Iterative Reconstruction Scheme Research Articles

A thresholding based iterative reconstruction method for limited-angle tomography data

Limited-angle computed tomography is often imposed by in-situ experiments combining tomography with sample environments. The missing projection data causes artifacts in the tomographic reconstruction. We demonstrate that the correction of these numerical artifacts can be achieved by restoring the missing projections using an iterative reconstruction scheme. The reconstruction is regularized using segmentation, and thresholds determined from the histogram of reconstructed gray levels. The missing projections are simulated by forward projection and incorporated into the original measured dataset to give a complete angular span. This scheme typically converges within a few iterations. Results are presented for several measurements using parallel-beam synchrotron X-ray tomography and 165 degrees of valid projection data. A simple numerical simulation is used to verify the validity of the experimental results.

Fast iterative reconstruction for photoacoustic tomography using learned physical model: Theoretical validation

Iterative reconstruction has demonstrated superior performance in medical imaging under compressed, sparse, and limited-view sensing scenarios. However, iterative reconstruction algorithms are slow to converge and rely heavily on hand-crafted parameters to achieve good performance. Many iterations are usually required to reconstruct a high-quality image, which is computationally expensive due to repeated evaluations of the physical model. While learned iterative reconstruction approaches such as model-based learning (MBLr) can reduce the number of iterations through convolutional neural networks, it still requires repeated evaluations of the physical models at each iteration. Therefore, the goal of this study is to develop a Fast Iterative Reconstruction (FIRe) algorithm that incorporates a learned physical model into the learned iterative reconstruction scheme to further reduce the reconstruction time while maintaining robust reconstruction performance. We also propose an efficient training scheme for FIRe, which releases the enormous memory footprint required by learned iterative reconstruction methods through the concept of recursive training. The results of our proposed method demonstrate comparable reconstruction performance to learned iterative reconstruction methods with a 9x reduction in computation time and a 620x reduction in computation time compared to variational reconstruction.

Joint Optimization of k-t Sampling Pattern and Reconstruction of DCE MRI for Pharmacokinetic Parameter Estimation

This work proposes to develop and evaluate a deep learning framework that jointly optimizes k-t sampling patterns and reconstruction for head and neck dynamic contrast-enhanced (DCE) MRI aiming to reduce bias and uncertainty of pharmacokinetic (PK) parameter estimation. 2D Cartesian phase encoding k-space subsampling patterns for a 3D spoiled gradient recalled echo (SPGR) sequence along a time course of DCE MRI were jointly optimized in a deep learning-based dynamic MRI reconstruction network by a loss function concerning both reconstruction image quality and PK parameter estimation accuracy. During training, temporal k-space data sharing scheme was optimized as well. The proposed method was trained and tested by multi-coil complex digital reference objects of DCE images (mcDROs). The PK parameters estimated by the proposed method were compared with two published iterative DCE MRI reconstruction schemes using normalized root mean squared errors (NRMSEs) and Bland-Altman analysis at temporal resolutions of [Formula: see text] = 2s, 3s, 4s, and 5s, which correspond to undersampling rates of R = 50, 34, 25, and 20. The proposed method achieved low PK parameter NRMSEs at all four temporal resolutions compared with the benchmark methods on testing mcDROs. The Bland-Altman plots demonstrated that the proposed method reduced PK parameter estimation bias and uncertainty in tumor regions at temporal resolution of 2s. The proposed method also showed robustness to contrast arrival timing variations across patients. This work provides a potential way to increase PK parameter estimation accuracy and precision, and thus facilitate the clinical translation of DCE MRI.

Fast Sampling and Reconstruction for Linear Inverse Problems: From Vectors to Tensors

Signals typical in the real world have different modes, expressed as vectors, matrices, or higher-order tensors. In practice, a target signal is commonly assumed to be linear in the residing factor mode(s) with low-dimensional parameters, and thus can be recovered from partial samples by solving a linear inverse problem (LIP). Sampling for LIPs is to partially query signals for better recovery. There exist many fast sampling methods for vector signals, but they are not applicable for higher-order tensors due to high computation and storage costs. In this paper, we propose a fast sampling algorithm for vector signals first and then extend it to sample tensors with low complexity. Specifically, a tensor signal with <inline-formula><tex-math notation="LaTeX">$R$</tex-math></inline-formula> modes is modelled as multilinear on <inline-formula><tex-math notation="LaTeX">$R$</tex-math></inline-formula> factor matrices <inline-formula><tex-math notation="LaTeX">${\mathbf \{U_{i}\}}^{R}_{i=1}$</tex-math></inline-formula>, whose vectorized version is linear on a matrix <inline-formula><tex-math notation="LaTeX">$\pmb {\Phi }$</tex-math></inline-formula>&#x2014;the Kronecker product of <inline-formula><tex-math notation="LaTeX">$\mathbf {U}_{i}$</tex-math></inline-formula>&#x0027;s. Thus, it can be estimated from partial noisy samples via the least-squares (LS) method, where the mean square error (MSE) depends on chosen samples and matrix <inline-formula><tex-math notation="LaTeX">$\pmb {\Phi }$</tex-math></inline-formula>. Instead of MSE, we minimize a modified MSE problem and prove it has the same optimal and greedy solutions to a problem with a sample-dependent sub-matrix of <inline-formula><tex-math notation="LaTeX">$\pmb {\Phi }\pmb {\Phi }^\top$</tex-math></inline-formula>. For one-order vector signals, where <inline-formula><tex-math notation="LaTeX">$\pmb {\Phi }=\mathbf {U}_{1}$</tex-math></inline-formula> is given, we propose a fast algorithm to solve the sampling problem via simple vector-vector multiplications based on a matrix inversion formula and greedy solution reuse. For higher-order tensors, we extend our sampling algorithm to select entries by addressing matrices <inline-formula><tex-math notation="LaTeX">$\mathbf {U}_{i}$</tex-math></inline-formula>&#x0027;s directly, thus removing the burden of obtaining <inline-formula><tex-math notation="LaTeX">$\pmb {\Phi }$</tex-math></inline-formula> explicitly. Accompanying our sampling strategy, we propose an iterative reconstruction scheme to estimate the LS solution. Experiments on synthetic and real-world signals from vectors to tensors validated the performance and efficacy of our strategy compared to previous methods.

Feasibility analysis on simultaneous electron density and attenuation coefficient reconstruction.

Creating a viable reconstruction method for Compton scatter tomography remains challenging. Accounting for scatter attenuation when the underlying attenuation map is not known is particularly challenging, and current mathematical approaches to this vary widely. This work explores a novel approach to joint scatter and attenuation image reconstruction, which leverages the underlying structural similarity between the two images and incorporates a deep learning model in an alternating iterative reconstruction scheme. A single-view computed tomography (CT) imaging procedure for recording Compton scatter is first described. A joint reconstruction model, which iterates between algebraically reconstructing scatter images and estimating the attenuation via deep learning, is then proposed. This model is tested on both a generated dataset of 2D phantom images designed to mimic human tissues as well as a realistically simulated dataset based on real CT images. Testing results yield convergence of the model and decent reconstruction quality to distinguish crucial features such as tumors and lesions, demonstrating the potential principled utilities of this configuration and deep learning approach. The model achieved a structural similarity index measure of at least 0.82 for scatter and 0.88 for attenuation reconstructions with the realistically simulated dataset. The iterative, deep learning approach outlined in this work shows potential for future efficient medical imaging procedures, reconstructing images with limited scatter information.

Deep Learning With Adaptive Hyper-Parameters for Low-Dose CT Image Reconstruction

Low-dose CT (LDCT) imaging is preferred in many applications to reduce the object's exposure to X-ray radiation. In recent years, one promising approach to image reconstruction in LDCT is the so-called optimization-unrolling-based deep learning approach, which replaces pre-defined image prior by learnable adaptive prior in some model-based iterative image reconstruction scheme (MBIR). While it is known that setting appropriate hyper-parameters in MBIR is challenging yet important to the reconstruction quality, it does not receive enough attention in the development of deep learning methods. This paper proposed a deep learning method for LDCT reconstruction that unrolls a half-quadratic splitting scheme. The proposed method not only introduces learnable image prior built on framelet filter bank, but also learns a network that automatically adjusts the hyper-parameters to fit noise level and the data for processing. As a result, only one universal model needs to be trained in our method to process the data taken under different dose levels. Experimental evaluation on clinical patient dataset showed that the proposed method outperformed both conventional and deep-learning-based solutions by a large margin.

Bespoke Fractal Sampling Patterns for Discrete Fourier Space via the Kaleidoscope Transform

Sampling strategies are important for sparse imaging methodologies, especially those employing the discrete Fourier transform (DFT). Chaotic sensing is one such methodology that employs deterministic, fractal sampling in conjunction with finite, iterative reconstruction schemes to form an image from limited samples. Using a sampling pattern constructed entirely from periodic lines in DFT space, chaotic sensing was found to outperform traditional compressed sensing for magnetic resonance imaging; however, only one such sampling pattern was presented and the reason for its fractal nature was not proven. Through the introduction of a novel image transform known as the kaleidoscope transform, which formalises and extends upon the concept of downsampling and concatenating an image with itself, this paper: (1) demonstrates a fundamental relationship between multiplication in modular arithmetic and downsampling; (2) provides a rigorous mathematical explanation for the fractal nature of the sampling pattern in the DFT; and (3) leverages this understanding to develop a collection of novel fractal sampling patterns for the 2D DFT with customisable properties. The ability to design tailor-made fractal sampling patterns expands the utility of the DFT in chaotic imaging and may form the basis for a bespoke chaotic sensing methodology, in which the fractal sampling matches the imaging task for improved reconstruction.

Maximum likelihood reconstruction for grating-based X-ray microscopy.

The combination of grating-based phase-contrast imaging with X-ray microscopy can result in a complicated image formation. Generally, transverse shifts of the interference fringes are nonlinearly dependent on phase differences of the measured wave front. We present an iterative reconstruction scheme based on a regularized maximum likelihood cost function that fully takes this dependency into account. The scheme is validated by numerical simulations. It is particularly advantageous at low photon numbers and when the premises for deconvolution-based reconstructions are not met. Our reconstruction scheme hence enables a broader applicability of X-ray grating interferometry in imaging and wave front sensing.

Calderón's inverse problem with a finite number of measurements II: independent data

ABSTRACTWe prove a local Lipschitz stability estimate for Gel'fand-Calderón's inverse problem for the Schrödinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of . A similar result for Calderón's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based on the stability result is also presented, for which we prove exponential convergence in the number of iterations and stability with respect to noise in the data and to mismodeling errors.

An Adaptive Intelligence Algorithm for Undersampled Knee MRI Reconstruction

Adaptive intelligence aims at empowering machine learning techniques with the additional use of domain knowledge. In this work, we present the application of adaptive intelligence to accelerate MR acquisition. Starting from undersampled k-space data, an iterative learning-based reconstruction scheme inspired by compressed sensing theory is used to reconstruct the images. We developed a novel deep neural network to refine and correct prior reconstruction assumptions given the training data. The network was trained and tested on a knee MRI dataset from the 2019 fastMRI challenge organized by Facebook AI Research and NYU Langone Health. All submissions to the challenge were initially ranked based on similarity with a known groundtruth, after which the top 4 submissions were evaluated radiologically. Our method was evaluated by the fastMRI organizers on an independent challenge dataset. It ranked #1, shared #1, and #3 on respectively the 8× accelerated multi-coil, the 4× multi-coil, and the 4× single-coil tracks. This demonstrates the superior performance and wide applicability of the method.

Multi-Scale Learned Iterative Reconstruction.

Model-based learned iterative reconstruction methods have recently been shown to outperform classical reconstruction algorithms. Applicability of these methods to large scale inverse problems is however limited by the available memory for training and extensive training times, the latter due to computationally expensive forward models. As a possible solution to these restrictions we propose a multi-scale learned iterative reconstruction scheme that computes iterates on discretisations of increasing resolution. This procedure does not only reduce memory requirements, it also considerably speeds up reconstruction and training times, but most importantly is scalable to large scale inverse problems with non-trivial forward operators, such as those that arise in many 3D tomographic applications. In particular, we propose a hybrid network that combines the multiscale iterative approach with a particularly expressive network architecture which in combination exhibits excellent scalability in 3D. Applicability of the algorithm is demonstrated for 3D cone beam computed tomography from real measurement data of an organic phantom. Additionally, we examine scalability and reconstruction quality in comparison to established learned reconstruction methods in two dimensions for low dose computed tomography on human phantoms.

High-fidelity optical diffraction tomography of multiple scattering samples

We propose an iterative reconstruction scheme for optical diffraction tomography that exploits the split-step non-paraxial (SSNP) method as the forward model in a learning tomography scheme. Compared with the beam propagation method (BPM) previously used in learning tomography (LT-BPM), the improved accuracy of SSNP maximizes the information retrieved from measurements, relying less on prior assumptions about the sample. A rigorous evaluation of learning tomography based on SSNP (LT-SSNP) using both synthetic and experimental measurements confirms its superior performance compared with that of the LT-BPM. Benefiting from the accuracy of SSNP, LT-SSNP can clearly resolve structures that are highly distorted in the LT-BPM. A serious limitation for quantifying the reconstruction accuracy for biological samples is that the ground truth is unknown. To overcome this limitation, we describe a novel method that allows us to compare the performances of different reconstruction schemes by using the discrete dipole approximation to generate synthetic measurements. Finally, we explore the capacity of learning approaches to enable data compression by reducing the number of scanning angles, which is of particular interest in minimizing the measurement time.

Iterative image reconstruction for sparse-view CT via total variation regularization and dictionary learning.

Recently, low-dose computed tomography (CT) has become highly desirable due to the increasing attention paid to the potential risks of excessive radiation of the regular dose CT. However, ensuring image quality while reducing the radiation dose in the low-dose CT imaging is a major challenge. Compared to classical filtered back-projection (FBP) algorithms, statistical iterative reconstruction (SIR) methods for modeling measurement statistics and imaging geometry can significantly reduce the radiation dose, while maintaining the image quality in a variety of CT applications. To facilitate low-dose CT imaging, we in this study proposed an improved statistical iterative reconstruction scheme based on the penalized weighted least squares (PWLS) standard combined with total variation (TV) minimization and sparse dictionary learning (DL), which is named as a method of PWLS-TV-DL. To evaluate this PWLS-TV-DL method, we performed experiments on digital phantoms and physical phantoms, and analyzed the results in terms of image quality and calculation. The results show that the proposed method is better than the comparison methods, which indicates the potential of applying this PWLS-TV-DL method to reconstruct low-dose CT images.

A Joint Deconvolution Algorithm to Combine Single-dish and Interferometer Data for Wideband Multiterm and Mosaic Imaging

Abstract Imaging in radio astronomy is usually carried out with a single-dish radio telescope doing a raster scan of a region of the sky or with an interferometer that samples the visibility function of the sky brightness. Mosaic observations are the current standard for imaging large fields of view with an interferometer, and multifrequency observations are now routinely carried out with both types of telescopes to increase the continuum imaging sensitivity and probe spectral structure. This paper describes an algorithm to combine wideband data from these two types of telescopes in a joint iterative reconstruction scheme that can be applied to spectral cube or wideband multiterm imaging both for narrow fields of view and for mosaics. Our results demonstrate the ability to prevent instabilities and errors that typically arise when wideband or joint mosaicking algorithms are presented with spatial and spectral structure that is inadequately sampled by the interferometer alone. For comparable noise levels in the single-dish and interferometer data, the numerical behavior of this algorithm is expected to be similar to the idea of generating artificial visibilities from single-dish data. However, our discussed implementation is simpler and more flexible in terms of applying relative data weighting schemes to match noise levels while preserving flux accuracy, fits within standard iterative image reconstruction frameworks, is fully compatible with wide-field and joint mosaicking gridding algorithms that apply corrections specific to the interferometer data, and may be configured to enable spectral cube and wideband multiterm deconvolution for single-dish data alone.

Chaotic Sensing.

We propose a sparse imaging methodology called Chaotic Sensing (ChaoS) that enables the use of limited yet deterministic linear measurements through fractal sampling. A novel fractal in the discrete Fourier transform is introduced that always results in the artefacts being turbulent in nature. These chaotic artefacts have characteristics that are image independent, facilitating their removal through dampening (via image denoising) and obtaining the maximum likelihood solution. In contrast with existing methods, such as compressed sensing, the fractal sampling is based on digital periodic lines that form the basis of discrete projected views of the image without requiring additional transform domains. This allows the creation of finite iterative reconstruction schemes in recovering an image from its fractal sampling that is also new to discrete tomography. As a result, ChaoS supports linear measurement and optimisation strategies, while remaining capable of recovering a theoretically exact representation of the image. We apply the method to simulated and experimental limited magnetic resonance (MR) imaging data, where restrictions imposed by MR physics typically favour linear measurements for reducing acquisition time.

Spatial domain morphological filtering for interpolation of the Fourier domain

We establish here a method to partially recover the missing frequencies in data acquired through sub-sampling in the Fourier domain. Non-linear open and close operations are applied recursively to the raw spatial image estimated from its sparse Fourier samples. The mean of the open and closed images updates the estimated image after applying each structuring element (SE). The updated Fourier spectra are required at each stage to remain consistent with the known sample values. The SE size is incremented until the values estimated for the missing Fourier coefficients converge. The average image obtained from applying multiple sets of randomly selected SEs forms the final result. A denoising scheme is finally used to supplement the interpolated result. The denoising estimates the alias artefacts that arise from the initial sub-sampling by assuming that a narrow region around the borders of the reconstructed image has constant grey level. Convergence occurs in around 30 iterations that usually requires, in total a little less than one minute on a 64GB RAM, i7 machine. Since the computational complexity of the method is low, it can serve as a reliable initialization for more sophisticated iterative reconstruction schemes. Results for several simulated and real sub-sampled image data show an improvement of about 0.1 in Structural SIMilarity index (SSIM, whose maximum possible value is 1) in comparison to the raw inverse FFT image estimate. This method is more likely to underestimate the missing frequency values.

New software protocols for enabling laboratory based temporal CT.

Temporal micro-computed tomography (CT) allows the non-destructive quantification of processes that are evolving over time in 3D. Despite the increasing popularity of temporal CT, the practical implementation and optimisation can be difficult. Here, we present new software protocols that enable temporal CT using commercial laboratory CT systems. The first protocol drastically reduces the need for periodic intervention when making time-lapse experiments, allowing a large number of tomograms to be collected automatically. The automated scanning at regular intervals needed for uninterrupted time-lapse CT is demonstrated by analysing the germination of a mung bean (vigna radiata), whilst the synchronisation with an in situ rig required for interrupted time-lapse CT is highlighted using a shear cell to observe granular segregation. The second protocol uses golden-ratio angular sampling with an iterative reconstruction scheme and allows the number of projections in a reconstruction to be changed as sample evolution occurs. This overcomes the limitation of the need to know a priori what the best time window for each scan is. The protocol is evaluated by studying barite precipitation within a porous column, allowing a comparison of spatial and temporal resolution of reconstructions with different numbers of projections. Both of the protocols presented here have great potential for wider application, including, but not limited to, in situ mechanical testing, following battery degradation and chemical reactions.

In vivo myocardial tissue characterization of all four chambers using high-resolution quantitative MRI

Abstract Quantitative native T1 Mapping of the myocardium without the application of contrast agents can be used to detect fibrosis in the left ventricle. Spatial resolution of standard native T1 mapping is limited by cardiac motion and hence is not sufficient to resolve small myocardial structures, such as the right ventricle and the atria. Here, we present a novel MR approach which provides cardiac motion information and native T1 maps from the same data. Motion information is utilized to optimize data selection for T1 mapping and a model-based iterative reconstruction scheme ensures high-resolution T1 maps for the entire heart. Feasibility of the approach was demonstrated in three healthy volunteers. In the T1 maps, the myocardium of all four chambers can be visualized and T1 values of the left atrium and right chambers were comparable to left ventricular T1 values.

Enhancing sparse-view photoacoustic tomography with combined virtually parallel projecting and spatially adaptive filtering.

To fully realize the potential of photoacoustic tomography (PAT) in preclinical and clinical applications, rapid measurements and robust reconstructions are needed. Sparse-view measurements have been adopted effectively to accelerate the data acquisition. However, since the reconstruction from the sparse-view sampling data is challenging, both the effective measurement and the appropriate reconstruction should be taken into account. In this study, we present an iterative sparse-view PAT reconstruction scheme, where a concept of virtual parallel-projection matching the measurement condition is introduced to aid the “compressive sensing” in the reconstruction procedure, and meanwhile, the non-local spatially adaptive filtering exploring the a priori information of the mutual similarities in natural images is adopted to recover the unknowns in the transformed sparse domain. Consequently, the reconstructed images with the proposed sparse-view scheme can be evidently improved in comparison to those with the universal back-projection method, for the cases of same sparse views. The proposed approach has been validated by the simulations and ex vivo experiments, which exhibits desirable performances in image fidelity even from a small number of measuring positions.

On the data acquisition, image reconstruction, cone beam artifacts, and their suppression in axial MDCT and CBCT - A review.

In the clinic, computed tomography (CT) has evolved into an essential modality for diagnostic imaging by multidetector row CT (MDCT) and image guided intervention by cone beam CT (CBCT). Recognizing the increasing importance of axial MDCT/CBCT in clinical and preclinical applications, and the existence of CB artifacts in MDCT/CBCT images, we provide a review of CB artifacts' root causes, rendering mechanisms and morphology, and possible solutions for elimination and/or reduction of the artifacts. By examining the null space in Radon and Fourier domain, the root cause of CB artifacts (i.e., data insufficiency) in axial MDCT/CBCT is analytically investigated, followed by a review of the data sufficiency conditions and the "circle +" source trajectories. The rendering mechanisms and morphology of CB artifacts in axial MDCT/CBCT and their special cases (e.g., half/short scan and full scan with latitudinally displaced detector) are then analyzed, followed by a survey of the potential solutions to suppress the artifacts. The phenomenon of imaged zone indention and its variation over FBP, BPF/DBPF, two-pass and iterative CB reconstruction algorithms and/or schemes are discussed in detail. An interdomain examination of the null space provides an insightful understanding of the root cause of CB artifacts in axial MDCT/CBCT. The decomposition of CB artifacts rendering mechanisms facilitates understanding of the artifacts' behavior under different conditions and the potential solutions to suppress them. An inspection of the imaged zone intention phenomenon provides guidance on the design and implementation of CB image reconstruction algorithms and schemes for CB artifacts suppression in axial MDCT/CBCT. With increasing importance of axial MDCT/CBCT in clinical and preclinical applications, this review article can update the community with in-depth information and clarification on the latest progress in dealing with CB artifacts and thus increase clinical/preclinical confidence.