Abstract: In this paper, we present an application of Riemannian
geometry for processing non-Euclidean image data. We consider the
image as residing in a Riemannian manifold, for developing a new
method to brain edge detection and brain extraction. Automating this
process is a challenge due to the high diversity in appearance brain
tissue, among different patients and sequences. The main contribution, in this paper, is the use of an edge-based
anisotropic diffusion tensor for the segmentation task by integrating
both image edge geometry and Riemannian manifold (geodesic,
metric tensor) to regularize the convergence contour and extract
complex anatomical structures. We check the accuracy of the
segmentation results on simulated brain MRI scans of single
T1-weighted, T2-weighted and Proton Density sequences. We
validate our approach using two different databases: BrainWeb
database, and MRI Multiple sclerosis Database (MRI MS DB). We
have compared, qualitatively and quantitatively, our approach with
the well-known brain extraction algorithms. We show that using
a Riemannian manifolds to medical image analysis improves the
efficient results to brain extraction, in real time, outperforming the
results of the standard techniques.
Abstract: This paper deals with the formulation of Maxwell-s equations in a cavity resonator in the presence of the gravitational field produced by a blackhole. The metric of space-time due to the blackhole is the Schwarzchild metric. Conventionally, this is expressed in spherical polar coordinates. In order to adapt this metric to our problem, we have considered this metric in a small region close to the blackhole and expressed this metric in a cartesian system locally.
Abstract: Einstein vacuum equations, that is a system of nonlinear
partial differential equations (PDEs) are derived from Weyl metric
by using relation between Einstein tensor and metric tensor. The
symmetries of Einstein vacuum equations for static axisymmetric
gravitational fields are obtained using the Lie classical method. We
have examined the optimal system of vector fields which is further
used to reduce nonlinear PDE to nonlinear ordinary differential
equation (ODE). Some exact solutions of Einstein vacuum equations
in general relativity are also obtained.
Abstract: The scalar wave equation for a potential in a curved space time, i.e., the Laplace-Beltrami equation has been studied in this work. An action principle is used to derive a finite element algorithm for determining the modes of propagation inside a waveguide of arbitrary shape. Generalizing this idea, the Maxwell theory in a curved space time determines a set of linear partial differential equations for the four electromagnetic potentials given by the metric of space-time. Similar to the Einstein-s formulation of the field equations of gravitation, these equations are also derived from an action principle. In this paper, the expressions for the action functional of the electromagnetic field have been derived in the presence of gravitational field.