Smoothing of biomedical images should preserve gray-level transitions between adjacent tissues, while restoring contours consistent with anatomical structures. Anisotropic diffusion operators are based on image appearance discontinuities (either local or contextual) and might fail at weak inter-tissue transitions. Meanwhile, the output of block-wise and morphological operations is prone to pres...

Quadratic and higher order finite elements are interesting candidates for the numerical solution of (elliptic) partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. While the systems of equations that arise from the discretisation of the underlying PDEs are often solved by iterative schemes like preconditioned Krylow-space metho...

The solution of elliptic partial differential equations on composite overlapping grids using multigrid is discussed. An approach is described that provides a fast and memory efficient scheme for the solution of boundary value problems in complex geometries. The key aspects of the new scheme are an automatic coarse grid generation algorithm, an adaptive smoothing technique for adjusting residual...

We present an explicit fourth-order compact nite diierence scheme for approximating the three dimensional convection-diiusion equation with variable coeecients. This 19-point formula is deened on a uniform cubic grid. Fourier smoothing analysis is performed to show that the smoothing factor of some relaxation techniques with our scheme is smaller than 1. We further design a parallelization-orie...

on utilizing the spectral representation of selfadjoint operators in hilbert spaces, some error bounds in approximating $n$-time differentiable functions of selfadjoint operators in hilbert spaces via a taylor's type expansion are given.

We present a set of interactive, free-form editing operators for direct manipulation of level-set models that supports the creation and removal of surface detail. The mathematics, data structures and algorithms needed to implement numerous level-set modeling capabilities have been developed. The first component of these capabilities identifies the Region-of-Influence (ROI) on the surface to be ...

Splatting-based rendering techniques are currently the best choice for efficient high-quality rendering of point-based geometries. However, such techniques are not suitable for large magnification, especially when the object is undersampled. This paper improves the rendering quality of pure splatting techniques using a fast dynamic up-sampling algorithm for point-based geometry. Our algorithm i...

In this paper, we introduce two generalizations of midpoint subdivision and analyze the smoothness of the resulting subdivision surfaces at regular and extraordinary points. The smoothing operators used in midpoint and mid-edge subdivision connect the midpoints of adjacent faces or of adjacent edges, respectively. An arbitrary combination of these two operators and the refinement operator that ...

In a companion paper we characterized the class of scale-invariant convolution operators: the generalized fractional derivatives of order γ. We used these operators to specify regularization functionals for a series of Tikhonov-like least squares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of ...

We establish boundedness estimates for solutions of generalized porous medium equations the form $$ \partial_t u+(-\mathfrak{L})[u^m]=0\quad\quad\text{in $\mathbb{R}^N\times(0,T)$}, where $m\geq1$ and $-\mathfrak{L}$ is a linear, symmetric, nonnegative operator. The wide class operators we consider includes, but not limited to, L\'evy operators. Our quantitative take precise $L^1$--$L^\infty$-s...