Projects

Rational Approximation of Golden Angles

Duration: 2023 - 2024
Technologies: C, C++, IDEA, Shell Scripting, Python
Collaborators: Philip Schaten, Christina Graf, Daniel Mackner, H. Christian M. Holme, Moritz Blumenthal, Andrew Mao, Jakob Assländer, Martin Uecker

Illustration of radial golden ratio (left) and RAGA sampling pattern (right) including the RAGA increment approximating \(\Phi\). The first four spokes sampled in temporal order are color-coded and marked with \(t=1,2,3,4\).

Rational approximation of golden angles (RAGA) describes an innovative and simple principle to acquire radial samples in magnetic resonance imaging (MRI). It has been developed at the Graz University of Technology, the NYU Center for Biomedical Imaging, and University Medical Center Göttingen. RAGA sampling combines the advantages of golden ratio sampling with the advantages of repeating equidistant angular patterns, creating a versatile and simple tool for a broad variety of applications in MRI, i.e. dynamic and quantitative applications.

RT-NLINV reconstruction of an in vivo cardiac short-axis FLASH acquisition acquired with radial RAGA sampling and temporal resolution of 58 ms/frame.

At its core, RAGA sampling is based on a generalized Fibonacci formulation that has been exploited to compute a rational approximation of golden ratio based angles. This defines a sampling pattern with equidistant angular spacing, in which each spoke is sampled with an increment that approximates the golden ratio angle.
A crucial insight of this work was that with the appropriate choice of the spoke increment and total number of spokes, the RAGA sampling scheme not only approximates the right angle but also steps through all possible angles of the underlying equidistant sampling scheme bijectively. Its mathematical proof was added to the publication referenced below.

In our studies, we could show that the developed RAGA sampling sped up Toeplitz-based reconstructions and GROG-based gridding while preserving the image quality and retrospective binning characteristics of golden ratio based sampling. This was demonstrated in numerical and experimental phantom as well as cardiac in vivo studies.
The concept is not limited to radial sampling schemes, but can also be applied to create repeating patterns in other non-Cartesian trajectories.

Resources

A main focus of this work has been to make it fully open-source and easily reproducible. The manuscript, all scripts and data to recreate the individual figures, the code of all developed methods, and a tutorial have been published under a free CC-BY 4.0 license and can be accessed online by everyone.
Resources Location
Software Tools Berkeley Advanced Reconstruction Toolbox
Scripts DOI, Github
Data DOI
Tutorial DOI, Github
Poster DOI
Selected Presentation DOI

References

The technique and parts of this page have been published in:
  • Scholand N, Schaten P, Graf C, et al. Rational approximation of golden angles: Accelerated reconstructions for radial MRI. Magn Reson Med. 2025; 93(1): 51-66. doi: 10.1002/mrm.30247
under a CC-BY 4.0 license.