Nonuniform Fourier Research Articles

Augmented Lagrangian with variable splitting for faster non-Cartesian L1-SPIRiT MR image reconstruction.

SPIRiT (iterative self-consistent parallel imaging reconstruction), and its sparsity-regularized variant L1-SPIRiT, are compatible with both Cartesian and non-Cartesian magnetic resonance imaging sampling trajectories. However, the non-Cartesian framework is more expensive computationally, involving a nonuniform Fourier transform with a nontrivial Gram matrix. We propose a novel implementation of the regularized reconstruction problem using variable splitting, alternating minimization of the augmented Lagrangian, and careful preconditioning. Our new method based on the alternating direction method of multipliers converges much faster than existing methods because of the preconditioners' heightened effectiveness. We demonstrate such rapid convergence substantially improves image quality for a fixed computation time. Our framework is a step forward towards rapid non-Cartesian L1-SPIRiT reconstructions.

Hybrid-SPRITE MRI

In a FID based frequency encoding MRI experiment the central part of k-space is not generally accessible due to the probe dead time. This portion of k-space is however crucial for image reconstruction. SPRITE (Single Point Ramped Imaging with T1 Enhancement), SPI with a linearly ramped phase encode gradient, has been employed to image short relaxation time systems for many years with great success. It is a robust imaging method in significant measure because it provides acquisition of high quality k-space origin data. We propose a new sampling scheme, termed hybrid-SPRITE, combining phase and frequency encoding to ensure high quality images with reduced acquisition times, reduced gradient duty cycle and increased sensitivity. In hybrid-SPRITE, numerous time domain points are collected to assist image reconstruction. An Inverse Non-uniform Discrete Fourier Transform (INDFT) is employed in 1D applications. A pseudo-polar grid is exploited in 2D hybrid-SPRITE for rapid and accurate image reconstruction.

基于空域非均匀傅里叶变换的傅里叶望远镜

In order to reconstruct the image using sparse emitter array, a process based on spatial non-uniform Fourier transform (NDFT) method was proposed. According to the relationship between the extracted spatial frequency of target and the position of Fourier telescope emitters and considering the MATLAB program characteristics, we completed the inverse Fourier transform of non-uniform airspace and reconstructed the image of target. Sparse emitter array configuration is as follows: 11 launch telescopes are placed on the T-array single arm, eight low-frequency components are continuously extracted and then followed by three high-frequency components. We selected four satellites with different shape and gray distribution as the imaging target. Comparing our reconstructed images with those by fast Fourier transform with zero-padding, we found that in the presence of noise (100 dB signal-to-noise ratio), the Strehl ratios of the images reconstructed with NDFT method are improved and the highest one is enhanced by 0.159 8.

Plane-wave ultrasound beamforming using a nonuniform fast fourier transform

Beamforming of plane-wave ultrasound echo signals in the Fourier domain provides fast and accurate image reconstruction. Conventional implementations perform a k-space interpolation from the uniform sampled grid to a nonuniform acoustic dispersion grid. In this paper, we demonstrate that this step can be replaced by a nonuniform Fourier transform. We study the performance of the nonuniform fast Fourier transform (NUFFT) in terms of signal-to-noise ratio and computational cost, and show that the NUFFT offers an advantage in the trade-off between speed and accuracy, compared with other frequency-domain beamforming strategies.

Highly accelerated PSF-mapping for EPI distortion correction with improved fidelity

This study presents an improved point-spread-function (PSF) mapping-based distortion correction method and accelerated PSF acquisition for distortion correction in EPI without loss of quality or reliability compared to full encoding. To correct geometric distortions accurately, the PSF in the EPI phase-encoding coordinates (EPI-PSF) was measured and used as a kernel for distortion correction. FOV reduction was applied in the PSF mapping dimension for highly accelerated PSF acquisition. A novel approach for fold-over artifact correction in this reduced dimension is introduced. Conventional gradient-echo EPI and corresponding full PSF reference data were acquired in phantoms and in human brain at 7 T. The distortion corrected EPI data with the proposed acceleration were compared to result with full encoding. Previously published interpolation methods based on shift maps, non-uniform Fourier transformation and a b-spline interpolation were compared with the proposed method. The results demonstrate that the proposed method corrects geometric distortions in EPI with high accuracy and quality despite the high acceleration. In contrast to partial parallel imaging acceleration, no noise enhancement is introduced. The proposed EPI-PSF-based distortion correction improves correction of EPI and accelerates PSF reference data acquisition and computation.

Recovering Exponential Accuracy from Non-harmonic Fourier Data Through Spectral Reprojection

Spectral reprojection techniques make possible the recovery of exponential accuracy from the partial Fourier sum of a piecewise-analytic function, essentially conquering the Gibbs phenomenon for this class of functions. This paper extends this result to non-harmonic partial sums, proving that spectral reprojection can reduce the Gibbs phenomenon in non-harmonic reconstruction as well as remove reconstruction artifacts due to erratic sampling. We are particularly interested in the case where the Fourier samples form a frame. These techniques are motivated by a desire to improve the quality of images reconstructed from non-uniform Fourier data, such as magnetic resonance (MR) images.

Augmented Lagrangian based reconstruction of non-uniformly sub-Nyquist sampled MRI data

MRI has recently been identified as a promising application for compressed-sensing-like regularization because of its potential to speed up the acquisition while maintaining the image quality. Thereby non-uniform k-space trajectories, such as random or spiral trajectories, are becoming more and more important, because they are well suited to be used within the compressed-sensing (CS) acquisition framework. In this paper, we propose a new reconstruction technique for non-uniformly sub-Nyquist sampled k-space data. Several parts make up this technique, such as the non-uniform Fourier transform (NUFT), the discrete shearlet transform and a augmented Lagrangian based optimization algorithm. Because MRI images are real-valued, we introduce a new imaginary value suppressing prior, which attenuates imaginary components of MRI images during reconstruction, resulting in a better overall image quality. Further, a preconditioning based on the Voronoi cell size of each NUFT data point speeds up the conjugate gradient optimization used as part of the optimization algorithm. The resulting algorithm converges in a relatively small number of iterations and guarantees solutions that fully comply to the imposed constraints. The results show that the algorithm is applicable not only to sub-Nyquist sampled k-space reconstruction, but also to MR image fusion and/or resolution enhancement.

Detection of Edges from Nonuniform Fourier Data

Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing (e.g. MRI) where the data is given in this way, however. This paper introduces a method for determining the locations of jump discontinuities, or edges, in a one-dimensional periodic piecewise-smooth function from nonuniform Fourier coefficients. The technique employs the use of Fourier frames. Numerical examples are provided.

Three-dimensional MR-encephalography: Fast volumetric brain imaging using rosette trajectories

MR-Encephalography (MREG) is a technique that allows real time observation of functional changes in the brain that appears within 100 msec. The high sampling rate is achieved at the cost of some spatial resolution. The article describes a novel imaging method for fast three-dimensional-MR-encephalography whole brain coverage based on rosette trajectories and the use of multiple small receiver coils. The technique allows the observation of changes in brain physiology at very high temporal resolution. A highly undersampled three-dimensional rosette trajectory is chosen, to perform single shot acquisition of k-space data within 23 msec. By using a 32-channel head coil array and regularized nonuniform Fourier transformation reconstruction, the spatial resolution is sufficient to detect even subtle centers of activation (e.g. human MT+). The method was applied to visual block design paradigms and compared with echo planar imaging-based functional MRI. As a proof-of-principle of the method's ability to detect local differences in the hemodynamic response functions, the analyzed MR-encephalography data revealed a spatially dependent delay of the arrival of the blood oxygenation level dependent response within the visual cortex.

NDFT-based Image Steganographic Scheme with Discrimination of Tampers

A new and secure image steganographic scheme based on nonuniform discrete Fourier transform (NDFT) is proposed in this paper. First, the chaotic system is introduced to select embedding points randomly in NDFT domain suitable range, and NDFT is implemented on every non-overlapping block of eight consecutive pixels. Second, the secret messages are scrambled by chaotic systems, and embedded into frequency coefficients by quantization method. The stego-image is obtained by inverse NDFT (INDFT). Besides, in order to discriminate tampers, the low frequency wavelet coefficients of 7 most significant bits (MSBs) of the stego-image are converted into the binary sequence after nonuniform scalar quantization. Then the obtained binary sequence is scrambled by the chaotic systems, and embedded into the least significant bit (LSB) of the stego-image. Finally, the watermarked stego-image can be obtained by a new improved LSB steganographic method. The embedded secret messages can be extracted from the watermarked stego-image without the original cover image. Experimental results show the validity of the proposed scheme, and dual statistics attacks are also conducted to indicate the security.

Non-Iterative Regularized reconstruction Algorithm for Non-CartesiAn MRI: NIRVANA

We introduce a novel noniterative algorithm for the fast and accurate reconstruction of nonuniformly sampled MRI data. The proposed scheme derives the reconstructed image as the nonuniform inverse Fourier transform of a compensated dataset. We derive each sample in the compensated dataset as a weighted linear combination of a few measured k-space samples. The specific k-space samples and the weights involved in the linear combination are derived such that the reconstruction error is minimized. The computational complexity of the proposed scheme is comparable to that of gridding. At the same time, it provides significantly improved accuracy and is considerably more robust to noise and undersampling. The advantages of the proposed scheme makes it ideally suited for the fast reconstruction of large multidimensional datasets, which routinely arise in applications such as f-MRI and MR spectroscopy. The comparisons with state-of-the-art algorithms on numerical phantoms and MRI data clearly demonstrate the performance improvement.

Fast Computation of Frequency Warping Transforms

In this paper, we introduce an analytical approach for the frequency warping transform. Criteria for the design of operators based on arbitrary warping maps are provided and an algorithm carrying out a fast computation is defined. Such operators can be used to shape the tiling of time-frequency (TF) plane in a flexible way. Moreover, they are designed to be inverted by the application of their adjoint operator. According to the proposed model, the frequency warping transform is computed by considering two additive operators: the first one represents its nonuniform Fourier transform approximation and the second one suppresses aliasing. The first operator is fast computable by various interpolation approaches. A factorization of the second operator is found for arbitrary shaped nonsmooth warping maps. By properly truncating the operators involved in the factorization, the computation turns out to be fast without compromising accuracy.

Gridding and fast Fourier transformation on non-uniformly sparse sampled multidimensional NMR data

For multidimensional NMR method, indirect dimensional non-uniform sparse sampling can dramatically shorten acquisition time of the experiments. However, the non-uniformly sampled NMR data cannot be processed directly using fast Fourier transform (FFT). We show that the non-uniformly sampled NMR data can be reconstructed to Cartesian grid with the gridding method that has been wide applied in MRI, and sequentially be processed using FFT. The proposed gridding-FFT (GFFT) method increases the processing speed sharply compared with the previously proposed non-uniform Fourier Transform, and may speed up application of the non-uniform sparse sampling approaches.

Optimized Least-Square Nonuniform Fast Fourier Transform

The main focus of this paper is to derive a memory efficient approximation to the nonuniform Fourier transform of a support limited sequence. We show that the standard nonuniform fast Fourier transform (NUFFT) scheme is a shift invariant approximation of the exact Fourier transform. Based on the theory of shift-invariant representations, we derive an exact expression for the worst-case mean square approximation error. Using this metric, we evaluate the optimal scale-factors and the interpolator that provides the least approximation error. We also derive the upper-bound for the error component due to the lookup tablebased evaluation of the interpolator; we use this metric to ensure that this component is not the dominant one. Theoretical and experimental comparisons with standard NUFFT schemes clearly demonstrate the significant improvement in accuracy over conventional schemes, especially when the size of the uniform fast Fourier transform (FFT) is small. Since the memory requirement of the algorithm is dependent on the size of the uniform FFT, the proposed developments can lead to iterative signal reconstruction algorithms with significantly lower memory demands.

Convolution-Based Trigonometric Interpolation of Band-Limited Signals

This paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band-limited signal from a finite sequence of samples. The polynomial delivers accurate approximations in the range covered by the sequence, except for a short frame close to the range limits. Besides, its accuracy increases exponentially with the frame width. The method is based on using a band-limited window in order to reduce the truncation error of a convolution series. It is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete Fourier transform (DFT). Specifically, two basic procedures are presented, one based on the fast Fourier transform (FFT), and another based on a recursive update algorithm for the short-time FFT. The paper contains three applications. The first is a variable fractional delay (VFD) filter, which consists of a short-time FFT combined with the evaluation of a trigonometric polynomial. This filter has low complexity and can be implemented using CORDIC rotations. The second is the interpolation of nonuniform Fourier summations, where the proposed method eliminates the need to interpolate any kernel sample. Finally, the third can be viewed as a generalization of the FFT convolution algorithm and makes it possible to interpolate the output of an finite-impulse-response (FIR) filter efficiently.

Estimation of Interchannel Time Difference in Frequency Subbands Based on Nonuniform Discrete Fourier Transform

Binaural cue coding (BCC) is an efficient technique for spatial audio rendering by using the side information such as interchannel level difference (ICLD), interchannel time difference (ICTD), and interchannel correlation (ICC). Of the side information, the ICTD plays an important role to the auditory spatial image. However, inaccurate estimation of the ICTD may lead to the audio quality degradation. In this paper, we develop a novel ICTD estimation algorithm based on the nonuniform discrete Fourier transform (NDFT) and integrate it with the BCC approach to improve the decoded auditory image. Furthermore, a new subjective assessment method is proposed for the evaluation of auditory image widths of decoded signals. The test results demonstrate that the NDFT-based scheme can achieve much wider and more externalized auditory image than the existing BCC scheme based on the discrete Fourier transform (DFT). It is found that the present technique, regardless of the image width, does not deteriorate the sound quality at the decoder compared to the traditional scheme without ICTD estimation.

An accurate nonuniform fourier transform for SPRITE magnetic resonance imaging data

A new algorithm is proposed for computing the discrete Fourier Transform (DFT) of purely phase encoded data acquired during Magnetic Resonance Imaging (MRI) experiments. These experiments use the SPRITE (Single Point Ramped Imaging with T 1 Enhancement) method and multiple-point acquisition, sampling data in a nonuniform manner that prohibits reconstruction by fast Fourier transform. The chirp z-transform algorithm of Rabiner, Schafer, and Rader can be combined with phase corrections to compute the DFT of this data to extremely high accuracy. This algorithm outperforms the interpolation methods that are traditionally used to process nonuniform data, both in terms of execution time and in terms of accuracy as compared to the DFT.

Parabolic Radon transform, sampling and efficiency

A good choice of the sampling in the transform domain is essential for a successful application of the parabolic Radon transform. The parabolic Radon transform is computed for each temporal frequency and is essentially equivalent to the nonuniform Fourier transform. This leads to new and useful insights in the parabolic Radon transform. Using nonuniform Fourier theory, we derive a minimum sampling interval for the curvature parameter and a maximum curvature range for which stability is guaranteed for general (irregular) sampling. A significantly smaller sampling interval requires stabilization. If diagonal stabilization is used, no gain in resolution is obtained. In contrast to conventional implementations, the curvature sampling interval is proposed to be inversely proportional to the temporal frequency. This results in improved quality of the transform and yields significant savings in computation time.