A comparative study on the influence of α-centroids over Κ-means algorithm and its variants in DT-RMI segmentation

Abstract

Abstract The diffusion tensor magnetic resonance imaging (DT-MRI) is a non-invasive and effective technique that ables us to study the micro-structural integrity of white matter of fibers and detecting tumors or anomalies in living tissues. Image segmentation in DT-RMI is used to identify and separate a tissue in different regions which preserve similar properties. \(K\) -means and deep learning algorithms can be applied for this purpose, being the second class the most used currently. Despite actually deep learning algorithms are the most frequently approach in image segmentation, $K$-means algorithm performed an important role in a recent past, carrying implicitly lots of sophisticated mathematical results. Originally, $K$-means algorithm was proposed to cluster a dataset in subsets with similar characteristics. Our interesting consists in empirically analyzing this technique and their variants under both Euclidean and Riemannian setting when applied to image segmentation in DT-MRI. Pixels in the bidimensional case and voxels in the tridimensional one can be respectively represented by 2-by-2 and 3-by-3 symmetric positive definite matrices (SPD) in DT-RMI. Since the set of \(n\) -by-$n$ SPD matrices can be seen as both a non-pointed convex cone in the vector space of $n$-by-$n$ symmetric matrices as well as a Hadamard manifold, centroids are well-posed in both sense, for any positive integer $n$. Thus, we developed a comparative study on the influence of centroids under Euclidean and Riemannian settings on the image segmentation by using the $K$-means algorithm and its variants.