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Related Concept Videos

Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
Gauss's Law: Planar Symmetry01:27

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a uniform...
Gauss's Law01:07

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.
Gaussian Elimination: Problem Solving01:30

Gaussian Elimination: Problem Solving

Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
Gauss's Law: Problem-Solving01:10

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...

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Related Experiment Video

Updated: Jun 26, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
08:12

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

Published on: March 1, 2022

Gaussian profile estimation in two dimensions.

Nathan Hagen1, Eustace L Dereniak

  • 1Fitzpatrick Institute for Photonics, Duke University, Durham, North Carolina 27708, USA. nhagen@optics.arizona.edu

Applied Optics
|December 24, 2008
PubMed
Summary
This summary is machine-generated.

This study extends Gaussian profile estimation to 2D, providing exact and approximate covariance matrices for parameter estimation. It also addresses truncated sampling and bias removal for optimal pixel size selection.

Related Experiment Videos

Last Updated: Jun 26, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
08:12

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

Published on: March 1, 2022

Area of Science:

  • Astrophysics and computational imaging
  • Statistical analysis and parameter estimation

Background:

  • Accurate parameter estimation is crucial for analyzing scientific data, particularly in imaging.
  • Previous work focused on one-dimensional Gaussian profiles, leaving two-dimensional cases less understood.

Purpose of the Study:

  • To extend parameter estimation techniques for Gaussian profiles to two dimensions.
  • To derive and approximate the covariance matrix of estimated parameters.
  • To analyze the impact of truncated sampling and bias on parameter estimation.

Main Methods:

  • Derivation of the exact covariance matrix for 2D Gaussian profile parameters.
  • Development of analytic approximations for the covariance matrix.
  • Formulation of methods to calculate parameter variances under truncated sampling.
  • Calculation of bias expressions and a bias removal approach.

Main Results:

  • The exact covariance matrix for 2D Gaussian profile parameters is derived.
  • Analytic approximations offer simplified insights into parameter estimation behavior.
  • Methods for handling truncated sampling and bias are presented.
  • The influence of bias on optimal pixel size selection is clarified.

Conclusions:

  • The derived covariance matrices and approximations enhance the analysis of 2D Gaussian profiles.
  • The study provides a foundation for clarifying prior research in the field.
  • Addressing truncated sampling and bias is essential for robust parameter estimation and optimal detector design.