Gaussian blur

In , a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss). It is a widely used effect in graphics software, typically to reduce and reduce detail. The visual effect of this blurring technique is a smooth blur resembling that of viewing the image through a translucent screen, distinctly different from the bokeh effect produced by an out-of-focus lens or the shadow of an object under usual illumination. Gaussian smoothing is also used as a pre-processing stage in computer vision algorithms in order to enhance image structures at different scales—see scale space representation and scale space implementation. Mathematics Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle (i.e., a circular box blur) w

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Simulation of 1D ARPES Data and Measures to recover the underlying Signal

Nadine Gächter

Fourier deconvolution is a good tool to remove the blurring of images with a very high SNR. However it suffers under severe noise degradation and becomes impracticable at a SNR of 20dB. If the convolution width is very high this happens even earlier. The analysis of the frequency spectrum brings information on the data, especially on the periodicity of certain features. Furthermore it can be used to better understand other processing tools. Gaussian blur and smoothing can be used with the goal to enhance image structures at different scales. They have the effect of reducing high frequency information, corresponding to high pixel to pixel variation and usually a characteristic of noise. However it also reduces detail. In 2D better resultsmay be obtained, because diagonal pixels can be used in addition. Gaussian smoothing is certainly a useful tool in the analysis of ARPES data and it should be well integrated in other measures of data recovery. If the blurring of the signal is high it is beneficial apply some Gaussian blur before proceeding to other measures. If the blurring of the signal is low, it is better to apply Gaussian smoothing only after de-convolution measures in order to not loose information on the signal.

STOCHASTIC PROCESSES AND CONSTRUCTION OF BROWNIAN MOTION

Christophe-Aschkan Mery

This project offers a rigorous introduction to the tools needed to construct a continuous stochastic process. Among other things, we give a very detailed proof of the Kolmogorov continuity criterion. We then construct a Brownian Motion following the formalism of D. Revuz and M. Yor. That is, we see the BM as a linear isometry from a Hilbert space into a Gaussian space.

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Non-gaussianity of the critical 3d Ising model

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2017