of image pixels
according to some rules and under the imposition of several strict
constraints. It is usually desirable to obtain a maximum similarity
measure amongst a group of images with a minimal extent of distortion.
Even a small level of distortion may induce wrong assumptions or violate
some stern conditions which should otherwise be an unbreakable prerequisite.
It is possible to think of the transformations used as if they pertain
to different levels of ``interference'' - the interference to
the analysis and interference with the integrity of the data. A typical
categorisation of transformation types is as follows (ordered by increasing
interference)^{20}:

**Rigid**- Allows scaling (uniformal size changes, i.e. shrinkage and
enlargement), translation (location in space) and rotation. The normalised
shape attributes are altogether preserved and the process is usually
concerned merely with some common alignment and bias-neutralisation
*.*Such alignment usually aims to place all instances upright and centred in the space origin with a fixed size of maximum 1 unit. The instance is virtually confined to lie inside a bounding structure (a square, or sphere in 2-D and 3-D respectively)^{21}. In 3-D, for instance, there is a total of 6 degrees of freedom so a rigid transformation will be wholly characterised by a tuple of 6 values^{22}. **Affine**- Allows the instance (image) to
*stretch*along at least one axis or dimension, but not necessarily all (so that homogeneous scaling can be broken). Despite the fact that previously essential constraints are broken, all lines that were parallel remain parallel after the transformation is applied^{23}. Reconstruction is said to be possible so that this transformation is invertible. For all affine transformations where can be a vector representation of an image (or volume) and their inverse , the expression must hold. This relation must always be calculable and retain simplicity which makes it easy to resolve. This will prove to be an important constraint when the practicability of warps is debated. **Non-rigid**- All other valid transformations fall into this category.
This includes tapering, spiral warps, pinching, etc. In principle,
no inviolable constraints are in place, but quite clearly the transformation
attempts to preserve some of the primary structure
^{24}of the image while avoiding tearing and folding [27]. This means that each pixel in the range must map to another and no pixel is left undefined. A bit more on this is to be explained later.

Figure 5: Registration examples; from top to bottom:
rigid, affine and non-rigid registration.

As the figure suggests, the appearance of an object remains identical under rigid transformations. It is allowed strictly to grow, shrink, move, and rotate. Affine transformation allows an object to lose its original form and non-rigid registration is far more permissive so the object can be subjected to rather obscure deformations.

What follows in this section briefly explains some of the main concepts,
techniques and ideas currently devised. The actual key points, which
describe non-rigid registration in the context of the Structure and
Function Grand Challenge^{25}, are as follows:

- Warps
- Similarity
- Objective function