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Rapid MR Imaging with "Compressed Sensing" and Randomly Under-Sampled 3DFT Trajectories
Abstract
Introduction Recently a rapid imaging method was proposed [1] that exploits the fact that sparse or compressible signals, such as MR images, can be recovered from randomly under-sampled frequency data [1,2,3]. Because pure random sampling in 2D is impractical for MRI hardware, it was proposed to use randomly perturbed spirals to approximate random sampling. Indeed, pure 2D random sampling is impractical, however, randomly under-sampling the phase encodes in a 3D Cartesian scan (Fig. 1) is practical, involves no overhead, is simple to implement and is purely random in two dimensions. Moreover, scan-time reduction in 3D Cartesian scans is always an issue. We provide a method to evaluate the effective randomness of a randomly under-sampled trajectory by analyzing the statistics of aliasing in the sparse transform domain. Applying this method to MR angiography, where images are truly sparse, we demonstrate a 5-fold scan time reduction, which can be crucial in time-limited situations or can be used for time resolved imaging Theory Medical images in general, and specifically angiograms, often have a sparse representation using a linear transform (wavelets, DCT, finite differences, etc.)[1]. Under-sampling the Fourier domain results in aliasing. When the under-sampling is random, the aliasing is incoherent and acts as additional noise interference in the image, but more importantly, as incoherent interference of the sparse transform coefficients. Therefore, it is possible to recover the sparse transform coefficients using a non-linear reconstruction scheme [1-4] and consequently, recover the image itself. The interference in the sparse domain is a generalization of a point-spread function (PSF) and is computed by I(n,m)=<Sig(x n),Sig(x m)> where x n is the n th transform coefficient, and Sig{x n } is the normalized projection of the transform coefficient onto the under-sampled Fourier space. The success of the reconstruction will depend on the sparsity of the coefficients and that the interference I(n,m) be small and have random statistics [2,3]. The interference can be used as a design criteria or a test for a practical randomly under-sampled trajectory. As an example, we analyzed the statistics of the interference of wavelet coefficients (See Fig. 2), leading to a conclusion that for images sparsified by wavelets, random sampling should have variable density sampling, with increased density toward the center of k-space. Methods Angiograms are truly sparse, with high signal from blood vessels and low background signal. To test our proposed trajectory, we considered an SSFP angiogram data set[6]. By post processing, we simulated a randomly under-sampled 3D Cartesian trajectory by removing phase encodes (Fig. 1), sampling more densely towards the center of k-space. We reconstructed from 5%, 9%, 13%, 20%, 30%, 50%, 80% percent of the data respectively using L 1 Total Variation(TV) [1-4] and compared the results to zero-filling the missing data, and a low-resolution acquisition with the same number of phase encodes. Results and Discussion Fig 3. illustrates a region of interest in the maximum intensity projection (MIP) of the reconstructions for different under-sampling ratios. As expected, reconstruction by zero filling is severely degraded by aliasing artifacts and most vessels do not show in the MIP for high under-sampling ratio. The low resolution reconstruction also exhibits narrowing of vessels due to smoothing of the edges. On the other hand, the L 1 reconstruction was able to recover the sparse signal and produces a similar quality MIP to the fully sampled reconstruction starting from only 20% of the data. In conclusion, L 1 –penalized image reconstruction recovers sparse images even with severe undersampling. We also showed a method to evaluate random sampling schemes. Our method is computationally intensive. In the current, Matlab TM implementation we are able to reconstruct a 128x128x256 image in a matter of 120 minutes, this can be improve by newly proposed reconstruction algorithms [4,5]. This type of approach can be used either to speed scan time or gain more spatial or temporal resolution.
... The CS technique has been explored for a wide range of engineering applications, among others are: medical image processing [5] and imaging [6], channel coding [7], [8], and single pixel camera [9]. Related to our present paper, a concept of (time-domain) compressive radar has been introduced in [10]. ...
... For a suitable number of measured data M given by (6), CS guarantees to recover perfectly the time domain signal through optimization ...
... Since the quality of the reconstruction is determined by the number of minimum sample defined in (6), signal degradation starts to emerge when smaller number of sample is used or if the number of the monocycles is increased due to increasing the degree of freedom S. For the same simulated A-scan with double reflections, decreasing the number of random sample into 53 (10× compression) reduces the PSNR to 78.5 dB, and reducing further to 44 random samples (12× compression) degrades the PSNR to just about 8.7 dB. ...
Data acquisition speed is an inherent problem of the stepped-frequency continuous wave (SFCW) radars, which discouraging further usage and development of this technolo-gy. We propose an emerging paradigm called the compressed sensing (CS), to manage this problem. In the CS, a signal can be reconstructed exactly based on only a few samples below the Nyquist rate. Accordingly, the data acquisition speed can be increased significantly. A novel design of SFCW ground penetrating radar (GPR) with a high acquisition speed capa-bility is proposed and evaluated. Simulation by a monocycle waveform and actual measurement by a Vector Network Ana-lyzer in a GPR test-range confirm the implementability of the proposed system.
... This is important, since completely random Fourier sampling is generally impractical in MRI. Recent results indicate that CS theory can be applied to MRI using undersampled radial [11][12] [13][14] [16], spiral [8], and 3DFT trajectories [10]. When used in MRI, the measurement matrix M in Equation (9) is an 2 K nN (undersampled) Fourier matrix whose entries are given by , kj i kj Me kr (10) In addition, it was shown that the discrete wavelet transform and finite differences (i.e. ...
... Recent results indicate that CS theory can be applied to MRI using undersampled radial [11][12] [13][14] [16], spiral [8], and 3DFT trajectories [10]. When used in MRI, the measurement matrix M in Equation (9) is an 2 K nN (undersampled) Fourier matrix whose entries are given by , kj i kj Me kr (10) In addition, it was shown that the discrete wavelet transform and finite differences (i.e. Total Variation (TV)) are good choices for the sparsity transform Ψ in MRI applications [15]. ...
... The combined parallel imaging and CS (PICS) reconstruction can be formulated as an unconstrained optimization problem: 2 21 min( ) f g -Ef Ψf (13) By comparing Equation (9) to Equation (13), it can be seen that the difference between CS and PICS is in the measurement matrix. In the CS case, the entries of the measurement matrix are given in Equation (10). The entries of the measurement matrix for the PICS case are given in Equation (5). ...
Although magnetic resonance imaging (MRI) is routinely used in clinical practice, long acquisition times limit its practical utility in many applications. To increase the data acquisition speed of MRI, parallel MRI (pMRI) techniques have recently been proposed. These techniques utilize multi-channel receiver arrays and are based on simultaneous acquisition of data from multiple receiver coils. Recently, a novel framework called Compressed Sensing (CS) was introduced. Since this new framework illustrates how signals can be reconstructed from much fewer samples than suggested by the Nyquist theory, it has the potential to significantly accelerate data acquisition in MRI. This paper illustrates that CS and pMRI techniques can be combined and such joint processing yields results that are superior to those obtained from independent utilization of each technique.
... Proof: See Appendix A. Figure 2 displays the power-law approximations of means and standard deviations of U k (n) and V k (n), respectively, where L = 500. In the figure, lines indicate the approximations of (12) and (13), while markers represent the numerical results of the parameters. Figure 2 shows that the means and the standard deviations are well described by power functions, which validates our approximations on the statistical parameters. ...
... Then, (33) becomes (12) and (13). Finally, ...
... The CS reconstruction is accomplished by solving an l 1 -minimization problem with convex optimization or greedy algorithms [2]. With efficient measurement and stable reconstruction, the CS technique has been of interest in a variety of research fields, e.g., communications [6]− [8], sensor networks [9]− [11], image processing [12]− [15], radar [16], etc. ...
... Proof : Since the S-OTS cryptosystem renews its measurement matrix at each encryption, the probability that the S-OTS cryptosystem is secure against T ref repeated CPA is given by (1 − P key ) T ref , which yields (16) immediately from (15). ✷ In the proof of Theorem 7, we assumed that an adversary has no benefits by exploiting multiple plaintext-ciphertext pairs against the S-OTS cryptosystem, due to the usage of keystreams in one-time manner. ...
In this paper, we study the security of a compressed sensing (CS) based cryptosystem called a sparse one-time sensing (S-OTS) cryptosystem, which encrypts each plaintext with a sparse measurement matrix. To generate the secret matrix and renew it at each encryption, a bipolar keystream and a random permutation pattern are employed as cryptographic primitives, which are obtained by a keystream generator of stream ciphers. With a small number of nonzero elements in the measurement matrix, the S-OTS cryptosystem achieves an efficient CS encryption process in terms of data storage and computational cost. For security analysis, we show that the S-OTS cryptosystem can be computationally secure against ciphertext only attacks (COA) in terms of the indistinguishability, as long as each plaintext has constant energy. Also, we consider a chosen plaintext attack (CPA) against the S-OTS cryptosystem, which consists of two stages of keystream and key recovery attacks. Then, we show that it can achieve the security against the CPA of keystream recovery with overwhelmingly high probability, as an adversary needs to distinguish a prohibitively large number of candidate keystreams. Finally, we conduct an information-theoretic analysis to demonstrate that the S-OTS cryptosystem has sufficient resistance against the CPA of key recovery by guaranteeing the extremely low probability of success. In conclusion, the S-OTS cryptosystem can be indistinguishable and secure against a CPA, while providing efficiency in CS encryption.
... For ex-This research is supported in part by the National Science Foundation, grant number DMS-1015346. ample, in the nuclear magnetic resonance (NMR) imaging application, compressive sensing can help reduce the radiation time [8,9]. Moreover, the compressive sensing technique has been successfully applied to many other practical scenarios including sub-Nyquist sampling [10,11], compressive imaging [12,13], and compressive sensor networks [14,15], to name just a few. ...
... For example, when the noise level is known, the goodness of fit function can be set as d(Y, A, X) = Y − AX 2 F . We propose an efficient method (Algorithm 3) to solve (8), where the constant τ x is chosen according to criterion in (3). ...
In this paper we consider the dictionary learning problem for sparse representation. We first show that this problem is NP-hard and then propose an efficient dictionary learning scheme to solve several practical formulations of this problem. Unlike many existing algorithms in the literature, such as K-SVD, our proposed dictionary learning scheme is theoretically guar-anteed to converge to the set of stationary points under cer-tain mild assumptions. For the image denoising application, the performance and the efficiency of the proposed dictionary learning scheme are comparable to that of K-SVD algorithm in simulation.
... Compressed sensing (CS) is an efficient tool that accelerates the data acquisition in MRI through the significant reduction of required measurements for image reconstruction. The performance of this technique highly depends on the sparsifying transformation that provides a basis in which the image has a sparse presentation [4,5]. In this approach three conditions must be considered 1) Sparsity: A sparse representation of the image must be possible in some known transform domain 2) Incoherent under-sampling: Kspace under-sampling should be randomized so it generates noise-like aliasing interference in that transform domain. ...
... Since the vicinity of the center of the Kspace determines the contrast of the MRI image, thus the sampling pattern contains highly dense sampling near the center of K-space. The image reconstruction in CS comprises solving a constrained optimization problem in which the sparsity is maximized while the error between K-space undersampled data and the Fourier transform of the reconstructed image is kept limited [4,5]. In CS we minimize an objective function defined as weighted sum of non-zero elements (to emphasize sparsity), error (to underline fidelity) and total variation (TV) (to denoise the image). ...
Compressed Sensing (CS) is a theory with potential to reconstruct sparse images from a small number of random samples in the frequency domain, and with the aim of increasing achievable acceleration factors along with improved SNR and fidelity. Therefore we minimize an objective function defined as weighted sum of non-zero elements, error and total variation (TV). The accuracy and speed of the reconstruction depends on how we choose and update the aforementioned weights and how to solve the minimization problem. In this study, we proposed the Principle Component Analysis (PCA) for weighting the sparsity in the CS formulation. Considering the dimension reduction property of PCA, it is suitable for weighting the sparse transform in the CS algorithm. In the proposed implementation the weight of sparsity is updated at each iteration using the norm L1 of PCA significant components, which quantifies the sparsity at that stage. Results were compared with the zero-filling (ZF) and low resolution (LR) techniques. Compared to CS without using the sparse weighted, the proposed method took 15% higher SNR and reached 10% higher correlation with the original image.
... Compressive imaging also can be used to increase the speed of imaging in some applications like magnetic resonance imaging (MRI). [4][5][6][7] Applications for combining HDR imaging and compressive imaging include scenarios where the scenes and/or objects have high contrast ratios, the radiation wavelength is beyond the range to which conventional cameras are sensitive, or the image capture speed is important. ...
... Lustig et al. have a series of papers that propose methods for rapid MRI imaging. [4][5][6][7] Their method is based on randomly selected pixels in k-space and reconstruction with l 1 minimization or TV minimization. With these techniques the image can be acquired faster and with relatively better quality. ...
Some scenes and objects have a wide range of brightness that cannot be captured with a conventional camera. This limitation, which degrades the dynamic range of an imaged scene or object, is addressed by use of high dynamic range (HDR) imaging techniques. With HDR imaging techniques, images of a very broad range of intensity can be obtained with conventional cameras. Another limitation of conventional cameras is the range of wavelength that they can capture. Outside the visible wavelengths, the responsivity of conventional cameras drops; therefore, a conventional camera cannot capture images in nonvisible wavelengths. Compressive imaging is a solution for this problem. Compressive imaging reduces the number of pixels of a camera to one, so a camera can be replaced by a detector with one pixel. The range of wavelengths to which such detectors are responsive is much wider than that of a conventional camera. A combination of HDR imaging and compressive imaging is introduced and is benefitted from the advantages of both techniques. An algorithm that combines these two techniques is proposed, and results are presented.
... Step 2: q l = (AW l ) † y, [9] Step 3: x l = W l q l , [10] where the parameter p is given by 1/2 ≤ p ≤ 1 (16). After x l is obtained in Step 3, the weighting matrix is recalculated, and FOCUSS iteration, Step 1 to Step 3, is reapplied. ...
... Otherwise, increase l and go to (a). both angular sampling and angular aliasing artifacts, while maintaining spatial resolution, we employ FOCUSS, as described in Eqs. [8]–[10]. Note that the main computational burden for FOCUSS comes from the following pseudo-inverse calculation: [20] because of the computationally expensive inversion of the ...
The focal underdetermined system solver (FOCUSS) was originally designed to obtain sparse solutions by successively solving quadratic optimization problems. This article adapts FOCUSS for a projection reconstruction MR imaging problem to obtain high resolution reconstructions from angular under-sampled radial k-space data. We show that FOCUSS is effective for projection reconstruction MRI, since medical images are usually sparse in some sense and the center region of the undersampled radial k-space samples still provides a low resolution, yet meaningful, image essential for the convergence of FOCUSS. The new algorithm is successfully applied for synthetic data as well as in vivo brain imaging obtained by under-sampled radial spin echo sequence.
... Then, (33) becomes (12) and (13). Finally, ...
In multiple measurement vectors (MMV) problems, L measurement vectors each of which has lengthM are available for recovering jointly sparse signals that have a common support set of sizeK. In this paper, a fast and noise-robust greedy algorithm is proposed for joint sparse recovery in MMV problems, by exploiting a posteriori probability ratios for every index of sparse input signals. The essence of the algorithm is to transfer the information through iterations, which contributes to performance improvement of support detection. When L is sufficiently large atM =K+1, we investigate the asymptotic performance of exact support recovery using a Gaussian assumption, power-law approximations, and order statistics, where the techniques are inspired by experimental results. In this case, we also present a sufficient condition on the number of measurements, or M=K+1= Ω(logN), for theoretical support recovery guarantee. The theoretical analysis reveals that the proposed algorithm can achieve reliable joint sparse recovery asymptotically at the theoretical limit of M=K+1 with high probability. By examining the performance for various M;K, and L, simulation results demonstrate that if M is not too small, the proposed algorithm can be reliable, fast, and noise-robust, compared to conventional ones such as simultaneous orthogonal matching pursuit (SOMP), subspace augmented MUSIC (SA-MUSIC), and rank-aware order recursive matching pursuit (RA-ORMP).
... see [1], [2]) have become very popular for compressing and processing highdimensional data. In particular, they have found widespread applications in data acquisition [3], machine learning [4], [5], medical imaging [6]- [9], and networking [10]- [12]. ...
... The original signal can be faithfully recovered from the measurement samples, if it is sparse with respect to a particular basis and sampled via a random projection. With efficient measurement and stable reconstruction, the CS technique has been of interest in a variety of research fields, e.g., communications [5][6][7], sensor networks [8][9][10], image processing [11][12][13], and radar [14]. Recently, a great deal of attention has been paid to the CS technique for data confidentiality in information security field. ...
The principle of compressed sensing (CS) can be applied in a cryptosystem by providing the notion of security. In this paper, we study the computational security of a CS-based cryptosystem that encrypts a plaintext with a partial unitary sensing matrix embedding a secret keystream. The keystream is obtained by a keystream generator of stream ciphers, where the initial seed becomes the secret key of the CS-based cryptosystem. For security analysis, the total variation distance, bounded by the relative entropy and the Hellinger distance, is examined as a security measure for the indistinguishability. By developing upper bounds on the distance measures, we show that the CS-based cryptosystem can be computationally secure in terms of the indistinguishability, as long as the keystream length for each encryption is sufficiently large with low compression and sparsity ratios. In addition, we consider a potential chosen plaintext attack (CPA) from an adversary, which attempts to recover the key of the CS-based cryptosystem. Associated with the key recovery attack, we show that the computational security of our CS-based cryptosystem is brought by the mathematical intractability of a constrained integer least-squares (ILS) problem. For a sub-optimal, but feasible key recovery attack, we consider a successive approximate maximum-likelihood detection (SAMD) and investigate the performance by developing an upper bound on the success probability. Through theoretical and numerical analyses, we demonstrate that our CS-based cryptosystem can be secure against the key recovery attack through the SAMD.
... To address the logistical and computational challenges involved in dealing with such high-dimensional data, we often depend on compression, which aims attending the most concise representation of a signal that is able to achieve a target level of acceptable distortion. One of the most popular techniques for signal compression is known as transform coding, and typically relies on finding a basis or frame that provides sparse or compressible representations for signals in a class of interest [1,39,28,29,24,41,25,33]. Compressed sensing (CS) has emerged as a new framework for signal acquisition and sensor design. ...
Compressive sensing is an efficient way to represent signal with less number of samples. Shannon’s theorem which states that the sampling rate must be at least twice the maximum frequency present in the signal (the so-called Nyquist rate) is a common practice and conventional approach to sampling signals or images. Compressive sensing reveals that signals can be sensed or recovered from lesser data than required by Shannon’s theorem. This paper presents a brief historical background, mathematical foundation, and a theory behind compressive sensing and its emerging applications with a special emphasis on communication, network design, signal processing, and image processing.
Keywords
Sampling theorem Compressive sampling Sparsity Incoherence
... However, research is still going on for finding the optimal sampling technique and sparsifying transform. [10][11][12][13][14][15][16][17]. Compressive Sensing can be effectively utilised in Parallel Imaging, making the process even less time consuming [18][19][20][21][22]. ...
Magnetic Resonance Imaging (MRI) is a widely used technique for acquiring images of human organs/tissues. Due to its complex imaging process, it consumes a lot of time to produce a high quality image. Compressive Sensing (CS) has been used by researchers for rapid MRI. It uses a global sparsity constraint with variable density random sampling and L1 minimisation. This work intends to speed up the imaging process by exploiting the non-uniform sparsity in the MR images. Locally Sparsified CS suggests that the image can be even better sparsified by applying local sparsity constraints. The image produced by local CS can further reduce the sample set. This paper establishes the basis for a methodology to exploit non-uniform nature of sparsity and to make the MRI process time efficient by using local sparsity constraints.
... In addition to the theoretical advances, compressive sensing has shown great potential in various applications. For example, in the nuclear magnetic resonance (NMR) imaging application, compressive sensing can help reduce the radiation time [8], [9]. Moreover, the compressive sensing technique has been successfully applied to many other practical scenarios including sub-Nyquist sampling [10], [11], compressive imaging [12], [13], and compressive sensor networks [14], [15], to name just a few. ...
In this paper we consider the dictionary learning problem for sparse
representation. We first show that this problem is NP-hard by polynomial time
reduction of the densest cut problem. Then, using successive convex
approximation strategies, we propose efficient dictionary learning schemes to
solve several practical formulations of this problem to stationary points.
Unlike many existing algorithms in the literature, such as K-SVD, our proposed
dictionary learning scheme is theoretically guaranteed to converge to the set
of stationary points under certain mild assumptions. For the image denoising
application, the performance and the efficiency of the proposed dictionary
learning scheme are comparable to that of K-SVD algorithm in simulation.
... implementation of the CS technology resulted in a speedup factor of seven times in obtaining a pediatric magnetic resonance image while preserving the diagnostic quality (Lustig et al., 2005, 2006; Trzasko and Manduca, 2009). ...
An approximate subsurface reflectivity distribution of the earth is usually obtained through the migration process. However, conventional migration algorithms, including those based on the least-squares approach, yield structure descriptions that are slightly smeared and of low resolution caused by the common migration artifacts due to limited aperture, coarse sampling, band-limited source, and low subsurface illumination. To alleviate this problem, we use the fact that minimizing the L1-norm of a signal promotes its sparsity. Thus, we formulated the Kirchhoff migration problem as a compressed-sensing (CS) basis pursuit denoise problem to solve for highly focused migrated images compared with those obtained by standard and least-squares migration algorithms. The results of various subsurface reflectivity models revealed that solutions computed using the CS based migration provide a more accurate subsurface reflectivity location and amplitude. We applied the CS algorithm to image synthetic data from a fault model using dense and sparse acquisition geometries. Our results suggest that the proposed approach may still provide highly resolved images with a relatively small number of measurements. We also evaluated the robustness of the basis pursuit denoise algorithm in the presence of Gaussian random observational noise and in the case of imaging the recorded data with inaccurate migration velocities.
... It greatly improves the efficiency of the signal sampling and compression. CS is applicable to many situations, such as medical imaging [4][5][6][7], compressed imaging [8][9][10], and so on. The framework of CS combines the notions of sparse representation, measurement, and reconstruction which is shown in Fig.1 [11]. ...
The compressed sensing (CS) theory makes sample rate relate to signal structure and content. CS samples and compresses the signal with far below Nyquist sampling frequency simultaneously. However, CS only considers the intra-signal correlations, without taking the correlations of the multi-signals into account. Distributed compressed sensing (DCS) is an extension of CS that takes advantage of both the inter- and intra-signal correlations, which is wildly used as a powerful method for the multi-signals sensing and compression in many fields. In this paper, the characteristics and related works of DCS are reviewed. The framework of DCS is introduced. As DCS’s main portions, sparse representation, measurement matrix selection, and joint reconstruction are classified and summarized. The applications of DCS are also categorized and discussed. Finally, the conclusion remarks and the further research works are provided.
... He was surprised to discover that he could reconstruct a test image exactly even though the available data seemed insufficient according to the Nyquist-Shannon criterion [1,2]. Many interesting contributions in very different areas derived from this novel theory [5,10], being the most representative example the single-pixel camera developed at the Rice University [14]. ...
The novel theory of compressive sensing takes advantage of the sparsity or compressibility of a signal in a specific domain allowing the assessment of its full representation from fewer measurements. In this work we tailored the concept of compressive sensing to assess the intrinsic discriminative capability of this method to distinguish human faces from objects. Afterwards we enrolled through a feature selection study to empirically determine the minimum amount of measurements required to properly detect human faces. This work was concluded with a comparative experiment against the SIFT descriptor. We determined that using only 40 measurements conducted by compressing sensing one is capable of capturing the relevant information that enable one to properly discriminate human faces from objects.
... However, research is still going on for finding the optimal sampling technique and sparsifying transform. [9]- [14]. Compressive Sensing can be effectively utilized in parallel imaging making the process even less time consuming [15]- [19]. ...
Magnetic Resonance Imaging (MRI) is a widely used technique for acquiring images of human organs/tissues. Due to its complex imaging process, it consumes a lot of time to produce a high quality image. Compressive Sensing(CS) has been used by researchers for rapid MRI. It uses a global sparsity constraint with variable density random sampling and L1 minimization. This work intends to speed up the imaging process by exploiting the non-uniform sparsity in the MR images. Local CS suggests that the image can be sparsified even better by applying local sparsity constraints. The image produced by local CS can further reduce the sample set. This paper establishes the basis for a methodology to exploit non-uniform nature of sparsity and to make the MRI process time efficient by using local sparsity constraints.
... It builds a fundamentally novel approach to data acquisition and compression which overcomes drawbacks of the traditional method. Nowadays, compressive sensing has been widely studied and applied to various fields, such as radar imaging [35], magnetic resonance imaging [36,37,38], analog-to-information conversion [39], sensor networks [40,41] and even homeland security [42]. ...
... implementation of the CS technology resulted in a speedup factor of seven times in obtaining a pediatric magnetic resonance image while preserving the diagnostic quality (Lustig et al., 2005, 2006; Trzasko and Manduca, 2009). ...
Least-squares migration is a linearized form of waveform inversion that aims to enhance the spatial resolution of the subsurface reflectivity distribution and reduce the migration artifacts due to limited recording aperture, coarse sampling of sources and receivers, and low subsurface illumination. Least-squares migration, however, due to the nature of its minimization process, tends to produce smoothed and dispersed versions of the reflectivity of the subsurface. Assuming that the subsurface reflectivity distribution is sparse, we propose the addition of a non-quadratic L1-norm penalty term on the model space in the objective function. This aims to preserve the sparse nature of the subsurface reflectivity series and enhance resolution. We further use a compressed-sensing algorithm to solve the linear system, which utilizes the sparsity assumption to produce highly resolved migrated images. Thus, the Kirchhoff migration implementation is formulated as a Basis Pursuit denoise (BPDN) problem to obtain the sparse reflectivity model. Applications on synthetic data show that reflectivity models obtained using this compressed-sensing algorithm are highly accurate with optimal resolution.
... Compressive Sensing in MRI is also based on this idea that most of the image data is redundant and can be discarded. To acquire a diagnostic quality image from an under-sampled image different techniques are used in [8][9][10][11][12][13][14][15][16][17][18][19]. ...
Magnetic Resonance Imaging (MRI) is an important imaging techniques. However, it is a time-consuming process. The aim of this study is to make the imaging process efficient. MR images are sparse in the sensing domain and Compressive Sensing exploits this sparsity. Locally sparsified Compressed Sensing is a specialized case of CS which sub-divides the image and sparsifies each region separately; later samples are taken based on sparsity level in that region. In this paper, a new structured approach is presented for defining the size and locality of sub-regions in image. Experiments were done on the regions defined by proposed framework and local sparsity constraints were used to achieve high sparsity level and to reduce the sample set. Experimental results and their comparison with global CS is presented in the paper.
... Magnets are dependent on their slew rates and other physical properties and hardware cannot be modified for speeding up the image acquisition process. Researchers are working on speeding up MR Image acquisition process by other means, some research has been done using parallel imaging [1]- [5] whereas other work focuses towards reducing the number of samples or measurements that are required for image reconstruction [6]- [12]. ...
The fact that medical images have redundant information is exploited by researchers for faster image acquisition. Sample set or number of measurements were reduced in order to achieve rapid imaging. However, due to inadequate sampling, noise artefacts are inevitable in Compressive Sensing (CS) MRI. CS utilizes the transform sparsity of MR images to regenerate images from under-sampled data. Locally sparsified Compressed Sensing is an extension of simple CS. It localises sparsity constraints for sub-regions rather than using a global constraint. This paper, presents a framework to use local CS for improving image quality without increasing sampling rate or without making the acquisition process any slower. This was achieved by exploiting local constraints. Localising image into independent sub-regions allows different sampling rates within image. Energy distribution of MR images is not even and most of noise occurs due to under-sampling in high energy regions. By sampling sub-regions based on energy distribution, noise artefacts can be minimized. Experiments were done using the proposed technique. Results were compared with global CS and summarized in this paper.
... Undergoing significant advances, CS has proved to be far reaching and has enabled several applications in many fields, such as: distributed source coding in sensor networks [7,8], coding, analog-digital (A/D) conversion, remote wireless sensing [1,9] and inverse problems, such as those presented by MRI [10]. ...
Compressive sensing (CS) has recently emerged and is now a subject of increasing research and discussion, undergoing significant advances at an incredible pace. The novel theory of CS provides a fundamentally new approach to data acquisition which overcomes the common wisdom of information theory, specifically that provided by the Shannon-Nyquist sampling theorem. Perhaps surprisingly, it predicts that certain signals or images can be accurately, and sometimes even exactly, recovered from what was previously believed to be highly incomplete measurements (information). As the requirements of many applications nowadays often exceed the capabilities of traditional imaging architectures, there has been an increasing deal of interest in so-called computational imaging (CI). CI systems are hybrid imagers in which computation assumes a central role in the image formation process. Therefore, in the light of CS theory, we present a transmissive single-pixel camera that integrates a liquid crystal display (LCD) as an incoherent random coding device, yielding CS-typical compressed observations, since the beginning of the image acquisition process. This camera has been incorporated into an optical microscope and the obtained results can be exploited towards the development of compressive-sensing-based cameras for pixel-level adaptive gain imaging or fluorescence microscopy.
... However, this non-convex prior results in an intractable and non-deterministic polynomial-time hard (NP-hard) optimization problem. For this reason, the l 1 norm has been widely used as a convex surrogate to the l 0 norm and has gained popularity in conjunction with wavelet [20,21] and discrete gradient transforms [22]. The latter is known as total variation (TV) regularization [23][24][25][26] and has been shown to outperform l 2 -based regularizations in CS-(p)MRI [27,28]. ...
Compressed sensing (CS) provides a promising framework for MR image reconstruction from highly undersampled data, thus reducing data acquisition time. In this context, sparsity-promoting regularization techniques exploit the prior knowledge that MR images are sparse or compressible in a given transform domain. In this work, a new regularization technique was introduced by iterative linearization of the non-convex smoothly clipped absolute deviation (SCAD) norm with the aim of reducing the sampling rate even lower than it is required by the conventional l1 norm while approaching an l0 norm.
The CS-MR image reconstruction was formulated as an equality-constrained optimization problem using a variable splitting technique and solved using an augmented Lagrangian (AL) method developed to accelerate the optimization of constrained problems. The performance of the resulting SCAD-based algorithm was evaluated for discrete gradients and wavelet sparsifying transforms and compared with its l1-based counterpart using phantom and clinical studies. The k-spaces of the datasets were retrospectively undersampled using different sampling trajectories. In the AL framework, the CS-MRI problem was decomposed into two simpler sub-problems, wherein the linearization of the SCAD norm resulted in an adaptively weighted soft thresholding rule with a sparsity enhancing effect.
It was demonstrated that the proposed regularization technique adaptively assigns lower weights on the thresholding of gradient fields and wavelet coefficients, and as such, is more efficient in reducing aliasing artifacts arising from k-space undersampling, when compared to its l1-based counterpart.
The SCAD regularization improves the performance of l1-based regularization technique, especially at reduced sampling rates, and thus might be a good candidate for some applications in CS-MRI.
... Other possible application areas of CS include imaging [33], medical imaging [22,96], and RF environments (where high-bandwidth signals may contain low-dimensional structures such as radar chirps) [97]. As research continues into practical methods for signal recovery (see Section 2.8.3), additional work has focused on developing physical devices for acquiring random projections. ...
... Interestingly, most of real-world signals, including the radar echoes, are sparse. The CS technique has been explored for a wide range of engineering applications, among others are the following: medical image processing [5] and imaging [6], channel coding [7], [8], and single-pixel cameras [9]. Related to this letter, a concept of (time-domain) compressive radar has been introduced in [10]. ...
Data acquisition speed is an inherent problem of stepped-frequency continuous-wave (SFCW) radars, which may discourage further usage and development of this technology. We propose an emerging paradigm called compressed sensing (CS) to overcome this problem. In CS, a signal can be reconstructed exactly based on only a few samples below the Nyquist rate. Accordingly, the data acquisition speed can be increased significantly. A novel design of an SFCW ground-penetrating radar (GPR) with high acquisition speed is proposed and evaluated. Simulation by a monocycle waveform and actual measurement by a vector network analyzer at a GPR test range indicate the applicability of the proposed system.
... CS has already had notable impact on several applications. One example is medical imaging178179180 227], where it has enabled speedups by a factor of seven in pediatric MRI while preserving diagnostic quality [236]. Moreover, the broad applicability of this framework has inspired research that extends the CS framework by proposing practical implementations for numerous applications , including sub-Nyquist sampling systems [125, 126, 186–188, 219, 224, 225, 228], compressive imaging architectures [99, 184, 205], and compressive sensor networks [7, 72, 141]. ...
Compressed sensing (CS) is an exciting, rapidly growing, field that has attracted considerable attention in signal processing, statistics, and computer science, as well as the broader scientific community. Since its initial development only a few years ago, thousands of papers have appeared in this area, and hundreds of conferences, workshops, and special sessions have been dedicated to this growing research field. In this chapter, we provide an up-to-date review of the basics of the theory underlying CS. This chapter should serve as a review to practitioners wanting to join this emerging field, and as a reference for researchers. We focus primarily on the theory and algorithms for sparse recovery in finite dimensions. In subsequent chapters of the book, we will see how the fundamentals presented in this chapter are expanded and extended in many exciting directions, including new models for describing structure in both analog and discrete-time signals, new sensing design techniques, more advanced recovery results and powerful new recovery algorithms, and emerging applications of the basic theory and its extensions. Introduction We are in the midst of a digital revolution that is driving the development and deployment of new kinds of sensing systems with ever-increasing fidelity and resolution. The theoretical foundation of this revolution is the pioneering work of Kotelnikov, Nyquist, Shannon, and Whittaker on sampling continuous-time bandlimited signals [162, 195, 209, 247]. Their results demonstrate that signals, images, videos, and other data can be exactly recovered from a set of uniformly spaced samples taken at the so-called Nyquist rate of twice the highest frequency present in the signal of interest.
... The simple consequence of (1) is that, when the acquisition of each measurement is " expensive, " we benefit by sensing only M values rather than N . One example of such a situation is magnetic resonance imaging (MRI) [19]. We seek to minimize the amount of time to image a patient; however, each measurement is time-consuming, leading to a total acquisition time that is currently on the order of tens of minutes. ...
The recently introduced compressive sensing (CS) framework enables digital
signal acquisition systems to take advantage of signal structures beyond
bandlimitedness. Indeed, the number of CS measurements required for stable
reconstruction is closer to the order of the signal complexity than the Nyquist
rate. To date, the CS theory has focused on real-valued measurements, but in
practice, measurements are mapped to bits from a finite alphabet. Moreover, in
many potential applications the total number of measurement bits is
constrained, which suggests a tradeoff between the number of measurements and
the number of bits per measurement. We study this situation in this paper and
show that there exist two distinct regimes of operation that correspond to
high/low signal-to-noise ratio (SNR). In the measurement compression (MC)
regime, a high SNR favors acquiring fewer measurements with more bits per
measurement; in the quantization compression (QC) regime, a low SNR favors
acquiring more measurements with fewer bits per measurement. A surprise from
our analysis and experiments is that in many practical applications it is
better to operate in the QC regime, even acquiring as few as 1 bit per
measurement.
... The third method is sampling from a Gaussian distribution (Fig. 3c). In practice it may not be efficient to sample points from the K-space following a Gaussian distribution, but previous works in CS based MR image reconstruction have reported results based on sampling trajectories based on probability distributions [17,27,28]. For the CS based reconstruction, it is claimed that the best image reconstruction is obtained if a combination of wavelet and TV regularization (10) is used [11,6,17]. ...
SENSitivity Encoding (SENSE) is a mathematically optimal parallel magnetic resonance (MRI) imaging technique when the coil sensitivities are known. In recent times, compressed sensing (CS)-based techniques are incorporated within the SENSE reconstruction framework to recover the underlying MR image. CS-based techniques exploit the fact that the MR images are sparse in a transform domain (e.g., wavelets). Mathematically, this leads to an l(1)-norm-regularized SENSE reconstruction. In this work, we show that instead of reconstructing the image by exploiting its transform domain sparsity, we can exploit its rank deficiency to reconstruct it. This leads to a nuclear norm-regularized SENSE problem. The reconstruction accuracy from our proposed method is the same as the l(1)-norm-regularized SENSE, but the advantage of our method is that it is about an order of magnitude faster.
Synopsis Smoothed Random-like Trajectory (SRT) is a promising MR imaging method, but the data acquisitions have some hardware limitations of the gradient amplitude and slew rate. In order to realize the SRT in k-space, we proposed a new multiple-leaf SRT in k-space to reduce hardware requirements for applying the Compressed Sensing (CS) theory. To guarantee the constrains of the gradient amplitude and slew rate and reduce readout, the proposed multiple-leaf gradient waveforms were optimized by the time-optimal method for arbitrary k-space trajectories. The simulations have showed that the proposed method could greatly improve the reconstruction image quality, comparing to spiral trajectories.
Purpose
Compressive sensing (CS)‐based image reconstruction methods have proposed random undersampling schemes that produce incoherent, noise‐like aliasing artifacts, which are easier to remove. The denoising process is critically assisted by imposing sparsity‐enforcing priors. Sparsity is known to be induced if the prior is in the form of the L p (0 ≤ p ≤ 1) norm. CS methods generally use a convex relaxation of these priors such as the L 1 norm, which may not exploit the full power of CS. An efficient, discrete optimization formulation is proposed, which works not only on arbitrary L p ‐norm priors as some non‐convex CS methods do, but also on highly non‐convex truncated penalty functions, resulting in a specific type of edge‐preserving prior. These advanced features make the minimization problem highly non‐convex, and thus call for more sophisticated minimization routines.
Theory and methods
The work combines edge‐preserving priors with random undersampling, and solves the resulting optimization using a set of discrete optimization methods called graph cuts. The resulting optimization problem is solved by applying graph cuts iteratively within a dictionary, defined here as an appropriately constructed set of vectors relevant to brain MRI data used here.
Results
Experimental results with in vivo data are presented.
Conclusion
The proposed algorithm produces better results than regularized SENSE or standard CS for reconstruction of in vivo data.
In this paper, we study the security of a compressed sensing (CS) based cryptosystem called a
sparse one-time sensing (S-OTS)
cryptosystem, which encrypts a plaintext with a sparse measurement matrix. To construct the secret matrix and renew it at each encryption, a bipolar keystream and a random permutation pattern are employed as cryptographic primitives, which can be obtained by a keystream generator of stream ciphers. With a small number of nonzero elements in the measurement matrix, the S-OTS cryptosystem achieves efficient CS encryption in terms of memory and computational cost. In security analysis, we show that the S-OTS cryptosystem can be indistinguishable as long as each plaintext has constant energy, which formalizes computational security against ciphertext only attacks (COA). In addition, we consider a chosen plaintext attack (CPA) against the S-OTS cryptosystem, which consists of two sequential stages, keystream and key recovery attacks. Against keystream recovery under CPA, we demonstrate that the S-OTS cryptosystem can be secure with overwhelmingly high probability, as an adversary needs to distinguish a prohibitively large number of candidate keystreams. Finally, we conduct an information-theoretic analysis to show that the S-OTS cryptosystem can be resistant against key recovery under CPA by guaranteeing that the probability of success is extremely low. In conclusion, the S-OTS cryptosystem can be computationally secure against COA and the two-stage CPA, while providing efficiency in CS encryption.
This chapter presents a new phase unwrapping algorithm for the 3D Interferometric Synthetic Aperture Radar (3D InSAR) volumes. The proposed approach is based on the relationship between the gradient vectors of the observed wrapped phase and the true phase respectively, when the Itoh condition is satisfied. Since this relationship is violated by the residue pixels in the observed wrapped phase, a general problem formulation which takes into account the estimation error due to these residue values is proposed. This approach exploits the temporal inter correlation between the interferometric frames within a compressive sensing framework. The 3D discrete curvelet transform is used in order to ensure a suitable sparse representation of the phase volume. The performance of the proposed 3D phase unwrapping algorithm is tested on simulated and real SAR 3D datasets.
The principle of compressed sensing (CS) can be applied in a cryptosystem by providing the notion of security. The Gaussian one-time sensing (G-OTS) CS-based cryptosystem employing a random Gaussian matrix and renewing the elements at each encryption is known to be perfectly secure, as long as each plaintext has constant energy. A random Bernoulli matrix can replace the Gaussian one for encrypting each plaintext efficiently in the Bernoulli one-time sensing (B-OTS) cryptosystem. In this paper, we analyze the security of G-OTS and B-OTS cryptosystems, respectively, where each cryptosystem may have unequal plaintext energy. By means of probability metrics, we study the indistinguishability of each CS-based cryptosystem, which formalizes the notion of computational security. Moreover, we investigate how much the indistinguishability is sensitive to energy variation of plaintexts in each cryptosystem. For the B-OTS cryptosystem, we analyze the indistinguishability and the energy sensitivity in a non-asymptotic manner for a finite plaintext length. In conclusion, this paper confirms that G-OTS and B-OTS cryptosystems can be strictly and asymptotically indistinguishable, respectively, as long as each plaintext has constant energy, but the indistinguishability is highly sensitive to energy variation of plaintexts.
In this letter, we study the security of a cryptosystem over wireless channels that employs the asymptotically Gaussian compressed encryption. We investigate the indistinguishability and the energy sensitivity of the cryptosystem, where the total variation (TV) distance is examined as a statistical measure for the indistinguishability. To characterize the TV distance, we compute the Hellinger distance between probability distributions of ciphertexts, each of which can be modeled as a circularly-symmetric complex Gaussian (CSCG) random vector with a constraint on plaintexts. Using the distance metrics, we show that the cryptosystem can be a promising option for secure wireless communications by guaranteeing the indistinguishability against an eavesdropper, as long as each plaintext has constant energy.
The electrocardiogram signal consists in a character of smaller amplitude together with a larger interference range and the reconstructed signal, according to the classical compressed sensing theory, cannot be accurately conveyed by the signal. To solve this problem, compressed sensing based on the wavelet transform was stressed on. We carry out a compressed sensing algorithm based on wavelet transform, thus is to use the wavelet decomposition to separate the electrocardiogram, to reduce the noise pollution, to compress and reconstruct the high-frequency coefficient and to recover the signal by inversing the wavelet transform. Meanwhile, analysis on the data effect was also made. The result of the simulation shows that it obviously proves the noise suppressing effect on combining wavelet transform with compressed sensing to recover the signal. The integrity of useful information is enhanced, as well as obtaining a higher signal-to-noise ratio.
Most compressive imaging architectures rely on programmable light-modulators to obtain coded linear measurements of a signal. As a consequence, the properties of the light modulator place fundamental limits on the cost, performance, practicality, and capabilities of the compressive camera. For example, the spatial resolution of the single pixel camera is limited to that of its light modulator, which is seldom greater than 4 megapixels. In this paper, we describe a novel approach to compressive imaging that avoids the use of spatial light modulator. In its place, we use novel cylindrical optics and a rotation gantry to directly sample the Radon transform of the image focused on the sensor plane. We show that the reconstruction problem is identical to sparse tomographic recovery and we can leverage the vast literature in compressive magnetic resonance imaging (MRI) to good effect.
The proposed design has many important advantages over existing compressive cameras. First, we can achieve a resolution of N × N pixels using a sensor with N photodetectors; hence, with commercially available SWIR line-detectors with 10k pixels, we can potentially achieve spatial resolutions of 100 megapixels, a capability that is unprecedented. Second, our design is scalable more gracefully across wavebands of light since we only require sensors and optics that are optimized for the wavelengths of interest; in contrast, spatial light modulators like DMDs require expensive coatings to be effective in non-visible wavebands. Third, we can exploit properties of line-detectors including electronic shutters and pixels with large aspect ratios to optimize light throughput. On the ip side, a drawback of our approach is the need for moving components in the imaging architecture.
Magnetic Resonance Parametric Imaging is a recently-proposed method that permits quantitative determination of MR parameters such as the T1 and T2 relaxation times. In contrast to conventional MRI, one or more encoding parameters in the RF excitation are randomly varied over the scan and tissue parameters are inferred from the temporal response to the excitation. This work presents a novel low-rank model-based parametric matrix estimation method for joint reconstruction and parameter estimation suitable for highly accelerated (i.e. highly undersampled) scans. The method is demonstrated on T2 cardiac breath-hold imaging with varying spin echo times.
The work of this thesis concerns the proposal of algorithms for the integration of prior knowledge in the reconstruction of sparse signals. The purpose is mainly to improve the reconstruction of these signals from a set of measurements well below what is requested by the famous theorem of Shannon-Nyquist. In the first part we propose, in the context of the new theory of "compressed sensing" (CS), the algorithm NNOMP (Neural Network Orthogonal Matching Pursuit), which is a modified version of the algorithm OMP in which we replaced the correlation step by a properly trained neural network. The goal is to better reconstruct sparse signals with additional structures, i.e. belonging to a particular model of sparse signals. For the experimental validation of NNOMP three simulated models of sparse signals with additional structures were considered and a practical application in an arrangement similar to the "single pixel imaging". In the second part, we propose a new method for under sampling in multidimensional NMR spectroscopy (including NMR spectroscopic imaging), when the corresponding spectra of lower dimensional acquisitions, e.g. one-dimensional, are intrinsically sparse. In this method, we model the whole process of data acquisition and reconstruction of multidimensional spectra, by a system of linear equations. We then use a priori knowledge about the non-zero locations in multidimensional spectra, to remove the under-determinacy induced by data under sampling. This a priori knowledge is obtained from the lower dimensional acquisition spectra, e.g. one-dimensional. The possibility of under sampling increases proportionally with the sparsity of these one dimensional spectra. The proposed method is evaluated on synthetic, experimental in vitro and in vivo data.
Sampling is the bridge between analog source signal and digital signal. With the rapid progress of information technologies. The demands for information are increasing dramatically. So the existing systems are very difficult to meet the challenges of high speed sampling, large volume data transmission and storage. How to acquire information in signal efficiently is an urgent problem in electronic information fields. In recent years, an emerging theory of signal acquirement-compressed sensing (CS) provides an opportunity for solving this problem. CS is a research focus rising in the last few years. It is a new sampling theory and points out that if a signal can be compressed under some condition, a very accurate reconstruction can be obtained from a relatively small number of non-traditional samples. In this paper, the CS framework is introduced firstly, and then the approximate sparsity in the wavelet domain of male and female speech signals is analyzed. Secondly, the CS algorithm preserves the low frequency wavelet transform coefficients but compresses the high frequency wavelet transform coefficients of the speech signal. Two methods are proposed to compress the high frequency wavelet transform coefficients of the speech signal. One is compressing them separately, and the other is compressing them together. Finally, the high frequency wavelet transform coefficients are recovered by using Orthogonal Matching Pursuit algorithm, and then the reconstruction of the speech signal can be achieved by the inverse wavelet transform. Simulation results show that whether male or female speech signals, the first method can acquire better reconstruction performance and it needs less time than the second one at the same measurement number.
Magnetic Resonance Imaging (MRI) is one of the prominent medical imaging techniques. This process is time-consuming and can take several minutes to acquire one image. The aim of this research is to reduce the imaging process time of MRI. This issue is addressed by reducing the number of acquired measurements using theory of Compressive Sensing (CS). Compressive Sensing exploits sparsity in MR images. Randomly under sampled k-space generates incoherent noise which can be handled using a nonlinear image reconstruction method. In this paper, a new framework is presented based on the idea to exploit non-uniform nature of sparsity in MR images, where local sparsity constrains were used instead of traditional global constraint, to further reduce the sample set. Experimental results and comparison with CS using global constraint are demonstrated.
In this paper, we explores a rapid imaging method based on a proposed random-like trajectory for compressed sensing (CS) which requires the sampling trajectory should satisfy the Restricted Isometry Property (RIP) condition. In the existing CS literature, the attentions are on randomly sampling points on the conventional trajectories. However, the proposed trajectory is a random-like trajectory generated based on the High Order Chirp (HOC) sequences, which use the Traveling Salesman Problem (TSP) solver to choose a "short" trajectory and design a time optimal gradient waveforms to satisfy the gradient amplitude and slew rate limitation. The MR physical feasibility of the proposed method is verified by the Bloch simulation, and the simulations show that the proposed method can reduce artifacts than conventional Spiral trajectory under the CS framework.
This paper is concerned about high resolution reconstruction of projection reconstruction MR imaging from angular under-sampled k-space data. A similar problem has been recently addressed in the framework of com-pressed sensing theory. Unlike the existing algorithms used in compressed sensing theory, this paper employs the FOCal Underdetermined System Solver(FOCUSS), which was originally designed for EEG and MEG source localization to obtain sparse solution by successively solving quadratic optimization. We show that FOCUSS is very effective for the projection reconstruction MRI, because the medical images are usually sparse in image domain, and the center region of the under-sampled radial k-space data still provides a meaningful low resolution image, which is essential for the convergence of FOCUSS. We applied FOCUSS for projection reconstruction MR imaging using single coil. Extensive experiments confirms that high resolution reconstruction with virtually free of angular aliasing artifacts can be obtained from severely under-sampled k-space data.
In parallel magnetic resonance imaging (MRI), the problem is to reconstruct an image given the partial K-space scans from all the receiver coils. Depending on its position within the scanner, each coil has a different sensitivity profile. All existing parallel MRI techniques require estimation of certain parameters pertaining to the sensitivity profile, e.g., the sensitivity map needs to be estimated for the SENSE and SMASH and the interpolation weights need to be calibrated for GRAPPA and SPIRiT. The assumption is that the estimated parameters are applicable at the operational stage. This assumption does not always hold, consequently the reconstruction accuracies of existing parallel MRI methods may suffer. We propose a reconstruction method called Calibration-Less Multi-coil (CaLM) MRI. As the name suggests, our method does not require estimation of any parameters related to the sensitivity maps and hence does not require a calibration stage. CaLM MRI is an image domain method that produces a sensitivity encoded image for each coil. These images are finally combined by the sum-of-squares method to yield the final image. It is based on the theory of Compressed Sensing (CS). During reconstruction, the constraint that "all the coil images should appear similar" is introduced within the CS framework. This leads to a CS optimization problem that promotes group-sparsity. The results from our proposed method are comparable (at least for the data used in this work) with the best results that can be obtained from state-of-the-art methods.
Fast multidimensional NMR is important in chemical shift assignment and for studying structures of large proteins. We present the first method which takes advantage of the sparsity of the wavelet representation of the NMR spectra and reconstructs the spectra from partial random measurements of its free induction decay (FID) by solving the following optimization problem: min ∥x∥ 1 subject to ∥y-SF τ W τ x∥ 2 ≤ϵ, where y is a given n×1 observation vector, S a random sampling operator, F denotes the Fourier transform, and W an orthogonal 2D wavelet transform. The matrix A=SF τ W τ is a given n×p matrix such that n<p. This problem can be solved by general-purpose solvers; however, these can be prohibitively expensive in large-scale applications. In the settings of interest, the underlying solution is sparse with a few nonzeros. We show here that for large practical systems, a good approximation to the sparsest solution is obtained by iterative thresholding algorithms running much more rapidly than general solvers. We demonstrate the applicability of our approach to fast multidimensional NMR spectroscopy. Our main practical result estimates a four-fold reduction in sampling and experiment time without loss of resolution while maintaining sensitivity for a wide range of existing settings. Our results maintain the quality of the peak list of the reconstructed signal which is the key deliverable used in protein structure determination.
Can we recover a signal f 2 RN from a small number of linear measurements? A series of recent papers developed a collection of results showing that it is surprisingly possible to reconstruct certain types of signals accurately from limited measurements. In a nutshell, suppose that f is compressible in the sense that it is well-approximated by a linear combination of M vectors taken from a known basis . Then not knowing anything in advance about the signal, f can (very nearly) be recovered from about M logN generic nonadaptive measurements only. The recovery procedure is concrete and consists in solving a simple convex optimization program. In this paper, we show that these ideas are of practical significance. Inspired by theoretical devel- opments, we propose a series of practical recovery procedures and test them on a series of signals and images which are known to be well approximated in wavelet bases. We demonstrate empirically that it is possible to recover an object from about 3M -5M projections onto generically chosen vectors with the same accuracy as the ideal M -term wavelet approximation. We briefly discuss possible implications in the areas of data compression and medical imaging.
Shrinkage is a well known and appealing denoising technique, introduced originally by Donoho and Johnstone in 1994. The use of shrinkage for denoising is known to be optimal for Gaussian white noise, provided that the sparsity on the signal's representation is enforced using a unitary transform. Still, shrinkage is also practiced with nonunitary, and even redundant representations, typically leading to very satisfactory results. In this correspondence we shed some light on this behavior. The main argument in this work is that such simple shrinkage could be interpreted as the first iteration of an algorithm that solves the basis pursuit denoising (BPDN) problem. While the desired solution of BPDN is hard to obtain in general, we develop a simple iterative procedure for the BPDN minimization that amounts to stepwise shrinkage. We demonstrate how the simple shrinkage emerges as the first iteration of this novel algorithm. Furthermore, we show how shrinkage can be iterated, turning into an effective algorithm that minimizes the BPDN via simple shrinkage steps, in order to further strengthen the denoising effect
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