Statistical mechanics of permanent random atomic and molecular networks: Structure and heterogeneity of the amorphous solid state

Under sufficient permanent random covalent bonding, a fluid of atoms or small molecules is transformed into an amorphous solid network. Being amorphous, local structural properties in such networks vary across the sample. A natural order parameter, resulting from a statistical-mechanical approach, captures information concerning this heterogeneity via a certain joint probability distribution. This joint probability distribution describes the variations in the positional and orientational localization of the particles, reflecting the random environments experienced by them, as well as further information characterizing the thermal motion of particles. A complete solution, valid in the vicinity of the amorphous solidification transition, is constructed essentially analytically for the amorphous solid order parameter, in the context of the random network model and approach introduced by Goldbart and Zippelius [Europhys. Lett. 27, 599 (1994)]. Knowledge of this order parameter allows us to draw certain conclusions about the stucture and heterogeneity of randomly covalently bonded atomic or molecular network solids in the vicinity of the amorphous solidification transition. Inter alia, the positional aspects of particle localization are established to have precisely the structure obtained perviously in the context of vulcanized media, and results are found for the analogue of the spin glass order parameter describing the orientational freezing of the bonds between particles.Comment: 31 pages, 5 figure

A multigrid approach to image processing

A second order partial differential operator is applied to an image function. By using a multigrid operator known from the so-called approximation property, we derive a new type of multiresolution decomposition of the image. As an example, the Poisson case is treated in-depth. Using the new transform we devise an algorithm for image fusion. The actual recombination is performed on the image functions on which the partial differential operator has been applied first. A fusion example is elaborated upon. Other applications can be envisaged as well

A multigrid approach to image processing.

A second order partial differential operator is applied to an image function. By using a multigrid operator known from the so-called approximation property, we derive a new type of multiresolution decomposition of the image. As an example, the Poisson case is treated in-depth. Using the new transform we devise an algorithm for image fusion. The actual recombination is performed on the imagefunctions on which the partial differential operator has been applied first. A fusion example is elaborated upon. Other applications can be envisaged as wel

A toolbox for the lifting scheme on quincunx grids (LISQ)

A collection of functions written in MATLAB is presented. The functions include second generation wavelet decomposition and reconstruction tools for images as well as functions for the computation of moments. The wavelet schemes rely on the lifting scheme of Sweldens and use the splitting of rectangular grids into quincunx grids, also known as red-black ordering. The prediction filters include the Neville filters as well as a nonlinear maxmin filter. Custom-made filters can be used too. The various functions are described and examples are given. The toolbox is provided with appliances for the visualization of data on quincunx grids. The software can be downloaded from a website and is publicly available

The Multigrid Image Transform

A second order partial differential operator is applied to an image function. To this end we consider both the Laplacian and a more general elliptic operator. By using a multigrid operator known from the so-called approximation property, we derive a multiresolution decomposition of the image without blurring of edges at coarser levels. We investigate both a linear and a nonlinear variant and compare to some established methods

Development of semi-coarsening techniques

Departing from Mulder's semi-coarsening technique for first order PDEs, the notion of a grid of grids is introduced and a multi-level finite-volume technique for second order elliptic PDEs is developed. Various grid transfer operators are investigated, in combination with damped Jacobi relaxation. Convergence rates as they are predicted by Fourier local mode analysis are compared with practical measurements. The wide variety of grids at our disposal leads to the notion of coherent representations of a function on different grids. A sawtooth multi-level algorithm is proposed for the case of multiple semi-coarsening. A hierarchical set of basis functions for finite volumes on sparse grids is briefly discussed

What multigrid and Poisson do to one's image

Though the pun in the title is intended, it is not quite fair to Piet Wesseling as he is a person who promoted the development of multigrid to far more complicated equations than the Poisson equation. Instead, the title should be taken more literally as it truly relates to the contents of this note. It is shown that while multigrid is renowned for his efficiency in solving partial differential equations, integral equations and what not, it can also, maybe surprisingly, be used for the multiresolution of images

The multigrid image transform.

A second order partial differential operator is applied to an image function. To this end we consider both the Laplacian and a more general elliptic operator. By using a multigrid operator known from the so-called approximation property, we derive a multiresolution decomposition of the image without blurring of edges at coarser levels. We investigate both a linear and a nonlinear variant and compare to some established method