# Robust Registration of Medical Images in the Presence of Spatially-Varying Noise
## 3 Results
### Evaluation of the Proposed Algorithms on the MRIs from the BrainWeb Dataset
#### 3.1.3 Registration Performance in the Presence of Bias Field  
In this section, we study the performance of the registration algorithms across the experiments that have converged. First, we explore the case that each of the reference and floating images are corrupted by one Gaussian function (\(K=1\)). The top two panels in Figure 5 show the T-RMSE and I-RMSE for the benchmark intensity-based algorithm as well as LR-EMD and AFR-EMD. For the SSD similarity measure, intensity-based registration has never converged, therefore, no corresponding error value is reported. For the MI similarity measure, both LR-EMD and AFR-EMD are remarkably more accurate compared to the intensity-based registration (50% lower T-RMSE and 40% lower I-RMSE). MI is a particularly interesting case because registration algorithms based on MI generally achieve a higher accuracy compared to the other three similarity measures and are often faster (Figure 5C). For the CC similarity measure, both EMD-based registrations have \(\sim\)20% lower average error rate. Using RC as the similarity measure, LR-EMD has a slightly higher average error rate compared to two other methods. The bottom panel in Figure 5 presents a comparison of the running time between the benchmark and two proposed algorithms. The intensity-based registration using CC similarity measure converges \(\sim\)50% faster compared to our two methods. All three algorithms have a similar running time when using MI as the similarity measure. Overall, both LR-EMD and AFR-EMD with the MI similarity measure achieve the highest accuracy with the lowest running time and are the best performing algorithms in the presence of one Gaussian kernel in the images. Figure 4: The convergence rate for baseline hierarchical registration with downsampling and our proposed algorithms based on EMD. Each panel corresponds to one similarity measure (SSD, CC, RC and MI). Here, the convergence is defined as T-RMSE error lower than four pixels. To present a more comprehensive comparison between our EMD-based registration algorithms and the benchmark method, one set of registered images is visualized in Figure 6. The image is randomly chosen from the BrainWeb dataset, then geometrically distorted using a random perturbation, and finally contaminated with \(K=1\) Gaussian function. In this case, MI is used as the similarity measure. Two images in the left panel of Figure 6 are the moving and the target images, both contaminated with bias field noise. The denoised registered images are shown on the right for our two proposed algorithms and the intensity-based benchmark algorithm. Visually, both LR-EMD and AFR-EMD perform a more accurate registration compared to the intensity-based method for this image. The improvement is visible both in the cerebral peduncle (red circle) and within the top right area of the skull (red arrow). Figure 7 shows the accuracy and running time for registrations in the presence of two Gaussian kernels in each of the floating and reference images (\(K=2\) case). Two Gaussian kernels simulate a more intense bias field noise. Using any of the SSD, CC, or MI similarity measures, EMD-based registrations achieve higher accuracy both in T-RMSE and I-RMSE. For MI, EMD-based registrations have \(\sim\)60% lower T-RMSE and \(\sim\)50% lower I-RMSE compared to the benchmark method. For all similarity measures, LR-EMD has slightly higher accuracy compared to AFR-EMD. Figure 5: Registration performance in the presence of one Gaussian kernel. Mean and variance of T-RMSE and I-RMSE for images in BrainWeb dataset are shown in (**A**,**B**) panels. (**C**) panel compares the running time for all methods. Figure 6: Target and reference images from BrainWeb MR dataset and registered images using our two methods and intensity-based hierarchical algorithm. MI is used as the similarity measure. Visually, both LR-EMD and AFR-EMD improve the registration accuracy in the center of image (red circle) and at the edge (red arrow).