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Representing Image Deformation

There are various image deformation methods which produce a full transformation that affects an entire image plane or volume [,,]. Sometimes, the transformations (also known as ``warps'') are also subjected to conditions that ensure they remain valid. A transformation might, for example, be carried out under the imposition of strict constraints. These prevent the transformation from collapsing upon itself - something which leads to problems that are explained later.

Transformations can be subdivided into different types. Each type of transformation has a different effect on the data it is applied to and a typical classification of transformation types is as follows (ordered by increasing level of versatility):

Euclidean Transformation. Permits translation (relocation in space), rotation, and scaling^2.1. In space, normalised shape attributes are preserved, so it generally provides an approximate alignment. Such alignment is ordinarily intended to position all data instances (images or volumes) upright and centred at the origin of a space, with a fixed size of 1 unit at most. All images are confined to lie inside a bounding structure (for example, a circle in 2-D, or a sphere in 3-D). In 3-D, for instance, there is a total of 7 degrees of freedom so a rigid transformation [] will be wholly characterised by a tuple of 7 parameters (scale included).

 

Affine Transformation. Allows an image to stretch and skew along at least one axis (corresponding to a single dimension). Despite the fact that dimensions can be rescaled at different magnitudes, lines which were parallel before an affine transformation is applied will remain parallel after this transformation is applied^2.2. This transformation is also invertible. Essentially, for a given affine transformation $T_{a}(x)$, where $x$ is a vectorised representation of an image (or volume), and its inverse $T_{a}^{-1}(x)$, the expression $T_{a}(T_{a}^{-1}(x))=Id(x)$, where $Id(x)$ is the identity operator, must hold as valid. It retains a level of simplicity, which makes it easy to determine and resolve. This proves to be an important constraint when the practicability of warps is further debated.

Non-rigid Transformation. There is a large variety of ways of transforming images more flexibly. They do not comply with the constraints imposed by Euclidean and affine transformations. Among the methods for transforming biomedical images, there is a fluid flow framework, which models deformations in terms of the flow of a fluid throughout the image region []. Hence, it potentially allows large-scale deformations. Clamped-plate splines [], on the other hand, make use of the composition of rounded [] and localised warps. When these warps are aggregated, they facilitate flexible deformation at multiple scales. Thin-plate splines [] are another way of formulating the transformation of pixels. Another common approach, for instance, models the motion of organs, which can be handled using free-form deformations (FFDs). These can be based on B-splines [].

The images of an apple in Figure illustrate the effect that each transformation type is permitted to have on the original image shown on the left.

Figure: Transformation examples. On the left column: the original, unwarped image; on the right column, from top to bottom: rigid, affine, and non-rigid transformations.

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