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.

/pdf/a-three-dimensional-statistical-approach-to-improved-image-1jddsdidms.pdf

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

In order to reduce CT radiation dose, there have been numerous efforts to develop low-dose acquisition protocols as well as noise reduction methods such as data denoising and iterative reconstruction. In this study, we analyze the first and second order statistics of post-log CT data and the resulting impact on iterative image reconstruction for extremely low-dose CT acquisitions. We performed a CT simulation incorporating polychromatic forward projection and realistic levels of quantum and electronic noise. We performed N=1000 simulations of a chest phantom to analyze the impact of processing steps on the statistics of post-log data. We investigated the impact of two non-positivity correction methods, threshold and mean-preserving filter. And, we analyzed the bias and variance of different weighting terms and performed weighted least squares reconstruction with these different weights. For the simulation of an extremely low dose chest acquisition with 80 kVp and 0.5 mAs, the mean-preserving filter reduced the mean bias of post-log sinogram by roughly seven times compared to the threshold method. The WLS reconstructed images using simple weighting terms that ignored the effect of non-positive correction lead to limited improvements in image quality. Accurate weighting terms including electronic noise and the variance change from MPF provided superior images, especially in highly attenuating regions where bias reductions of ∼17% were achieved compared to simple weighting matrices. Appropriate selection of the non-positivity correction method is essential for low flux CT data processing. The proposed method for estimating the weighting matrix with electronic noise and the effect of pre-corrections leads to some improvements in variance estimation for post-log CT data, although it has potential for further improvement.

Analysis of statistical models for iterative reconstruction of extremely low-dose CT data

A method for normalizing a calibration of a computed tomographic (CT) imaging apparatus having a plurality of detector rows includes utilizing a prestored, predetermined inversion matrix and CT numbers obtained from images of a phantom to determine normalized calibration vectors for each row of a plurality of detector rows, and storing the determined normalized calibration vectors for each row of the plurality of detector rows in a memory for use in subsequent image reconstructions.

Methods and apparatus for CT calibration

Compressed sensing can recover a signal that is sparse in some way from a small number of samples. For computed tomography (CT) imaging, this has the potential to obtain good reconstruction from a smaller number of projections or views, thereby reducing the amount of radiation that a patient is exposed to In this work, we applied compressed sensing to fan beam CT image reconstruction, which is a special case of an important 3-D CT problem (cone beam CT). We compared the performance of two compressed sensing algorithms, denoted as the LP and the QP, in simulation. Our results indicate that the LP generally provides smaller reconstruction error and converges faster; therefore, it is preferable.

Compressed sensing algorithms for fan-beam computed tomography image reconstruction

Various methods and systems are provided for estimating and compensating for table deflection in reconstructed images. In one embodiment, a method for computed tomography (CT) imaging comprises reconstructing images from data acquired during a helical CT scan where table deflection parameters are estimated and the reconstruction is adjusted based on the table deflection parameters. In this way, images may be reconstructed without artifacts caused by table deflection.

Methods and systems for correcting table deflection

Medical imaging typically requires the reconstruction of a limited region of interest (ROI) to obtain a high
resolution image of the anatomy of interest. Although targeted reconstruction is straightforward for analytical
reconstruction methods, it is more complicated for statistical iterative techniques, which must reconstruct all
objects in the field of view (FOV) to account for all sources of attenuation along the ray paths from x-ray
source to detector. A brute force approach would require the reconstruction of the full field of view in high-resolution,
but with prohibitive computational cost. In this paper, we propose a multi-resolution approach to
accelerate targeted iterative reconstruction using the non-homogeneous ICD (NH-ICD) algorithm. NH-ICD aims
at speeding up convergence of the coordinate descent algorithm by selecting preferentially those voxels most in
need of updating. To further optimize ROI reconstruction, we use a multi-resolution approach which combines
three separate improvements. First, we introduce the modified weighted NH-ICD algorithm, which weights the
pixel selection criteria according to the position of the voxel relative to the ROI to speed up convergence within
the ROI. Second, we propose a simple correction to the error sinogram to correct for inconsistencies between
resolutions when the forward model is not scale invariant. Finally, we leverage the flexibility of the ICD algorithm
to add selected edge pixels outside the ROI to the ROI reconstruction in order to minimize transition artifacts
at the ROI boundary. Experiments on clinical data illustrate how each component of the method improves
convergence speed and image quality.

/pdf/non-homogeneous-icd-optimization-for-targeted-reconstruction-5frinqyvqv.pdf

Jean-Baptiste Thibault

Papers

Analysis of statistical models for iterative reconstruction of extremely low-dose CT data

Methods and apparatus for CT calibration

Compressed sensing algorithms for fan-beam computed tomography image reconstruction

Methods and systems for correcting table deflection

Non-homogeneous ICD Optimization for Targeted Reconstruction of Volumetric CT