6,758 research outputs found

    Density-equalizing maps for simply-connected open surfaces

    Full text link
    In this paper, we are concerned with the problem of creating flattening maps of simply-connected open surfaces in R3\mathbb{R}^3. Using a natural principle of density diffusion in physics, we propose an effective algorithm for computing density-equalizing flattening maps with any prescribed density distribution. By varying the initial density distribution, a large variety of mappings with different properties can be achieved. For instance, area-preserving parameterizations of simply-connected open surfaces can be easily computed. Experimental results are presented to demonstrate the effectiveness of our proposed method. Applications to data visualization and surface remeshing are explored

    Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT

    Get PDF
    We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients σ(x)\sigma(x) in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions s(x)s(x) over the domain Ω\Omega. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of Ω\Omega when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we obtain an optimally robust homogenization algorithm for arbitrary rough coefficients. Next, we consider inverse homogenization and show how to decompose it into a linear ill-posed problem and a well-posed non-linear problem. We apply this new geometric approach to Electrical Impedance Tomography (EIT). It is known that the EIT problem admits at most one isotropic solution. If an isotropic solution exists, we show how to compute it from any conductivity having the same boundary Dirichlet-to-Neumann map. It is known that the EIT problem admits a unique (stable with respect to GG-convergence) solution in the space of divergence-free matrices. As such we suggest that the space of convex functions is the natural space in which to parameterize solutions of the EIT problem
    • …
    corecore