A new linear convergence result for the iterative soft thresholding algorithm

Magnetic resonance imaging systems usually feature linear and shift-invariant (stationary) transform characteristics. The point spread function or equivalently the modulation transfer function may thus be used for an objective quality assessment of imaging modalities. The recently introduced theory of compressed sensing, however, incorporates nonlinear and nonstationary reconstruction algorithms into the magnetic resonance imaging process which prohibits the usage of the classical point spread function and therefore the according evaluation. In this work, a local point spread function concept was applied to assess the quality of magnetic resonance images which were reconstructed by an iterative soft thresholding algorithm for compressed sensing. The width of the main lobe of the local point spread function was used to perform studies on the spatial and temporal resolution properties of both numerical phantom and in vivo images. The impact of k-space sampling patterns as well as additional sparsifying transforms on the local spatial image resolution was investigated. In addition, the local temporal resolution of image series, which were reconstructed by exploiting spatiotemporal sparsity, was determined. Finally, the dependency of the local resolution on the thresholding parameter of the algorithm was examined. The sampling patterns as well as the additional sparsifying transform showed a distinct impact on the local image resolution of the phantom image. The reconstructions, which were using x-f-space as a sparse transform domain showed slight temporal blurring for dynamic parts of the imaged object. The local image resolution had a dependence on the thresholding parameter, which allowed for optimizing the reconstruction. Local point spread functions enable the evaluation of the local spatial and temporal resolution of images reconstructed with the nonlinear and nonstationary iterative soft thresholding algorithm. By determining the influence of thresholding parameter and sampling pattern chosen on this model-based reconstruction, the method allows selecting appropriate acquisition parameters and thus improving the results.

The low-rank plus sparse (L+S) decomposition model enables the reconstruction of undersampled dynamic magnetic resonance imaging (MRI) data. Solving for the L and S components is anonsmooth composite convex minimization problem. While current techniques for this model are based on the classical iterative soft thresholding algorithm (ISTA), accelerated methods can be applied to obtain faster rate of convergence of the algorithm. This paper proposes two alternative methods for solving the L+S problem, one based on the fast iterative shrinkage-thresholding algorithm (FISTA), and the other based on the recent proximal optimized gradient method (POGM). Numerical results suggest faster convergence than the traditional ISTA, while preserving its computational simplicity.

The low-rank plus sparse (L+S) decomposition model enables the reconstruction of under-sampled dynamic parallel magnetic resonance imaging (MRI) data. Solving for the low-rank and the sparse components involves non-smooth composite convex optimization, and algorithms for this problem can be categorized into proximal gradient methods and variable splitting methods. This paper investigates new efficient algorithms for both schemes. While current proximal gradient techniques for the L+S model involve the classical iterative soft thresholding algorithm (ISTA), this paper considers two accelerated alternatives, one based on the fast iterative shrinkage-thresholding algorithm (FISTA), and the other with the recent proximal optimized gradient method (POGM). In the augmented Lagrangian (AL) framework, we propose an efficient variable splitting scheme based on the form of the data acquisition operator, leading to simpler computation than the conjugate gradient (CG) approach required by existing AL methods. Numerical results suggest faster convergence of the efficient implementations for both frameworks, with POGM providing the fastest convergence overall and the practical benefit of being free of algorithm tuning parameters.

Synthetic aperture radar (SAR) is widely used for ground observations owing to its all-weather and all-time capabilities. Sparse SAR reconstruction imaging, which can enhance SAR imaging performance, has emerged as a significant research area. The traditional sparse SAR reconstruction imaging method was successfully applied to raw SAR data processing. However, the computational complexity makes sparse reconstruction imaging difficult for large scenes. Meanwhile, because raw data are usually kept confidential, only limited data are available. To address the issue of how to use publicly available complex image data for sparse reconstruction and effectively improve image quality, this study proposes a joint L1- and F-norm edge-preservation-based sparse SAR reconstruction algorithm for complex images. The algorithm uses the gradient descent (GD) technique to optimize the L1-norm regularization and reconstruct the overall image information; the iterative soft thresholding algorithm (ISTA) to optimize the F-norm regularization and reconstruct the high-frequency image information; and fuses the two sets of reconstructed information to obtain SAR reconstructed images with characteristics such as edge preservation, clear texture, and high quality. The results show that, compared with traditional matched filtering (MF) imaging results, the joint L1- and F-norm results obtained for complex images have lower sidelobes, higher signal-to-noise ratio, and better target resolution ability; compared with the F-norm-based sparse SAR reconstruction imaging results, the joint L1- and F-norm method provides images with better edge texture clarity.

In this paper we continue to study the problem of efficient video coding based on compressive sensing. In our previous work we proposed CS-JPEG video codec, where dragon noiselet transform is applied to a differential frame computed as a residual between an input frame and its downsampled and upsampled version. The downsampled version of the frame is compressed via JPEG baseline, while the noiselet coefficients are subsampled, quantized, entropy encoded and embedded into the application header of the JPEG file. At the decoder side, an iterative soft thresholding algorithm (ISTA) with video block-matching and 3D filtering (VBM3D) used as the thresholding operator is applied. In this paper, we first introduce a fast ISTA based on simple temporal and spatial transforms with parameters which are randomly selected at each iteration, and show that the proposed approach reduces the recovery time by around 300 times. Second, we consider the codec parameters selection providing the best reconstruction quality for a given bit rate. Finally, we evaluate the codec performance on HD video sequences recorded by real-life street surveillance systems without camera motion and show that CS-JPEG with the fast decoding and the selected parameters provides 54.3% and 22.4% bit rate savings, in average, comparing with the fast implementations of H.264/AVC Intra or H.265/HEVC Intra, respectively.

Various deconvolution algorithms for acoustic source are developed to improve spatial resolution and suppress sidelobe of the conventional beamforming. To improve the computational efficiency and solution convergence of deconvolution, this paper proposes a Fourier-based improved fast iterative shrinkage thresholding algorithm. Simulations and experiments show that Fourier-based improved fast iterative shrinkage thresholding algorithm can achieve excellent acoustic identification performance, with high computational efficiency and good convergence. For Fourier-based improved fast iterative shrinkage thresholding algorithm, the larger the weight coefficient, the narrower the mainlobe width, and the better the convergence, but the spurious source also increases. The recommended weight coefficient for the array described herein is 3. In addition, like other Fourier-based deconvolution algorithms, Fourier-based improved fast iterative shrinkage thresholding algorithm using irregular focus grid can obtain better acoustic source identification performance than using the conventional regular focus grid.

Recovering static signals from compressed measurements is an important problem that has been extensively studied in modern signal processing. However, only recently have methods been proposed to tackle the problem of recovering a time-varying sequence from streaming online compressed measurements. In this paper, we study the capacity of the standard iterative soft-thresholding algorithm (ISTA) to perform this task in real-time. In previous work, ISTA has been shown to recover static sparse signals. The present paper demonstrates its ability to perform this recovery online in the dynamical setting where measurements are constantly streaming. Our analysis shows that the l 2 -distance between the output and the target signal decays according to a linear rate, and is supported by simulations on synthetic and real data.

The recently proposed compressed sensing theory equips us with methods to recover exactly or approximately, high resolution images from very few encoded measurements of the scene. The traditional ill-posed problem of MRI image recovery from heavily under-sampled κ-space data can be thus solved using CS theory. Differing from the soft thresholding methods that have been used earlier in the case of CS MRI, we suggest a simple iterative hard thresholding algorithm which efficiently recovers diagnostic quality MRI images from highly incomplete κ-space measurements. The new multi-scale redundant systems, curvelets and contourlets having high directionality and anisotropy, and thus best suited for curved-edge representation are used in this iterative hard thresholding framework for CS MRI reconstruction and their performance is compared. The κ-space under-sampling schemes such as the variable density sampling and the more conventional radial sampling are experimented at the same sampling rate and the effect of encoding scheme on iterative hard thresholding compressed sensing reconstruction is studied.

Multiphase flow measurements, based on tomographic imaging techniques, have made substantial progress in the past several years. In particular, ultrasonic tomographic reconstruction has shown its high potential in accurately measuring fluid fractions in a flow cross-section based on direct flow measurements. These types of measurements are very attractive as the system is nonradioactive, has direct contact to fluid, yet without flow obstruction. An ultrasonic tomographic multiphase flow meter (UTM) is being developed to capture the fluid flow rates and phase compositions in producing oil wells. The UTM aims to generate cross-sectional images of the flow inside the pipe. To recover these images, complex computations and high processing time are required, based on a limited amount of available data. For this purpose, compressive sensing (CS) reconstruction methods have shown great potential in overcoming the mentioned data quality and quantity issues. CS builds upon the fact that one can recover certain data of interest from potentially very few measurements. Many CS algorithms have been proposed in the literature. In this paper, we seek to find a suitable algorithm by comparing different CS algorithms in terms of the resulting recovered image quality. Specifically, we survey three algorithms in CS, namely basis pursuit, iterative soft thresholding (IST), and orthogonal matching pursuit (OMP). We also compare the results of the CS algorithms to that of a least-square based conventional algorithm. The numerical results show the superiority of CS recovery algorithms over the conventional approach, both qualitatively and quantitatively. The results also suggest that high image quality could potentially be achieved with less complex CS algorithms, i.e., the IST and OMP algorithms. As presented in the paper, our goal is to adapt CS reconstruction algorithms for images generated by the UTM, for the purpose of accurate quantification of oil, gas, and water phase fractions produced from different wells.

Compressed sensing (CS) is a novel developed theoretical framework for information acquisition and processing. Taking advantages of the sparsity or compressibility inherent in real world signals, CS can collect compressed data at the sampling rate much lower than that needed in Shannon's theorem. When it has been used for medical imaging techniques, CS theory can fast the scanning speed of MRI, reduce the radiation doses and alleviate the patients' suffering. In order to obtain the effective sparse representation, the nonsubsampled contourlet transform in the frequency domain (NSCT-FD) is adopted, then an improved fast iterative soft thresholding algorithm (FISTA) is applied for CS-MRA reconstruction. This novel method not only inherits the simplicity and effectiveness of the original FISTA, and the sparse curvy representation ability of the contourlet transform, but also has the advantages of the shift-invariant property and much less computational burden. The performance is evaluated qualitatively and quantitatively both in noiseless and noisy situations, compared to the classic wavelet and the sharp frequency localization contourlet transform (SFLCT) method. Three quantitative indices are employed including the peak signal to noise ratio (PSNR), mutual information (MI) and relative l 2 norm error (RLNE) and qualitative performance evaluations use the profiles in horizontal direction and local region magnification comparison. Experimental results demonstrate the superiority of the NSCT-FD algorithm, displaying higher PSNR and MI, and lower RLNE indices in both noiseless and noisy MRA images, which have good reconstruction accuracy with reasonable real time computation speed. It is manifest that the NSCT-FD algorithm can be applied in fast medical imaging fields to achieve quality images in a relatively fast process, which will benefit cardiac and carotid MRI, dynamic MRI and MRA.

Sparse prior to image denoising is a classical research field with a long history in computer vision. We propose an end-to-end supervised neural network, named DnMLCSC-net, which is inspired via multi-layer convolutional sparse coding model embedded with symbiotic analysis–synthesis priors for natural image denoising. Unfolding a multi-layer, learned iterative soft thresholding algorithm (ML-LISTA) and developing into a convolutional recurrent neural network, all parameters in the model are updated adaptively to minimize mixed loss via gradient descent using backpropagation. In addition, a combined ReLU function is taken as the activation function. Inconsistent dilated convolution and batch normalization were empirically introduced into the encoding layers corresponding to the first iteration of ML-LISTA. Experimental results show that our network achieves a competitive denoising effect in comparison with several state-of-the-art denoising methods.

Debiasing of seismic reflectivity inversion using basis pursuit de-noising algorithm

Microphone arrays have long been used to characterize and locate sound sources. However, existing algorithms for processing the signals are computationally expensive and, consequently, different methods need to be explored. Recently, the trained iterative soft thresholding algorithm (TISTA), a data-driven solver for inverse problems, was shown to improve on existing approaches. Here, a more in-depth analysis of its robustness and frequency dependence is provided using synthesized as well as real measurement data. It is demonstrated that TISTA yields favorable results in comparison to a covariance matrix fitting inverse method, especially for large numbers of sources.

Synthetic aperture radio telescopes are important in the field of radio astronomy. However, the inverse imaging process of these telescopes is an ill-posed inverse problem. Although the compressed sensing technology represented by the sparsity averaging reweighted analysis (SARA) algorithm has been successfully applied to the imaging of the synthetic aperture radio telescope, large reconstruction errors remain in the traditional SARA algorithm due to its difficulty in selecting soft threshold parameters. Therefore, an improved SARA algorithm is proposed. This algorithm uses an improved projected fast iterative soft thresholding algorithm to solve the minimization model, adaptively updates soft threshold parameters by fully utilizing data fidelity and regularization terms, and adopts restarting and adaptive strategies to accelerate convergence, thereby improving the overall accuracy and speed of inversion imaging. The simulation results demonstrate that compared with the traditional compressed sensing algorithm, the improved SARA algorithm can effectively reduce reconstruction error and increase calculation speed, thus proving its effectiveness.

Magnetotelluric (MT) sounding data are easily contaminated by various noise sources, especially noise with a long duration (even full-time noise), which makes it difficult to obtain accurate values when calculating the weight factors of the response function, resulting in distortion of the response results. Based on blind source separation theory, fast independent component analysis (FastICA) can separate this kind of noise. However, this method is challenged by the number of separated field sources and the unequal signal amplitude before and after processing. We have developed a novel signal noise separation method, improving the traditional FastICA, where a correlation coefficient is used to compute the number of field sources, and the fast iterative shrinkage threshold algorithm (FISTA) is used to adjust the signal amplitude problem before and after FastICA decomposition. Compared with common field source division methods and the traditional FastICA, the experimental results indicate that our method can separate and remove noise with a long duration, increase the signal-to-noise ratio of the data, and improve the MT response curves. Meanwhile, case studies of measured data illustrate that our method obtains a more robust magnetotelluric response than the conventional robust method and FastICA.