Fast and accurate Polar Fourier transform

Averbuch, Amir; Coifman, Ronald R.; Donoho, David L.; Elad, Michael; Israeli, M.

doi:10.1016/j.acha.2005.11.003

Cited by 177 publications

(133 citation statements)

References 31 publications

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“…For the non-Cartesian approaches such as radial and spiral trajectories, the non-uniform sampling density agrees with the k-space distribution to gain higher SNR and collect more important data points for a faithful reconstruction. Nevertheless, as mentioned in section 1.1, there is no fast and accurate Fourier transform for a non-Cartesian grid to connect the frequency and image domains [82]. This limitation restricts the extent of applications of non-Cartesian trajectories in CS MRI.…”

Section: Pseudo-random Under-sampling Patternmentioning

confidence: 99%

“…Based on the definition of a polar-like 2D grid, the points of pseudo-polar grid in the k-space domain of normalized region of [−1,1] can be separated into two subsets, that is, basically vertical (BV) and basically horizontal (BH), which can be mathematically described as [82]: …”

Section: Theory 521 Pseudo-polar Gridmentioning

confidence: 99%

“…Here, we use Fractional Fourier Transform (FRFT) to calculate the summation in equation (5-7) [82,101,138]. The results will be stacked in an array with 2N rows and N columns.…”

Section: Pseudo-polar Fourier Transformmentioning

confidence: 99%

“…where i stands for the current iteration step, D is a positive-definite matrix to control the descent of each iteration step (for details, see [82]). As shown in [82], a proper D can ensure good convergence of the corresponding algorithm.…”

Section: Pseudo-polar Fourier Transformmentioning

confidence: 99%

“…In recent years, a radial-like, pseudo-polar (PP) trajectory has been developed to avoid the above regridding / inverse-regridding operation [82,139,140]. This efficient pseudo-polar FFT (PPFFT) operation completely avoids the 2D interpolation, and it involves only a PPFFT operation, which has similar computational complexity as a Cartesian 2D-FFT.…”

Section: Introductionmentioning

confidence: 99%

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Improving the image quality in compressed sensing MRI by the exploitation of data properties

Yang¹

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Minimizing scan time without compromising image quality has been the main thrust of magnetic resonance imaging (MRI) since the establishment of the field. Tremendous effort has been put into MRI systems in pursuit of faster MR imaging techniques, which can substantially facilitate clinical diagnoses and improve the patient experience. Traditional MRI accelerated approaches mainly focused on hardware improvements to improve the speed of scanning. These methods are still governed by Nyquist-Shannon sampling rate. Hence, when the acceleration rate is pushed beyond the theoretical limit, aliasing artefacts are introduced which degrade the reconstructed image quality and are difficult to eliminate via current methods. Recently, it has been found that a newly developed signal processing theory, compressed sensing (CS), can be used to create high-resolution signal data sets from sparse samples, despite violating the classical Nyquist-Shannon sampling criteria. The application of CS to MRI opens up an appealing avenue to offer potentially significant scan time reductions. CS reconstructs MR images from incomplete k-space measurements. Taking advantage of the fact that MR images have sparse representations under certain mathematical transformations, CS overcomes the distortions resulting from the under-sampled k-space data. Significant progress has been made in the development of CS MRI. However, there are two major remaining challenges to its full adoption in clinical applications: (1) the small amount of under-sampled data (for example, less than 1/8 of the whole k-space data) will inherently introduce artefacts in the reconstructed image and compromise diagnosis accuracy for the reconstructed images; (2) owing to hardware limits, it is difficult to realise two-dimension random under-sampling, which is favoured by CS operation.Practically, the random phase-encode under-sampling pattern has been widely implemented.However, it introduces coherent aliasing artefacts that are hard to eliminate using the CS theory.In this thesis, several techniques are proposed that can improve the quality of images reconstructed by CS MRI. These techniques have the potential to facilitate faithful reconstruction of CS MRI in clinical applications. A novel two-stage reconstruction scheme is introduced in Chapter 3. In this method, the under-sampled k-space data is segmented into low-frequency and high-frequency parts.Then, in stage one, using dense measurements, the low-frequency region of k-space data is faithfully reconstructed. The fully reconstituted low-frequency k-space data from the first stage is then combined with the under-sampled high-frequency k-space data to complete the second stage reconstruction of the whole image. With this two-stage approach, each reconstruction inherently incorporates a lower data under-sampling rate than conventional approaches. Because the restricted isometric property is easier to satisfy than conventional CS method, the reconstruction consequently ii produces lower residual errors in each step...

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Section: Pseudo-random Under-sampling Patternmentioning

confidence: 99%

Section: Theory 521 Pseudo-polar Gridmentioning

confidence: 99%

“…Here, we use Fractional Fourier Transform (FRFT) to calculate the summation in equation (5-7) [82,101,138]. The results will be stacked in an array with 2N rows and N columns.…”

Section: Pseudo-polar Fourier Transformmentioning

confidence: 99%

Section: Pseudo-polar Fourier Transformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Improving the image quality in compressed sensing MRI by the exploitation of data properties

Yang¹

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Goodness‐of‐fit measures based on the Mellin transform for beta generalized lifetime data

Vasconcelos

Cintra

Nascimento

2021

Math Methods in App Sciences

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In recent years various probability models have been proposed for describing lifetime data. Increasing model flexibility is often sought as a means to better describe asymmetric and heavy tail distributions. Such extensions were pioneered by the beta-G family. However, efficient (GoF) measures for the beta-G distributions are sought. In this paper, we combine probability weighted moments (PWMs) and the Mellin transform (MT) in order to furnish new qualitative and quantitative GoF tools for model selection within the beta-G class. We derive PWMs for the Fréchet and Kumaraswamy distributions, and we provide expressions for the MT, and for the log-cumulants (LC) of the beta-Weibull, beta-Fréchet, beta-Kumaraswamy, and beta-log-logistic distributions. Subsequently, we construct LC diagrams and, based on the Hotelling's T 2 statistic, we derive confidence ellipses for the LCs. Finally, the proposed GoF measures are applied on five real data sets in order to demonstrate their applicability.

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Fast direct fourier reconstruction of radial and PROPELLER MRI data using the chirp transform algorithm on graphics hardware

Feng

Song

Xin

et al. 2012

Magnetic Resonance in Med

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Fast and accurate Polar Fourier transform

Cited by 177 publications

References 31 publications

Improving the image quality in compressed sensing MRI by the exploitation of data properties

Improving the image quality in compressed sensing MRI by the exploitation of data properties

Goodness‐of‐fit measures based on the Mellin transform for beta generalized lifetime data

Fast direct fourier reconstruction of radial and PROPELLER MRI data using the chirp transform algorithm on graphics hardware

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