Explicit formulae for Chern-Simons invariants of the hyperbolic J(2n,−2m)J(2n,-2m) knot orbifolds

We calculate the Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds using the Schl\"{a}fli formula for the generalized Chern-Simons function on the family of cone-manifold structures of $J(2n,-2m)$ knot. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham and Lee's methods to a bi-infinite family. We dealt with even slopes just as easily as odd ones. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic $J(2n,-2m)$ knot orbifolds. For the fundamental group of $J(2n, -2m)$ knot, we take and tailor Hoste and Shanahan's. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert's canonical 2-bridge diagram or not.Comment: 9 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1601.00723, arXiv:1607.0804

Development of Texture Weighted Fuzzy C-Means Algorithm for 3D Brain MRI Segmentation

The segmentation of human brain Magnetic Resonance Image is an essential component in the computer-aided medical image processing research. Brain is one of the fields that are attracted to Magnetic Resonance Image segmentation because of its importance to human. Many algorithms have been developed over decades for brain Magnetic Resonance Image segmentation for diagnosing diseases, such as tumors, Alzheimer, and Schizophrenia. Fuzzy C-Means algorithm is one of the practical algorithms for brain Magnetic Resonance Image segmentation. However, Intensity Non- Uniformity problem in brain Magnetic Resonance Image is still challenging to existing Fuzzy C-Means algorithm. In this paper, we propose the Texture weighted Fuzzy C-Means algorithm performed with Local Binary Patterns on Three Orthogonal Planes. By incorporating texture constraints, Texture weighted Fuzzy C-Means could take into account more global image information. The proposed algorithm is divided into following stages: Volume of Interest is extracted by 3D skull stripping in the pre-processing stage. The initial Fuzzy C-Means clustering and Local Binary Patterns on Three Orthogonal Planes feature extraction are performed to extract and classify each cluster’s features. At the last stage, Fuzzy C-Means with texture constraints refines the result of initial Fuzzy C-Means. The proposed algorithm has been implemented to evaluate the performance of segmentation result with Dice’s coefficient and Tanimoto coefficient compared with the ground truth. The results show that the proposed algorithm has the better segmentation accuracy than existing Fuzzy C-Means models for brain Magnetic Resonance Image