Image Processing Techniques in Emission Tomography - Correction Techniques in Emission Tomography - page 129

(a) Original (b) NN (c) Linear (d) Spline

FIGURE 7.8: Detail of the scaling example from Figure 7.7. (a) Detail loca-
tion indicated by black rectangle. Interpolation results for detail region using
(b) NN, (c) linear, and (d) spline interpolation.

To determine the advantages of spline interpolation we look at a detail in
Figure 7.8(a) indicated by the rectangle. The inferiority of NN interpolation
gets even more evident. Furthermore, the spline interpolation looks smoother
compared to the linear interpolation which is only piecewise differentiable.
Many tasks in medical image processing need continuously differentiable im-
ages as input, e.g., minimization of the distance functional for registration (see
Equation (7.15)). Hence, from a theoretical point of view it is essential to use
spline interpolation for such tasks as the differentiability is given. Yet, fast
and simple methods like linear interpolation are often sucient in practice.
Apart from the standard interpolation methods discussed in this section
a large variety of algorithms exists. A diffusion-based interpolation technique
with the intention to reduce the partial volume effect (see Section 7.5) was
presented in [53]. In [68] a PDE-based framework for interpolation and regular-
ization of scalar- and tensor-valued images is presented that does not require a
regular grid. The method based on anisotropic diffusion allows discontinuity-
preserving interpolation without additional oscillations.

7.4 Registration
Correction techniques in emission tomography images often include reg-
istration. Registration denotes the spatial alignment of two corresponding
images. In medical imaging these corresponding images are obtained from the
same or different patients, acquired at the same or at a different time, using
the same or different scanning techniques. One image is transformed in order
to match the other image as well as possible.
This simple definition raises some fundamental questions: How can the
similarity of two images be measured in order to find a good match? How can
the optimal spatial transformation, maximizing the similarity, be found?
Before answering these questions, an introductory example of registration
in emission tomography is discussed (Figure 7.9). Attenuation correction in