Sparse Representation Over Learned Dictionaries on the Riemannian Manifold for Automated Grading of Nuclear Pleomorphism in Breast Cancer

Breast cancer is found to be the most pervasive type of cancer among women. Computer aided detection and diagnosis of cancer at the initial stages can increase the chances of recovery and thus reduce the mortality rate through timely prognosis and adequate treatment planning. The nuclear atypia scoring or histopathological breast tumor grading remains to be a challenging problem due to the various artifacts and variabilities introduced during slide preparation and also because of the complexity in the structure of the underlying tissue patterns. Inspired by the success of symmetric positive definite (SPD) matrices in many of the challenging tasks in machine learning and computer vision, a sparse coding and dictionary learning on SPD matrices is proposed in this paper for the breast tumor grading. The proposed covariance-based SPD matrices form a Riemannian manifold and are represented as the sparse combination of Riemannian dictionary atoms. Non-linearity of the SPD manifold is tackled by embedding into the reproducing kernel Hilbert space using kernels derived from log-Euclidean metric, Jeffrey and Stein divergences and compared with the non-kernel-based affine invariant Riemannian metric. The novelty of the work lies in exploiting the kernel approach for the Hilbert space embedding of the Riemannian manifold, that can achieve a better discrimination of the breast cancer tissues, following a sparse representation over learned dictionaries and henceforth it outperforms many of the state-of-the-art algorithms in breast cancer grading in terms of quantitative and qualitative analysis.