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25 pages, 999 KiB

Open AccessArticle

Perspective Shape-from-Shading Problem: A Unified Convergence Result for Several Non-Lambertian Models

by Silvia Tozza

J. Imaging 2022, 8(2), 36; https://doi.org/10.3390/jimaging8020036 - 1 Feb 2022

Cited by 1 | Viewed by 2313

Abstract

Shape-from-Shading represents the problem of computing the three-dimensional shape of a surface given a single gray-value image of it as input. In a recent paper, we showed that the introduction of an attenuation factor in the brightness equations related to various perspective Shape-from-Shading [...] Read more.

Shape-from-Shading represents the problem of computing the three-dimensional shape of a surface given a single gray-value image of it as input. In a recent paper, we showed that the introduction of an attenuation factor in the brightness equations related to various perspective Shape-from-Shading models allows us to ensure the well-posedness of the corresponding differential problems. Here, we propose a unified convergence result of a numerical scheme for several non-Lambertian reflectance models. This result is interesting since it can be easily extended to other non-Lambertian models in a unified and, therefore, powerful framework. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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35 pages, 5964 KiB

Open AccessArticle

Nearly Exact Discrepancy Principle for Low-Count Poisson Image Restoration

by Francesca Bevilacqua, Alessandro Lanza, Monica Pragliola and Fiorella Sgallari

J. Imaging 2022, 8(1), 1; https://doi.org/10.3390/jimaging8010001 - 23 Dec 2021

Cited by 7 | Viewed by 2771

Abstract

The effectiveness of variational methods for restoring images corrupted by Poisson noise strongly depends on the suitable selection of the regularization parameter balancing the effect of the regulation term(s) and the generalized Kullback–Liebler divergence data term. One of the approaches still commonly used [...] Read more.

The effectiveness of variational methods for restoring images corrupted by Poisson noise strongly depends on the suitable selection of the regularization parameter balancing the effect of the regulation term(s) and the generalized Kullback–Liebler divergence data term. One of the approaches still commonly used today for choosing the parameter is the discrepancy principle proposed by Zanella et al. in a seminal work. It relies on imposing a value of the data term approximately equal to its expected value and works well for mid- and high-count Poisson noise corruptions. However, the series truncation approximation used in the theoretical derivation of the expected value leads to poor performance for low-count Poisson noise. In this paper, we highlight the theoretical limits of the approach and then propose a nearly exact version of it based on Monte Carlo simulation and weighted least-square fitting. Several numerical experiments are presented, proving beyond doubt that in the low-count Poisson regime, the proposed modified, nearly exact discrepancy principle performs far better than the original, approximated one by Zanella et al., whereas it works similarly or slightly better in the mid- and high-count regimes. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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14 pages, 1990 KiB

Open AccessArticle

Learned Primal Dual Reconstruction for PET

by Alessandro Guazzo and Massimiliano Colarieti-Tosti

J. Imaging 2021, 7(12), 248; https://doi.org/10.3390/jimaging7120248 - 24 Nov 2021

Cited by 7 | Viewed by 2284

Abstract

We have adapted, implemented and trained the Learned Primal Dual algorithm suggested by Adler and Öktem and evaluated its performance in reconstructing projection data from our PET scanner. Learned Primal Dual reconstructions are compared to Maximum Likelihood Expectation Maximisation (MLEM) reconstructions. Different strategies [...] Read more.

We have adapted, implemented and trained the Learned Primal Dual algorithm suggested by Adler and Öktem and evaluated its performance in reconstructing projection data from our PET scanner. Learned Primal Dual reconstructions are compared to Maximum Likelihood Expectation Maximisation (MLEM) reconstructions. Different strategies for training are also compared. Whenever the noise level of the data to reconstruct is sufficiently represented in the training set, the Learned Primal Dual algorithm performs well on the recovery of the activity concentrations and on noise reduction as compared to MLEM. The algorithm is also shown to be robust against the appearance of artefacts, even when the images that are to be reconstructed present features were not present in the training set. Once trained, the algorithm reconstructs images in few seconds or less. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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27 pages, 2345 KiB

Open AccessArticle

Conditional Invertible Neural Networks for Medical Imaging

by Alexander Denker, Maximilian Schmidt, Johannes Leuschner and Peter Maass

J. Imaging 2021, 7(11), 243; https://doi.org/10.3390/jimaging7110243 - 17 Nov 2021

Cited by 32 | Viewed by 4557

Abstract

Over recent years, deep learning methods have become an increasingly popular choice for solving tasks from the field of inverse problems. Many of these new data-driven methods have produced impressive results, although most only give point estimates for the reconstruction. However, especially in [...] Read more.

Over recent years, deep learning methods have become an increasingly popular choice for solving tasks from the field of inverse problems. Many of these new data-driven methods have produced impressive results, although most only give point estimates for the reconstruction. However, especially in the analysis of ill-posed inverse problems, the study of uncertainties is essential. In our work, we apply generative flow-based models based on invertible neural networks to two challenging medical imaging tasks, i.e., low-dose computed tomography and accelerated medical resonance imaging. We test different architectures of invertible neural networks and provide extensive ablation studies. In most applications, a standard Gaussian is used as the base distribution for a flow-based model. Our results show that the choice of a radial distribution can improve the quality of reconstructions. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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13 pages, 1403 KiB

Open AccessArticle

Discretization of Learned NETT Regularization for Solving Inverse Problems

by Stephan Antholzer and Markus Haltmeier

J. Imaging 2021, 7(11), 239; https://doi.org/10.3390/jimaging7110239 - 15 Nov 2021

Cited by 8 | Viewed by 1845

Abstract

Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Network Tikhonov Regularization), which contains a trained neural network as regularizer in [...] Read more.

Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Network Tikhonov Regularization), which contains a trained neural network as regularizer in generalized Tikhonov regularization. The existing analysis of NETT considers fixed operators and fixed regularizers and analyzes the convergence as the noise level in the data approaches zero. In this paper, we extend the frameworks and analysis considerably to reflect various practical aspects and take into account discretization of the data space, the solution space, the forward operator and the neural network defining the regularizer. We show the asymptotic convergence of the discretized NETT approach for decreasing noise levels and discretization errors. Additionally, we derive convergence rates and present numerical results for a limited data problem in photoacoustic tomography. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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10 pages, 616 KiB

Open AccessArticle

Recovering the Magnetic Image of Mars from Satellite Observations

by Igor Kolotov, Dmitry Lukyanenko, Inna Stepanova, Yanfei Wang and Anatoly Yagola

J. Imaging 2021, 7(11), 234; https://doi.org/10.3390/jimaging7110234 - 9 Nov 2021

Cited by 10 | Viewed by 2092

Abstract

One of the possible approaches to reconstructing the map of the distribution of magnetization parameters in the crust of Mars from the data of the Mars MAVEN orbiter mission is considered. Possible ways of increasing the accuracy of reconstruction of the magnetic image [...] Read more.

One of the possible approaches to reconstructing the map of the distribution of magnetization parameters in the crust of Mars from the data of the Mars MAVEN orbiter mission is considered. Possible ways of increasing the accuracy of reconstruction of the magnetic image of Mars are discussed. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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29 pages, 25475 KiB

Open AccessArticle

An Optimization-Based Meta-Learning Model for MRI Reconstruction with Diverse Dataset

by Wanyu Bian, Yunmei Chen, Xiaojing Ye and Qingchao Zhang

J. Imaging 2021, 7(11), 231; https://doi.org/10.3390/jimaging7110231 - 31 Oct 2021

Cited by 16 | Viewed by 3351

Abstract

This work aims at developing a generalizable Magnetic Resonance Imaging (MRI) reconstruction method in the meta-learning framework. Specifically, we develop a deep reconstruction network induced by a learnable optimization algorithm (LOA) to solve the nonconvex nonsmooth variational model of MRI image reconstruction. In [...] Read more.

This work aims at developing a generalizable Magnetic Resonance Imaging (MRI) reconstruction method in the meta-learning framework. Specifically, we develop a deep reconstruction network induced by a learnable optimization algorithm (LOA) to solve the nonconvex nonsmooth variational model of MRI image reconstruction. In this model, the nonconvex nonsmooth regularization term is parameterized as a structured deep network where the network parameters can be learned from data. We partition these network parameters into two parts: a task-invariant part for the common feature encoder component of the regularization, and a task-specific part to account for the variations in the heterogeneous training and testing data. We train the regularization parameters in a bilevel optimization framework which significantly improves the robustness of the training process and the generalization ability of the network. We conduct a series of numerical experiments using heterogeneous MRI data sets with various undersampling patterns, ratios, and acquisition settings. The experimental results show that our network yields greatly improved reconstruction quality over existing methods and can generalize well to new reconstruction problems whose undersampling patterns/trajectories are not present during training. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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The pictures (from top to bottom) display the reconstruction results, zoomed-in details, point-wise errors with a color bar, and associated radial masks for meta-learning, conventional learning, and ISTA-Net<math display="inline"><semantics> <msup> <mrow/> <mo>+</mo> </msup> </semantics></math> with four different CS ratios of 10%, 20%, 30%, 40% (from left to right). The top-right image is the ground truth fully-sampled image. Full article ">Figure 2
The pictures (from top to bottom) display the T2 brain image reconstruction results, zoomed-in details, point-wise errors with a color bar, and associated radial masks for meta-learning and conventional learningwith four different CS ratios of 10%, 20%, 30%, 40% (from left to right). The top-right iage is the ground truth fully-sampled image. Full article ">Figure 3
The pictures (from top to bottom) display the T1 brain image reconstruction results, zoomed-in details, point-wise errors with a color bar, and associated radial masks for meta-learning and conventional learning. Meta-learning was trained with CS ratios of 10%, 20%, 30%, and 40% and tested with three different unseen CS ratios of 15%, 25%, and 35% (from left to right). Conventional learning was trained and tested with the same CS ratios of 15%, 25%, and 35%. The top-right image is the ground truth fully-sampled image. Full article ">Figure 4
The pictures (from top to bottom) display the T2 brain image reconstruction results, zoomed-in details, point-wise errors with a color bar, and associated radial masks for meta-learning and conventional learning. Meta-learning was trained with CS ratios of 10%, 20%, 30%, and 40% and test edwith three different unseen CS ratios of 15%, 25%, and 35% (from left to right). Conventional learning was trained and tested with the same CS ratios of 15%, 25%, and 35%. The top-right image is the ground truth fully-sampled image. Full article ">Figure 5
The pictures (from top to bottom) display the T2 brain image reconstruction results, zoomed-in details, point-wise errors with a color bar, and associated Cartesian masks for meta-learning and conventional learningwith four different CS ratios of 10%, 20%, 30%, and 40% (from left to right). The top-right image is the ground truth fully-sampled image. Full article ">Figure 6
Convergence behavior of Algorithm 1 on T1 weighted MRI image reconstruction with four different CS ratios using radial mask. Left: Objective function value <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> versus phase number. Right: PSNR value versus phase number. Full article ">

15 pages, 16522 KiB

Open AccessArticle

On a Variational and Convex Model of the Blake–Zisserman Type for Segmentation of Low-Contrast and Piecewise Smooth Images

by Liam Burrows, Anis Theljani and Ke Chen

J. Imaging 2021, 7(11), 228; https://doi.org/10.3390/jimaging7110228 - 28 Oct 2021

Cited by 3 | Viewed by 2254

Abstract

This paper proposes a new variational model for segmentation of low-contrast and piecewise smooth images. The model is motivated by the two-stage image segmentation work of Cai–Chan–Zeng (2013) for the Mumford–Shah model. To deal with low-contrast images more effectively, especially in treating higher-order [...] Read more.

This paper proposes a new variational model for segmentation of low-contrast and piecewise smooth images. The model is motivated by the two-stage image segmentation work of Cai–Chan–Zeng (2013) for the Mumford–Shah model. To deal with low-contrast images more effectively, especially in treating higher-order discontinuities, we follow the idea of the Blake–Zisserman model instead of the Mumford–Shah. Two practical ideas are introduced here: first, a convex relaxation idea is used to derive an implementable formulation, and second, a game reformulation is proposed to reduce the strong dependence of coupling parameters. The proposed model is then analysed for existence and further solved by an ADMM solver. Numerical experiments can show that the new model outperforms the current state-of-the-art models for some challenging and low-contrast images. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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26 pages, 4540 KiB

Open AccessArticle

Fast Fiber Orientation Estimation in Diffusion MRI from kq-Space Sampling and Anatomical Priors

by Marica Pesce, Audrey Repetti, Anna Auría, Alessandro Daducci, Jean-Philippe Thiran and Yves Wiaux

J. Imaging 2021, 7(11), 226; https://doi.org/10.3390/jimaging7110226 - 27 Oct 2021

Cited by 2 | Viewed by 2326

Abstract

High spatio-angular resolution diffusion MRI (dMRI) has been shown to provide accurate identification of complex neuronal fiber configurations, albeit, at the cost of long acquisition times. We propose a method to recover intra-voxel fiber configurations at high spatio-angular resolution relying on a 3D [...] Read more.

High spatio-angular resolution diffusion MRI (dMRI) has been shown to provide accurate identification of complex neuronal fiber configurations, albeit, at the cost of long acquisition times. We propose a method to recover intra-voxel fiber configurations at high spatio-angular resolution relying on a 3D kq-space under-sampling scheme to enable accelerated acquisitions. The inverse problem for the reconstruction of the fiber orientation distribution (FOD) is regularized by a structured sparsity prior promoting simultaneously voxel-wise sparsity and spatial smoothness of fiber orientation. Prior knowledge of the spatial distribution of white matter, gray matter, and cerebrospinal fluid is also leveraged. A minimization problem is formulated and solved via a stochastic forward–backward algorithm. Simulations and real data analysis suggest that accurate FOD mapping can be achieved from severe kq-space under-sampling regimes potentially enabling high spatio-angular resolution dMRI in the clinical setting. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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24 pages, 2863 KiB

Open AccessArticle

Flexible Krylov Methods for Edge Enhancement in Imaging

by Silvia Gazzola, Sebastian James Scott and Alastair Spence

J. Imaging 2021, 7(10), 216; https://doi.org/10.3390/jimaging7100216 - 18 Oct 2021

Cited by 2 | Viewed by 2567

Abstract

Many successful variational regularization methods employed to solve linear inverse problems in imaging applications (such as image deblurring, image inpainting, and computed tomography) aim at enhancing edges in the solution, and often involve non-smooth regularization terms (e.g., total variation). Such regularization methods can [...] Read more.

Many successful variational regularization methods employed to solve linear inverse problems in imaging applications (such as image deblurring, image inpainting, and computed tomography) aim at enhancing edges in the solution, and often involve non-smooth regularization terms (e.g., total variation). Such regularization methods can be treated as iteratively reweighted least squares problems (IRLS), which are usually solved by the repeated application of a Krylov projection method. This approach gives rise to an inner–outer iterative scheme where the outer iterations update the weights and the inner iterations solve a least squares problem with fixed weights. Recently, flexible or generalized Krylov solvers, which avoid inner–outer iterations by incorporating iteration-dependent weights within a single approximation subspace for the solution, have been devised to efficiently handle IRLS problems. Indeed, substantial computational savings are generally possible by avoiding the repeated application of a traditional Krylov solver. This paper aims to extend the available flexible Krylov algorithms in order to handle a variety of edge-enhancing regularization terms, with computationally convenient adaptive regularization parameter choice. In order to tackle both square and rectangular linear systems, flexible Krylov methods based on the so-called flexible Golub–Kahan decomposition are considered. Some theoretical results are presented (including a convergence proof) and numerical comparisons with other edge-enhancing solvers show that the new methods compute solutions of similar or better quality, with increased speedup. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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23 pages, 7547 KiB

Open AccessArticle

Mitral Valve Segmentation Using Robust Nonnegative Matrix Factorization

by Hannah Dröge, Baichuan Yuan, Rafael Llerena, Jesse T. Yen, Michael Moeller and Andrea L. Bertozzi

J. Imaging 2021, 7(10), 213; https://doi.org/10.3390/jimaging7100213 - 16 Oct 2021

Cited by 5 | Viewed by 3004

Abstract

Analyzing and understanding the movement of the mitral valve is of vital importance in cardiology, as the treatment and prevention of several serious heart diseases depend on it. Unfortunately, large amounts of noise as well as a highly varying image quality make the [...] Read more.

Analyzing and understanding the movement of the mitral valve is of vital importance in cardiology, as the treatment and prevention of several serious heart diseases depend on it. Unfortunately, large amounts of noise as well as a highly varying image quality make the automatic tracking and segmentation of the mitral valve in two-dimensional echocardiographic videos challenging. In this paper, we present a fully automatic and unsupervised method for segmentation of the mitral valve in two-dimensional echocardiographic videos, independently of the echocardiographic view. We propose a bias-free variant of the robust non-negative matrix factorization (RNMF) along with a window-based localization approach, that is able to identify the mitral valve in several challenging situations. We improve the average f1-score on our dataset of 10 echocardiographic videos by 0.18 to a f1-score of 0.56. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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27 pages, 9722 KiB

Open AccessArticle

Bayesian Activity Estimation and Uncertainty Quantification of Spent Nuclear Fuel Using Passive Gamma Emission Tomography

by Ahmed Karam Eldaly, Ming Fang, Angela Di Fulvio, Stephen McLaughlin, Mike E. Davies, Yoann Altmann and Yves Wiaux

J. Imaging 2021, 7(10), 212; https://doi.org/10.3390/jimaging7100212 - 14 Oct 2021

Cited by 3 | Viewed by 2619

Abstract

In this paper, we address the problem of activity estimation in passive gamma emission tomography (PGET) of spent nuclear fuel. Two different noise models are considered and compared, namely, the isotropic Gaussian and the Poisson noise models. The problem is formulated within a [...] Read more.

In this paper, we address the problem of activity estimation in passive gamma emission tomography (PGET) of spent nuclear fuel. Two different noise models are considered and compared, namely, the isotropic Gaussian and the Poisson noise models. The problem is formulated within a Bayesian framework as a linear inverse problem and prior distributions are assigned to the unknown model parameters. In particular, a Bernoulli-truncated Gaussian prior model is considered to promote sparse pin configurations. A Markov chain Monte Carlo (MCMC) method, based on a split and augmented Gibbs sampler, is then used to sample the posterior distribution of the unknown parameters. The proposed algorithm is first validated by simulations conducted using synthetic data, generated using the nominal models. We then consider more realistic data simulated using a bespoke simulator, whose forward model is non-linear and not available analytically. In that case, the linear models used are mis-specified and we analyse their robustness for activity estimation. The results demonstrate superior performance of the proposed approach in estimating the pin activities in different assembly patterns, in addition to being able to quantify their uncertainty measures, in comparison with existing methods. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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14 pages, 826 KiB

Open AccessArticle

Sparsity-Based Recovery of Three-Dimensional Photoacoustic Images from Compressed Single-Shot Optical Detection

by Dylan Green, Anne Gelb and Geoffrey P. Luke

J. Imaging 2021, 7(10), 201; https://doi.org/10.3390/jimaging7100201 - 2 Oct 2021

Cited by 5 | Viewed by 2669

Abstract

Photoacoustic (PA) imaging combines optical excitation with ultrasonic detection to achieve high-resolution imaging of biological samples. A high-energy pulsed laser is often used for imaging at multi-centimeter depths in tissue. These lasers typically have a low pulse repetition rate, so to acquire images [...] Read more.

Photoacoustic (PA) imaging combines optical excitation with ultrasonic detection to achieve high-resolution imaging of biological samples. A high-energy pulsed laser is often used for imaging at multi-centimeter depths in tissue. These lasers typically have a low pulse repetition rate, so to acquire images in real-time, only one pulse of the laser can be used per image. This single pulse necessitates the use of many individual detectors and receive electronics to adequately record the resulting acoustic waves and form an image. Such requirements make many PA imaging systems both costly and complex. This investigation proposes and models a method of volumetric PA imaging using a state-of-the-art compressed sensing approach to achieve real-time acquisition of the initial pressure distribution (IPD) at a reduced level of cost and complexity. In particular, a single exposure of an optical image sensor is used to capture an entire Fabry–Pérot interferometric acoustic sensor. Time resolved encoding as achieved through spatial sweeping with a galvanometer. This optical system further makes use of a random binary mask to set a predetermined subset of pixels to zero, thus enabling recovery of the time-resolved signals. The Two-Step Iterative Shrinking and Thresholding algorithm is used to reconstruct the IPD, harnessing the sparsity naturally occurring in the IPD as well as the additional structure provided by the binary mask. We conduct experiments on simulated data and analyze the performance of our new approach. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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31 pages, 6589 KiB

Open AccessArticle

Spatially Coherent Clustering Based on Orthogonal Nonnegative Matrix Factorization

by Pascal Fernsel

J. Imaging 2021, 7(10), 194; https://doi.org/10.3390/jimaging7100194 - 28 Sep 2021

Cited by 2 | Viewed by 2136

Abstract

Classical approaches in cluster analysis are typically based on a feature space analysis. However, many applications lead to datasets with additional spatial information and a ground truth with spatially coherent classes, which will not necessarily be reconstructed well by standard clustering methods. Motivated [...] Read more.

Classical approaches in cluster analysis are typically based on a feature space analysis. However, many applications lead to datasets with additional spatial information and a ground truth with spatially coherent classes, which will not necessarily be reconstructed well by standard clustering methods. Motivated by applications in hyperspectral imaging, we introduce in this work clustering models based on Orthogonal Nonnegative Matrix Factorization (ONMF), which include an additional Total Variation (TV) regularization procedure on the cluster membership matrix to enforce the needed spatial coherence in the clusters. We propose several approaches with different optimization techniques, where the TV regularization is either performed as a subsequent post-processing step or included into the clustering algorithm. Finally, we provide a numerical evaluation of 12 different TV regularized ONMF methods on a hyperspectral dataset obtained from a matrix-assisted laser desorption/ionization imaging measurement, which leads to significantly better clustering results compared to classical clustering models. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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14 pages, 6437 KiB

Open AccessArticle

A Green Prospective for Learned Post-Processing in Sparse-View Tomographic Reconstruction

by Elena Morotti, Davide Evangelista and Elena Loli Piccolomini

J. Imaging 2021, 7(8), 139; https://doi.org/10.3390/jimaging7080139 - 7 Aug 2021

Cited by 13 | Viewed by 2822

Abstract

Deep Learning is developing interesting tools that are of great interest for inverse imaging applications. In this work, we consider a medical imaging reconstruction task from subsampled measurements, which is an active research field where Convolutional Neural Networks have already revealed their great [...] Read more.

Deep Learning is developing interesting tools that are of great interest for inverse imaging applications. In this work, we consider a medical imaging reconstruction task from subsampled measurements, which is an active research field where Convolutional Neural Networks have already revealed their great potential. However, the commonly used architectures are very deep and, hence, prone to overfitting and unfeasible for clinical usages. Inspired by the ideas of the green AI literature, we propose a shallow neural network to perform efficient Learned Post-Processing on images roughly reconstructed by the filtered backprojection algorithm. The results show that the proposed inexpensive network computes images of comparable (or even higher) quality in about one-fourth of time and is more robust than the widely used and very deep ResUNet for tomographic reconstructions from sparse-view protocols. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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18 pages, 5669 KiB

Open AccessArticle

Directional TGV-Based Image Restoration under Poisson Noise

by Daniela di Serafino, Germana Landi and Marco Viola

J. Imaging 2021, 7(6), 99; https://doi.org/10.3390/jimaging7060099 - 16 Jun 2021

Cited by 17 | Viewed by 2553

Abstract

We are interested in the restoration of noisy and blurry images where the texture mainly follows a single direction (i.e., directional images). Problems of this type arise, for example, in microscopy or computed tomography for carbon or glass fibres. In order to deal [...] Read more.

We are interested in the restoration of noisy and blurry images where the texture mainly follows a single direction (i.e., directional images). Problems of this type arise, for example, in microscopy or computed tomography for carbon or glass fibres. In order to deal with these problems, the Directional Total Generalized Variation (DTGV) was developed by Kongskov et al. in 2017 and 2019, in the case of impulse and Gaussian noise. In this article we focus on images corrupted by Poisson noise, extending the DTGV regularization to image restoration models where the data fitting term is the generalized Kullback–Leibler divergence. We also propose a technique for the identification of the main texture direction, which improves upon the techniques used in the aforementioned work about DTGV. We solve the problem by an ADMM algorithm with proven convergence and subproblems that can be solved exactly at a low computational cost. Numerical results on both phantom and real images demonstrate the effectiveness of our approach. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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20 pages, 6782 KiB

Open AccessArticle

Calibration-Less Multi-Coil Compressed Sensing Magnetic Resonance Image Reconstruction Based on OSCAR Regularization

by Loubna El Gueddari, Chaithya Giliyar Radhakrishna, Emilie Chouzenoux and Philippe Ciuciu

J. Imaging 2021, 7(3), 58; https://doi.org/10.3390/jimaging7030058 - 19 Mar 2021

Cited by 5 | Viewed by 3245

Abstract

Over the last decade, the combination of compressed sensing (CS) with acquisition over multiple receiver coils in magnetic resonance imaging (MRI) has allowed the emergence of faster scans while maintaining a good signal-to-noise ratio (SNR). Self-calibrating techniques, such as ESPiRIT, have become the [...] Read more.

Over the last decade, the combination of compressed sensing (CS) with acquisition over multiple receiver coils in magnetic resonance imaging (MRI) has allowed the emergence of faster scans while maintaining a good signal-to-noise ratio (SNR). Self-calibrating techniques, such as ESPiRIT, have become the standard approach to estimating the coil sensitivity maps prior to the reconstruction stage. In this work, we proceed differently and introduce a new calibration-less multi-coil CS reconstruction method. Calibration-less techniques no longer require the prior extraction of sensitivity maps to perform multi-coil image reconstruction but usually alternate estimation sensitivity map estimation and image reconstruction. Here, to get rid of the nonconvexity of the latter approach we reconstruct as many MR images as the number of coils. To compensate for the ill-posedness of this inverse problem, we leverage structured sparsity of the multi-coil images in a wavelet transform domain while adapting to variations in SNR across coils owing to the OSCAR (octagonal shrinkage and clustering algorithm for regression) regularization. Coil-specific complex-valued MR images are thus obtained by minimizing a convex but nonsmooth objective function using the proximal primal-dual Condat-Vù algorithm. Comparison and validation on retrospective Cartesian and non-Cartesian studies based on the Brain fastMRI data set demonstrate that the proposed reconstruction method outperforms the state-of-the-art (

?_{1}

-ESPIRiT, calibration-less AC-LORAKS and CaLM methods) significantly on magnitude images for the T1 and FLAIR contrasts. Additionally, further validation operated on 8 to 20-fold prospectively accelerated high-resolution ex vivo human brain MRI data collected at 7 Tesla confirms the retrospective results. Overall, OSCAR-based regularization preserves phase information more accurately (both visually and quantitatively) compared to other approaches, an asset that can only be assessed on real prospective experiments. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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Map of (a) structural similarity (SSIM) (b) peak signal-to-noise ratio (pSNR) score as a function of hyperparameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>λ</mi> <mo>,</mo> <mi>γ</mi> <mo>)</mo> </mrow> </semantics></math> involved in OSCAR-band (b-OSCAR) regularization using 20-fold prospectively accelerated Sparkling sampling scheme. Full article ">Figure 2
Phase processing for better visualization of the brain structure and hence easier comparisons. (a) Raw wrapped phase image. (b) Unwrapped phase image. (c) High pass filtered phase image, notice that the structures in the brain are more visible here. (d) Contrast stretching of 2nd and 98th percentiles of intensity values that permits contrast enhancement and improved visualization. Full article ">Figure 3
Retrospective results for Cartesian mask (A) and non-Cartesian undersampling pattern (B) on T1 weighted (i) and FLAIR (ii) images of the brain fastMRI data set. The results are presented in the form of box plots, computed over <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> slices. From left to right in each boxplot, we compared subbandwise OSCAR (b-OSCAR), scalewise OSCAR (s-OSCAR), global OSCAR (g-OSCAR), CaLM, L1-ESPiRIT and AC-LORAKS, reconstruction methods. Full article ">Figure 4
Retrospective results for a single slice of FLAIR (top row) and T1 weighted (bottom row) images of the brain fastMRI data set obtained using the Cartesian mask shown in <a href="#jimaging-07-00058-f003" class="html-fig">Figure 3</a>A with <math display="inline"><semantics> <mrow> <mi>UF</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, corresponding to <math display="inline"><semantics> <mrow> <mi>AF</mi> <mo>≃</mo> <mn>4</mn> </mrow> </semantics></math>. The fully sampled Cartesian reference and the different methods (Zero filled Inverse, g-OSCAR, CaLM, L1-ESPiRIT and AC-LORAKS) are shown from left to right and the SSIM scores are indicated to reflect the performance of each method. Full article ">Figure 5
Retrospective results for a single slice of FLAIR (top row) and T1 weighted (bottom row) images of the brain fastMRI data set obtained using the Non-Cartesian sampling pattern shown in <a href="#jimaging-07-00058-f003" class="html-fig">Figure 3</a>B with <math display="inline"><semantics> <mrow> <mi>AF</mi> <mo>=</mo> <mn>16</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>UF</mi> <mo>=</mo> <mn>1.66</mn> </mrow> </semantics></math>. The fully sampled Cartesian reference and the different methods (Density Compensated (DC) adjoint NUFFT, s-OSCAR, CaLM, L1-ESPiRIT and AC-LORAKS) are shown from left to right and the SSIM scores are indicated to reflect the performance of each method. Full article ">Figure 6
(Top Row) Reconstructed MR images (magnitude) from 20-fold accelerated SPARKLING acquisitions using different methods. The scan time for Cartesian reference was 4min41s while the scan time of accelerated SPARKLING was 15s. (a) Cartesian reference. (b) The density compensated adjoint of raw k-space data (DC adj-NUFFT). (c) Reconstruction based on the b-OSCAR formulation. (d) calibration-less reconstruction based on CaLM or group-LASSO regularization. (e) Self-calibrating <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-ESPIRiT reconstruction. (f) Auto-calibrated (AC) LORAKS reconstruction. (Second Row) Respective zooms in the red frame. Reconstructed MR images. (Third Row) Enhanced structures in phase images obtained by the method described in <a href="#sec4dot5-jimaging-07-00058" class="html-sec">Section 4.5</a> on the virtual coil reconstructions of each method. The respective MSE of the phase images with respect to Cartesian reference are also reported. (Bottom Row) Respective zooms to highlight details. Full article ">

18 pages, 1343 KiB

Open AccessArticle

Data-Driven Regularization Parameter Selection in Dynamic MRI

by Matti Hanhela, Olli Gröhn, Mikko Kettunen, Kati Niinimäki, Marko Vauhkonen and Ville Kolehmainen

J. Imaging 2021, 7(2), 38; https://doi.org/10.3390/jimaging7020038 - 20 Feb 2021

Cited by 1 | Viewed by 2636

Abstract

In dynamic MRI, sufficient temporal resolution can often only be obtained using imaging protocols which produce undersampled data for each image in the time series. This has led to the popularity of compressed sensing (CS) based reconstructions. One problem in CS approaches is [...] Read more.

In dynamic MRI, sufficient temporal resolution can often only be obtained using imaging protocols which produce undersampled data for each image in the time series. This has led to the popularity of compressed sensing (CS) based reconstructions. One problem in CS approaches is determining the regularization parameters, which control the balance between data fidelity and regularization. We propose a data-driven approach for the total variation regularization parameter selection, where reconstructions yield expected sparsity levels in the regularization domains. The expected sparsity levels are obtained from the measurement data for temporal regularization and from a reference image for spatial regularization. Two formulations are proposed. Simultaneous search for a parameter pair yielding expected sparsity in both domains (S-surface), and a sequential parameter selection using the S-curve method (Sequential S-curve). The approaches are evaluated using simulated and experimental DCE-MRI. In the simulated test case, both methods produce a parameter pair and reconstruction that is close to the root mean square error (RMSE) optimal pair and reconstruction. In the experimental test case, the methods produce almost equal parameter selection, and the reconstructions are of high perceived quality. Both methods lead to a highly feasible selection of the regularization parameters in both test cases while the sequential method is computationally more efficient. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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Figure 1
Vascular and tumor regions and the three template signals used in the simulation. (Top left) The simulated image with the vascular region of interest (ROI) marked in blue and labeled ‘1’, and the tumor ROI marked in red and labeled ‘2’. (Top right) Simulated vascular ROI signal template. (Bottom left) Simulated tumor ROI signal template. (Bottom right) Simulated signal template in the rest of the tissue. Each image pixel was multiplied with the corresponding template and the result was added to the image to create the simulated time series. Full article ">Figure 2
Cartesian gradient-echo pulse sequence full data reconstructions with spatial total variation (TV) regularization from before and after contrast injection used as a reference. The two images have the same adjusted color scale. Full article ">Figure 3
The joint sparsity contour used in the S-surface parameter selection as well as all the parameter selection curves for Sequential S-curve, L-curve and MC-SURE in the simulation with noise level 5%. The red dots mark the computational grid of <math display="inline"><semantics> <mi>α</mi> </semantics></math>:s and <math display="inline"><semantics> <mi>β</mi> </semantics></math>:s in the contour, and the red squares mark the 1D grid of regularization parameters in the parameter selection curves. Note that the spatial TV norm for the Sequential S-curve is from a single image frame whereas the spatial TV norm for the L-curve is from the whole image series. Full article ">Figure 4
Joint RMSE contours with noise levels 2%, 5% and 10% with the parameters of the four different reconstructions marked and the jRMSE values of the chosen reconstructions. The images show that the regularization parameter pairs obtained with the sparsity based S-surface and Sequential S-curve methods are close to the MinRMSE parameter pair. Full article ">Figure 5
Top rows: Single frame images of the simulated true target and reconstructions of the simulation with noise level 5% at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>≈</mo> <mn>90</mn> </mrow> </semantics></math> s (marked in the signal curves with a dashed line) with parameters chosen with different methods; minimum joint RMSE (MinRMSE), simultaneous parameter selection according to joint sparsity (S-surface), sequential parameter selection according to sparsity (Seq. S-curve) and sequential parameter selection by the L-curve and MC-SURE methods. Bottom row: Single pixel signals from the vascular and tumour areas as well as healthy tissue from the right side of the cortex with the different methods compared to the phantom and RMS errors of the corresponding regions. Full article ">Figure 6
The joint sparsity contour used in the S-surface parameter selection as well as all the parameter selection curves for the Sequential S-curve, L-curve and MC-SURE methods in the experimental data case. The red dots mark the computational grid of <math display="inline"><semantics> <mi>α</mi> </semantics></math>:s and <math display="inline"><semantics> <mi>β</mi> </semantics></math>:s in the contour, and the red squares mark the 1D grid of regularization parameters in the parameter selection curves. Note that the spatial TV norm for the Sequential S-curve is from a single image frame whereas the spatial TV norm for the L-curve is from the whole image series. Full article ">Figure 7
Top rows: Single frame images of the cartesian reference image with spatial regularization from before contrast agent injection and the reconstructions of the experimental data at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>≈</mo> <mn>200</mn> </mrow> </semantics></math> s (marked below with a dashed line) with parameters chosen with the S-surface, Sequential S-curve, L-curve and MC-SURE methods. Bottom row: Single pixel signals from the tumour and vascular areas with the four different methods. Full article ">

21 pages, 9628 KiB

Open AccessArticle

A Model-Based Optimization Framework for Iterative Digital Breast Tomosynthesis Image Reconstruction

by Elena Loli Piccolomini and Elena Morotti

J. Imaging 2021, 7(2), 36; https://doi.org/10.3390/jimaging7020036 - 13 Feb 2021

Cited by 9 | Viewed by 3596

Abstract

Digital Breast Tomosynthesis is an X-ray imaging technique that allows a volumetric reconstruction of the breast, from a small number of low-dose two-dimensional projections. Although it is already used in the clinical setting, enhancing the quality of the recovered images is still a [...] Read more.

Digital Breast Tomosynthesis is an X-ray imaging technique that allows a volumetric reconstruction of the breast, from a small number of low-dose two-dimensional projections. Although it is already used in the clinical setting, enhancing the quality of the recovered images is still a subject of research. The aim of this paper was to propose and compare, in a general optimization framework, three slightly different models and corresponding accurate iterative algorithms for Digital Breast Tomosynthesis image reconstruction, characterized by a convergent behavior. The suggested model-based implementations are specifically aligned to Digital Breast Tomosynthesis clinical requirements and take advantage of a Total Variation regularizer. We also tune a fully-automatic strategy to set a proper regularization parameter. We assess our proposals on real data, acquired from a breast accreditation phantom and a clinical case. The results confirm the effectiveness of the presented framework in reconstructing breast volumes, with particular focus on the masses and microcalcifications, in few iterations and in enhancing the image quality in a prolonged execution. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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23 pages, 2380 KiB

Open AccessArticle

A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization

by Germana Landi, Fabiana Zama and Villiam Bortolotti

J. Imaging 2021, 7(2), 18; https://doi.org/10.3390/jimaging7020018 - 28 Jan 2021

Cited by 4 | Viewed by 2598

Abstract

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and [...] Read more.

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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26 pages, 3307 KiB

Open AccessArticle

A Computationally Efficient Reconstruction Algorithm for Circular Cone-Beam Computed Tomography Using Shallow Neural Networks

by Marinus J. Lagerwerf, Daniël M. Pelt, Willem Jan Palenstijn and Kees Joost Batenburg

J. Imaging 2020, 6(12), 135; https://doi.org/10.3390/jimaging6120135 - 8 Dec 2020

Cited by 11 | Viewed by 3505

Abstract

Circular cone-beam (CCB) Computed Tomography (CT) has become an integral part of industrial quality control, materials science and medical imaging. The need to acquire and process each scan in a short time naturally leads to trade-offs between speed and reconstruction quality, creating a [...] Read more.

Circular cone-beam (CCB) Computed Tomography (CT) has become an integral part of industrial quality control, materials science and medical imaging. The need to acquire and process each scan in a short time naturally leads to trade-offs between speed and reconstruction quality, creating a need for fast reconstruction algorithms capable of creating accurate reconstructions from limited data. In this paper, we introduce the Neural Network Feldkamp–Davis–Kress (NN-FDK) algorithm. This algorithm adds a machine learning component to the FDK algorithm to improve its reconstruction accuracy while maintaining its computational efficiency. Moreover, the NN-FDK algorithm is designed such that it has low training data requirements and is fast to train. This ensures that the proposed algorithm can be used to improve image quality in high-throughput CT scanning settings, where FDK is currently used to keep pace with the acquisition speed using readily available computational resources. We compare the NN-FDK algorithm to two standard CT reconstruction algorithms and to two popular deep neural networks trained to remove reconstruction artifacts from the 2D slices of an FDK reconstruction. We show that the NN-FDK reconstruction algorithm is substantially faster in computing a reconstruction than all the tested alternative methods except for the standard FDK algorithm and we show it can compute accurate CCB CT reconstructions in cases of high noise, a low number of projection angles or large cone angles. Moreover, we show that the training time of an NN-FDK network is orders of magnitude lower than the considered deep neural networks, with only a slight reduction in reconstruction accuracy. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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33 pages, 3286 KiB

Open AccessReview

Off-The-Grid Variational Sparse Spike Recovery: Methods and Algorithms

by Bastien Laville, Laure Blanc-Féraud and Gilles Aubert

J. Imaging 2021, 7(12), 266; https://doi.org/10.3390/jimaging7120266 - 6 Dec 2021

Cited by 7 | Viewed by 3086

Abstract

Gridless sparse spike reconstruction is a rather new research field with significant results for the super-resolution problem, where we want to retrieve fine-scale details from a noisy and filtered acquisition. To tackle this problem, we are interested in optimisation under some prior, typically [...] Read more.

Gridless sparse spike reconstruction is a rather new research field with significant results for the super-resolution problem, where we want to retrieve fine-scale details from a noisy and filtered acquisition. To tackle this problem, we are interested in optimisation under some prior, typically the sparsity i.e., the source is composed of spikes. Following the seminal work on the generalised LASSO for measures called the Beurling-Lasso (BLASSO), we will give a review on the chief theoretical and numerical breakthrough of the off-the-grid inverse problem, as we illustrate its usefulness to the super-resolution problem in Single Molecule Localisation Microscopy (SMLM) through new reconstruction metrics and tests on synthetic and real SMLM data we performed for this review. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging)

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