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Figure 2. Examples of fine scale displacement fields used in considering the effect of spatial blur: (a) uniform translation, (b) radial expansion, and (c) counter-clockwise rotation.

to represent a uniform rotation exactly, the displacement field would ideally be defined with infinitesimally small pixels. The poorer the resolution, the coarser is the approximation to uniform rotation.

In inter-modality and intra-modality applications, where images are acquired with different resolution, the undesirable effects on the displacement field may be reduced by blurring the higher resolution image to match the resolution of the other image although noise reduction is also an important consideration.20 In practice, a number of registration algorithms adopt a multi-scale approach, which allows for controlled blurring of both images. Note, however, that multi-scale properties of the displacement field are less well understood than those of the image itself. Resolution effects are further complicated by the fact that imaging systems are, at best, only approximately shift invariant and the resolution varies from one part of the imaged volume to another. An example where the approximation breaks down is the position-dependent resolution in SPECT. Where possible, such effects should be corrected prior to the application of the registration algorithm.

Temporal blur occurs where the duration of image acquisition is of the order of, or longer than, the time constant of any movement of tissues. In modalities such as CT and fast MRI, acquisition can be completed with breath hold, so that only cardiac motion contributes significantly to data blurring (non-blur artefacts may also occur). In slower acquisition modalities (e.g. SPECT), respiratory motion, peristaltic intestinal motion, and involuntary patient's motion, in addition to cardiac motion, can all contribute to temporal blur. The effect on the registration process depends on the application. In intra-modality problems, where the temporal effects in both images are similar, the principal consequence is that the desired deformation is defined with greater uncertainty, leading to a possible diminution of registration accuracy. In inter-modality applications, the temporal blurring may be very different in the two images, impeding the registration process. Unlike the case of spatial blur, we cannot bring the two images to a common temporal scale with a uniform spatial blur.

Noise can have a significant effect in that the random fluctuation of intensity values may confuse the registration algorithm. Nuclear medicine images are generally noisier than those obtained with other modalities and this can be a significant factor in registration problems. An additional complication is that the noise level in those images depends on local mean intensity.

Thus the aim of the registration algorithm is to determine the relative deformation between image domains given images g1(x,y,z) and g2(x,y,z), each subject to the corrupting influences of contrast generation, spatial and temporal blur, as well as noise.

3. Generic Registration Algorithm

Almost all registration algorithms follow the iterative scheme shown in Figure 3. One can distinguish between the deformable floating image and the fixed reference image. The floating image is first deformed using some initial values of the deformation parameters. The core of the algorithm is the evaluation of the similarity between the deformed floating image and the reference image. An iterative optimization algorithm is employed to find the deformation that maximizes similarity. The output is the optimum deformation defined by a set of parameters. The similarity measure may be evaluated for a part of the image at a time, instead of the entire image.

4. Types of Deformation

So far, the deformation U was represented by a displacement vector field, where deformation at each point r in the image is given by Dr = (Dx, Dy, Dz)T. There are two major issues arising from this representation:

Reference image

Floating image

Initialize deformation

Deform volume

Evaluate similarity measure

Optimum deformation parameters

Update deformation parameters

Optimum deformation parameters

Figure 3. A flowchart for image registration.

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