**organiser: ** Claude Comtat, Frédéric Joliot Hospital Facility, CEA, France

**course description**

This course will provide an overview of medical image reconstruction methods. For the first time, not only X-ray computed tomography (CT) and emission tomography (PET and SPECT), but also magnetic resonance imaging (MRI) will be covered. Both iterative and non-iterative reconstruction methods will be presented, starting from the basics, and progressing to cover more advanced reconstruction techniques. Several examples and reconstruction demonstrations will be presented. The course will start with the basics of analytical tomographic image reconstruction in two-dimension (2D), with a focus for X-ray CT. Advanced topics in three-dimensional (3D) analytical reconstruction will follow for cone-beam CT and PET. Then, the basics of iterative tomographic image reconstruction will be presented, followed by advanced topics in iterative image reconstruction in PET and CT. The course will end with MRI reconstruction.

Prerequisite knowledge includes basic knowledge of the physics of emission (PET & SPECT) and transmission (X-ray CT) imaging system, statistics, and elementary linear algebra. The MR reconstruction part is intended to scientists who are non-MRI specialists.

**Please note that this is a 2-day course, but it has been planned so that each day can be taken independently. You can register for this course for both days (SC5), for the first day only (SC5D1), or for the second day only (SC5D2). See the registration information page for details.**

**course outline**

*First Morning*

- Introduction (C. Comtat)
- Basics of analytic image reconstruction for a 2D setting (E. Sidky)
- Imaging Model
- The physics of imaging for 2D CT
- Parallel- and fan-beam scanning configurations
- Their relation to the Radon transform

- Reconstruction (inversion of the model)
- The Fourier central slice theorem
- Image reconstruction from 2D parallel-beam projection data
- Image reconstruction from 2D fan-beam projection data

- Incomplete data
- Discrete sampling
- Limited scanning angular range
- Limited scanning angular range

- Data inconsistency
- Non-ideal physical factors: quantum noise, polychromatic beam spectra, X-ray scatter, etc.
- The corresponding artifacts in the reconstructed images
- Exploitation of data redundancy to control image artifacts

- Advanced topics in 3D analytic image reconstruction in cone-beam CT (E. Sidky)
- Cone-beam CT data and inversion formulas
- The 3D Radon and X-ray transform
- 3D Radon transform inversion
- Tuy's cone-beam (3D X-ray transform) inversion formula

- Incomplete data
- Tuy's condition on the scanning trajectory and circular cone-beam CT
- Approximate cone-beam CT image reconstruction
- Helical CT and the long-object problem

- Advances in helical cone-beam CT image reconstruction
- Grangeat's relation between the 3D Radon and X-ray transform
- PI-line and m-line based inversion for cone-beam CT

- Incomplete data revisited
- A new perspective on 2D CT - ROI imaging with truncated projections
- ROI imaging in 3D CT
- A tiny a priori

- Ongoing research
- 3D data consistency conditions
- Exploitation of these conditions for improved exact cone-beam image reconstruction
- Image quality metrics
- Model observers for image reconstruction algorithm development (Ideal observer vs. human observer)

- Advanced topics in 3D analytic image reconstruction in PET (C. Comtat)
- Image reconstruction from 3D parallel-beam projection data
- Incomplete data: the reprojection 3DRP algorithm
- Rebinning algorithms for 3D PET data: single slice rebinning algorithm (SSRB) and the Fourier rebinning algorithm (FORE)

*First Afternoon and Second Morning*

- Basics of iterative image reconstruction (A. Reader, C. Comtat)
- Data and object parameterization: list-mode data, sinograms, voxels and other spatial basis functions
- Data modelling: system matrix, resolution modeling, data corrections and factorization, model estimation (analytical - Monte Carlo – measured), noise (Poisson – Gaussian)
- Objective functions: maximum likelihood, least squares, maximum a posteriori
- Optimization algorithms: ML-EM, OSL, surrogate, gradient-based methods, subsets, convergence

- Advanced topics in iterative image reconstruction for PET (A. Reader and C. Comtat)
- Nested EM for linear or non-linear parameter estimation from tomographic data, used for 4D and direct parametric image reconstruction
- Anatomy-based image priors
- Mitigation of Gibbs artefacts (PSF modelling)
- Bias and negative values: NEG, AML
- Synergistic joint PET-MR reconstruction

- Advanced topics in iterative image reconstruction for CT (E. Sidky)

(Background material will be covered quickly with a CT focus)
- Implicit solution of an imaging model
- Inversion of complex physical models
- Image reconstruction from discretely sampled data
- A different perspective on the classical incomplete data sampling problems
- The trade-off between image representation and data fidelity

- Basic principles of iterative image reconstruction (IIR)
- Algebraic approach
- Likelihood maximization
- Exploitation of image sparsity
- Incorporation of prior knowledge

- Optimization for IIR in CT
- Types of optimization problems: convex/non-convex, unconstrained/constrained, smooth/non-smooth
- Standard solvers used for CT IIR
- Recent developments in solvers
- Early stopping and algorithm acceleration

- Ongoing research
- A utility-driven approach to IIR algorithm development
- The search for meaningful IIR algorithm parameterization
- Integrated IIR algorithm/tomographic system development

*Second Afternoon*

- MRI reconstruction (P. Ciuciu)

This course is intended to scientists who are non-MRI specialists but rather familiar with inverse problem solving in medical imaging (e.g. in PET or CT imaging). For this reason, the background on MR image acquisition/formation will be reviewed before focusing on image reconstruction. In particular, P. Ciuciu will explain how the data are collected in the Fourier (or k-space) along (piecewise) continuous or more regular trajectories in 2D or 3D will be explained. Next, he will provide an overview of the classical ways to perform MR image reconstruction for Cartesian and non-Cartesian acquisition scenarios and he will address how to correct for static field inhomogeneities within the reconstruction process.

In the following, P. Ciuciu will pay attention to the classical parallel imaging techniques (SENSE, GRAPPA, ...) that are available on the clinical scanners to fasten acquisitions up to a given acceleration factor R. He will outline the main limitations of these approaches that prevent from considering R ≥ 5 in 2D. This will motivate the presentation of compressed sensing (CS) for MRI in the last part of this course. He will briefly review the original CS theory, present the original applications to MRI using conventional acquisition schemes and sparsity-promoting reconstruction algorithms (proximal methods). Finally he will detail recent developments based on his own research that allows one to achieve 40-fold R values at very high resolution. Matlab code with different examples and reconstruction scenarios will be provided as part of the course.
- Background in MRI: Image formation, contrast
- Fourier-based approaches for MR image reconstruction
- Parallel imaging techniques
- Compressed sensing MRI

**instructors**

Philippe Ciuciu received his PhD in signal processing from the University of Paris-Sud in 2000. After a two-year postdoctoral fellowship in the Life Science Division of the Atomic Energy Commission (CEA), he has been hired by the same institute to develop signal processing methods for functional Magnetic Resonance Imaging (fMRI) data analysis. Since 2007, Dr Ciuciu has joined NeuroSpin, the CEA neuroimaging center dedicated to ultra-high field MRI and its applications to neuroscience. In 2008, he became principal investigator and then was promoted as CEA expert senior scientist for his contributions to biomedical research in the field of signal and image processing for neuroimaging. Dr Ciuciu has conducted an interdisciplinary research with a track record ranging from MRI data acquisition to analysis of functional neuroimaging data (fMRI, MEG). Dr Ciuciu has developed new 3D/3D+time MR image reconstruction algorithms for anatomical and functional MRI data acquired in parallel imaging. Such algorithms have been patented in Japan and Europe. Since 2011, he has collaborated with mathematicians (P. Weiss, J. Kahn) of the IMT institute (Univ. of Toulouse) on compressed sensing (CS) theory and applications to MRI. The main breakthrough achieved during this collaboration was new mathematically-principled and practically feasible CS sampling schemes specifically designed for MRI, which now permit to dramatically accelerate real acquisitions at 7 Tesla by a factor up to 40, a worldwide breakthrough never achieved before.

Claude Comtat is a researcher at the French Atomic Energy Commission (CEA), a government-funded technological research organization. He is with the Frédéric Joliot Hospital Facility (SHFJ) and IMIV, an In Vivo Molecular Imaging multidisciplinary research laboratory. He received his Ph.D. degree in High Energy Physics from the University of Lausanne in 1996. Prior to CEA, he worked as a postdoctoral fellow for two years at the PET Facility of the University of Pittsburgh Medical Center (UPMC) on PET Image Reconstruction with Paul Kinahan and David Townsend. He has coauthored more than 160 research outputs, including 47 peer-reviewed publications. His main research interest is in iterative reconstruction in PET, with an emphasis on data modelling and multi-modality imaging.

Andrew Reader is a Professor of Imaging Sciences at King’s College London (KCL, UK). He was previously a holder of a Canada Research Chair from 2008-2015 at McGill University (Montreal Neurological Institute, Canada) and prior to that was a senior lecturer at the University of Manchester in the UK. Andrew Reader teaches on master’s level courses for medical image reconstruction, and the physics and maths of radionuclide imaging, and has previously been involved with teaching at summer schools in the UK and France. He now has well over 170 research outputs, including 68 peer-reviewed publications. His primary research interest concerns image reconstruction and modelling for positron emission tomography (PET), and more recently its integration with simultaneous magnetic resonance imaging (MRI). Particular topics include joint PET-MR image reconstruction, fully 4D image reconstruction and direct kinetic parameter estimation, particularly applied to imaging of the brain.

Emil Sidky is a Research Associate Professor at the University of Chicago Department of Radiology and Graduate Program for Medical Physics. He received his Ph.D. in physics from The University of Chicago in 1993. He has coauthored approximately 80 publications in the area of his current interests: CT image reconstruction, large-scale optimization, and objective assessment of image quality. He has ongoing research projects in developing image reconstruction algorithms in Digital Breast Tomosynthesis and spectral CT.