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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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Directional radiation patterns are central to antenna analysis, as they illustrate how signal strength varies with direction. These patterns are often modeled using polar plots, where the radial distance from the origin represents signal intensity at a given angle. A commonly used idealized form is the four-lobed rose curve, which captures the concept of directional beams in a simplified mathematical form.The four-lobed rose curve, described by r = cos⁡(2θ), features four symmetric lobes, each...
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Phase-unwrapping algorithm for translation extraction from spherical navigator echoes.

Junmin Liu1, Maria Drangova

  • 1Imaging Research Laboratories, Robarts Research Institute, Ontario, Canada.

Magnetic Resonance in Medicine
|November 18, 2009
PubMed
Summary
This summary is machine-generated.

This study presents a novel phase-unwrapping algorithm for spherical navigator echoes, enabling accurate simultaneous measurement of rigid-body rotation and translation. The method achieves high precision for translation determination, crucial for various imaging applications.

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Area of Science:

  • Medical Imaging
  • Biophysics
  • Computational Physics

Background:

  • Spherical navigator echoes can determine simultaneous rigid-body rotation and translation.
  • Phase changes in echoes are used to calculate translations after rotation is known.
  • A phase-unwrapping algorithm is necessary due to the limited phase change interval (-pi, pi).

Purpose of the Study:

  • To develop and validate a phase-unwrapping algorithm for accurate translation determination from spherical navigator echoes.
  • To evaluate the algorithm's accuracy and precision using simulations and in vivo experiments.
  • To investigate the impact of phase noise on translation measurement accuracy.

Main Methods:

  • A novel phase-unwrapping algorithm based on a priori information of the translation-induced phase-change function was developed.
  • The algorithm was tested using simulated data and in vivo experimental data.
  • Accuracy and precision of translation determination were evaluated, considering background and off-resonance phase noise.

Main Results:

  • The proposed phase-unwrapping algorithm successfully measured translations up to 15 mm with accuracy better than 5%.
  • For translations up to 40 mm, the algorithm demonstrated an error of approximately 10%.
  • The study explored the effects of various phase noise sources on measurement accuracy.

Conclusions:

  • The developed phase-unwrapping algorithm enables accurate and precise simultaneous measurement of rigid-body translation and rotation using spherical navigator echoes.
  • The algorithm shows robust performance in the presence of phase noise, making it suitable for real-world applications.
  • This method advances the capabilities of motion correction and tracking in medical imaging and other fields.