# Image Resolution Enhancement of Highly Compressively Sensed CT/PET Signals
## 5 The Super-Resolution Algorithm vs. Image Registration Issues  
High accuracy and deformable image registration are key factors in the very challenging medical imaging field. The non-rigid registration algorithms prone to solving a highly non-convex optimisation problem with millions of degrees of freedom. Mathematically, in its domain, this issue is considered typically categorised to be a problem of continuous optimisation--being expressed as a non-convex cost function, which is inclined to get stuck at to local minimums, which leads to mImage Super-Resolution registration issues [7, 8]. Moreover, to make matters worse, the area of medical imaging is associated with one of the most challenging application fields for motion artefacts reduction [16]. Moreover, the deformable motion estimation is already nested at the core of the presented iterative super-resolution algorithm. Furthermore, the technique takes into account the blur kernel and noise estimate, see Figure 2. The application of this algorithm provides much lower registration errors while compared to the other procedures even if large motion trajectory distortions occur. The goal of the algorithm is to minimise the cost function consisting of the High-Resolution (HR) input image \(\Gamma_{H}\), deformable image registration parameters \(\{w_{i}\}\), noise level \(\{\theta_{i}\}\) as well blur kernel \(B\)[16]. The algorithm starts with taking one image from a set of LR observations as its reference input. The image registration, which is presented in this paper, is performed on a whole set of CT as well as PET low-resolution observations:  
\[\hat{\Gamma}_{H}^{n}=arg\min_{\Gamma_{H}^{n}}\theta_{0}\left\|DB\Gamma_{H}^{n} -\Gamma_{L}^{n,0}\right\|+g\left\|\
abla\Gamma_{H}^{n}\right\|+\sum_{i=-N,i \
eq 0}^{N}\left\|DBw_{i}\Gamma_{H}^{n}-\Gamma_{L}^{n,0}\right\| \tag{3}\]  
Note that the \(\Gamma_{L}^{n,0}\) represents the simulated reference initial guess related to \(n\)-th set of compressively sensed CT/PET images, \(\Gamma_{L}^{n,j}\) is the -th acquired LR image with regard to degradation parameters, \(\Gamma_{L}^{n}\) is HR estimate, \(\
abla\) is the gradient operator, \(\hat{\Gamma}_{H}^{n}\) is the \(n\)-th HR image estimate, \(D,B,w_{i}\) and g denote the down-sampling, blur kernel, motion parameters, additional degradation operator individually. To allow a gradient optimisation to be doable, L1 norms are supposed to be replaced with their differentiable approximations.  
Input: _set of Low-resolution input CT/PET_.  
1. high-resolution estimate.
2. repeat until convergence \(|\Gamma_{L}^{n}-\Gamma_{L}^{n,\text{observed}}|<\varepsilon\)  
1. Estimate noise parameters \(\theta_{i}=\frac{\alpha+N_{a}-1}{\beta+N_{b}\varepsilon}\)  
2. Calculate deformable image registration parameters and realign an image grid using them 3. Estimate blur kernel operator \(B_{x}=arg\min_{B_{x}}\theta_{0}\|AM_{y}B_{x}-\Gamma_{L}^{n}\|+\xi\|\
abla B_{x}\|\)  
3. Improve the High-Resolution estimate \(\Gamma_{H}^{n}\)  
4. Repeat steps a-d until convergence, see Figure 2  
Output: _high-resolution CT/PET image_.  
The algorithm minimises the cost function consisting of the High-Resolution CT/PET input image GH, motion vectors fwig, noise level qi as well blur kernel B. The flowchart starts with considering one image from a set of Low-Resolution observations as its reference input.