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Related Concept Videos

Fast Fourier Transform01:10

Fast Fourier Transform

The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
The computational efficiency of the FFT becomes...
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Continuous -time Fourier Transform01:11

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...
Convergence of Fourier Series01:21

Convergence of Fourier Series

The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
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Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...
Trigonometric Fourier series01:17

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Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
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ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis
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Regularization techniques on least squares non-uniform fast Fourier transform.

Fabio Gibiino1, Vincenzo Positano, Luigi Landini

  • 1Department of Energy and System Engineering, University of Pisa, Pisa, Italy. f.gibiino@gmail.com

International Journal for Numerical Methods in Biomedical Engineering
|February 15, 2013
PubMed
Summary

Regularization methods improve Magnetic Resonance Imaging (MRI) reconstruction quality by addressing ill-conditioning in least squares non-uniform fast Fourier transform (LS_NUFFT) gridding. Truncated singular value decomposition (TSVD) offers the best reconstruction but requires balancing processing time with image quality.

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Area of Science:

  • Medical Imaging
  • Magnetic Resonance Imaging (MRI)
  • Image Reconstruction

Background:

  • Non-Cartesian MRI accelerates data acquisition while maintaining image quality.
  • Least squares non-uniform fast Fourier transform (LS_NUFFT) is a gridding method for non-Cartesian MRI reconstruction.
  • Increasing interpolation kernel size in LS_NUFFT can lead to ill-conditioned problems.

Purpose of the Study:

  • To compare the effectiveness of three regularization methods for LS_NUFFT.
  • To evaluate reconstruction performance and processing time for different regularization techniques.
  • To determine the optimal regularization strategy for LS_NUFFT in MRI.

Main Methods:

  • Applied truncated singular value decomposition (TSVD), Tikhonov regularization, and L₁-regularization to LS_NUFFT.
  • Evaluated reconstruction performance against the direct summation method using simulated and experimental MRI data.
  • Assessed the processing time required for interpolator calculation.

Main Results:

  • Identified the interpolator size threshold requiring regularization for LS_NUFFT.
  • TSVD demonstrated superior reconstruction quality compared to Tikhonov and L₁-regularization for larger kernel sizes.
  • Processing time increased significantly with larger interpolator sizes, impacting computational feasibility.

Conclusions:

  • Regularization is necessary for LS_NUFFT when using larger interpolation kernels in MRI.
  • TSVD provides the best reconstruction accuracy but necessitates careful consideration of computational cost.
  • An optimal balance between reconstruction quality and processing time is crucial for practical LS_NUFFT applications.