Nonlinear Reconstruction Algorithm Research Articles

Into nonlinearity and beyond for Zernike-like wavefront sensors

Context. Telescopes like the Extremely Large Telescope (ELT) and the Giant Magellan Telescope (GMT) will be used together with extreme adaptive optics (AO) instruments to directly image Earth-like planets. The AO systems will need to perform at the fundamental limit in order to image Earth twins. A crucial component is the wavefront sensor. Interferometric wavefront sensors, such as the Zernike wavefront sensor (ZWFS), have been shown to perform close to the fundamental sensitivity limit. However, sensitivity comes at the cost of linearity; the ZWFS has strong nonlinear behavior. Aims. The aim of this work is to increase the dynamic range of Zernike-like wavefront sensors by using nonlinear reconstruction algorithms combined with phase sorting interferometry (PSI) and multi-wavelength measurements. Methods. The response of the ZWFS is explored analytically and numerically. Results. The proposed iterative (non)linear reconstructors reach the machine precision for small aberrations (<0.25 rad rms). Coupling the nonlinear reconstruction algorithm with PSI increases the dynamic range of the ZWFS by a factor of three to about 0.75 rad rms. Adding multiple wavebands doubles the dynamic range again, to 1.4 radians rms. Conclusions. The ZWFS is one of the most sensitive wavefront sensors, but has a limited dynamic range. The ZWFS will be an ideal second-stage wavefront sensor if it is combined with the proposed nonlinear reconstruction algorithm.

Image reconstruction based on nonlinear diffusion model for limited-angle computed tomography

The problem of limited-angle computed tomography (CT) imaging reconstruction has a wide range of practical applications. Due to various factors such as high x-ray absorption, structural characteristics of the scanned object, and equipment limitations, it is often impractical to obtain a complete angular scan, resulting in limited-angle scan data. In this paper, we propose an iterative image reconstruction algorithm for limited-angle CT. The algorithm carries out a traditional CT reconstruction and a nonlinear diffusion process alternatively. Specifically, a subtle partial differential equation is constructed to guide the nonlinear diffusion process to eliminate limited-angle artifacts in the reconstructed image. Numerical experiments on both analytic data and real data validate the efficacy of the proposed nonlinear diffusion reconstruction algorithm. Furthermore, a linear diffusion reconstruction algorithm which combines a traditional CT reconstruction algorithm and a linear diffusion process is also presented in the paper.

GROG-pCS: GRAPPA Operator Gridding with CS-based p-thresholding for Under-sampled Radially Encoded MRI

Major limitation of MRI is long scan time. Compressed Sensing (CS) is a contemporary technique used to accelerate MRI scan time. In CS, fully sampled MRI images are reconstructed from the partially acquired k-space data. In CS MRI, the utilization of a non-linear reconstruction algorithm is one of the key requirements for successful signal recovery. Numerous methods have been used in CS for solving the non-linear problems to get the solution image. In this paper, we proposed GRAPPA Operator gridding (GROG) with CS-based p-thresholding to reconstruct the artefact free MR images from the partially acquired radial k-space data. In this proposed scheme, initially radially acquired under-sampled k-space data is mapped onto Cartesian space using GROG gridding and then CS reconstruction is performed by using iterative p-thresholding. The proposed method is tested on four MRI data sets, (i) simulated Shepp-Logan phantom, (ii) 1.5T human brain data, (iii) 3T human brain, and (iv) 3T short-axial cardiac (SA) radial data. The reconstruction results are compared with the CS-based iterative hard-thresholding and soft-thresholding reconstructions. The quality of the solution images is evaluated by using (i) Artifact Power (AP), (ii) Root Mean Square Error (RMSE), and (iii) Peak Signal-to-Noise Ratio (PSNR).

Non-Linear Reconstruction for ERT Inverse Problem Based on Homotopy Algorithm

Due to the non-linear and non-convex property, classical ERT image reconstruction algorithms are less effective and less accuracy. In this paper, a non-linear and non-convex image reconstruction algorithm based on homotopy method was proposed. The proposed algorithm converted the ERT inverse problem to a multi-object non-convex optimization which promoted the reconstruction accuracy and avoid the local optimal. Experimental validations were conducted. The optimization process is studied which demonstrate the effectiveness. Moreover, the proposed algorithm is compared with 6 other representative algorithms (linear/non-linear, convex/non-convex) at the condition of different distribution and number of objects. The image quality parameters of the proposed algorithms are studied which show that the homotopy algorithm can provide image reconstruction result with higher quality and better stability than the other conventional algorithms.

Characterization of urban microclimates from reconstructed wind velocity and temperature fields

This work aims to study microclimate changes within urban microspaces using atmospheric acoustic tomography (AAT). Of particular interest are the imaging of wind velocity and temperature field variations. To reduce the inaccuracy of using the conventional AAT straight ray model, a nonlinear reconstruction algorithm is employed. Simulated and measurement results are presented and discussed.

Open structure magnetic particle imaging by nonlinear back projection tomography reconstruction.

In open structure MPI systems, the nonlinear variation of the field free lines in the large region of interest scanning process distorts the x-space image reconstruction. In this study, we propose a nonlinear field free line projection reconstruction algorithm to solve the edge distortion problem of open structure MPI imaging. First, we calculate the curvature change law of the field free line in the scanning process. Then, we design a nonlinear back projection reconstruction algorithm according to the nonlinear characteristics of the field free line in the scanning process. Finally, the nonlinear back projection reconstruction algorithm is used to complete the tomography of blood vessels. The numerical calculation and simulation results show that the open structure MPI combined with a nonlinear back projection reconstruction algorithm can accomplish vascular fault reconstruction. The reconstruction algorithm proposed in this paper suppresses the edge distortion of the image and improves the positioning accuracy of the image. The size of the region of interest where distortions are low is increased 16 times by allowing 10.9% degradation in the gradient. We provide a non-linear inverse projection reconstruction algorithm to reduce the structural artefacts caused by FFL distortion. It provides a reconstruction scheme for a large region of interest fine imaging of open structure FFL-MPI.

Non-linear reconstruction of features in the primordial power spectrum from large-scale structure

ABSTRACT Potential features in the primordial power spectrum have been searched for in galaxy surveys in recent years since these features can assist in understanding the nature of inflation. The null detection to date suggests that any such features should be fairly weak, and next-generation galaxy surveys, with their unprecedented sizes and precisions, are in a position to place stronger constraints than before. However, even if such primordial features once existed in the early Universe, they would have been significantly damped in the non-linear regime at low redshift due to structure formation, which makes them difficult to be directly detected in real observations. A potential way to tackle this challenge for probing the features is to undo the cosmological evolution, i.e. using reconstruction to obtain an approximate linear density field. By employing a set of N-body simulations, here we show that a recently proposed non-linear reconstruction algorithm can effectively retrieve damped oscillatory features from halo catalogues and improve the accuracy of the measurement of feature parameters (assuming that such primordial features do exist). We do a Fisher analysis to forecast how non-linear reconstruction affects the constraining power, and find that it can lead to significantly more robust constraints on the feature amplitude for a DESI-like survey. Comparing non-linear reconstruction with other ways of improving constraints, such as increasing the survey volume and range of scales, this shows that it is possible to achieve what the latter do, but at a lower cost.

Single-Shot Compressed Imaging via Random Phase Modulation

Compressed sensing (CS) provides an innovative framework for signal sampling, which enables accurate recovery of the sparse or compressible signal from a small set of linear measurements far fewer than the Nyquist rate in traditional signal processing. In compressed sensing, random modulation plays a key role, which can spread out the signal information more or less evenly across all locations. There are many modulation techniques, such as amplitude modulation, frequency modulation, phase modulation, spectrum modulation, and so on. Among these modulation techniques, phase modulation is vital due to the efficiency and convenience of modulation. In this paper, we review both the theoretical and application of compressed sensing and several compressed imaging systems using random phase modulation. First, we review the fundamentals of compressed sensing, dividing it into three parts: sparse representation, incoherent measurement, and nonlinear reconstruction algorithm. We then show how phase modulation can be applied to compressed sensing and compressed imaging, where the presentation can be divided into six main parts, corresponding to different aspects of phase modulation applied in compressed sensing or compressed imaging: (1) Fundamentals of compressed sensing. (2) Principles of phase modulation. (3) Single-shot compressed imaging with spatial-domain single random phase mask (CI-SSRPM). (4) Single-shot compressed imaging with a random convolution using a double random phase mask (CI-DRPM). (5) Single-shot compressed imaging with Fourier-domain single random phase mask (CI-FSRPM). (6) Single-shot compressed imaging with double random phase encoding (CI-DRPE).

Compressed Sensing in Parallel MRI: A Review

Magnetic resonance imaging (MRI) is a dynamic and safe imaging technique in medical imaging. Recently, parallel MRI (pMRI) is widely used for accelerating conventional MRI. Both frequency and image domain-based reconstructions are the most attractive methods for generating the image from multi-channel k-space data. Compressed sensing (CS) is a recently used procedure to reduce the acquisition time of conventional MRI. This reduction is achieved by taking fewer measurements from the fully sampled k-space data. Therefore, applying the CS technique in pMRI is the most emerging way for further improving the acquisition time that is a tremendous research interest. However, as the phase encoding plane may be perpendicular or parallel to the coil elements plane, finding the exact domain for CS in pMRI reconstruction is a major challenging issue. In this work, the application of the CS technique in pMRI in both domains is investigated. Later some widely used methodologies are presented as the nonlinear reconstruction algorithm of CS in pMRI. Finally, a discussion is performed based on CS in pMRI to perceive the reality of different reconstruction algorithms at a glance for finding preferred methodologies.

Nonlinear reconstruction algorithm for line-scanning microscopy via virtual structured-modulation

Line-scanning microscopy with virtually structured modulation (vLSM) provides an easy, low-cost, axial laminating ability, and phase-artifact-free strategy to achieve super-resolution imaging. In this paper, we apply the nonlinear reconstruction (NLR) algorithm to the vLSM and obtain rapid super-resolution imaging. The reconstruction algorithm can directly process raw data in the spatial domain without complicated processing. NLR algorithm is about 3 times faster than the conventional linear algorithm stemming from the advantage of spatial domain processing, providing ∼2.02 times the resolution of the wide-field imaging. Both simulated and experimental data were used to verify resolution enhancement.

Iterative removal of redshift-space distortions from galaxy clustering

ABSTRACT Observations of galaxy clustering are made in redshift space, which results in distortions to the underlying isotropic distribution of galaxies. These redshift-space distortions (RSDs) not only degrade important features of the matter density field, such as the baryonic acoustic oscillation (BAO) peaks, but also pose challenges for the theoretical modelling of observational probes. Here, we introduce an iterative non-linear reconstruction algorithm to remove RSD effects from galaxy clustering measurements, and assess its performance by using mock galaxy catalogues. The new method is found to be able to recover the real-space galaxy correlation function with an accuracy of $\sim \!1{{\ \rm per\ cent}}$, and restore the quadrupole accurately to 0, on scales $s\gtrsim 20\,h^{-1}\, {\rm Mpc}$. It also leads to an improvement in the reconstruction of the initial density field, which could help to accurately locate the BAO peaks. An ‘internal calibration’ scheme is proposed to determine the values of cosmological parameters, as a part of the reconstruction process, and possibilities to break parameter degeneracies are discussed. RSD reconstruction can offer a potential way to simultaneously extract the cosmological parameters, initial density field, real-space galaxy positions, and large-scale peculiar velocity field (of the real Universe), making it an alternative to standard perturbative approaches in galaxy clustering analysis, bypassing the need for RSD modelling.

Blind structured illumination as excitation for super-resolution photothermal radiometry

ABSTRACT Photothermal radiometry with an infrared camera allows the contactless temperature measurement of multiple surface pixels simultaneously. A short light pulse heats the sample. The heat propagates through the sample by diffusion and the corresponding temperature evolution is measured at the surface of the sample by an infrared camera. The main drawback in radiometric imaging is the loss of the spatial resolution with increasing depth due to heat diffusion, which results in blurred images for deeper lying structures. We circumvent this information loss due to the diffusion process by using blind structured illumination, combined with a nonlinear joint-sparsity reconstruction algorithm. The structured illumination is realised by parallel laser lines from a vertical-cavity surface-emitting laser (VCSEL) array controlled by a random binary pattern generator. By using 150 different patterns of structured illumination and our iterative joint-sparsity algorithm, it was possible to resolve 1 mm thick lines at a distance down to 0.5 mm, which results in a resolution enhancement of approximately a factor of four compared to the resolution of 5.9 mm for homogeneous illuminated thermographic reconstruction.

Incorporating prior knowledge via volumetric deep residual network to optimize the reconstruction of sparsely sampled MRI

For sparse sampling that accelerates magnetic resonance (MR) image acquisition, non-linear reconstruction algorithms have been developed, which incorporated patient specific a prior information. More generic a prior information could be acquired via deep learning and utilized for image reconstruction. In this study, we developed a volumetric hierarchical deep residual convolutional neural network, referred to as T-Net, to provide a data-driven end-to-end mapping from sparsely sampled MR images to fully sampled MR images, where cartilage MR images were acquired using an Ultra-short TE sequence and retrospectively undersampled using pseudo-random Cartesian and radial acquisition schemes. The network had a hierarchical architecture that promoted the sparsity of feature maps and increased the receptive field, which were valuable for signal synthesis and artifact suppression. Relatively dense local connections and global shortcuts were established to facilitate residual learning and compensate for details lost in hierarchical processing. Additionally, volumetric processing was adopted to fully exploit spatial continuity in three-dimensional space. Data consistency was further enforced. The network was trained with 336 three-dimensional images (each consisting of 32 slices) and tested by 24 images. The incorporation of a priori information acquired via deep learning facilitated high acceleration factors (as high as 8) while maintaining high image fidelity (quantitatively evaluated using the structural similarity index measurement). The proposed T-Net had an improved performance as compared to several state-of-the-art networks.

BAO reconstruction: a swift numerical action method for massive spectroscopic surveys

A new fully non-linear reconstruction algorithm for the accurate recovery of the Baryonic Acoustic Oscillations (BAO) scale in two-point correlation functions is proposed, based on the least-action principle and extending the Fast Action Minimisation method by Nusser & Branchini (2000). Especially designed for massive spectroscopic surveys, it is tested on dark-matter halo catalogues extracted from the DEUS-FUR $\Lambda$CDM simulation to trace the trajectories of up to $\sim207,000$ haloes backward-in-time, well beyond the first-order Lagrangian approximation. The new algorithm successfully recovers the BAO feature in real and redshift-space in both the monopole and the anisotropic two-point correlation function, also for anomalous samples showing misplaced or absent signature of BAO. In redshift-space the non-linear displacement parameter $\Sigma_\mathrm{NL}$ is reduced from $11.8\pm0.3h^{-1}$Mpc at redshift $z=0$ (pre-reconstruction) to $4.0\pm0.5h^{-1}$Mpc at $z\simeq37$ after reconstruction. A comparison with the first-order Lagrangian reconstruction is presented, showing that this techniques outperforms the linear approximation in recovering an unbiased measurement of the acoustic scale.

Model-based material decomposition with a penalized nonlinear least-squares CT reconstruction algorithm

Spectral information in CT may be used for material decomposition to produce accurate reconstructions of material density and to separate materials with similar overall attenuation. Traditional methods separate the reconstruction and decomposition steps, often resulting in undesirable trade-offs (e.g. sampling constraints, a simplified spectral model). In this work, we present a model-based material decomposition algorithm which performs the reconstruction and decomposition simultaneously using a multienergy forward model. In a kV-switching simulation study, the presented method is capable of reconstructing iodine at 0.5 mg ml−1 with a contrast-to-noise ratio greater than two, as compared to 3.0 mg ml−1 for image domain decomposition. The presented method also enables novel acquisition methods, which was demonstrated in this work with a combined kV-switching/split-filter acquisition explored in simulation and physical test bench studies. This novel design used four spectral channels to decompose three materials: water, iodine, and gadolinium. In simulation, the presented method accurately reconstructed concentration value estimates with RMSE values of 4.86 mg ml−1 for water, 0.108 mg ml−1 for iodine and 0.170 mg ml−1 for gadolinium. In test-bench data, the RMSE values were 134 mg ml−1, 5.26 mg ml−1 and 1.85 mg ml−1, respectively. These studies demonstrate the ability of model-based material decomposition to produce accurate concentration estimates in challenging spatial/spectral sampling acquisitions.

CALDERÓN’S INVERSE PROBLEM WITH A FINITE NUMBER OF MEASUREMENTS

We prove that an$L^{\infty }$potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace${\mathcal{W}}$. As a corollary, we obtain a similar result for Calderón’s inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces${\mathcal{W}}$, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of$\dim {\mathcal{W}}$.

Reconstructing the baryon acoustic oscillations using biased tracers

The reconstruction of the initial conditions of the Universe is an important topic in cosmology, particularly in the context of sharpening the measurement of the baryon acoustic oscillation (BAO) peak. Nonlinear reconstruction algorithms developed in recent years, when applied to late-time matter fields, can recover to a substantial degree the initial density distribution, however, when applied to sparse tracers of the matter field, the performance is poorer. In this paper we apply the Shi et al. non-linear reconstruction method to biased tracers in order to establish what factors affect the reconstruction performance. We find that grid resolution, tracer number density and mass assignment scheme all have a significant impact on the performance of our reconstruction method, with triangular-shaped-cloud (TSC) mass assignment and a grid resolution of ${\sim}1{-}2h^{-1}$ Mpc being the optimal choice. We also show that our method can be easily adapted to include generic tracer biases up to quadratic order in the reconstruction formalism. Applying the reconstruction to halo and galaxy samples with a range of tracer number densities, we find that the linear bias is by far the most important bias term, while including nonlocal and nonlinear biases only leads to marginal improvements on the reconstruction performance. Overall, including bias in the reconstruction substantially improves the recovery of BAO wiggles, down to $k\sim0.25~h\text{Mpc}^{-1}$ for tracer number densities between $2\times10^{-4}$ and $2\times10^{-3}~(h^{-1}\text{Mpc})^{-3}$.

Accelerating quantitative susceptibility imaging acquisition using compressed sensing

To evaluate the feasibility of compressed sensing (CS) for accelerating quantitative susceptibility mapping (QSM) acquisition in MR histology, control, demyelination, and remyelination mice were scanned using a modified 3D gradient echo (GRE) pulse sequence (allowing undersampling the k-space in two phase dimensions) at 9.4 T. Fully sampled and CS data at various acceleration factors (AF) (4.0–16.0) were acquired with high spatial resolution of 22.5 µm3 isotropic. For prospective scans, the local phase was reconstructed slice-by-slice using a nonlinear reconstruction algorithm with phase cycling, and the QSM was calculated using the iLSQR method. Susceptibility values of major white matter tracts at various CS AF were compared with the fully sampled data as ground truth. The reconstructed QSM preserved major information with negligible artifacts, even at CS AF of 8.0 and 16.0. The maximum variation of susceptibility value of control mice was 5.8% in the fimbria (FI) area with the acceleration factor of 16.0, compared to the ground truth. The maximum variation was 9.5% in corpus callosum (CC) area of the demyelination mouse brain with the acceleration factor of 8.0. The maximum variation was 8.2% in AC area of remyelination mouse brain with 16.0 acceleration factor. This preliminary study shows that CS can reduce the acquisition time of GRE MRI in MR histology without losing significant accuracy in quantitative susceptibility values.

288 Evaluation of iterative reconstruction algorithms in CT by means of low contrast detectability: A feasibility study

Purpose With non-linear iterative reconstruction algorithms (IRA) conventional Fourier metrics are not applicable. Newly proposed metrics are complicated. This study aims to test the usefulness of straightforward low contrast detectability (LCD) evaluations based on a statistical method (1) for a task specific comparison of two IRA: raw data-based +knowledge based SAFIRE and knowledge-based (both Siemens, Erlangen). Methods Two recent CT systems of the same manufacturer with different IRA and different detector technology have been studied (Table 1). A homogeneous phantom (Siemens calibration phantom) has been acquired at 120 kV with exposure values ranging from 0.2 to 1.5 times the standard dose. Reconstruction has been carried out with three different clinical algorithms and slice thicknesses: smooth J30 (3.0 mm), medium J40 (1.5 mm), sharp J70 (1.5 mm). Based on a statistical approach a contrast-detail curve for 10 subsequent reconstructed slices has been determined, the mean curve calculated and interpolated. To summarize the data an overall index Image Quality Figure (IQF) has been defined. Results In both systems LCD improves steadly with CTDIvol. For filter J30 there is no difference in LCD between the two systems at clinical doses. At very low doses system 2 with ADMIRE scores slightly better. For filter J40 system 2 performs better at all dose levels, with an average dose reduction of −17%, which includes detector and IRA advantage. Conclusions The LCD method with adoption of the overall index IQF revealed sensitive to subtle variations in the image texture produced by different IRA. It has demonstrated to be a reliable tool for simple image quality assessment in CT quality control tasks. The data confirm that the novel algorithm ADMIRE does not bring a generalized image quality improvement or dose saving potential over SAFIRE, but that it acts selectively on the image quality, depending on the noise content of the region. In decreasing order of importance the possible gains depend on image kernel, IRA strength, IRA type. Download : Download high-res image (230KB) Download : Download full-size image Download : Download high-res image (295KB) Download : Download full-size image

Learning-Based Compressive MRI.

In the area of magnetic resonance imaging (MRI), an extensive range of non-linear reconstruction algorithms has been proposed which can be used with general Fourier subsampling patterns. However, the design of these subsampling patterns has typically been considered in isolation from the reconstruction rule and the anatomy under consideration. In this paper, we propose a learning-based framework for optimizing MRI subsampling patterns for a specific reconstruction rule and anatomy, considering both the noiseless and noisy settings. Our learning algorithm has access to a representative set of training signals, and searches for a sampling pattern that performs well on average for the signals in this set. We present a novel parameter-free greedy mask selection method and show it to be effective for a variety of reconstruction rules and performance metrics. Moreover, we also support our numerical findings by providing a rigorous justification of our framework via statistical learning theory.