M.Sc Thesis | |

M.Sc Student | Reani Yair |
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Subject | Functional Tracing of Discrete Vector Fields |

Department | Department of Computer Science |

Supervisor | ASSOCIATE PROF. Mirela Ben-Chen |

Full Thesis text |

We propose a method for approximating the flowlines of a discrete tangent vector field on a triangle mesh. Our method makes use of the recently proposed discrete representation of a vector field as a derivation operator. This representation allows us to state the problem of flowlines computation as the advection of the Euclidean coordinate functions by the vector field. By representing the vector field as a linear derivation operator, and discretizing both the vector field operator and the coordinate functions using Lagrange linear elements (or "hat functions"), the spatial discretization of the flowline equations leads to three linear systems of ordinary differential equations (ODEs), one system for each Euclidean coordinate function. These linear ODEs have a closed form solution as a function of time, thus the system can be solved without explicit time discretization, using an exponential integrator.

Our approach requires only the
construction of the derivative operator that represents the vector field and
multiplying the exponential of a sparse matrix by a vector, which can both be
efficiently computed. For a given equally spaced time vector, we compute the
flowlines from *all* the vertices of the mesh simultaneously. With this
global definition of the problem, our method is characterized by making use of
mostly global solutions, as opposed to algorithms that analyse local geometric
details through the explicit generation of curves and intersecting line
segments. We compare our approach to analytical solutions in cases where these
are known, and to an iterative simple tracing algorithm. In addition, we
examine our solution from other aspects, such as invariance to global
transformations, the distance of the flowlines from the mesh, and other local
characteristics. Finally, we use our method for the simple, robust and efficient
visualization of discrete tangent vector fields on triangle meshes.