significant modification, so the chances of smoothing out important information in the name of rendering quality is reduced.

11.4 Volumetric Image Registration

An important field in medical imaging is registration, which is the process of aligning multiple images. The images may be of the same modality (e.g. MRI to MRI), known as intra-modality registration, or of different modalities (e.g. MRI to CT), known as inter-modality registration. They may be images of the same patient at different times, for example longitudinal studies looking at normal or pathological growth. Alternatively they may be images of different patients, for example group studies which attempt to quantify similarities and differences within a particular patient cohort. Registering images of different patients can also be useful in segmentation, as we will see in Sect. 11.5.2.1.

11.4.1 A Hierarchy of Transformations

The first consideration when describing the problem of registration is the form of the aligning transformation. How many degrees of freedom are appropriate? This will very much depend on the given situation. Are the images of the same patient? Is there likely to be change of shape, for example due to growth, either natural or due to disease or by deformation of the tissue.

Here we consider the most common transformations and they can be thought to exist in a hierarchy, ranging from simple transformations with few degrees of freedom to more general transformations with more degrees of freedom. An interesting point to note is that, as the generality of the transformation increases, fewer properties remain unchanged (e.g. lengths, angles). Such properties are known as invariance properties of the transformations. See [31] for further details on transformations.

11.4.1.1 Rigid Body Transformation

In aligning 3D images, the simplest form is the rigid body transformation. This is the same as a change of coordinate systems and consists of a simple translation, t = [tx , ty , tz]T , and rotation:

where det(R) = 1 and RT R = I and I is the identity matrix. In 3D there are three degrees of freedom for each making six in total. This can be represented by a matrix in homogeneous form:

Alternatively, the rotations can also be represented by quaternions. Note that all lengths and angles are invariants under a rigid body transformation and, therefore, properties derived from length and angle, such as volume, are also invariant.

11.4.1.2 Similarity Transformations and Anisotropic Scaling

Adding a further degree of freedom we have the similarity transformation. This allows for a uniform scale factor, s, in all dimensions (i.e. isotropic scaling), making 7 degrees of freedom in all. In a similarity transformation, angles and length ratios are invariant, but lengths are not. In effect, shape remains the same, but size changes.

In anisotropic scaling, the scaling factors, sx , sy and sz , along each of the axes x, y and z of the coordinate system can be different, providing 9 degrees of freedom. This can be easily expressed in homogeneous matrix form:

11.4.1.3 Affine Transformations

The affine transformation enables skew to be taken into account. These can be expressed as a linear skew of one axis in the direction of another; for example, Sx y. The skews Sx y and Sy x do not represent the same transformation so, at first sight, there are 6 skews. However, these are not independent and, again, 3 degrees of freedom are added. The overall 12 degrees of freedom for the affine transformation can be represented by a single homogeneous matrix:

which has 12 variables. In the general affine transformation, neither lengths nor angles are invariant, but parallelism is preserved.

11.4.1.4 Perspective Transformations

Finally there is the set of perspective projections, in which lines remain lines and any features that are collinear remain collinear but parallelism is not invariant:

The division by λ to find (x , y , z )T makes the transformation non-linear (in inhomogeneous coordinates) and 3 further degrees of freedom are added. These transformations are very rarely used for mapping 3D datasets to each other, but perspective projections are key to mapping 3D data onto 2D images, for example when aligning a 3D preoperative model to video or X-ray images. As a final observation note that, as we move to more general transformations, the more specific transformations are specialized subsets within these, i.e. perspective transformations include all affine transformations, which include all similarity transformations, which include all rigid body transformations.

11.4.1.5 Non-rigid Transformations

When moving into the realm of non-linear or non-rigid transformations there are a huge number of possibilities to choose from. It is also possible to use global polynomials, piecewise linear transformations over triangulations, or piecewise polynomials. Desirable functions for a non-linear mapping are that it is one-to-one, continuous (at least C1 and preferably in higher dimensions). When matching anatomical shapes, one of the earliest examples was the thin-plate spline [13], which is a radial basis function technique that can interpolate between given landmark points. It is highly dependent on the location and accuracy of the landmarks and is therefore very much a point-based technique.

As a more general scheme for describing non-linear transformations, the freeform deformation has proved very successful in medical imaging [71]. Here, the transformation is described by a cubic B-spline surface controlled by the position of a regular grid of node points. The cubic B-spline interpolation function is designed to approximate a Gaussian and has the following four basis functions:

B0(u) = (1 − u)3/6

B1(u) = 3u3 − 6u2 + 4 /6

B2(u) = −3u3 + 3u2 + 3u + 1 /6

B3(u) = u3/6.