A fast and accurate Fourier algorithm for iterative parallel-beamtomography

We use a series-expansion approach and an operator framework to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction. This algorithm is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires O(N²logN) floating-point operations per iteration to reconstruct an N×N image from P view angles, as compared to O(N ²P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary     

Fast,iterative image reconstruction for MRI in the presence of field inhomogeneities

In magnetic resonance imaging, magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast Fourier trasform (FFT) methods. Several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities, but these methods assume that the field map that characterizes the off-resonance frequencies is spatially smooth. Recently, iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off-resonance effects. However, straightforward implementations of such iterative methods suffer from inconveniently long computation times. This paper describes a tool for accelerating iterative reconstruction of field-corrected MR images: a novel time-segmented approximation to the MR signal equation. We use a min-max formulation to derive the temporal interpolator. Speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast Fourier transform with normalized root mean squared approximation errors of 0.07%. The proposed method provides fast, accurate, field-corrected image reconstruction even when the field map is not smooth.     

A fast wavelet-based reconstruction method for magnetic resonance imaging

In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary k-space trajectories. Reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage/thresholding algorithm (ISTA) which, unfortunately, converges slowly. To make the approach more practical, we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k-space trajectory. We present a mathematical analysis that explains the performance of the algorithms. Using simulated and in vivo data, we show that our nonlinear method is fast, as it accelerates ISTA by almost two orders of magnitude. We also show that it remains competitive with TV regularization in terms of image quality.     

An EM algorithm for wavelet-based image restoration

This paper introduces an expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in the wavelet coefficients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, the resulting criteria are solved approximately or require demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation offered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. Thus, it is a general-purpose approach to wavelet-based image restoration with computational complexity comparable to that of standard wavelet denoising schemes or of frequency domain deconvolution methods. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an efficient iterative process requiring O(NlogN) operations per iteration. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach performs competitively with, in some cases better than, the best existing methods in benchmark tests.     

一种车载毫米波FMCW MIMO雷达快速成像方法

针对车载毫米波FMCW MIMO雷达现有的常规波束形成算法的旁瓣效应造成的方位向分辨率低以及高分辨算法的工程实时性低的问题,提出了一种迭代自适应算法（IAA）高分辨成像的快速实现方法。该方法首先利用快速傅里叶变换（FFT）获取目标一维距离像,然后对每一距离单元利用FFT算子和Gohberg-Semencul（GS）因子分解计算迭代自适应算法（IAA）的数据协方差矩阵和其逆矩阵,利用快速Toeplitz矩阵向量乘法计算IAA迭代值,从整体上提升了IAA估计各角度散射系数的实时性。仿真和实验结果验证了该方法的可行性和有效性。     

Reconstruction algorithms: Transform methods

Transform methods for image reconstruction from projections are based on analytic inversion formulas. In this tutorial paper, the inversion formula for the case of two-dimensional (2-D) reconstruction from line integrals is manipulated into a number of different forms, each of which may be discretized to obtain different algorithms for reconstruction from sampled data. For the convolution-backprojection algorithm and the direct Fourier algorithm the emphasis is placed on understanding the relationship between the discrete operations specified by the algorithm and the functional operations expressed by the inversion formula. The performance of the Fourier algorithm may be improved, with negligible extra computation, by interleaving two polar sampling grids in Fourier space. The convolution-backprojection formulas are adapted for the fan-beam geometry, and other reconstruction methods are summarized, including the rho-filtered layergram method, and methods involving expansions in angular harmonics. A standard mathematical process leads to a known formula for iterative reconstruction from projections at a finite number of angles. A new iterative reconstruction algorithm is obtained from this formula by introducing one-dimensional (1-D) and 2-D interpolating functions, applied to sampled projections and images, respectively. These interpolating functions are derived by the same Fourier approach which aids in the development and understanding of the more conventional transform methods.     

Inverse scattering of two-dimensional dielectric objects buried ina lossy earth using the distorted Born iterative method

An efficient inverse-scattering algorithm is developed to reconstruct both the permittivity and conductivity profiles of two-dimensional (2D) dielectric objects buried in a lossy earth using the distorted Born iterative method (DBIM). In this algorithm, the measurement data are collected on (or over) the air-earth interface for multiple transmitter and receiver locations at single frequency. The nonlinearity due to the multiple scattering of pixels to pixels, and pixels to the air-earth interface has been taken into account in the iterative minimization scheme. At each iteration, a conjugate gradient (CG) method is chosen to solve the linearized problem, which takes the calling number of the forward solver to a minimum. To reduce the CPU time, the forward solver for buried dielectric objects is implemented by the CG method and fast Fourier transform (FFT). Numerous numerical examples are given to show the convergence, stability, and error tolerance of the algorithm     

Conjugate phase MRI reconstruction with spatially variant sample density correction

A new image reconstruction method to correct for the effects of magnetic field inhomogeneity in non-Cartesian sampled magnetic resonance imaging (MRI) is proposed. The conjugate phase reconstruction method, which corrects for phase accumulation due to applied gradients and magnetic field inhomogeneity, has been commonly used for this case. This can lead to incomplete correction, in part, due to the presence of gradients in the field inhomogeneity function. Based on local distortions to the k-space trajectory from these gradients, a spatially variant sample density compensation function is introduced as part of the conjugate phase reconstruction. This method was applied to both simulated and experimental spiral imaging data and shown to produce more accurate image reconstructions. Two approaches for fast implementation that allow the use of fast Fourier transforms are also described. The proposed method is shown to produce fast and accurate image reconstructions for spiral sampled MRI.     

Fast MR image reconstruction for partially parallel imaging with arbitrary k-space trajectories

Both acquisition and reconstruction speed are crucial for magnetic resonance (MR) imaging in clinical applications. In this paper, we present a fast reconstruction algorithm for SENSE in partially parallel MR imaging with arbitrary k-space trajectories. The proposed method is a combination of variable splitting, the classical penalty technique and the optimal gradient method. Variable splitting and the penalty technique reformulate the SENSE model with sparsity regularization as an unconstrained minimization problem, which can be solved by alternating two simple minimizations: One is the total variation and wavelet based denoising that can be quickly solved by several recent numerical methods, whereas the other one involves a linear inversion which is solved by the optimal first order gradient method in our algorithm to significantly improve the performance. Comparisons with several recent parallel imaging algorithms indicate that the proposed method significantly improves the computation efficiency and achieves state-of-the-art reconstruction quality.     

Labeling of MR brain images using Boolean neural network

Presents a knowledge-based approach for labeling two-dimensional (2-D) magnetic resonance (MR) brain images using the Boolean neural network (BNN), which has binary inputs and outputs, integer weights, fast learning and classification, and guaranteed convergence. The approach consists of two components: a BNN clustering algorithm and a constraint satisfying Boolean neural network (CSBNN) labeling procedure. The BNN clustering algorithm is developed to initially segment an image into a number of regions. Then the segmented regions are labeled with the CSBNN, which is a modified version of BNN. The CSBNN uses a knowledge base that contains information on image-feature space and tissue models as constraints. The method is tested using sets of MR brain images. The regions of the different brain tissues are satisfactorily segmented and labeled. A comparison with the Hopfield neural network and the traditional simulated annealing method for image labeling is provided. The comparison results show that the CSBNN approach offers a fast, feasible, and reliable alternative to the existing techniques for medical image labeling.     

An efficient method for dynamic magnetic resonance imaging

Many magnetic resonance imaging applications require the acquisition of a time series of images. In conventional Fourier transform based imaging methods, each of these images is acquired independently so that the temporal resolution possible is limited by the number of spatial encodings (or data points in the Fourier space) collected, or one has to sacrifice spatial resolution for temporal resolution. Here, a generalized series based imaging technique is proposed to address this problem. This technique makes use of the fact that, in most time-sequential imaging problems, the high-resolution image morphology does not change from one image to another, and it improves imaging efficiency (and temporal resolution) over the conventional Fourier imaging methods by eliminating the repeated encodings of this stationary information. Additional advantages of the proposed imaging technique include a reduced number of radio frequency (RF) pulses for data collection, and thus lower RF power deposition. This method should prove useful for a variety of dynamic imaging applications, including dynamic studies of contrast agents and functional brain imaging.     

基于AR模型的最大熵磁共振成像重建算法

本文提出了一种基于ＡＲ模型的磁共振成像算法，通过所获得的自回归系数及线性预测误差，替代了传统的快速傅里叶（ＦＦＴ）重建方法，获得了满意的重建图像，由于在实际系统中，只能得到有限的频谱数据，利用传统的ＦＦＴ方法重建磁共振图像，将导致截断伪影和低的分辨率。本算法利用ＡＲ模型外推未知频谱数据，替代ＦＦＴ方法的填零法重建，并利用ＢＵＲＧ算法中的ＡＲ模型参数计算的有效性，不仅消除了截断伪影，抑制噪声提高了分     

Fast l₁-SPIRiT compressed sensing parallel imaging MRI: scalable parallel implementation and clinically feasible runtime

We present l?-SPIRiT, a simple algorithm for auto calibrating parallel imaging (acPI) and compressed sensing (CS) that permits an efficient implementation with clinically-feasible runtimes. We propose a CS objective function that minimizes cross-channel joint sparsity in the wavelet domain. Our reconstruction minimizes this objective via iterative soft-thresholding, and integrates naturally with iterative self-consistent parallel imaging (SPIRiT). Like many iterative magnetic resonance imaging reconstructions, l?-SPIRiT's image quality comes at a high computational cost. Excessively long runtimes are a barrier to the clinical use of any reconstruction approach, and thus we discuss our approach to efficiently parallelizing l?-SPIRiT and to achieving clinically-feasible runtimes. We present parallelizations of l?-SPIRiT for both multi-GPU systems and multi-core CPUs, and discuss the software optimization and parallelization decisions made in our implementation. The performance of these alternatives depends on the processor architecture, the size of the image matrix, and the number of parallel imaging channels. Fundamentally, achieving fast runtime requires the correct trade-off between cache usage and parallelization overheads. We demonstrate image quality via a case from our clinical experimentation, using a custom 3DFT spoiled gradient echo (SPGR) sequence with up to 8× acceleration via Poisson-disc undersampling in the two phase-encoded directions.     

Extension of finite-support extrapolation using the generalizedseries model for MR spectroscopic imaging

In magnetic resonance (MR) imaging, limited data sampling in k-space leads to the well-known Fourier truncation artifact, which includes ringing and blurring. This problem is particularly severe for MR spectroscopic imaging, where only 16-24 points are typically acquired along each spatial dimension. Several methods have been proposed to overcome this problem by incorporating prior information in the image reconstruction. These include the generalized series (GS) model and the finite-support extrapolation method. This paper shows the connection between finite-support extrapolation and the GS model. In particular, finite-support extrapolation is a limiting case of the GS model, when the only available prior information is the support region. The support region refers to those image portions with nonzero intensities, and it can be estimated in practice as the nonbackground region of an image. By itself, the support region constitutes a rather weak constraint that may not lead to considerable resolution gain. This situation can be improved by using additional prior information, which can be incorporated systematically with the GS model. Examples of such additional prior information include intensity estimates of anatomical structures inside the support region.     

基于压缩感知的二维联合超分辨ISAR成像算法

在ISAR成像中,距离和方位分辨率分别受发射信号带宽和成像积累角的限制。基于压缩感知(CS)理论,该文提出了一种2维联合超分辨ISAR成像算法。首先建立ISAR观测信号模型并构造2维超分辨字典,然后利用ISAR图像的稀疏先验信息将2维联合超分辨成像建模为最小l1范数的优化问题,最后提出一种快速算法求解该优化问题。该方法进行距离维和方位维的联合处理,有效利用了回波数据的2维耦合信息;通过共轭梯度(CG)运算,快速傅里叶变换(FFT),Hadamard乘积等操作,有效提高了算法的实现效率。仿真和实测实验验证了该算法的有效性。     

Model-Based Iterative Reconstruction for Radial Fast Spin-Echo MRI

In radial fast spin-echo magnetic resonance imaging (MRI), a set of overlapping spokes with an inconsistent T2 weighting is acquired, which results in an averaged image contrast when employing conventional image reconstruction techniques. This work demonstrates that the problem may be overcome with the use of a dedicated reconstruction method that further allows for T2 quantification by extracting the embedded relaxation information. Thus, the proposed reconstruction method directly yields a spin-density and relaxivity map from only a single radial data set. The method is based on an inverse formulation of the problem and involves a modeling of the received MRI signal. Because the solution is found by numerical optimization, the approach exploits all data acquired. Further, it handles multicoil data and optionally allows for the incorporation of additional prior knowledge. Simulations and experimental results for a phantom and human brain in vivo demonstrate that the method yields spin-density and relaxivity maps that are neither affected by the typical artifacts from TE mixing, nor by streaking artifacts from the incomplete k-space coverage at individual echo times.      

Computation of the mid-sagittal plane in 3-D brain images

We present a new method to automatically compute, reorient, and recenter the mid-sagittal plane in anatomical and functional three-dimensional (3-D) brain images. This iterative approach is composed of two steps. At first, given an initial guess of the mid-sagittal plane (generally, the central plane of the image grid), the computation of local similarity measures between the two sides of the head allows to identify homologous anatomical structures or functional areas, by way of a block matching procedure. The output is a set of point-to-point correspondences: the centers of homologous blocks. Subsequently, we define the mid-sagittal plane as the one best superposing the points on one side and their counterparts on the other side by reflective symmetry. Practically, the computation of the parameters characterizing the plane is performed by a least trimmed squares estimation. Then, the estimated plane is aligned with the center of the image grid, and the whole process is iterated until convergence. The robust estimation technique we use allows normal or abnormal asymmetrical structures or areas to be treated as outliers, and the plane to be mainly computed from the underlying gross symmetry of the brain. The algorithm is fast and accurate, even for strongly tilted heads, and even in presence of high acquisition noise and bias field, as shown on a large set of synthetic data. The algorithm has also been visually evaluated on a large set of real magnetic resonance (MR) images. We present a few results on isotropic as well as anisotropic anatomical (MR and computed tomography) and functional (single photon emission computed tomography and positron emission tomography) real images, for normal and pathological subjects.     

An improved algorithm for rotational motion artifact suppression inMRI

An improved algorithm for planar rotational motion artifact suppression in standard two-dimensional Fourier transform magnetic resonance images is presented. It is shown that interpolation of acquired view data on the uncorrupted k-space create data overlap and void regions. The authors present a method of managing overlap data regions, using weighted averaging of redundant data. The weights are assigned according to a priority ranking based on the minimum distance between the data set and the k-space grid points. An iterative estimation technique for filling the data void regions, using projections onto convex sets (POCS), is also described. The method has been successfully tested using computer simulations     

Imaging heart motion using harmonic phase MRI

This paper describes a new image processing technique for rapid analysis and visualization of tagged cardiac magnetic resonance (MR) images. The method is based on the use of isolated spectral peaks in spatial modulation of magnetization (SPAMM)-tagged magnetic resonance images. We call the calculated angle of the complex image corresponding to one of these peaks a harmonic phase (HARP) image and show that HARP images can be used to synthesize conventional tag lines, reconstruct displacement fields for small motions, and calculate two-dimensional (2-D) strain. The performance of this new approach is demonstrated using both real and simulated tagged MR images. Potential for use of HARP images in fast imaging techniques and three-dimensional (3-D) analyses are discussed.     

Application of autoregressive moving average parametric modeling in magnetic resonance image reconstruction

The modeling of data is an alternative to conventional use of the fast Fourier transform (FFT) algorithm in the reconstruction of magnetic resonance (MR) images. The application of the FFT leads to artifacts and resolution loss in the image associated with the effective window on the experimentally-truncated phase encoded MR data. The transient error modeling method treats the MR data as a subset of the transient response of an infinite impulse filter (H(z) = B(z)IA(z)). Thus, the data are approximated by a deterministic autoregressive moving average (ARMA) model. The algorithm for calculating the filter coefficients is described. It is demonstrated that using the filter coefficients to reconstruct the image removes the truncation artifacts and improves the resolution. However, determining the autoregressive (AR) portion of the ARMA filter by algorithms that minimize the forward and backward prediction errors (e.g., Burg) leads to significant image degradation. The moving average (MA) portion is determined by a computationally efficient method of solving a finite difference equation with initial values. Special features of the MR data are incorporated into the transient error model. The sensitivity to noise and the choice of the best model order are discussed. MR images formed using versions of the transient error reconstruction (TERE) method and the conventional FFT algorithm are compared using data from a phantom and a human subject. Finally, the computational requirements of the algorithm are addressed.