Applying the uniform resampling (URS) algorithm to a lissajous trajectory: fast image reconstruction with optimal gridding

Resolution adjustable Lissajous scanning with piezoelectric MEMS mirrors

We previously designed a dual-axis piezoelectric MEMS mirror with a low crosstalk gimbal structure, which is utilized as the key device for further research for laser beam scanning. This paper mainly focuses on studying the Lissajous scanning resolution of this MEMS mirror with frequency ratio and phase modulation. For accurately evaluating the scanning resolution, the center angular resolution of Lissajous scanning is redefined by theoretical calculation and verified with experimental measurement. Meanwhile, the scanning nonlinearity of MEMS mirror is studied carefully. Finally, the MEMS mirror works at the state of pseudo-resonance, and the center angular resolution better than 0.16° (H) × 0.03° (V) is achieved at a scanning Field of view (FoV) of 35.0° (H) × 16.5° (V). Moreover, a feasible route of resolution adjustable Lissajous scanning is provided by optimization of frequency ratio and phase modulation, which is helpful for high definition and high frame rate (HDHF) laser scanning imaging with the dual-axis mirror.

Improved sampling scheme for LiDAR in Lissajous scanning mode

MEMS light detection and ranging (LiDAR) is becoming an indispensable sensor in vehicle environment sensing systems due to its low cost and high performance. The beam scanning trajectory, sampling scheme and gridding are the key technologies of MEMS LiDAR imaging. In Lissajous scanning mode, this paper improves the sampling scheme, through which a denser Cartesian grid of point cloud data at the same scanning frequency can be obtained. By summarizing the rules of the Cartesian grid, a general sampling scheme independent of the beam scanning trajectory patterns is proposed. Simulation and experiment results show that compared with the existing sampling scheme, the resolution and the number of points per frame are both increased by 2 times with the same hardware configuration and scanning frequencies for a MEMS scanning mirror (MEMS-SM). This is beneficial for improving the point cloud imaging performance of MEMS LiDAR.

Adaptive Lissajous scanning pattern design by phase modulation

This paper proposes a phase modulation method for Lissajous scanning systems, which provides adaptive scan pattern design without changing the frame rate or the field of view. Based on a rigorous analysis of Lissajous scanning, phase modulation constrains and a method for pixel calculation are derived. An accurate and simple metric for resolution calculation is proposed based on the area spanned by neighboring pixels and used for scan pattern optimization also considering the scanner dynamics. The methods are implemented using MEMS mirrors for verification of the adaptive pattern shaping, where a 5-fold resolution improvement in a defined region of interest is demonstrated.

Evaluation of partial k-space strategies to speed up time-domain EPR imaging

Narrow-line spin probes derived from the trityl radical have led to the development of fast in vivo time-domain EPR imaging. Pure phase-encoding imaging modalities based on the single-point imaging scheme have demonstrated the feasibility of three-dimensional oximetric images with functional information in minutes. In this article, we explore techniques to improve the temporal resolution and circumvent the relatively short biological half-lives of trityl probes using partial k-space strategies. There are two main approaches: one involves the use of the Hermitian character of the k-space by which only part of the k-space is measured and the unmeasured part is generated using the Hermitian symmetry. This approach is limited in success by the accuracy of numerical estimate of the phase roll in the k-space that corrupts the Hermiticy. The other approach is to measure only a judicially chosen reduced region of k-space (a centrosymmetric ellipsoid region) that more or less accounts for >70% of the k-space energy. Both of these aspects were explored in Fourier transform-EPR imaging with a doubling of scan speed demonstrated by considering ellipsoid geometry of the k-space. Partial k-space strategies help improve the temporal resolution in studying fast dynamics of functional aspects in vivo with infused spin probes.

High-speed Lissajous-scan atomic force microscopy: scan pattern planning and control design issues

Tracking of triangular or sawtooth waveforms is a major difficulty for achieving high-speed operation in many scanning applications such as scanning probe microscopy. Such non-smooth waveforms contain high order harmonics of the scan frequency that can excite mechanical resonant modes of the positioning system, limiting the scan range and bandwidth. Hence, fast raster scanning often leads to image distortion. This paper proposes analysis and design methodologies for a nonlinear and smooth closed curve, known as Lissajous pattern, which allows much faster operations compared to the ordinary scan patterns. A simple closed-form measure is formulated for the image resolution of the Lissajous pattern. This enables us to systematically determine the scan parameters. Using internal model controllers (IMC), this non-raster scan method is implemented on a commercial atomic force microscope driven by a low resonance frequency positioning stage. To reduce the tracking errors due to actuator nonlinearities, higher order harmonic oscillators are included in the IMC controllers. This results in significant improvement compared to the traditional IMC method. It is shown that the proposed IMC controller achieves much better tracking performances compared to integral controllers when the noise rejection performances is a concern.

Efficient sample density estimation by combining gridding and an optimized kernel

The reconstruction of non-Cartesian k-space trajectories often requires the estimation of nonuniform sampling density. Particularly for 3D, this calculation can be computationally expensive. The method proposed in this work combines an iterative algorithm previously proposed by Pipe and Menon (Magn Reson Med 1999;41:179-186) with the optimal kernel design previously proposed by Johnson and Pipe (Magn Reson Med 2009;61:439-447). The proposed method shows substantial time reductions in estimating the densities of center-out trajectories, when compared with that of Johnson. It is demonstrated that, depending on the trajectory, the proposed method can provide reductions in execution time by factors of 12 to 85. The method is also shown to be robust in areas of high trajectory overlap, when compared with two analytical density estimation methods, producing a 10-fold increase in accuracy in one case. Initial conditions allow the proposed method to converge in fewer iterations and are shown to be flexible in terms of the accuracy of information supplied. The proposed method is not only one of the fastest and most accurate algorithms, it is also completely generic, allowing any arbitrary trajectory to be density compensated extemporaneously. The proposed method is also simple and can be implemented on parallel computing platforms in a straightforward manner.

A least squares quantization table method for direct reconstruction of MR images with non-Cartesian trajectory

The direct Fourier transform method is a straightforward solution with high accuracy for reconstructing magnetic resonance (MR) images from nonuniformly sampled k-space data, given that the optimal density compensation function is selected and the underlying magnetic field is sufficiently uniform. The computation however is very time-consuming, making it impractical especially for large-size images. In this paper, the least squares quantization table (LSQT) method is proposed to accelerate the direct Fourier transform computation, similar to the recently proposed methods such as using look-up table (LUT) or equal-phase-line (EPL). With LSQT, all the image pixels are first classified into several groups where the Lloyd-Max quantization scheme is used to ensure the minimal classification error. The representative value of each group is stored in a small-size LSQT in advance to reduce the computational load. The pixels in the same group receive the same contribution, which is calculated only once for each group instead of for each pixel, resulting in the reduction of computation because the number of groups is far smaller than the number of pixels. Finally, each image pixel is mapped into the nearest group and its representative value is used to reconstruct the image. The experimental results show that the LSQT method requires far smaller memory size than the LUT method and fewer multiplication operations than the LUT and EPL methods. Moreover, the LSQT method can perform large-size reconstructions that achieve comparable or higher accuracy as compared to the EPL and gridding methods when the appropriate parameters are given. The inherent parallel structure also makes the LSQT method easily adaptable to a multiprocessor system.

Methods for quantitative image quality evaluation of MRI parallel reconstructions: detection and perceptual difference model

Many reconstruction algorithms are being proposed for parallel magnetic resonance imaging (MRI), which uses multiple coils and subsampled k-space data, and a quantitative method for comparison of algorithms is sorely needed. On such images, we compared three methods for quantitative image quality evaluation: human detection, computer detection model and a computer perceptual difference model (PDM). One-quarter sampling and three different reconstruction methods were investigated: a regularization method developed by Ying et al., a simplified regularization method and an iterative method proposed by Pruessmann et al. Images obtained from a full complement of k-space data were also included as reference images. Detection studies were performed using a simulated dark tumor added on MR images of fresh bovine liver. Human detection depended strongly on reconstruction methods used, with the two regularization methods achieving better performance than the iterative method. Images were also evaluated using detection by a channelized Hotelling observer model and by PDM scores. Both predicted the same trends as observed from human detection. We are encouraged that PDM gives trends similar to that for human detection studies. Its ease of use and applicability to a variety of MRI situations make it attractive for evaluating image quality in a variety of MR studies.

Deconvolution-interpolation gridding (DING): accurate reconstruction for arbitrary k-space trajectories

A simple iterative algorithm, termed deconvolution-interpolation gridding (DING), is presented to address the problem of reconstructing images from arbitrarily-sampled k-space. The new algorithm solves a sparse system of linear equations that is equivalent to a deconvolution of the k-space with a small window. The deconvolution operation results in increased reconstruction accuracy without grid subsampling, at some cost to computational load. By avoiding grid oversampling, the new solution saves memory, which is critical for 3D trajectories. The DING algorithm does not require the calculation of a sampling density compensation function, which is often problematic. DING's sparse linear system is inverted efficiently using the conjugate gradient (CG) method. The reconstruction of the gridding system matrix is simple and fast, and no regularization is needed. This feature renders DING suitable for situations where the k-space trajectory is changed often or is not known a priori, such as when patient motion occurs during the scan. DING was compared with conventional gridding and an iterative reconstruction method in computer simulations and in vivo spiral MRI experiments. The results demonstrate a stable performance and reduced root mean square (RMS) error for DING in different k-space trajectories.

Progressive magnetic resonance image reconstruction based on iterative solution of a sparse linear system

Image reconstruction from nonuniformly sampled spatial frequency domain data is an important problem that arises in computed imaging. Current reconstruction techniques suffer from limitations in their model and implementation. In this paper, we present a new reconstruction method that is based on solving a system of linear equations using an efficient iterative approach. Image pixel intensities are related to the measured frequency domain data through a set of linear equations. Although the system matrix is too dense and large to solve by direct inversion in practice, a simple orthogonal transformation to the rows of this matrix is applied to convert the matrix into a sparse one up to a certain chosen level of energy preservation. The transformed system is subsequently solved using the conjugate gradient method. This method is applied to reconstruct images of a numerical phantom as well as magnetic resonance images from experimental spiral imaging data. The results support the theory and demonstrate that the computational load of this method is similar to that of standard gridding, illustrating its practical utility.

Optimal design of k-space trajectories using a multi-objective genetic algorithm

Spiral, radial, and other nonrectilinear k-space trajectories are an area of active research in MRI due largely to their typically rapid acquisition times and benign artifact patterns. Trajectory design has commonly proceeded from a description of a simple shape to an investigation of its properties, because there is no general theory for the derivation of new trajectories with specific properties. Here such a generalized methodology is described. Specifically, a multi-objective genetic algorithm (GA) is used to design trajectories with beneficial flow and off-resonance properties. The algorithm converges to a well-defined optimal set with standard spiral trajectories on the rapid but low-quality end, and a new class of trajectories on the slower but high-quality end. The new trajectories all begin with nonzero gradient amplitude at the k-space origin, and curve gently outward relative to standard spirals. Improvements predicted in simulated imaging experiments were found to correlate well with improvements in actual experimental measures of image quality. The impact of deviations from the desired k-space trajectory is described, as is the impact of using different phantoms.

Iterative Next-Neighbor Regridding (INNG): improved reconstruction from nonuniformly sampled k-space data using rescaled matrices

The reconstruction of MR images from nonrectilinearly sampled data is complicated by the fact that the inverse 2D Fourier transform (FT) cannot be performed directly on the acquired k-space data set. k-Space gridding is commonly used because it is an efficient reconstruction method. However, conventional gridding requires optimized density compensation functions (DCFs) to avoid profile distortions. Oftentimes, the calculation of optimized DCFs presents an additional challenge in obtaining an accurately gridded reconstruction. Another type of gridding algorithm, the block uniform resampling (BURS) algorithm, often requires singular value decomposition (SVD) regularization to avoid amplification of data imperfections, and under some conditions it is difficult to adjust the regularization parameters. In this work, new reconstruction algorithms for nonuniformly sampled k-space data are presented. In the newly proposed algorithms, high-quality reconstructed images are obtained from an iterative reconstruction that is performed using matrices scaled to sizes greater than that of the target image matrix. A second version partitions the sampled k-space region into several blocks to avoid limitations that could result from performing multiple 2D-FFTs on large data matrices. The newly proposed algorithms are a simple alternative approach to previously proposed optimized gridding algorithms.

New approach to gridding using regularization and estimation theory

When sampling under time-varying gradients, data is acquired over a non-equally spaced grid in k-space. The most computationally efficient method of reconstruction is first to interpolate the data onto a Cartesian grid, enabling the subsequent use of the inverse fast Fourier transform (IFFT). The most commonly used interpolation technique is called gridding, and is comprised of four steps: precompensation, convolution with a Kaiser-Bessel window, IFFT, and postcompensation. Recently, the author introduced a new gridding method called Block Uniform ReSampling (BURS), which is both optimal and efficient. The interpolation coefficients are computed by solving a set of linear equations using singular value decomposition (SVD). BURS requires neither the pre- nor the postcompensation steps, and resamples onto an n x n grid rather than the 2n x 2n matrix required by conventional gridding. This significantly decreases the computational complexity. Several authors have reported that although the BURS algorithm is very accurate, it is also sensitive to noise. As a consequence, even in the presence of a low level of measurement noise, the resulting image is often highly contaminated with noise. In this work, the origin of the noise sensitivity is traced back to the potentially ill-posed matrix inversion performed by BURS. Two approaches to the solution are presented. The first uses regularization theory to stabilize the inversion process. The second formulates the interpolation as an estimation problem, and employs estimation theory for the solution. The new algorithm, called rBURS, contains a regularization parameter, which is used to trade off the accuracy of the result against the signal-to-noise ratio (SNR). The results of the new method are compared with those obtained using conventional gridding via simulations. For the SNR performance of conventional gridding, it is shown that the rBURS algorithm exhibits equal or better accuracy. This is achieved at a decreased computational cost compared to conventional gridding.

Modified block uniform resampling (BURS) algorithm using truncated singular value decomposition: fast accurate gridding with noise and artifact reduction

The block uniform resampling (BURS) algorithm is a newly proposed regridding technique for nonuniformly-sampled k-space MRI. Even though it is a relatively computationally intensive algorithm, since it uses singular value decomposition (SVD), its procedure is simple because it requires neither a pre- nor a postcompensation step. Furthermore, the reconstructed image is generally of high quality since it provides accurate gridded values when the local k-space data SNR is high. However, the BURS algorithm is sensitive to noise. Specifically, inaccurate interpolated data values are often generated in the BURS algorithm if the original k-space data are corrupted by noise, which is virtually guaranteed to occur to some extent in MRI. As a result, the reconstructed image quality is degraded despite excellent performance under ideal conditions. In this article, a method is presented which avoids inaccurate interpolated k-space data values from noisy sampled data with the BURS algorithm. The newly proposed technique simply truncates a series of singular values after the SVD is performed. This reduces the computational demand when compared with the BURS algorithm, avoids amplification of noise resulting from small singular values, and leads to image SNR improvements over the original BURS algorithm.