The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformly-spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

any reduction in scan time offers a number of potential benefits ranging from high-temporal-rate observation of physiological processes to improvements in patient comfort. Following recent developments in Compressive Sensing (CS) theory, several authors have demonstrated that certain classes of MR images which possess sparse representations in some transform domain can be accurately reconstructed from very highly undersampled K-space data by solving a convex ℓ1-minimization problem. Although ℓ1-based techniques are extremely powerful, they inherently require a degree of over-sampling above the theoretical minimum sampling rate to guarantee that exact reconstruction can be achieved. In this paper, we propose a generalization of the Compressive Sensing paradigm based on homotopic approximation of the ℓ0 quasi-norm and show how MR image reconstruction can be pushed even further below the Nyquist limit and significantly closer to the theoretical bound. Following a brief review of standard Compressive Sensing methods and the developed theoretical extensions, several example MRI reconstructions from highly undersampled K-space data are presented.

Abstract-Parallel imaging methods such as SMASH and SENSE reduce imaging time by using receiver coil sensitivity patterns to reduce requirements for gradient based image localization. This paper describes the first use of a coil array to completely eliminate gradient phase encoding. By using a 64 element planar array and a custom built 64 channel receiver system, 64x256 resolution images were constructed from a single line of k-space, demonstrating the ability to form complete images from a single echo acquisition. Together, the receiver and array are capable of generating complete images during successive echo acquisitions, potentially enabling extremely rapid frame rates.

Parallel imaging is a powerful technique to speed up Magnetic Resonance (MR) image acquisition via multiple coils. Both the received signal of each coil and its sensitivity map, which describes its spatial response, are needed during reconstruction. Widely used schemes such as SENSE assume that sensitivity maps of the coils are noiseless while the only errors are due to a noisy signal. In practice, however sensitivity maps are subject to a wide variety of errors. At first glance, sensitivity noise appears to result in an errors-in-variables problem of the kind that is typically solved using Total Least Squares (TLS). However, existing TLS algorithms are inappropriate for the specific type of block structure that arises in parallel imaging. In this paper we take a maximum likelihood approach to the problem of parallel imaging in the presence of independent Gaussian sensitivity noise. This results in a quasi-quadratic objective function, which can be efficiently minimized. Experimental evidence suggests substantial gains over conventional SENSE, especially in non-ideal imaging conditions like low SNR, high g-factors, large acceleration and misaligned sensitivity maps.

Parallel imaging techniques for MRI use differences in spatial sensitivity of multiple receiver coils to achieve additional encoding effect and significantly reduce data acquisition time. Recently, a projection onto convex sets (POCS) based method for reconstruction from sensitivity-encoded data (POCSENSE) has been proposed. The main advantage of the POCSENCE in comparison with other iterative reconstruction techniques is that it offers a straightforward and computationally efficient way to incorporate non-linear constraints into the reconstruction that can lead to improved image quality and/or reliable reconstruction for underdetermined problems. However, POCSENSE algorithm demonstrates slow convergence in cases of badly conditioned problems. In this work, we propose a novel method for image reconstruction from sensitivity encoded MRI data that overcomes the limitation of the original POCSENSE technique. In the proposed method, the convex combination of projections onto convex sets is used to obtain an updated estimate of the solution via relaxation. The new method converges very efficiently due to the use of an iterationdependent relaxation parameter that may extend far beyond the theoretical limits of POCS. The developed method was validated with phantom and volunteer MRI data and was demonstrated to have a much higher convergence rate than that of the original POCSENSE technique.

Image reconstruction from sensitivity encoded MRI data using extrapolated iterations of parallel projections

SENSE (SENSitivity Encoding) imaging provides significant acquisition speedups in MRI. The main drawback of the method is that it generates images that have increased and spatially nonuniform noise levels and, hence, will often require retrospective filtering. In this paper, we show that standard anisotropic diffusion filtering, while being an effective technique for edgepreserving denoising of images with uniform noise levels, is often non-optimal for SENSE-reconstructed data. We have developed a modification of this filter for SENSE images using a robust statistical analysis of the anisotropic diffusion process. The new method utilizes the image noise matrix that is available from the SENSE reconstruction to automatically adjust filtering parameters with local noise levels. The effectiveness of the method and its advantage over standard anisotropic diffusion filtering for SENSE images were demonstrated with phantom and patient MRI data. 1.

Since the 1990-s, parallel Magnetic Resonance Imaging (pMRI) has emerged as a powerful 3D imaging technique for reducing scanning time. To speed up acquisition, the acquired k-space is sampled R times under the Nyquist rate. Full Field of View images are then reconstructed from the aliased data acquired along complementary coils, by applying for instance the Sensitivity Encoding (SENSE) algorithm [1]. However, SENSE-based reconstructed images suffer from several artifacts because of noise and inaccurate coil sensitivity profiles. The inverse pMRI reconstruction problem being ill-conditioned, regularization tools allowed us to obtain a significant enhancement of the reconstructed image quality even at high reduction factors (e.g. R = 4) and low magnetic field (1.5 Tesla) [2,3]. In this talk, we summarize our recent advances for regularizing the pMRI reconstruction in the Wavelet Transform (WT) domain, which gives access to sparse image representations. Here, a special attention has to be paid about the regularization model since the data and the unknown image are complex-valued. Several penalty functions, which assume independence or not between real and imaginary parts of the wavelet coefficients, have been successively

Abstract. Two rather similar historical evolutions are evoked, each one originating in fundamental spin studies by physicists, and ending as magnetic resonance imaging (MRI), a set of invaluable tools for clinical diagnosis in the hands of medical doctors. The first one starts with the early work on nuclear magnetic resonance, the founding stone of the usual proton-based MRI, of which the basic principles are described. The second one starts with the optical pumping developments made to study the effects of spin polarization in various fundamental problems. Its unexpected outcome is a unique imaging modality, also based on MRI, for the study of lung physiology and pathologies. 1. Historical introduction Magnetic Resonance Imaging (MRI), now widely known for its usefulness as a medical diagnosis tool and for the variety of clear pictures of the body’s interior obtained in a harmless and non-invasive manner, had its foundations laid more than 60 years ago in physics experiments designed to measure properties of the nuclear spins of hydrogen atoms. In even earlier experiments, Rabi had shown that

We propose an extension of the BM4D volumetric filter to the denoising of data corrupted by spatially nonuniform noise. BM4D implements the grouping and collaborative filtering paradigm, where similar cubes of voxels are stacked into a four-dimensional “group”. Each group undergoes a sparsifying four-dimensional transform, that exploits the local correlation among voxels in each cube and the nonlocal correlation between corresponding voxels of different cubes. Thus, signal and noise are effectively separated in transform domain. In this work we take advantage of the sparsity induced by the four-dimensional transform to provide a spatially adaptive estimation of the local noise variance by applying a robust median estimator of the absolute deviation to the spectrum of each filtered group. The adaptive variance estimates are then used during coefficients shrinkage. Finally, the inverse four-dimensional transform is applied to the filtered group, and each individual cube estimate is adaptively aggregated at its original location. Experiments on medical data corrupted by spatially varying Gaussian and Rician noise demonstrate the efficacy of the proposed approach in volumetric data denoising. In case of magnetic resonance signals, the adaptive variance estimate can be also used to compensate the estimation bias due to the non-zero-mean errors of the Rician-distributed data.