Introduction to linear and nonlinear programming

Image reconstruction is essential for imaging applications across the physical and life sciences, including optical and radar systems, magnetic resonance imaging, X-ray computed tomography, positron emission tomography, ultrasound imaging and radio astronomy. During image acquisition, the sensor encodes an intermediate representation of an object in the sensor domain, which is subsequently reconstructed into an image by an inversion of the encoding function. Image reconstruction is challenging because analytic knowledge of the exact inverse transform may not exist a priori, especially in the presence of sensor non-idealities and noise. Thus, the standard reconstruction approach involves approximating the inverse function with multiple ad hoc stages in a signal processing chain, the composition of which depends on the details of each acquisition strategy, and often requires expert parameter tuning to optimize reconstruction performance. Here we present a unified framework for image reconstruction-automated transform by manifold approximation (AUTOMAP)-which recasts image reconstruction as a data-driven supervised learning task that allows a mapping between the sensor and the image domain to emerge from an appropriate corpus of training data. We implement AUTOMAP with a deep neural network and exhibit its flexibility in learning reconstruction transforms for various magnetic resonance imaging acquisition strategies, using the same network architecture and hyperparameters. We further demonstrate that manifold learning during training results in sparse representations of domain transforms along low-dimensional data manifolds, and observe superior immunity to noise and a reduction in reconstruction artefacts compared with conventional handcrafted reconstruction methods. In addition to improving the reconstruction performance of existing acquisition methodologies, we anticipate that AUTOMAP and other learned reconstruction approaches will accelerate the development of new acquisition strategies across imaging modalities.

Image reconstruction by domain-transform manifold learning

X-ray computer tomography (CT) is fast becoming an accepted tool within the materials science community for the acquisition of 3D images. Here the authors review the current state of the art as CT transforms from a qualitative diagnostic tool to a quantitative one. Our review considers first the image acquisition process, including the use of iterative reconstruction strategies suited to specific segmentation tasks and emerging methods that provide more insight (e.g. fast and high resolution imaging, crystallite (grain) imaging) than conventional attenuation based tomography. Methods and shortcomings of CT are examined for the quantification of 3D volumetric data to extract key topological parameters such as phase fractions, phase contiguity, and damage levels as well as density variations. As a non-destructive technique, CT is an ideal means of following structural development over time via time lapse sequences of 3D images (sometimes called 3D movies or 4D imaging). This includes information nee...

https://www.tandfonline.com/doi/pdf/10.1179/1743280413Y.0000000023

Quantitative X-ray tomography

Multislice helical computed tomography scanning offers the advantages of faster acquisition and wide organ coverage for routine clinical diagnostic purposes. However, image reconstruction is faced with the challenges of three-dimensional cone-beam geometry, data completeness issues, and low dosage. Of all available reconstruction methods, statistical iterative reconstruction (IR) techniques appear particularly promising since they provide the flexibility of accurate physical noise modeling and geometric system description. In this paper, we present the application of Bayesian iterative algorithms to real 3D multislice helical data to demonstrate significant image quality improvement over conventional techniques. We also introduce a novel prior distribution designed to provide flexibility in its parameters to fine-tune image quality. Specifically, enhanced image resolution and lower noise have been achieved, concurrently with the reduction of helical cone-beam artifacts, as demonstrated by phantom studies. Clinical results also illustrate the capabilities of the algorithm on real patient data. Although computational load remains a significant challenge for practical development, superior image quality combined with advancements in computing technology make IR techniques a legitimate candidate for future clinical applications.

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A three-dimensional statistical approach to improved image quality for multislice helical CT.

Model-based reconstruction is a powerful framework for solving a variety of inverse problems in imaging. In recent years, enormous progress has been made in the problem of denoising, a special case of an inverse problem where the forward model is an identity operator. Similarly, great progress has been made in improving model-based inversion when the forward model corresponds to complex physical measurements in applications such as X-ray CT, electron-microscopy, MRI, and ultrasound, to name just a few. However, combining state-of-the-art denoising algorithms (i.e., prior models) with state-of-the-art inversion methods (i.e., forward models) has been a challenge for many reasons. In this paper, we propose a flexible framework that allows state-of-the-art forward models of imaging systems to be matched with state-of-the-art priors or denoising models. This framework, which we term as Plug-and-Play priors, has the advantage that it dramatically simplifies software integration, and moreover, it allows state-of-the-art denoising methods that have no known formulation as an optimization problem to be used. We demonstrate with some simple examples how Plug-and-Play priors can be used to mix and match a wide variety of existing denoising models with a tomographic forward model, thus greatly expanding the range of possible problem solutions.

/pdf/plug-and-play-priors-for-model-based-reconstruction-51xolwckvh.pdf

Plug-and-Play priors for model based reconstruction

Recent applications of model-based iterative reconstruction (MBIR) algorithms to multislice helical CT reconstructions have shown that MBIR can greatly improve image quality by increasing resolution as well as reducing noise and some artifacts However, high computational cost and long reconstruction times remain as a barrier to the use of MBIR in practical applications Among the various iterative methods that have been studied for MBIR, iterative coordinate descent (ICD) has been found to have relatively low overall computational requirements due to its fast convergence This paper presents a fast model-based iterative reconstruction algorithm using spatially nonhomogeneous ICD (NH-ICD) optimization The NH-ICD algorithm speeds up convergence by focusing computation where it is most needed The NH-ICD algorithm has a mechanism that adaptively selects voxels for update First, a voxel selection criterion VSC determines the voxels in greatest need of update Then a voxel selection algorithm VSA selects the order of successive voxel updates based upon the need for repeated updates of some locations, while retaining characteristics for global convergence In order to speed up each voxel update, we also propose a fast 1-D optimization algorithm that uses a quadratic substitute function to upper bound the local 1-D objective function, so that a closed form solution can be obtained rather than using a computationally expensive line search algorithm We examine the performance of the proposed algorithm using several clinical data sets of various anatomy The experimental results show that the proposed method accelerates the reconstructions by roughly a factor of three on average for typical 3-D multislice geometries

Fast Model-Based X-Ray CT Reconstruction Using Spatially Nonhomogeneous ICD Optimization

For various reasons, a projection dataset acquired on a computed tomography (CT) scanner can be truncated. That is, a portion of the scanned object is positioned outside the scan field-of-view (SFOV) and the line integrals corresponding to those regions are not measured. A projection truncation problem causes imaging artifacts that lead to suboptimal image quality. In this paper, we propose a reconstruction algorithm that enables an adequate estimation of the projection outside the SFOV. We make use of the fact that the total attenuation of each ideal projection in a parallel sampling geometry remains constant over views. We use the magnitudes and slopes of the projection samples at the location of truncation to estimate water cylinders that can best fit to the projection data outside the SFOV. To improve the robustness of the algorithm, continuity constraints are placed on the fitting parameters. Extensive phantom and patient experiments were conducted to test the robustness and accuracy of the proposed algorithm.

A novel reconstruction algorithm to extend the CT scan field-of-view.

PURPOSE Assess the effect of filtered back projection (FBP) and hybrid (adaptive statistical iterative reconstruction [ASIR]) and pure (model-based iterative reconstruction [MBIR]) iterative reconstructions on abdominal computed tomography (CT) acquired with 75% radiation dose reduction. MATERIALS AND METHODS In an institutional review board-approved prospective study, 10 patients (mean [standard deviation] age, 60 (8) years; 4 men and 6 women) gave informed consent for acquisition of additional abdominal images on 64-slice multidetector-row CT (GE 750HD, GE Healthcare). Scanning was repeated over a 10-cm scan length at 200 and 50 milliampere second (mA s), with remaining parameters held constant at 120 kilovolt (peak), 0.984:1 pitch, and standard reconstruction kernel. Projection data were deidentified, exported, and reconstructed to obtain 4 data sets (200-mA s FBP, 50-mA s FBP, 50-mA s ASIR, 50-mA s MBIR), which were evaluated by 2 abdominal radiologists for lesions and subjective image quality. Objective noise and noise spectral density were measured for each image series. RESULTS Among the 10 patients, the maximum weight recorded was 123 kg, with maximum transverse diameter measured as 43.7 cm. Lesion conspicuity at 50-mA s MBIR was better than on 50-mA s FBP and ASIR images (P < 0.01). Image noise was rated as suboptimal on low-dose FBP and ASIR but deemed acceptable in MBIR images. Objective noise with 50-mA s MBIR was 2 to 3 folds lower compared to 50-mA s ASIR, 50-mA s FBP, and 200-mA s FBP (P < 0.0001). Noise spectral density analyses demonstrated that ASIR retains the noise spectrum signature of FBP, whereas MBIR has much lower noise with a more regularized noise spectrum pattern. CONCLUSION Model-based iterative reconstruction renders acceptable image quality and diagnostic confidence in 50- mA s abdominal CT images, whereas FBP and ASIR images are associated with suboptimal image quality at this radiation dose level.

Comparison of hybrid and pure iterative reconstruction techniques with conventional filtered back projection: dose reduction potential in the abdomen.

Over the past two decades, rapid system and hardware development of x-ray computed tomography (CT) technologies has been accompanied by equally exciting advances in image reconstruction algorithms. The algorithmic development can generally be classified into three major areas: analytical reconstruction, model-based iterative reconstruction, and application-specific reconstruction. Given the limited scope of this chapter, it is nearly impossible to cover every important development in this field; it is equally difficult to provide sufficient breadth and depth on each selected topic. As a compromise, we have decided, for a selected few topics, to provide sufficient high-level technical descriptions and to discuss their advantages and applications.

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Jean-Baptiste Thibault

Papers

A three-dimensional statistical approach to improved image quality for multislice helical CT.

Fast Model-Based X-Ray CT Reconstruction Using Spatially Nonhomogeneous ICD Optimization

A novel reconstruction algorithm to extend the CT scan field-of-view.

Comparison of hybrid and pure iterative reconstruction techniques with conventional filtered back projection: dose reduction potential in the abdomen.

Recent Advances in CT Image Reconstruction