3D Image Reconstruction for CT and PET

A Practical Guide with Python

Daniele Panetta, Niccolo Camarlinghi

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780367539818
• eBook ISBN: 9781000175882
• Edition: 1
• Topic: Medicine
• Subtopic: Radiology, Radiotherapy & Nuclear Medicine

This is a practical guide to tomographic image reconstruction with projection data, with strong focus on Computed Tomography (CT) and Positron Emission Tomography (PET). Classic methods such as FBP, ART, SIRT, MLEM and OSEM are presented with modern and compact notation, with the main goal of guiding the reader from the comprehension of the mathematical background through a fast-route to real practice and computer implementation of the algorithms. Accompanied by example data sets, real ready-to-run Python toolsets and scripts and an overview the latest research in the field, this guide will be invaluable for graduate students and early-career researchers and scientists in medical physics and biomedical engineering who are beginners in the field of image reconstruction.

• A top-down guide from theory to practical implementation of PET and CT reconstruction methods, without sacrificing the rigor of mathematical background
• Accompanied by Python source code snippets, suggested exercises, and supplementary ready-to-run examples for readers to download from the CRC Press website
• Ideal for those willing to move their first steps on the real practice of image reconstruction, with modern scientific programming language and toolsets

Daniele Panetta is a researcher at the Institute of Clinical Physiology of the Italian National Research Council (CNR-IFC) in Pisa. He earned his MSc degree in Physics in 2004 and specialisation diploma in Health Physics in 2008, both at the University of Pisa. From 2005 to 2007, he worked at the Department of Physics "E. Fermi" of the University of Pisa in the field of tomographic image reconstruction for small animal imaging micro-CT instrumentation. His current research at CNR-IFC has as its goal the identification of novel PET/CT imaging biomarkers for cardiovascular and metabolic diseases. In the field micro-CT imaging, his interests cover applications of three-dimensional morphometry of biosamples and scaffolds for regenerative medicine. He acts as reviewer for scientific journals in the field of Medical Imaging: Physics in Medicine and Biology, Medical Physics, Physica Medica, and others. Since 2012, he is adjunct professor in Medical Physics at the University of Pisa.

Niccolò Camarlingh i is a researcher at the University of Pisa. He obtained his MSc in Physics in 2007 and his PhD in Applied Physics in 2012. He has been working in the field of Medical Physics since 2008 and his main research fields are medical image analysis and image reconstruction. He is involved in the development of clinical, pre-clinical PET and hadron therapy monitoring scanners. At the time of writing this book he was a lecturer at University of Pisa, teaching courses of life-sciences and medical physics laboratory. He regularly acts as a referee for the following journals: Medical Physics, Physics in Medicine and Biology, Transactions on Medical Imaging, Computers in Biology and Medicine, Physica Medica, EURASIP Journal on Image and Video Processing, Journal of Biomedical and Health Informatics.

CHAPTER 1

Preliminary notions

The purpose of this chapter is to provide a concise list of the most important theoretical and practical elements involved in CT and PET reconstruction, along with a brief description for each of them. The list does not follow a precise logical order, and it is constructed with the ultimate goal to make the reading of the following chapters as fluent and goal-oriented as possible (with the “goal” being the reconstruction of tomographic images).

Those readers who are already familiar with the definitions and concepts provided in this chapter, can skip reading it. For those who are still at the preliminary stage of their study in PET/CT technology and reconstruction, the reading of this chapter is mandatory for a full understanding of what comes next. Moreover, it will serve as a quick reference during reading of the subsequent chapters, for all those concepts that are referenced therein. The beginners can then come back to this chapter at any time to refresh their knowledge.

1.1Image reconstruction from projection

1.1.1Purpose of image reconstruction

Given an unknown function f describing the spatio-temporal distribution of some chemical-physical property of a patient or object under examination, and a set p of projections of f (the formal definition of projection will be given below when discussing the concepts of line integral and ray-sum), the purpose of image reconstruction is to recover the best estimate of f from p. This problem is referred to as an inverse problem, in which the final goal is to retrieve an unknown input data back from the result (output) of a physical measurement performed on it. In the domain of tomographic imaging, the corresponding forward problem is precisely the process of acquiring the projection data, by means of specific instrumentation for which technical description is well beyond of the scope of this book. Even though any forward problem has a defined solution, this is not true for inverse problems. That is, several possible images are compatible with a given set of projection measurements. Hence, image reconstruction is referred to as an ill-posed inverse problem. For a thorough discussion on inverse problems in imaging, refer to Bertero and Boccaccio [2].

1.1.2Families of reconstruction methods

Reconstruction methods in tomographic imaging can be grouped into two families:

•Analytical reconstruction methods

•Iterative reconstruction methods

Analytical methods treat both images and projections as continuous functions. The process of image acquisition is ideally represented as a linear and continuous operator acting on the space of object functions representing the patient (or object) under examination. Hence, the acquisition process gives rise to a integral transform of the unknown function describing the patient; image reconstruction must be carried out by inverting such an integral transform. The discrete nature of projection data and reconstructed image (in digital form) is taken into account only at the very final stage of image reconstruction, when a closed analytical inversion formula has been derived starting from the continuous assumption.

Iterative algorithms are instead based on the assumption that both images and projection data are discrete. In this view, the forward problem of projection data acquisition can be conveniently set in the form of a system of linear equations. Solving such a system of equations is not a trivial task in real-life situations, given that both the number of unknowns (the voxel values) and the number of equations (their projections) can easily go beyond several millions or b...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• About the Authors
• CHAPTER 1: Preliminary notions
• CHAPTER 2: Short guide to Python samples
• CHAPTER 3: Analytical reconstruction algorithms
• CHAPTER 4: Iterative reconstruction algorithms
• CHAPTER 5: Overview of methods for generation projection data
• Bibliography
• Index