Kernel-based Fisher discriminant analysis on the Riemannian manifold for nuclear atypia scoring of breast cancer

Abstract

Breast carcinoma is the most prevalent type of malignancy among women worldwide. Breast cancer grading often termed as Nuclear Atypia Scoring (NAS) forms a significant factor in determining individualized treatment plans and in the prognosis of the disease. For addressing the problem of breast cancer grading, we attempt to model the variations in features between histopathological images of different cancer grades and thereby explore the discriminative information concealed in these variations. In this regard, we aggregate multiple correlated features from the images using the geodesic geometric mean of the region covariances, to obtain the gmRC descriptors. As these gmRC descriptors are symmetric positive definite (SPD) matrices lying on the non-Euclidean Riemannian manifold, the discriminant analysis techniques developed for the Euclidean framework may not be appropriate. Hence, we propose a kernel-based Fisher discriminant analysis on the Riemannian manifold (KFDAR), that exploits the kernel trick for embedding the non-linear Riemannian manifold M into a higher dimensional linear Hilbert space H, which are then reduced to a low-dimensional and more discriminative subspace, where the samples become linearly separable. The kernel approach formulated for the Hilbert space embedding and for the kernel discriminant analysis is based on three Riemannian distance metrics: the log-Euclidean metric and the two symmetrized Bregman divergences – Stein and Jeffrey divergences. The experimental results show that this mapping to a highly discriminative space has succeeded in well-separating the histopathological images belonging to different cancer grades and hence it qualitatively and quantitatively outperforms the existing algorithms for cancer grading.