Algorithm to extract direction in 2D discrete distributions and a continuous Frobenius norm

TL;DR

A first-order approximation of the CFND between two similar Gaussian distributions takes the form of an absolute sine function, offering a simple analytical form with potential applications in specialized areas such as segmented inverse beta decay neutrino detectors, astronomy, machine learning, and more.

Abstract

In this study, we present a novel algorithm for determining directionality in 2D distributions of discrete data. We compare a reference dataset with a known direction to a measured dataset with an unknown direction by the Frobenius norm of the difference (FND) to find the unknown direction. To generalize this concept, we develop a continuous Frobenius norm of the difference (CFND) as a continuous analog of the FND and derive its analytical expression. By relating fitted and normalized 2D Gaussian distributions, we show that the CFND approximates the FND, and we validate this relationship with computer simulations. We find that a first-order approximation of the CFND between two similar Gaussian distributions takes the form of an absolute sine function, offering a simple analytical form with potential for specialized applications in segmented inverse beta decay (IBD) neutrino detectors, astronomy, machine learning, and more. Although this method may easily extend to 3D scalar fields, our focus here is on 2D real-valued fields as it directly applies to directionality. Our methodology consists of modeling a 2D Gaussian distribution, binning the data into a histogram, and encoding it as a square matrix. Rotating this matrix around its geometric center and comparing it to a measured dataset using the FND gives us rotational data that we fit with an absolute sine function. The location of the minimum of this fit is the angle closest to the true angle of the direction in the measured dataset. We present the derivation and discuss initial applications of the CFND in our novel algorithm, demonstrating its success in approximating directionality in 2D distributions.

Figures (11)

• Figure 1: Example of a Gaussian fit to a histogram of Gaussian data with standard deviation $\sigma=1$ and centroid $\mu=0$.
• Figure 2: Bivariate normal distribution with standard deviation $\sigma_x=\sigma_y=1$ and centroid $\bm \mu=(0,0)$.
• Figure 3: 2D Gaussian histogram with standard deviation $\sigma_x=\sigma_y=1$ and centroid $\bm \mu=(0,0)$. The grid size is $16\times 16$ and the bin width $\Delta x=1/2$.
• Figure 4: Shows affine rotation of 2D histogram bin grid rotation by $\vartheta=30^\circ$ in the $3\times3$ case. The Gaussian data is plotted in the top row with the respective coordinate system and the binned result is shown in the bottom row. This figure highlights the 2D histogram binning process after rotation of the dataset. The centroid of the distribution is marked by a red crosshair.
• Figure 5: Simulation of algorithm on dataset with true or unknown angle $\vartheta_0=0^\circ$. Red is simulated FND data for $n=1000$ and $\Delta x=16$. The distribution parameters were $\sigma=10$ and $\mu=2$. The black curve is the theoretical CFND curve described in (\ref{['eq:finalfitcurve']}). The minimum of the FND data indicates the reference angle of the dataset.