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Image registration is an important process for removing motion artefacts and incorporating multimodal image analysis in this study. The registration aligns an image to a template of the same object acquired, either at different times, from different scanning directions or by different imaging devices (115). Basic components of registration algorithms comprise three parts: 1) transformation, 2) similarity measures and cost function, and 3) optimisation (116, 117), as described in the following sections.

Transformations 3.2.3.1

Transformation specifies a mapping function in order to overlay a floating image (𝑋) on a reference image (𝑌) geometrically. The transformation model provides two main operations: control the spatial mapping and interpolate missing features between reference and floating images (116). The complexity of transformation in terms of elasticity, reliability and computational load is explained by the degrees of freedom (DOF). DOF is the number of independent transformation in a specified mapping function. For example, translation in 3D space encompasses 3 independent translation, which is equivalent to 3 DOF transformation model (117). The transformation model can be classified into three main types: rigid, affine and non-linear transformation.

Rigid transformations are linear operations, which preserve the shape and size of an object. These techniques are suitable for intramodal imaging data of the same subject that has no distortion and no anatomical changes (118). A rigid transformation comprises 6 DOF: three rotations and three translations, and provides the fastest calculation.

Affine transformations offer higher DOF of linear transformations than rigid transformations. They allow linear coordinate changes up to 12 DOF including rotation, translation, scaling and skews (shear). Affine transformations are typically used as an initialisation to nonrigid transformations, either for primary approximation of location, or compensation for geometric image distortions affected by eddy currents in diffusion MRI, for instance (119).

Nonrigid, non-linear, deformable or elastic transformations are more commonly used for images of different subjects, images with distortions during acquisition or images with actual physical differences such as biological changes (116). The non-linear transformations have at least 12 to millions of DOF. The choice of DOF affects the computational load and the accuracy of the registration. Generally a low DOF or affine transformations are more useful than a very high DOF because they are less sensitive to artefacts and poor image quality, hence a more robust model. A high DOF may be more beneficial for an image, having moderate to high resolution with good contrast and little to no artefacts (117).

An interpolation is considered as part of the transformation process and is used to fill the missing values between the reference and floating images. The interpolation method is calculated by convolving the image data with a continuous kernel. The kernel functions, which have been commonly used, are for example trilinear, spline and sinc kernels. Sinc interpolation was showed to provide more accuracy than the trilinear approach, as recommended by Jenkinson et al. (120). Therefore sinc interpolation is used with a default window width 7. The optimisation adopts a multi-resolution approach with a local

optimisation method (95, 121, 122). Two methods for dealing with the local minima problem are cost function apodization, which is used to reduce or eliminate small-scale dips, and the hybrid global-local optimisation technique. The details of the optimisation are beyond the scope of this thesis.

Similarity and Cost Functions 3.2.3.2

Similarity and cost functions measure the quality of a transformation model, orientating two images on the same spatial location. Cost functions can be broadly divided to geometric- and intensity-based cost function. Intensity-based cost functions offer higher accuracy and reliability than geometric-based cost functions (120). Intensity-based cost functions can be categorised by intramodal and intermodal registration. For registration of intramodal imaging, the most commonly used functions are least squares and normalised correlation. For registration of intermodal imaging, the most commonly used functions are woods, mutual information, normalised mutual information, and correlation ratio.

In FLIRT, the correlation ratio (123) is a default similarity function, while normalised mutual information is an alternative cost function. The correlation ratio (CR) and normalised mutual information (NMI) are mathematically defined in equation (3.1) and (3.2) respectively. 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛⁡𝑅𝑎𝑡𝑖𝑜 = ⁡ 1 𝑉𝑎𝑟(𝐼_𝐵)⁡∑ 𝑛_𝑖 𝑁𝑉𝑎𝑟(𝐼𝐵𝑖) 𝑖 (3.1) 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑠𝑒𝑑⁡𝑀𝑢𝑡𝑢𝑎𝑙⁡𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 = ⁡ 𝐻(𝐼𝐴, 𝐼𝐵) 𝐻(𝐼_𝐴)⁡+ ⁡𝐻(𝐼_𝐵)⁡ (3.2)

where 𝐼𝐴 and 𝐼𝐵 represent a set of image intensity of image 𝐴 and 𝐵; 𝐼𝐵𝑖 is the 𝑖𝑡ℎ set of intensities of image 𝐼𝐵 at positions where the intensity in 𝐼𝐴 is the⁡𝑖𝑡ℎ intensity bin; 𝑉𝑎𝑟(𝐼𝐵𝑖) is the variance of the 𝐼𝐵 in area 𝑖; 𝑉𝑎𝑟(𝐼𝐵) is the variance of set 𝐼𝐵; 𝑁 = ∑ 𝑛𝑖 𝑖⁡; 𝑛_𝑖⁡ is the number of elements in the set 𝐼𝐵𝑖; 𝐻(𝐼𝐴, 𝐼𝐵) = ⁡ − ∑

𝑛𝑖𝑗 𝑁 𝑙𝑜𝑔 (

𝑛𝑖𝑗 𝑁)⁡

𝑖𝑗 ⁡ where 𝑛𝑖𝑗 is the number of voxels that are assigned to the bin pair (𝑖, 𝑗); The marginal entropies 𝐻(𝐼𝐴) and 𝐻(𝐼𝐵) are defined similarly, but using the individual image histograms rather than the joint histogram.

Optimisation 3.2.3.3

After a cost function is chosen, the transformation (𝑇∗) is optimised by minimising the cost function as mathematically defined by equation (3.3) (120).

𝑇∗_{= arg min}

T∈STC(𝐼𝐵, T(𝐼𝐴)) (3.3)

where 𝑆𝑇 is the transformable space,⁡𝐼𝐵 is the reference images, 𝐼𝐴 is the floating images, C(𝐼_𝐵, T(𝐼_𝐴)) is the cost function and T(𝐼_𝐴) represents the transformed image 𝐼_𝐴 by the transformation⁡T.

Intensity Normalisation