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Image registration ordinarily involves the manipulation of image pixels
according to some rules and under the imposition of several strict
constraints. It is commonly desirable to obtain a maximum similarity
estimation [] or simply an overlap 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 pre-requisite.
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)3.1:
- Rigid
- Allows translation (relocation in space), rotation and scaling
(uniformal size changes, i.e. shrinkage and enlargement)3.2. The normalised shape attributes are altogether preserved and the
process is usually concerned merely with some common alignment.
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 circle
or sphere, in 2-D and 3-D respectively)3.3. 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 values3.4. This does in fact fully describe a rigid transform.
- Affine
- Allows the instance (image for example) to stretch
and skew 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 applied3.5. 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 a non-rigid transformation attempts to preserve some of the
primary structure3.6 of the image while avoiding tearing and folding [,].
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.
The images of an apple in Figure
illustrates the effect that each transformation type has on the image
on the left.
Figure:
Registration examples;
from top to bottom: rigid, affine and non-rigid transformations.
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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.
Figure:
CPS non-rigid warp example.
Warp is shown on the right-hand side.
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What follows in this chapter briefly explains some of the main concepts,
techniques and ideas currently employed. The actual key points, which
describe non-rigid registration in the context of current investigative
work on the issue3.7, are as follows:
- Warps
- Similarity
- Objective function
These three points will be explained in more detail with reference
to current work, practical considerations and attempts already made.
For now, a concise introduction would do. The approach often taken
is that an image3.8 needs to be warped (equivalent to transformation) until it matches
another. The match is estimated with the assistance of similarity
measures and this process of warping and similarity is sometimes wrapped
up and put under one generic objective function. In that sense, the
objective function bridges warps and similarity. Objective functions
are then handled by an optimiser - a term which is further explained
in sec:Optimisation.
Next: Diffeomorphism
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2004-08-02