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Polar Coordinates

Polar Coordinates

The polar coordinate system offers an alternative to the Cartesian coordinate system for specifying points in a plane, using a distance and an angle instead of x and y coordinates. This system is particularly advantageous in situations involving circular or rotational symmetry, such as in physics or engineering problems involving waves, oscillations, or orbital paths.Defining Polar CoordinatesIn polar coordinates, a point is represented as P(r, ��), where r is the radial distance from a fixed...

Curvilinear Motion: Polar Coordinates

Curvilinear Motion: Polar Coordinates

In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
The particle's location is described using a unit vector along the radial direction. Deriving the particle's position with respect to time...

Imaging Studies IV: Magnetic Resonance Imaging

Imaging Studies IV: Magnetic Resonance Imaging

Introduction:Magnetic Resonance Imaging, or MRI, can include a specialized imaging technique of the urinary system known as Magnetic Resonance Urography (MRU). This radiation-free technique uses strong magnetic fields and radio waves to produce detailed images with the help of a computer. MRU is particularly effective for visualizing fluid-filled structures like the kidneys, ureters, and bladder.Applications of MRI in the Genitourinary SystemKidneys and Ureters: MRI detects tumors, cysts,...

Polar and Cylindrical Coordinates

Polar and Cylindrical Coordinates

The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.

Magnetic Resonance Imaging

Magnetic Resonance Imaging

Magnetic resonance imaging (MRI) is a noninvasive medical imaging technique based on a phenomenon of nuclear physics discovered in the 1930s, in which matter exposed to magnetic fields and radio waves was found to emit radio signals. In 1970, a physician and researcher named Raymond Damadian noticed that malignant (cancerous) tissue gave off different signals than normal body tissue. He applied for a patent for the first MRI scanning device in clinical use by the early 1980s. The early MRI...

Graphs of Polar Equations

Graphs of Polar Equations

The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...

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Related Experiment Video

Updated: May 23, 2026

An Experimental Protocol for Assessing the Performance of New Ultrasound Probes Based on CMUT Technology in Application to Brain Imaging

An Experimental Protocol for Assessing the Performance of New Ultrasound Probes Based on CMUT Technology in Application to Brain Imaging

Published on: September 24, 2017

[MRI image reconstruction using polar-coordinates conversion of k-space data].

Atsushi Tachibana¹, Takeyuki Hashimoto, Kazuya Sakaguchi

¹Department of Radiology, The Jikei University Hospital.

Nihon Hoshasen Gijutsu Gakkai Zasshi

|April 21, 2012

Summary

New magnetic resonance imaging (MRI) reconstruction techniques using filtered back projection (FBP) and simultaneous reconstruction technique (SIRT) were developed. These methods accurately reconstruct images from k-space data converted to polar coordinates.

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Area of Science:

Medical Imaging
Image Reconstruction
Signal Processing

Context:

Conventional magnetic resonance imaging (MRI) acquisition involves Cartesian k-space sampling.
Image reconstruction from k-space data is crucial for diagnostic quality.
Developing efficient and accurate reconstruction algorithms is an ongoing challenge in MRI.

Purpose:

To propose and evaluate novel MRI reconstruction techniques using filtered back projection (FBP) and simultaneous reconstruction technique (SIRT).
To convert k-space data from Cartesian to polar coordinates for projection creation.
To assess the accuracy of reconstructed images using linear and sinc interpolation methods.

Summary:

K-space data was converted from Cartesian to polar coordinates, creating projections for reconstruction.
Filtered back projection (FBP) and simultaneous reconstruction technique (SIRT) were applied to these projections.
Sinc interpolation demonstrated high accuracy, with a relative error of 0.013 compared to standard 2D Fourier Transform (2DFT) reconstruction.
Visual comparison showed no discernible difference between standard and projection-reconstructed images.

Impact:

Introduces efficient MRI reconstruction methods with high fidelity.
Sinc interpolation proves effective for k-space conversion in projection-based MRI reconstruction.
Potential to improve image quality and reduce acquisition time in MRI.
Offers a viable alternative to conventional 2DFT reconstruction for specific MRI applications.