image processing · numerical methods · Python · Software

From numbers to images – Raw image processing with Python

Modern digital system cameras and also some digital compact cameras allow to create and to store images in a so-called raw format. A raw image contains the minimally pre-processed sensor data, meta-data about the picture such as camera model, lens model and camera settings under which the picture has been taken and possibly a fully… Continue reading From numbers to images – Raw image processing with Python

numerical methods

Calculating the Hermite functions

The Hermite functions appear as the solutions of the quantum mechanical harmonic oscillator. But they have applications in many other fields and applications, e.g., pseudospectral methods. The Hermite functions $h_n(x)$ are defined as\begin{equation}\label{eq:h}h_n(x) = \frac{1}{\sqrt{\sqrt{\pi} 2^n n!}} \mathrm{e}^{-x^2/2} H_n(x) \,,\end{equation} where $H_n(x)$ denotes the $n$th Hermite polynomial defined via the recurrence relation\begin{equation}H_{n}(x) = 2xH_{n-1}(x)-2(n-1)H_{n-2}(x)\end{equation} with… Continue reading Calculating the Hermite functions

numerical methods

The Lanczos algorithm

Finding the eigenvalues and eigenvectors of large hermitian matrices is a key problem of (numerical) quantum mechanics. Often, however, the matrices of interest are much too large to employ exact methods. A popular and powerful approximation method is based on the Lanczos algorithm. The Lanczos algorithm determines an orthonormal basis of the Kyrlov sub-space $\mathcal{K}_k(\Psi,… Continue reading The Lanczos algorithm

numerical methods · Python

Eigenvalues and eigenfunctions

Computing the eigenvectors $\psi(x)$ and eigenvalues $E$ of some Hamiltonian $H$ belongs to the key problems of quantum mechanics. For the one-dimensional Schrödinger equation we have to solve \begin{equation} E\psi(x) = H\psi(x) = -\frac{1}{2}\frac{\mathrm{d}^2\psi(x)}{\mathrm{d} x^2} + V(x)\psi(x) \end{equation}for some given potential $V(x)$. Analytical solutions, however, are scare goods. Thus, one has to rely on approximations… Continue reading Eigenvalues and eigenfunctions