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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...
Area Computation by the Alternative Coordinate Method01:24

Area Computation by the Alternative Coordinate Method

The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
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...
Spherical Coordinates01:23

Spherical Coordinates

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
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Field Application of Global Positioning System

The Global Positioning System (GPS) has become an indispensable tool in fieldwork, offering unparalleled precision and efficiency for surveying, navigation, and infrastructure development. By harnessing signals from a constellation of satellites, GPS receivers determine the location of objects with remarkable speed and accuracy, often completing calculations within a second.Advantages of Modern GPS TechnologyContemporary GPS receivers are designed to meet the practical demands of field...
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,...

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Four-Dimensional CT Analysis Using Sequential 3D-3D Registration
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Four-Dimensional CT Analysis Using Sequential 3D-3D Registration

Published on: November 23, 2019

Geodesic active fields--a geometric framework for image registration.

Dominique Zosso1, Xavier Bresson, Jean-Philippe Thiran

  • 1Signal Processing Laboratory, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland. dominique.zosso@epfl.ch

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|November 25, 2010
PubMed
Summary
This summary is machine-generated.

We introduce geodesic active fields, a novel geometric framework for image registration. This method offers a data-dependent regularization and reparametrization invariance for accurate image mapping.

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Multimodal Cross-Device and Marker-Free Co-Registration of Preclinical Imaging Modalities
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Multimodal Cross-Device and Marker-Free Co-Registration of Preclinical Imaging Modalities
07:13

Multimodal Cross-Device and Marker-Free Co-Registration of Preclinical Imaging Modalities

Published on: October 27, 2023

Area of Science:

  • Computer Vision
  • Geometric Deep Learning
  • Medical Imaging Analysis

Background:

  • Image registration is crucial for aligning images but is an ill-posed inverse problem.
  • Existing methods often require specific regularization terms, limiting their generalizability.
  • Current approaches may lack invariance to parameterization or adaptive regularization.

Purpose of the Study:

  • To present a novel geometric framework, geodesic active fields, for general image registration.
  • To introduce a multiplicative coupling for data-dependent regularization strength.
  • To achieve reparametrization invariance in image registration.

Main Methods:

  • Developed a geometric framework based on weighted minimal surfaces and minimization flows.
  • Employed a multiplicative coupling between image discrepancy and regularization terms.
  • Investigated different weighting functions: squared error, approximated absolute error, and local joint entropy.

Main Results:

  • The framework generalizes to various surfaces, including non-flat and multiscale images, registering all scales simultaneously.
  • Achieved reparametrization invariance, a first in image registration literature.
  • Demonstrated data-dependent regularization strength and tunable anisotropic regularization.

Conclusions:

  • Geodesic active fields provide a versatile and robust geometric approach to general image registration.
  • The method offers significant advantages over specialized techniques, particularly in handling complex image types and achieving intrinsic regularization.
  • This framework advances the field by introducing reparametrization invariance and adaptive regularization for improved image alignment.