# Robust Registration of Medical Images in the Presence of Spatially-Varying Noise
## 2 Materials and Methods
### Bias Field Noise and Empirical Mode Decomposition  
Spatially varying noise or bias field is a common source of noise in MR images. Bias field noise can potentially cause the registration process to diverge, particularly when the similarity measure is not robust to the noise. Sum-of-squared-differences (SSD), correlation coefficient (CC), and mutual information (MI) are some of the non-robust similarity measures. Our results suggest that even Residual Complexity (RC) similarity measure has limited capacity in overcoming the bias field noise. In this section, we investigate the ability of EMD in reducing and removing this noise during the registration process.  
EMD is a powerful tool to extract the bias field noise from the image. Here, we provide an intuitive explanation for this statement. EMD provides a decomposition of the signal into its principle components in different levels of granularity. The principle patterns or the IMFs in each level (scale) contain a portion of the signal with a particular level of detail. The remaining components of the signal are explained in the residual part. Defined by the sifting process, each IMF is constructed by iteratively subtracting the average of the minimum and maximum envelopes from the original signal. In a two-dimensional space, the bias field noise could be estimated as an additive local spatial variation in the signal. The average of the minimum and the maximum envelopes are highly affected by this local spatial variation because both signals carry low-frequency patterns. Therefore, the average of the minimum and the maximum envelopes absorbs the bias field noise. By subtracting this average envelope from the signal, IMF becomes noise-free.  
Mathematically, the bias field noise can be represented by a mixture of two-dimensional Gaussian functions (kernels) [22]. We distort the image \(I(x,y)\) with an additive bias field noise according to the following equation:  
\[I_{distorted}(x,y)=I(x,y)+\frac{1}{K}\sum_{k=1}^{K}e^{-||([x,y]-\mu_{k}||^{2})/ 2\
u^{2}}, \tag{1}\]  
where \(K\) is the number of Gaussian functions. Figure 1A shows a sample Gaussian kernel. We manually add one Gaussian kernel (\(K=1\)) to both floating and reference MR images. The coordination of the center of Gaussian kernel is selected at random and the standard deviation is set to be \(\frac{1}{16}\) of the image width. Figure 1B shows a sample MR image from the BrainWeb dataset [43] corrupted with the bias field noise. To explore the ability of the EMD in extracting the bias field noise, we decomposed both noisy floating and reference images into IMFs and residuals. Figure 2 illustrates the set of IMFs and residuals for three EMD levels. The Gaussian kernel is highly expressed in residuals and IMFs are noise-free. This observation suggests that the registration algorithms based on the IMFs may be robust to the bias field noise. In the following two subsections, we propose two registration algorithms based on this observation.  
Figure 1: Simulated bias field noise in MR image. (**A**). Sample two-dimensional Gaussian function. (**B**). We manually add A Gaussian function with random mean to both floating and reference MR images.