Regularized Reconstruction Method Research Articles

Learned Tensor Low-CP-Rank and Bloch Response Manifold Priors for Non-Cartesian MRF Reconstruction.

Magnetic resonance fingerprinting (MRF) can rapidly perform simultaneous imaging of multiple tissue parameters. However, the rapid acquisition schemes used in MRF inevitably introduce aliasing artifacts in the recovered tissue fingerprints, reducing the accuracy of the predicted parameter maps. Current regularized reconstruction methods are based on iterative procedures which are usually time-consuming. In addition, most of the current deep learning-based methods for MRF often lack interpretability owing to the black-box nature, and most deep learning-based methods are not applicable for non-Cartesian scenarios, which limits the practical applications. In this paper, we propose a joint reconstruction model incorporating MRF-physics prior and the data correlation constraint for non-Cartesian MRF reconstruction. To avoid time-consuming iterative procedures, we unroll the reconstruction model into a deep neural network. Specifically, we propose a learned CANDECOMP/PARAFAC (CP) decomposition module to exploit the tensor low-rank priors of high-dimensional MRF data, which avoids computationally burdensome singular value decomposition. Inspired by the MRF-physics, we also propose a Bloch response manifold module to learn the mapping between reconstructed MRF data and the multiple parameter maps. Numerical experiments show that the proposed network can reconstruct high-quality MRF data and multiple parameter maps within significantly reduced computational time.

Regularized reconstruction based on joint smoothly clipped absolute deviation regularization and graph manifold learning for fluorescence molecular tomography

Objective. Fluorescence molecular tomography (FMT) is an optical imaging modality that provides high sensitivity and low cost, which can offer the three-dimensional distribution of biomarkers by detecting the fluorescently labeled probe noninvasively. In the field of preclinical cancer diagnosis and treatment, FMT has gained significant traction. Nonetheless, the current FMT reconstruction results suffer from unsatisfactory morphology and location accuracy of the fluorescence distribution, primarily due to the light scattering effect and the ill-posed nature of the inverse problem. Approach. To address these challenges, a regularized reconstruction method based on joint smoothly clipped absolute deviation regularization and graph manifold learning (SCAD-GML) for FMT is presented in this paper. The SCAD-GML approach combines the sparsity of the fluorescent sources with the latent manifold structure of fluorescent source distribution to achieve more accurate and sparse reconstruction results. To obtain the reconstruction results efficiently, the non-convex gradient descent iterative method is employed to solve the established objective function. To assess the performance of the proposed SCAD-GML method, a comprehensive evaluation is conducted through numerical simulation experiments as well as in vivo experiments. Main results. The results demonstrate that the SCAD-GML method outperforms other methods in terms of both location and shape recovery of fluorescence biomarkers distribution. Siginificance. These findings indicate that the SCAD-GML method has the potential to advance the application of FMT in in vivo biological research.

Joint Bi-Modal Tomographic Reconstruction from Observations with Missing Data

A method for tomographic reconstruction of bi-modal data sets using cross modal regularization was developed. The method constrains the Simultaneous Algebraic Reconstruction Technique by using clustering-based segmented images and quality controlled fused images, to jointly regularize the reconstruction of structures imaged with different methods. The joint regularized reconstruction method was tested using a synthetic X-Ray and neutron data set with a mix of attenuation driven data starvation. Assuming consistent material boundaries in the different imaging modalities, it was shown that improvements in terms of the Feature Similarity Index Measure are made compared to a traditional SART-TV reconstruction. UK Ministry of Defence ©Crown Owned Copyright 2023/AWE

Distortion-corrected image reconstruction with deep learning on an MRI-Linac.

MRI is increasingly utilized for image-guided radiotherapy due to its outstanding soft-tissue contrast and lack of ionizing radiation. However, geometric distortions caused by gradient nonlinearities (GNLs) limit anatomical accuracy, potentially compromising the quality of tumor treatments. In addition, slow MR acquisition and reconstruction limit the potential for effective image guidance. Here, we demonstrate a deep learning-based method that rapidly reconstructs distortion-corrected images from raw k-space data for MR-guided radiotherapy applications. We leverage recent advances in interpretable unrolling networks to develop a Distortion-Corrected Reconstruction Network (DCReconNet) that applies convolutional neural networks (CNNs) to learn effective regularizations and nonuniform fast Fourier transforms for GNL-encoding. DCReconNet was trained on a public MR brain dataset from 11 healthy volunteers for fully sampled and accelerated techniques, including parallel imaging (PI) and compressed sensing (CS). The performance of DCReconNet was tested on phantom, brain, pelvis, and lung images acquired on a 1.0T MRI-Linac. The DCReconNet, CS-, PI-and UNet-based reconstructed image quality was measured by structural similarity (SSIM) and RMS error (RMSE) for numerical comparisons. The computation time and residual distortion for each method were also reported. Imaging results demonstrated that DCReconNet better preserves image structures compared to CS- and PI-based reconstruction methods. DCReconNet resulted in the highest SSIM (0.95 median value) and lowest RMSE (<0.04) on simulated brain images with four times acceleration. DCReconNet is over 10-times faster than iterative, regularized reconstruction methods. DCReconNet provides fast and geometrically accurate image reconstruction and has the potential for MRI-guided radiotherapy applications.

Resolution aware nonconvex quasinorm iterative digital breast tomosynthesis imaging

Digital breast tomosynthesis is a promising modality providing 3D cross sectional images of breast. However, reconstructing the image from few projections is still a challenging task. We introduce an anisotropic total variation regularized reconstruction method which exploits varying resolution along transverse and sagittal planes by adapting the weights of TV gradients. Moreover, we also investigate if non-convex ℓp quasinorm with p ∈ (0,1) can help to improve the performance of anisotropic TV in detecting micro calcifications and spheroidal masses. The contrast to noise ratio (CNR) for in-plane quality, full with at half maximum (FWHM) of artefact spread function (ASF) for resolution in the z-axis, signal difference to noise ratio (SdNR), and quality factor (QF) for a comprehensive evaluation were used in quantitative analyses and comparisons. For low contrast analysis, QF value of the proposed method for p = 0.9 is 0.18 whereas QF value of isotropic TV with ℓ1 and ℓ2 norms are 0.15 and 0.14, respectively. For high contrast features such as micro-calcifications, although the best CNR value changes with different p values, the majority of the best CNR values are calculated for values 0.8 and 0.9. Overall, the results show that the proposed method, anisotropic TV with non-convex ℓp quasinorm, outperforms isotropic TV with ℓ1 and ℓ2 norms by recovering small micro calcifications and spheroidal masses with higher CNR and increased depth resolution. The proposed method shows the best performance for p values of 0.8 ≤ p < 1.

A content-adaptive unstructured grid based regularized CT reconstruction method with a SART-type preconditioned fixed-point proximity algorithm

The goal of this study is to develop a new computed tomography (CT) image reconstruction method, aiming at improving the quality of the reconstructed images of existing methods while reducing computational costs. Existing CT reconstruction is modeled by pixel-based piecewise constant approximations of the integral equation that describes the CT projection data acquisition process. Using these approximations imposes a bottleneck model error and results in a discrete system of a large size. We propose to develop a content-adaptive unstructured grid (CAUG) based regularized CT reconstruction method to address these issues. Specifically, we design a CAUG of the image domain to sparsely represent the underlying image, and introduce a CAUG-based piecewise linear approximation of the integral equation by employing a collocation method. We further apply a regularization defined on the CAUG for the resulting ill-posed linear system, which may lead to a sparse linear representation for the underlying solution. The regularized CT reconstruction is formulated as a convex optimization problem, whose objective function consists of a weighted least square norm based fidelity term, a regularization term and a constraint term. Here, the corresponding weighted matrix is derived from the simultaneous algebraic reconstruction technique (SART). We then develop a SART-type preconditioned fixed-point proximity algorithm to solve the optimization problem. Convergence analysis is provided for the resulting iterative algorithm. Numerical experiments demonstrate the superiority of the proposed method over several existing methods in terms of both suppressing noise and reducing computational costs. These methods include the SART without regularization and with the quadratic regularization, the traditional total variation (TV) regularized reconstruction method and the TV superiorized conjugate gradient method on the pixel grid.

On Hallucinations in Tomographic Image Reconstruction.

Tomographic image reconstruction is generally an ill-posed linear inverse problem. Such ill-posed inverse problems are typically regularized using prior knowledge of the sought-after object property. Recently, deep neural networks have been actively investigated for regularizing image reconstruction problems by learning a prior for the object properties from training images. However, an analysis of the prior information learned by these deep networks and their ability to generalize to data that may lie outside the training distribution is still being explored. An inaccurate prior might lead to false structures being hallucinated in the reconstructed image and that is a cause for serious concern in medical imaging. In this work, we propose to illustrate the effect of the prior imposed by a reconstruction method by decomposing the image estimate into generalized measurement and null components. The concept of a hallucination map is introduced for the general purpose of understanding the effect of the prior in regularized reconstruction methods. Numerical studies are conducted corresponding to a stylized tomographic imaging modality. The behavior of different reconstruction methods under the proposed formalism is discussed with the help of the numerical studies.

An Efficient One-Step Method for Spectral CT Based on an Approximate Linear Model

Spectral computed tomography (spectral CT) is a medical and biomedical imaging technique which uses the spectral information of the attenuated X-ray beam. Energy-resolved photon-counting detector is a promising technique for improved spectral CT imaging and allows to obtain material selective images. Two different kind of approaches can be used to solve the problem of spectral reconstruction consisting of material decomposition and tomographic reconstruction: the “two steps” methods which are most often projection-based methods, and the “one step” methods. While the projection-based methods are interesting for the fast computational time, the one-step methods link directly the material image domain and the multienergy sinograms domain, which is an advantage to introduce spatial prior information in the material image domain. In this work we propose a proximal operator associated to a data fidelity term (based on an empirical linear forward model) for multimaterial decomposition. This proximal operator, which has a closed-form solution, makes it possible to propose fast regularized reconstruction methods for spectral CT, in particular for data of photon-counting detectors with many energy bins. We propose a one-step method based on the alternating direction method of multipliers (ADMM) and this proximal operator. We present promising results on dual material image reconstructions obtained from simulated and experimental data. We show qualitative and quantitative comparisons with two previously published multimaterial decomposition methods.

Study of the Effect of Reconstruction Parameters for Myocardial Perfusion Imaging in PET With a Novel Flow Phantom

Myocardial perfusion imaging (MPI) with positron emission tomography (PET) allows quantitative temporal measurements of the radioactive tracer distribution in tissue. The quantification for myocardial blood flow (MBF) is conducted with kinetic modelling of the image-derived time-activity curves (TACs) allowing derivation for MBF in units of mL/min per gram of tissue. The ordered-subset expectation maximization (OSEM) reconstruction algorithm with time-of-flight (TOF) and point spread function (PSF) modelling is now routinely employed in cardiac imaging. However, the varying counting statistics of the MPI measurements conducted with short-lived tracers present a challenge for the PET image reconstruction methods. Thus, the effect of the reconstruction methods on the flow quantification needs to be evaluated in a standardized manner. Recently, a novel PET flow phantom modelling the MBF has been developed for investigation of the standardization of the MBF measurements. In this study, the effect of the reconstruction parameters on the image-derived flow values against a known reference flow of the flow phantom was studied with [15O]H2O. The effects were studied by comparison of TACs and relative errors of the image-derived flow values with respect to the phantom-derived reference flow value using 5 repeated PET scans with fixed acquisition parameters using a digital Discovery MI PET/CT system. The reconstruction methods applied were OSEM using both TOF and PSF (OSEM-TOF-PSF) with several matrix sizes (128x128, 192x192, 256x256, 384x384), Gaussian filter sizes (4 mm, 8mm) and OSEM without TOF and PSF (OSEM), with TOF (OSEM-TOF) and with PSF (OSEM-PSF) in addition to recently introduced regularized reconstruction method based on Bayesian-penalized maximum likelihood (Q.Clear). Between repeated measurements, the image-derived flow values showed high repeatability with a SD less than 2 mL/min as well as high accuracy with the maximum error of 7 % with respect to the reference flow for all reconstructions. Overall, reconstruction settings had only a small impact on the resulting flow values. In conclusion, due to the small differences detected, any of the implemented reconstruction algorithms on the system can be applied in MPI studies for accurate flow quantification.

A Total Fractional-Order Variation Regularized Reconstruction Method for CT

The total variation (TV) regularized reconstruction methods for computed tomography (CT) may lead to staircase effects in the reconstructed images because of using the TV regularization. This paper develops a total fractional-order variation regularized CT reconstruction method, aiming at overcoming the weakness of the reconstruction methods based on the TV. Specifically, we propose an optimization model for CT reconstruction, including a fidelity term, a regularization term, and a constraint term. Here, the regularization is a total fractional-order variation arising from the fractional derivative of the underlying solution. To address the nondifferentiability of the resulting model, we introduce a fixed-point characterization for its solution through the proximity operators of the nondifferentiable functions. Based on the characterization, we further develop a fixed-point iterative scheme to solve the resulting model and provide convergence analysis of the developed scheme. Numerical experiments are presented to demonstrate that the developed method outperforms the TV regularized reconstruction method in terms of suppressing noise for CT reconstruction.

Quantitative X-ray fluorescence imaging of gold nanoparticles using joint L1 and total variation regularized reconstruction.

This work proposed a joint L1 and total variation (TV) regularized reconstruction method for X-ray fluorescence tomography (XFT), and investigated the performance of this method in quantitative imaging of gold nanoparticles (GNPs). We developed a dual-modality XFT/CT imaging system which consisted of a benchtop X-ray source, a translation/rotation stage, a silicon drift detector for X-ray fluorescence (XRF) detection, and a flat panel detector for transmission X-ray detection. A pencil-beam collimator was 3D printed with steel and employed in sample excitation. The sensitivity of the XFT imaging system was determined by imaging water phantoms with multiple inserts containing GNP solutions of various concentrations (0.02-0.16 wt.%). A joint L1 and total variation (TV) regularized algorithm was developed for XFT reconstruction, where the L1 regularization was used to reduce image artifacts and the TV regularization was used to preserve the shape of targets. Nonlinear conjugate gradient (NCG) descent algorithm with backtracking line search was adopted to solve the reconstruction problem. We compared the L1 + TV regularization method with filtered back projection (FBP), maximum likelihood expectation maximization (ML-EM), L1 regularization, and TV regularization methods. Contrast-to-noise ratio (CNR), Dice similarity coefficient (DSC) and localization error (LE) metrics were used to compare the performance of different methods. The CT and XFT imaging doses were also measured using EBT2 radiochromic films. The 3D printed pencil-beam collimator shaped an excitation beam with a 2 mm full width at half maximum at the imaging isocenter. Based on the phantom imaging experiments, the joint L1 and TV regularization method performed better than FBP, ML-EM, L1 regularization and TV regularization methods, with higher localization accuracy (offset <0.6 mm), CNR and DSC values. Compared with CT, XFT with L1 + TV regularized reconstruction demonstrated higher sensitivity in GNP imaging, and could detect GNP at a concentration of 0.02 wt.% or lower. Moreover, there existed a significant linear correlation (R2>0.99) between the reconstructed and true GNP concentration. The estimated XFT imaging dose is about 41.22 cGy under current setting. The joint L1 + TV regularized reconstruction algorithm performed better in noise suppression and shape preservation. Using the L1 + TV regularized reconstruction, the XFT system is able to localize GNP targets with submillimeter accuracy and quantify GNP distribution at a concentration of 0.02 wt.% or lower.

Sparsity adaptive reconstruction for highly accelerated cardiac MRI.

To enable parameter-free, accelerated cardiovascular magnetic resonance (CMR). Regularized reconstruction methods, such as compressed sensing (CS), can significantly accelerate MRI data acquisition but require tuning of regularization weights. In this work, a technique, called Sparsity adaptive Composite Recovery (SCoRe) that exploits sparsity in multiple, disparate sparsifying transforms is presented. A data-driven adjustment of the relative contributions of different transforms yields a parameter-free CS recovery process. SCoRe is validated in a dynamic digital phantom as well as in retrospectively and prospectively undersampled cine CMR data. The results from simulation and 6 retrospectively undersampled datasets indicate that SCoRe with auto-tuned regularization weights yields lower root-mean-square error (RMSE) and higher structural similarity index (SSIM) compared to state-of-the-art CS methods. In 45 prospectively undersampled datasets acquired from 15 volunteers, the image quality was scored by 2 expert reviewers, with SCoRe receiving a higher average score (p<0.01) compared to other CS methods. SCoRe enables accelerated cine CMR from highly undersampled data. In contrast to other acceleration techniques, SCoRe adapts regularization weights based on noise power and level of sparsity in each transform, yielding superior performance without admitting any free parameters.

Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology.

Reconstruction of phase objects is a central problem in digital holography, whose various applications include microscopy, biomedical imaging, and fluid mechanics. Starting from a single in-line hologram, there is no direct way to recover the phase of the diffracted wave in the hologram plane. The reconstruction of absorbing and phase objects therefore requires the inversion of the non-linear hologram formation model. We propose a regularized reconstruction method that includes several physically-grounded constraints such as bounds on transmittance values, maximum/minimum phase, spatial smoothness or the absence of any object in parts of the field of view. To solve the non-convex and non-smooth optimization problem induced by our modeling, a variable splitting strategy is applied and the closed-form solution of the sub-problem (the so-called proximal operator) is derived. The resulting algorithm is efficient and is shown to lead to quantitative phase estimation on reconstructions of accurate simulations of in-line holograms based on the Mie theory. As our approach is adaptable to several in-line digital holography configurations, we present and discuss the promising results of reconstructions from experimental in-line holograms obtained in two different applications: the tracking of an evaporating droplet (size ∼ 100μm) and the microscopic imaging of bacteria (size ∼ 1μm).

On the computational implementation of forward and back-projection operations for cone-beam computed tomography.

Forward- and back-projection operations are the main computational burden in iterative image reconstruction in computed tomography. In addition, their implementation has to be accurate to ensure stable convergence to a high-quality image. This paper reviews and compares some of the variations in the implementation of these operations in cone-beam computed tomography. We compare four algorithms for computing the system matrix, including a distance-driven algorithm, an algorithm based on cubic basis functions, another based on spherically symmetric basis functions, and a voxel-driven algorithm. The focus of our study is on understanding how the choice of the implementation of the system matrix will influence the performance of iterative image reconstruction algorithms, including such factors as the noise strength and spatial resolution in the reconstructed image. Our experiments with simulated and real cone-beam data reveal the significance of the speed-accuracy trade-off in the implementation of the system matrix. Our results suggest that fast convergence of iterative image reconstruction methods requires accurate implementation of forward- and back-projection operations, involving a direct estimation of the convolution of the footprint of the voxel basis function with the surface of the detectors. The required accuracy decreases by increasing the resolution of the projection measurements beyond the resolution of the reconstructed image. Moreover, reconstruction of low-contrast objects needs more accurate implementation of these operations. Our results also show that, compared with regularized reconstruction methods, the behavior of iterative reconstruction algorithms that do not use a proper regularization is influenced more significantly by the implementation of the forward- and back-projection operations.

Spatial adaptive regularized MAP reconstruction for LD-based night vision

Target recognition plays a vital role in video surveillance, land and ocean exploration, in which imaging detection especially night vision is widely used. It is known that imagery detection is affected not only by absorption and scattering properties of atmosphere or water, but also the optical systems including lenses and sensors. In order to enhance the visual quality of the night vision detecting images to a best possible level, applications such as restoration and super-resolution reconstruction can be applied. The presented effort applies a spatial adaptive regularized MAP reconstruction method to a night vision laser diode (LD) imaging detection system. Blind, objective image quality metrics are adopted for determining the iteration time. Experimental results of the target recognition show adequate improvement in the detecting range as well as the detecting quality provided by the proposed approaches.

Reconstruction From Limited-Angle Projections Based on $\delta-u$ Spectrum Analysis

This paper proposes a sparse representation of an image using discrete delta-u functions. A delta-u function is defined as the product of a Kronecker delta function and a step function. Based on the sparse representation, we have developed a novel and effective method for reconstructing an image from limited-angle projections. The method first estimates the parameters of the sparse representation from the incomplete projection data, and then directly calculates the image to be reconstructed. Experiments have shown that the proposed method can effectively recover the missing data and reconstruct images more accurately than the total-variation (TV) regularized reconstruction method.

Normalization of a spatially variant image reconstruction problem in electrical impedance tomography using system blurring properties

The electrical impedance tomography (EIT) image reconstruction problem is ill posed and spatially variant. Because of the problem's ill-posed nature, small amounts of measurement noise can corrupt reconstructed images. The problem must be regularized to reduce image artifacts. In this paper, we focus on the spatially variant characteristics of the problem. Correcting errors due to spatial variance should improve reconstruction accuracy. In this paper, we present methods to normalize the spatially variant image reconstruction problem by equalizing the point spread function (PSF). In order to equalize the PSF, we used the reconstruction blurring properties obtained from the sensitivity matrix. We compared three mathematical normalization schemes: pixel-wise scaling (PWS), weighted pseudo-inversion (WPI) and weighted minimum norm method (WMNM) to equalize images. The quantity index (QI), defined as the integral of pixel values of an EIT conductivity image, was considered in investigating spatial variance. The QI values along with reconstructed images are presented for cases of two-dimensional full array and hemiarray electrode topologies. We found that a spatially invariant QI could be obtained by applying normalization methods based on equalization of the PSF using conventional regularized reconstruction methods such as truncated singular value decomposition (TSVD) and WMNM. We found that WMNM normalization applied to WMNM regularized reconstruction was the best of the methods tested overall, for both hemiarray and full array electrode topologies.

Analysis of observer performance in known-location tasks for tomographic image reconstruction

We consider the task of detecting a statistically varying signal of known location on a statistically varying background in a reconstructed tomographic image. We analyze the performance of linear observer models in this task. We show that, if one chooses a suitable reconstruction method, a broad family of linear observers can exactly achieve the optimal detection performance attainable with any combination of a linear observer and linear reconstructor. This conclusion encompasses several well-known observer models from the literature, including models with a frequency-selective channel mechanism and certain types of internal noise. Interestingly, the "optimal" reconstruction methods are unregularized and in some cases quite unconventional. These results suggest that, for the purposes of designing regularized reconstruction methods that optimize lesion detectability, known-location tasks are of limited use.

Regularized reconstruction in electrical impedance tomography using a variance uniformization constraint.

This paper describes a new approach to reconstruction of the conductivity field in electrical impedance tomography. Our goal is to improve the tradeoff between the quality of the images and the numerical complexity of the reconstruction method. In order to reduce the computational load, we adopt a linearized approximation to the forward problem that describes the relationship between the unknown conductivity and the measurements. In this framework, we focus on finding a proper way to cope with the ill-posed nature of the problem, mainly caused by strong attenuation phenomena; this is done by devising regularization techniques well suited to this particular problem. First, we propose a solution which is based on Tikhonov regularization of the problem. Second, we introduce an original regularized reconstruction method in which the regularization matrix is determined by space-uniformization of the variance of the reconstructed condictivities. Both methods are nonsupervised, i.e., all tuning parameters are automatically determined from the measured data. Tests performed on simulated and real data indicate that Tikhonov regularization provides results similar to those obtained with iterative methods, but with a much smaller amount of computations. Regularization using a variance uniformization constraint yields further improvements, particularly in the central region of the unknown object where attenuation is most severe. We anticipate that the variance uniformization approach could be adapted to iterative methods that preserve the nonlinearity of the forward problem. More generally, it appears as a useful tool for solving other severely ill-posed reconstruction problems such as eddy current tomography.