Anisotropic diffusion

Summary

In and computer vision, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing without removing significant parts of the image content, typically edges, lines or other details that are important for the interpretation of the image. Anisotropic diffusion resembles the process that creates a scale space, where an image generates a parameterized family of successively more and more blurred images based on a diffusion process. Each of the resulting images in this family are given as a convolution between the image and a 2D isotropic Gaussian filter, where the width of the filter increases with the parameter. This diffusion process is a linear and space-invariant transformation of the original image. Anisotropic diffusion is a generalization of this diffusion process: it produces a family of parameterized images, but each resulting image is a combination betw

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Numerical methods for nonlinear reaction-diffusion equations with traveling wave solutions

Antoine Giovanni Imboden

In this report, we first introduce operator splitting as a way to solve nonlinear reaction-diffusion PDEs. We then present a flux-limiting technique for anisotropic diffusion problems that allows to recover the discrete maximum principle. Finally, we implement a simple action potential model, which allows us to represent the evolution of the transmembrane potential of a stimulated cell, and investigate the effect of anisotropic diffusion on the positivity of this potential.

2014

Simulation of silicon KOH and dry etching methods for micromould fabrication

Laurène Tribolet

The purpose of this project is to investigate the suitability of wet etching methods such as KOH etching for silicon micromoulds fabrication. Those moulds are intended for preceramic polymer (PCP) casting and specific requirements regarding the profile of the etched structures should be achieved. A process allowing to create deep structures with a high aspect ratio and smooth sidewalls would be the most appropriate. Moreover, obtaining slanted sidewalls could be useful as it would allow the resulting ceramic part to unmould itself during the shrinkage induced by pyrolysis. This is why KOH etching has to be considered, as it is an anisotropic process based on the etching along specific crystalline planes. Indeed, the etching rate along the planes of silicon is the slowest one, allowing therefore to etch structures with atomically smooth tilted walls. A mask with many different openings was then designed and fabricated in order to define the limits of KOH etching in peculiar situations. The consequences of using detailed structures containing reflex angles or the effects of tilting them on the mask with respect to the flat of the wafer will be then determined. The etching simulation was then realised thanks to the etching simulation software IntelliEtch, provided by IntelliSense. This tool allows to simulate both KOH and dry etching methods before visualising the results in 3D. It was therefore possible to firstly show that KOH is indeed a simple method to obtain smooth slanted and vertical walls by playing with the alignment of the mask openings. However, underetching of the mask can also easily occur, leading therefore to the loss of details contained in the original shapes design. Solutions can be found to overcome those issues by modifying the masks or by playing with other wafer orientations, but the problem of etching detailed structures nevertheless remains. An idea is therefore to use deep reactive ion etching methods such as Bosch process, as its advantages are mainly the ability to obtain high aspect ratio structures without underetching and therefore to etch detailed mask openings. Many unwanted effects can however occur with Bosch process, but by comparing both KOH etching and DRIE, it can be noticed that the drawbacks linked to both methods can compensate each other. It is therefore interesting to study the extent to which a process combining both techniques could be appropriate for micromoulds fabrication. Through a literature study, it will finally be shown that it is possible to obtain high aspect ratio structures containing many details with straight smooth sidewalls and which could be easily unmoulded. Different ways to combine KOH and dry etching methods that allow reaching all the requirements for an optimal micromould fabrication will also be presented.

2020

A theoretical framework describing the hydrodynamic interactions between a passive particle and an active medium in out-of-equilibrium systems predicts long-range Levy flights for the diffusing particle driven by the density of the active component. Brownian motion is widely used as a model of diffusion in equilibrium media throughout the physical, chemical and biological sciences. However, many real-world systems are intrinsically out of equilibrium owing to energy-dissipating active processes underlying their mechanical and dynamical features(1). The diffusion process followed by a passive tracer in prototypical active media, such as suspensions of active colloids or swimming microorganisms(2), differs considerably from Brownian motion, as revealed by a greatly enhanced diffusion coefficient(3-10) and non-Gaussian statistics of the tracer displacements(6,9,10). Although these characteristic features have been extensively observed experimentally, there is so far no comprehensive theory explaining how they emerge from the microscopic dynamics of the system. Here we develop a theoretical framework to model the hydrodynamic interactions between the tracer and the active swimmers, which shows that the tracer follows a non-Markovian coloured Poisson process that accounts for all empirical observations. The theory predicts a long-lived Levy flight regime(11) of the loopy tracer motion with a non-monotonic crossover between two different power-law exponents. The duration of this regime can be tuned by the swimmer density, suggesting that the optimal foraging strategy of swimming microorganisms might depend crucially on their density in order to exploit the Levy flights of nutrients(12). Our framework can be applied to address important theoretical questions, such as the thermodynamics of active systems(13), and practical ones, such as the interaction of swimming microorganisms with nutrients and other small particles(14) (for example, degraded plastic) and the design of artificial nanoscale machines(15).

2020

Simulation of silicon KOH and dry etching methods for micromould fabrication

Laurène Tribolet

2020