# Robust Registration of Medical Images in the Presence of Spatially-Varying Noise
## 1 Introduction  
The accurate registration and alignment of two images has been a challenging problem in a wide variety of applications such as medical image processing [1, 2], remote sensing [3], biology [4, 5, 6], and computer vision [7, 8]. Particularly, the registration of the medical images has been widely used in tumor localization and targeting [9], organ growth studies [10] and brain atlas reconstruction [11]. Overall, the registration algorithms can be categorized into two classes [12]. One class is concerned with the registration based on the intensity of the images. In this approach, a similarity measure is defined to quantify the similarity of both floating (or moving) and reference (or target) images. Then, an optimization process identifies an optimal map for the floating image to achieve the highest similarity to the reference image. In the second class of registration algorithms, a set of features such as landmarks [13], a histogram of intensity [14] or responses to Gabor [15] and Alpha stable filters [16, 17] are first extracted from both floating and reference images. Then, two images are aligned through the mapping of these features. A similarity measure between two images is a crucial component of intensity-based image registration. A wide variety of measures such as the sum-of-squared-differences (SSD), correlation coefficient (CC), or mutual information (MI) [18, 19, 20], have been used inthe registration process. However, none of these measures are robust against the spatially varying intensity distortion. The bias field noise is one of the common instances of the spatially varying distortion [21] and is often present in magnetic resonance (MR) or retina images. Examples of spatially varying noise in MRI are standing wave artifacts, \(B_{1}\) and \(B_{0}\) inhomogeneity artifacts, susceptibility artifact, and surface coils artifacts. Therefore the traditional registration algorithms may easily fail for the MR or retina images when these images are distorted. Traditionally, two distinct types of algorithms have been studied to reduce the bias-field noise effect in image registration. One type is focused on defining robust similarity measures. Residual complexity (RC) [22], rank-induced [23; 24] and sparsity-based [25; 26] similarity measures are some examples. The second type of algorithms is based on simply reducing the noise effect and then registration of the denoised images [27; 28]. Our two proposed algorithms in this paper could be classified in the second group. We examine the ability of state-of-the-art hierarchical signal decomposition techniques in removing additive spatially varying noise in MR images. Specifically, we propose to use empirical mode decomposition (EMD) [29] to overcome the effect of bias field noise in MR images. EMD is a multi-resolution computational technique to decompose the image into several principle patterns and residual components. We show that the bias field noise is highly expressed in the residual components and not in the principle patterns. Therefore, it is possible to remove the noise by removing the EMD residuals. EMD-based algorithms have been previously studied for the basic image registration task [30; 31; 32]. However, to our knowledge, the advantage of the EMD-based registration algorithms in the presence of spatially varying noise is still unknown. In this study, we explore this problem by proposing robust EMD-based registration algorithms. Our contributions to this paper are twofold: First, we show that the EMD is able to extract the bias field noise from the image through multi-scale decomposition of the image. Second, we introduce two multi-resolution [33] EMD-based registration algorithms robust to the bias field noise. In one algorithm (LR-EMD), we hierarchically register EMD feature maps from both floating and reference images. The feature maps are constructed from various resolutions of EMD-based principle patterns.