Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data (original) (raw)
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2008
Previous compressive sensing papers have considered the example of recovering an image with sparse gradient from a surprisingly small number of samples of its Fourier transform. The samples were taken along radial lines, this being equivalent to a tomographic reconstruction problem. The theory of compressive sensing, however, considers random sampling instead. We perform numerical experiments to compare the two approaches, in terms of the number of samples necessary for exact recovery, algorithmic performance, and robustness to noise. We use a nonconvex approach, this having previously been shown to allow reconstruction with fewer measurements and greater robustness to noise, as confirmed by our results here.
NON-CONVEX ALGORITHM FOR SPARSE AND LOW-RANK RECOVERY: APPLICATION TO DYNAMIC MRI RECONSTRUCTION
In this work we exploit two assumed properties of dynamic MRI in order to reconstruct the images from under-sampled K-space samples. The first property assumes the signal is sparse in the x-f space and the second property assumes the signal is rank-deficient in the x-t space. These assumptions lead to an optimization problem that requires minimizing a combined lp-norm and Schatten-p norm. We propose a novel FOCUSS based approach to solve the optimization problem. Our proposed method is compared with state-of-the-art techniques in dynamic MRI reconstruction. Experimental evaluation carried out on three real datasets show that for all these datasets, our method yields better reconstruction both in quantitative and qualitative evaluation.
Iterative ℓ1 minimization for non-convex compressed sensing
2016
An algorithmic framework, based on the difference of convex functions algorithm, is proposed for minimizing a class of concave sparse metrics for compressed sensing problems. The resulting algorithm iterates a sequence of l1 minimization problems. An exact sparse recovery theory is established to show that the proposed framework always improves on the basis pursuit (l1 minimization) and inherits robustness from it. Numerical examples on success rates of sparse solution recovery illustrate further that, unlike most existing non-convex compressed sensing solvers in the literature, our method always out-performs basis pursuit, no matter how ill-conditioned the measurement matrix is. Moreover, the iterative l1 algorithms lead by a wide margin the state-of-the-art algorithms on l1/2 and logarithimic minimizations in the strongly coherent (highly ill-conditioned) regime, despite the same objective functions.
IJERT-Compressive Sensing Reconstruction for Sparse Signals with Convex Optimization
International Journal of Engineering Research and Technology (IJERT), 2014
https://www.ijert.org/compressive-sensing-reconstruction-for-sparse-signals-with-convex-optimization https://www.ijert.org/research/compressive-sensing-reconstruction-for-sparse-signals-with-convex-optimization-IJERTV3IS090011.pdf The theory of compressive sampling (CS), also known as compressed sensing. It is a modern sensing scheme that goes against the common theory in data acquisition. The CS theory claims that one can recover images or signals from fewer samples or measurements than the traditional methods use. To achieve this recovery, CS theory depends on two basic principles: the first is the sparsity of signal, which relates to the signals of interest, and the incoherence, which relates to the sensing method. In this paper we will give a simple review on the CS theory and the analog to information (AIC) system will be discussed briefly supported with two examples of signal reconstruction from undersampled signals. Simulation results show the powerful of the CS reconstruction for both sparse in time and spars in frequency signals.
Compressive sensing in medical imaging
Applied Optics, 2015
The promise of compressive sensing, exploitation of compressibility to achieve high quality image reconstructions with less data, has attracted a great deal of attention in the medical imaging community. At the Compressed Sensing Incubator meeting held in April 2014 at OSA Headquarters in Washington, DC, presentations were given summarizing some of the research efforts ongoing in compressive sensing for x-ray computed tomography and magnetic resonance imaging systems. This article provides an expanded version of these presentations. Sparsityexploiting reconstruction algorithms that have gained popularity in the medical imaging community are studied, and examples of clinical applications that could benefit from compressive sensing ideas are provided. The current and potential future impact of compressive sensing on the medical imaging field is discussed.
Convex Feasibility Methods for Compressed Sensing
We present a computationally-efficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS). CS theory requires solving a convex constrained minimization problem. We propose to transform this optimization problem into a convex feasibility problem (CFP), and solve it using subgradient projection methods, which are iterative, fast, robust and convergent schemes for solving CFPs. As opposed to some of the recentlyintroduced CS algorithms, such as Bayesian CS and gradient projections for sparse reconstruction, which become prohibitively inefficient as the problem dimension and sparseness degree increase, the newly-proposed methods exhibit a marked robustness with respect to these factors. This renders the subgradient projection methods highly viable for large-scale compressible scenarios.
2019 Computing in Cardiology Conference (CinC), 2019
Magnetic Resonance (MR) imaging is a multiparametric imaging technique allowing the diagnosis of a wide spectrum of cardiovascular diseases. Unfortunately, MR acquisitions tend to be slow, limiting patient throughput and limiting potential indications for use while driving up costs. Compressed sensing (CS) is a method for reducing MR scan time, increasing image reconstruction time. In this study we formulated a novel CS-based approach to speed up reconstruction procedure. A fidelity term that constrains the solution to be similar to the acquired samples was embedded in a nonconvex weighted total variation-based approach starting from highly subsampled k-space data. This approach was tested for the reconstruction of cardiac images in 10 delayed contrast enhanced MR (DCE-MR) acquisitions, using different k-space masks. Fully sampled MR images and the reconstructed images were compared by means of peakand signal-to-noise ratio (PSNR and SNR) metrics. Compared to other k-space filling trajectories, radial mask allowed the reconstruction of images of comparable quality (PSNR in [30 40]) but using less information. Overall, in all the test images we obtained a good reconstruction with similar SNR of the corresponding fully sampled images but using less than 20% of the original samples.
Non-Convex Compressed Sensing from Noisy Measurements
This paper proposes solution to the following non-convex optimization problem: min || x ||p subject to || y .. Ax ||q Such an optimization problem arises in a rapidly advancing branch of signal processing called ‘Compressed Sensing’ (CS). The problem of CS is to reconstruct a k-sparse vector xnX1, from noisy measurements y = Ax +.. , where AmXn (m<n) is the measurement matrix and ..mX1 is additive noise. In general the optimization methods developed for CS minimizes a sparsity promoting l1-norm (p=1) for Gaussian noise (q=2). This is restrictive for two reasons: i) theoretically it has been shown that, with positive fractional norms (0<p<1), the sparse vector x can be reconstructed by fewer measurements than required by l1-norm; and ii) Noises other than Gaussian require the norm of the misfit (q) to be something other than 2. To address these two issues an Iterative Reweighted Least Squares based algorithm is proposed here to solve the aforesaid optimization problem.
An algorithm for sparse MRI reconstruction by Schatten p-norm minimization
In recent years, there has been a concerted effort to reduce the MR scan time. Signal processing research aims at reducing the scan time by acquiring less K-space data. The image is reconstructed from the subsampled K-space data by employing compressed sensing (CS)-based reconstruction techniques. In this article, we propose an alternative approach to CS-based reconstruction. The proposed approach exploits the rank deficiency of the MR images to reconstruct the image. This requires minimizing the rank of the image matrix subject to data constraints, which is unfortunately a nondeterministic polynomial time (NP) hard problem. Therefore we propose to replace the NP hard rank minimization problem by its nonconvex surrogate - Schatten p-norm minimization. The same approach can be used for denoising MR images as well. Since there is no algorithm to solve the Schatten p-norm minimization problem, we derive an efficient first-order algorithm. Experiments on MR brain scans show that the reconstruction and denoising accuracy from our method is at par with that of CS-based methods. Our proposed method is considerably faster than CS-based methods.
Frequency extrapolation by nonconvex compressive sensing
Proceedings - International Symposium on Biomedical Imaging, 2011
Tomographic imaging modalities sample subjects with a discrete, finite set of measurements, while the underlying object function is continuous. Because of this, inversion of the imaging model, even under ideal conditions, necessarily entails approximation. The error incurred by this approximation can be important when there is rapid variation in the object function or when the objects of interest are small. In this work, we investigate this issue with the Fourier transform (FT), which can be taken as the imaging model for magnetic resonance imaging (MRI) or some forms of wave imaging. Compressive sensing has been successful for inverting this data model when only a sparse set of samples are available. We apply the compressive sensing principle to a somewhat related problem of frequency extrapolation, where the object function is represented by a super-resolution grid with many more pixels than FT measurements. The image on the super-resolution grid is obtained through nonconvex minimization. The method fully utilizes the available FT samples, while controlling aliasing and ringing. The algorithm is demonstrated with continuous FT samples of the Shepp-Logan phantom with additional small, high-contrast objects.