Andreas Maier · Stefan Steidl Vincent Christlein Joachim Hornegger (Eds.) Tutorial LNCS 11111 An Introductory Guide Medical Imaging Systems Lecture Notes in Computer Science 11111 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, Lancaster, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Friedemann Mattern ETH Zurich, Zurich, Switzerland John C. Mitchell Stanford University, Stanford, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel C. Pandu Rangan Indian Institute of Technology Madras, Chennai, India Bernhard Steffen TU Dortmund University, Dortmund, Germany Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbr ü cken, Germany More information about this series at http://www.springer.com/series/7412 Andreas Maier • Stefan Steidl Vincent Christlein Joachim Hornegger (Eds.) Medical Imaging Systems An Introductory Guide Editors Andreas Maier Lehrstuhl f ü r Mustererkennung Friedrich-Alexander-Universit ä t Erlangen-N ü rnberg Erlangen Germany Stefan Steidl Lehrstuhl f ü r Mustererkennung Friedrich-Alexander-Universit ä t Erlangen-N ü rnberg ä t Erlangen Germany Vincent Christlein Lehrstuhl f ü r Mustererkennung Friedrich-Alexander-Universit ä t Erlangen-N ü rnberg Erlangen Germany Joachim Hornegger Lehrstuhl für Mustererkennung Friedrich-Alexander-Universit ä t Erlangen-N ü rnberg ä t Erlangen Germany ISSN 0302-9743 ISSN 1611-3349 (electronic) Lecture Notes in Computer Science ISBN 978-3-319-96519-2 ISBN 978-3-319-96520-8 (eBook) https://doi.org/10.1007/978-3-319-96520-8 Library of Congress Control Number: 2018948380 LNCS Sublibrary: SL6 – Image Processing, Computer Vision, Pattern Recognition, and Graphics © The Editor(s) (if applicable) and The Author(s) 2018. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speci fi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional af fi liations. Cover illustration: Graphical visualization of the Fourier slice theorem. LNCS 11111, p. 154. Used with permission. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland The present book is the result of four years of work that started in Winter 2014/15 and was fi nally concluded in Summer 2018. As such, numerous hours of work went into this manuscript by several authors, who were all af fi liated with the Pattern Recognition Lab of the Friedrich-Alexander-University Erlangen-Nuremberg. I truly appreciate the dedication and the hard work of my colleagues that led to this fi nal manuscript and, although many already left the lab to take positions in academia and industry, they still supported the fi nalization of this book. While major parts of the book were already completed in Winter 2016/17, Springer gave us the opportunity to rework the book with new concepts like the geek boxes and new fi gures in order to adapt the book to a broader audience. With the present concepts, we hope that the book is suited to early-stage undergraduate students as well as stu- dents who already completed fundamental math classes and want to deepen their knowledge on medical imaging. We believe, the time to improve the manuscript was well spent and the fi nal polish gave rise to a textbook with a coherent story line. In particular, we break with the historical development of the described imaging devices and present, e. g., magnetic resonance imaging before computed tomography, although they were developed in opposite order. A closer look reveals that this change of order is reasonable for didactical purposes: magnetic resonance imaging relies mainly on the Fourier transform, while computed tomography requires understanding of the Fourier slice theorem discovered by Johann Radon. These observations then also mend the apparent historical disorder, as we celebrate Joseph Fourier ’ s 250 th birthday this year and celebrated the 100 th birthday of the Radon transform last year. We also tried to fi nd many graphical explanations for many of the mathematical operations such that the book does not require complete understanding of all mathe- matical details. Yet, we also offer details and references to further literature in the previously mentioned geek boxes as students in the later semesters also need to be familiar with these concepts. In conclusion, we hope that we created a useful textbook that will be accessible to many readers. In order to improve this ease of access further, we chose to publish the entire manuscript as open access book under Creative Com- mons Attribution 4.0 International License. Thus, any information in this book can shared, copied, adapted, or remixed even for commercial purposes as long as the original source is appropriately referenced and a link to the license is provided. June 2018 Andreas Maier Preface 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 System Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.1 Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Types of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Convolution Theorem & Properties . . . . . . . . . . . . . . . . . 25 2.4 Discrete System Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.2 Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.3 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 Images and Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1 Images as Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 Histograms of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Image Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 Window and Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 Gamma Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.3 Histogram Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Edge Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Image Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4.1 Filtering – Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4.2 Linear Shift-invariant Filters in Image Processing . . . . . 44 3.4.3 Nonlinear Filters – the Median Filter . . . . . . . . . . . . . . . 47 3.5 Morphological Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Contents 3.6 Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4 Contents 4 Endoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1 Minimally Invasive Surgery and Open Surgery . . . . . . . . . . . . . 57 4.2 Minimally Invasive Abdominal Surgery . . . . . . . . . . . . . . . . . . . . 58 4.3 Assistance Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Range Imaging in Abdominal Surgery . . . . . . . . . . . . . . . . . . . . . 63 4.4.1 Stereo Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4.2 Structured Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.4.3 Time-of-Flight (TOF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5 Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1 Image Formation in a Thin Lens . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Compound Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3 Bright Field Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.4 Fluorescence Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.5 Phase Contrast Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.6 Quantitative Phase Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.7 Limitation of Light Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.8 Beyond Light Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.9 Light Microscopy Beyond the Diffraction Limit . . . . . . . . . . . . . 88 6 Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.1 Nuclear Magnetic Resonance (NMR) . . . . . . . . . . . . . . . . . . . . . . 91 6.1.1 Genesis of the Resonance Effect . . . . . . . . . . . . . . . . . . . . 91 6.1.2 Relaxation and Contrasts . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.2 Principles of Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . 100 6.2.1 Slice Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2.2 Spatial Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2.3 k -space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2.4 Slice-selective vs. Volume-selective 3-D Imaging . . . . . . 105 6.3 Pulse Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.3.1 Spin Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.3.2 Gradient Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.1 Parallel Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.2 Spectrally Selective Excitation . . . . . . . . . . . . . . . . . . . . . 114 6.4.3 Non-contrast Angiography . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4.4 The BOLD Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7 X-ray Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.1.1 Definition of X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.1.2 History and Present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.2 X-ray Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Contents 5 7.3.1 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.3.2 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.3.3 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.3.4 Rayleigh scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4 X-ray Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.4.1 Image Intensifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.4.2 Flat Panel Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.4.3 Sources of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.5 X-ray Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.5.1 Radiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.5.2 Fluoroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.5.3 Digital Subtraction Angiography . . . . . . . . . . . . . . . . . . . 143 8 Computed Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.1.2 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.2 Mathematical Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.2.1 Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2.2 Fourier Slice Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.3 Image Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.3.1 Analytic Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.3.2 Algebraic Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.3.3 Acquisition Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.4 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.4.1 Spatial Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.4.2 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.4.3 Image Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.5 X-ray Attenuation with Polychromatic Attenuation . . . . . . . . . 176 8.5.1 Mono- vs. Polychromatic Attenuation . . . . . . . . . . . . . . . 176 8.5.2 Single, Dual, and Spectral CT . . . . . . . . . . . . . . . . . . . . . 179 8.5.3 Beam Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.6 Spectral CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.6.1 Different Spectral CT Measurements . . . . . . . . . . . . . . . . 182 8.6.2 Basis Material Decomposition . . . . . . . . . . . . . . . . . . . . . . 186 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.2 Talbot-Lau Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 9.2.1 Talbot-Lau Interferometer Setup . . . . . . . . . . . . . . . . . . . 195 9.2.2 Phase Stepping and Reconstruction . . . . . . . . . . . . . . . . . 197 9.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 9.4 Research Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.3 X-ray Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9 X-ray Phase Contrast: Modality . . . . . . . . . 191 Research on a Future Imaging . . . . . . . . . 6 Contents 10 Emission Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.2 Physics of Emission Tomography . . . . . . . . . . . . . . . . . . . . . . . . . 208 10.2.1 Photon Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 10.2.2 Photon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 10.3 Acquisition Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10.3.1 SPECT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10.3.2 PET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 10.4 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10.4.1 Filtered Back-Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10.4.2 Iterative Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 10.4.3 Quantitative Reconstructions . . . . . . . . . . . . . . . . . . . . . . 222 10.4.4 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 10.5 Clinical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 10.5.1 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 10.5.2 Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 10.6 Hybrid Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 10.6.1 Clinical Need . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 10.6.2 Advent und Acceptance of Hybrid Scanners . . . . . . . . . . 231 10.6.3 Further Benefits of Hybrid Imaging . . . . . . . . . . . . . . . . . 232 11 Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 11.2 Physics of Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 11.2.1 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 11.2.2 Sound Wave Characteristics at Boundaries . . . . . . . . . . 239 11.2.3 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 11.3 Image Acquisition for Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . 243 11.3.1 Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 11.3.2 Piezoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 11.3.3 Spatial Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 11.3.4 Imaging Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 11.4 Safety Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 12 Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 12.1 Working Principle of OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 12.1.1 Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.1.2 Coherence Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.2 Time Domain OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 12.3 Fourier Domain OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 12.4 OCT Angiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 12.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Chapter 1 Introduction Author: Andreas Maier The design and manufacturing of modern medical devices requires knowl- edge of several disciplines, ranging from physics, over material science, to computer science. Thus, designing a single lecture as an introduction to med- ical engineering faces a lot of challenges. Nonetheless, the manuscript Medi- cal Imaging Systems – An Introductory Guide aims at being a complete and comprehensive introduction to this field for students in the early semesters. Medical imaging devices are by now an integral part of modern medicine, and have probably already been encountered by all students in their personal life. This book does not simply summarize the content of the lecture held in Erlangen. Instead, it should be understood as additional material to gain a better understanding of the theory that is covered in the lecture. To give a complete introduction, the lecture notes also cover basic math and physics that are required to understand the underlying principles of the imaging devices. However, we try to limit this to the very basics. Obviously, this is not sufficient to describe everything in the appropriate level of detail. For this reason, we introduced geek boxes (cf. Geek Box 1.1) that contain optional additional background information. This concept will be used in all chapters of the book which are summarised in the following sections. Chap. 2 and 3 of this book cover an introduction to signal and image pro- cessing. Chap. 2 introduces the concepts of filtering, convolution, and Fourier transforms for 1-D signals, all of which are fundamental tools that are later on used across the entire book. We try to explain why these concepts are required and as most image processing is digital also emphasize the discrete algorithmic counter parts. At the beginning of Chap. 3, the transition to im- ages is made, and therefore also the transition from 1-D to 2-D. The chapter c © The Author(s) 2018 A. Maier et al. (Eds.): Medical Imaging Systems, LNCS 11111, pp. 7–12, 2018. https://doi.org/10.1007/978-3-319-96520-8 _ 1 8 1 Introduction Geek Box 1.1: Geek Boxes We designed the manuscript to be readable from the first semester on. However, we felt that we need to demonstrate that there is much more depth that we could go into. In order not to confuse a less experienced reader, we omitted most equations and math from the main text and relocated them to geek boxes that go into more detail and give references to further reading. In addition, we also refresh concepts that are already known to most readers. Nonetheless, the important concepts are already mentioned in the main text. This way, the reader can return to this book at a time when these concepts are introduced, e. g., in more advanced math courses seemingly unrelated to medical imaging. As such this book can be read twice: once omitting all geek boxes to get an overview on the field and a second time with a more throrough focus on the mathematical details. covers the basics of image processing and explains how different image trans- formations such as edge detection and blurring are implemented as image filters using convolution. The following chapters cover examples for imaging devices using stan- dard optics. In this book, endoscopy and microscopy are discussed as typical modalities of this genre. Endoscopes, see Chap. 4, were among the first med- ical imaging devices that were used. Images can be acquired by using long and flexible optical fibers that are able to transport visible light through the body of a patient. Microscopes also use visible light. However, tissue samples or cells have to be extracted from the body first, e. g., in a biopsy. Then the microscope’s op- tics are used to acquire images at high magnifications that allow the imaging of individual cells and even smaller structures. Microscopes and the principles of optics are described in Chap. 5. Magnetic resonance imaging (MRI), see Chap. 6 uses electromagnetic waves to excite water atoms inside the human body. Once the excitation is stopped, the atoms return to their normal state and by doing so emit the same electromagnetic radio wave that was used to excite them. This ef- fect is called nuclear magnetic resonance. Using this effect, an MRI image is obtained. Fig. 1.1 shows a state-of-the-art MR scanner. X-ray imaging devices, see Chap. 7, use light of very high energy. However, the light is no longer visible for the human eye. The higher energy of the light allows for a deeper penetration of the body. Due to different absorption rates of X-rays, different body tissues can be distinguished on X-ray images. Tissues with high X-ray absorption, e. g., bones, become visible as bright structures in X-ray projection images. Today, X-rays are among the most widely spread 1 Introduction 9 Figure 1.1: MRI is based on nuclear magnetic resonance which does not involve ionizing radiation. For this reason MRI is often used in pediatric applications. Image courtesy of Siemens Healthineers AG. Figure 1.2: X-ray projection images are one of the most wide-spread imaging modalities. Image courtesy of Siemens Healthineers AG. 10 1 Introduction Figure 1.3: Modern CT systems allow even scanning of the beating heart. Image courtesy of Siemens Healthineers AG. medical imaging technologies. An example for an X-ray imaging device is shown in Fig. 1.2. Computed tomography (CT) uses X-rays to reconstruct slice and volume data as described in Chap. 8. The total absorption along the path of an X-ray through the body is actually given by the sum of absorptions by tissues with different absorption characteristics along its path. Thus, a measurement of the absorptions of X-rays from different directions allows for a reconstruction of slice images through the patient’s body. In doing so, much better contrast between types of soft tissue is obtained. One is even able to differentiate between different tissue types such as brain and brain tumor. Once several slices are combined, the entire volume can be reconstructed by stacking the slices, which is then referred to as a 3-D image. Fig. 1.3 shows a state-of-the- art CT system with a gantry that rotates at 4 Hz. X-rays essentially are electromagnetic waves that can be described by their amplitude, wavelength, and phase. Phase contrast imaging exploits the effect that an X-ray passing through tissue is not only influenced by absorption, but that also the phase of the electromagnetic wave is shifted. Chap. 9 shows that the phase shift of X-rays can be used to visualize the tissue the X- rays have passed. Today, phase contrast imaging is not yet used in clinical practice. In fact, due to the high requirements on the type of irradiation, such images often require a synchrotron as the source of the radiation. However, new developments in research now allow to generate phase contrast images using a normal clinical X-ray tube, which renders the application clinically feasible. At present, technical limitations allow only the scanning of small specimen such as peanuts and the mechanical design is still challenging. First image results indicate that the modality might be of high clinical relevance. Fig. 1.4 shows the reconstruction of peanut fibers that are in the range of 1 Introduction 11 a b c Figure 1.4: An X-ray dark-field setup can be used to reconstruct the ori- entation of fibers that are smaller than the detector resolution. The image on the left shows the reconstructed fiber orientation in different layers of a peanut. The image on the right shows a microscopic visualization of the waist of the peanut (picture courtesy of ECAP Erlangen). Figure 1.5: Modern SPECT/CT systems combine different modalities to achieve multi-modal imaging. Image courtesy of Siemens Healthineers AG. several micrometers. Phase contrast allows for a reconstruction of these fibers, although the resolution of the used imaging device based on the absorption of X-rays was only 0 1 mm. Emission tomography, described in Chap. 10, is used for imaging different bodily functions. It uses tracers , which are molecules that are marked with radioactive atoms. For example one can introduce a radioactive atom into a sugar molecule. When this tracer is consumed by the body it will follow the normal metabolism, and its path through the body can be followed. While sugar consumption is normal in certain parts of the body such as the muscles or the brain, tumors also require a lot of sugar for their growth. Thus, emis- 12 1 Introduction Figure 1.6: A typical ultrasound system as it can be found in clinics world- wide. Image courtesy of Siemens Healthineers AG. sion tomography enables us to see anomalies in sugar consumption within the body which is useful to spot tumors or metastases. Fig. 1.5 shows a com- bined single-photon emission computed tomography (SPECT) / CT system that combines emission tomography with X-ray CT. Ultrasound (US) uses high-frequency sound waves to penetrate bodily tis- sue. The sound waves are emitted from a probe that is in direct contact with the body. The same probe is then also used to measure the reflections of the sound waves. Given the time between the emission of the sound wave and the measurement of the reflection, one is able to reconstruct how deep the wave penetrated the tissue. US is one of the most wide-spread imaging modalities as it is rather inexpensive compared to other imaging modalities. Fig. 1.6 shows a clinical ultrasound system. The measurement principle of optical coherence tomography (OCT) is quite similar to US. However, light waves are used instead of sound waves. Thus, the measurement process needs to be performed at much higher speed and penetration depth is much lower than in the case of US. Most applications are in eye imaging where 3-D images of the eye are generated. 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Chapter 2 System Theory Authors: Peter Fischer, Klaus Sembritzki, and Andreas Maier 2.1 Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Discrete System Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 In the digital age, any medical image needs to be transformed from contin- uous domain to discrete domain (i. e. 1’s and 0’s) in order to be represented in a computer. To do so, we have to understand what a continuous and a discrete signal is. Both of them are handled by systems which will also be introduced in this chapter. Another fundamental concept is the Fourier transform as it allows us to represent any time domain signal in frequency space. In particular, we will find that both representations – time domain and frequency domain – are equivalent and can be converted into each other. Having found this important relationship, we can then determine conditions which will guarantee that also conversion from continuous to discrete domain and vice versa is possible without loss of information. On the way, we will introduce several other important concepts that will also find repeated use later in this book. c © The Author(s) 2018 A. Maier et al. (Eds.): Medical Imaging Systems, LNCS 11111, pp. 13–36, 2018. https://doi.org/10.1007/978-3-319-96520-8 _ 2 14 2 System Theory 2.1 Signals and Systems 2.1.1 Signals A signal is a function f ( t ) that represents information. Often, the indepen- dent variable t is a physical dimension, like time or space. The output f of the signal is also called the dependent variable. Signals are everywhere in every- day life, although we are mostly not aware of them. A very prominent example is the speech signal, where the independent variable is time. The dependent variable is the electric signal that is created by measuring the changes of air pressure using a microphone. The description of the speech generation pro- cess enables to do efficient speech processing, e. g., radio transmission, speech coding, denoising, speech recognition, and many more. In general, many do- mains can be described using system theory, e. g., biology, society, economy. For our application, we are mainly interested in medical signals. Both the dependent and the independent variable can be multidimen- sional. Multidimensional independent variables t are very common in images. In normal camera images, space is described using two spatial coordinates. However, medical images, e. g., CT volume scans, can also have three spatial dimensions. It is not necessary that all dimensions have the same meaning. Videos have two spatial coordinates and one time coordinate. In the medi- cal domain, we can also find higher-dimensional examples like time-resolved 4-D MR and CT with three spatial dimensions and one time dimension. To represent multidimensional values, i. e., vectors, we use bold-face letters t or multiple scalar values, e. g., t = ( x, y, z ) ᵀ . The medical field also contains examples of multidimensional dependent variables f . An example with many dimensions is the Electroencephalography (EEG). Electrodes are attached to the skull and measure electrical brain activity from multiple positions over time. To represent multidimensional dependent variables, we also use bold- face letters f The signals described above are all in continuous domain, e. g., time and space change continuously. Also, the dependent variables vary continuously in principle, like light intensity and electrical voltage. However, some sig- nals exist naturally in discrete domains w. r. t. the independent variable or the dependent variable. An example for a discrete signal in dependent and independent variable is the number of first semester students in medical en- gineering. The independent variable time is discrete in this case. The starting semesters are WS 2009, WS 2010, WS 2011, and so on. Other points in time are considered to be constant in this interval. The number of students is restricted to natural numbers. In general, it is also possible that only the de- pendent or the independent variable is discrete and the other one continuous. In addition to signals that are discrete by nature, other signals must be rep- resented discretely for processing with a digital computer, which means that the independent variable must be discretized before processing with a com- 2.1 Signals and Systems 15 Figure 2.1: A system H{ } with the input signal f ( t ) and the output signal g ( t ). puter. Furthermore, data storage in computers has limited precision, which means that the dependent variable must be discrete. Both are a direct conse- quence of the finite memory and processing speed of computers. This is the reason why discrete system theory is very important in practice. Signals can be further categorized into deterministic and stochastic signals. For a deterministic signal, the whole waveform is known and can be written down as a function. In contrast, stochastic signals depend randomly on the independent variable, e. g., if the signal is corrupted by noise. Therefore, for practical applications, the stochastic properties of signals are very important. Nevertheless, deterministic signals are important to analyze the behavior of systems. A short introduction into stochastic signals and randomness will be given in Sec. 2.4.3. This chapter is presents basic knowledge on how to represent, analyze, and process signals. The correct processing of signals requires some math and theory. A more in-depth introduction into the concepts presented here can be found in [3]. The application to medical data is treated in [2]. 2.1.2 Systems Signals are processed in processes or devices, which are abstracted as sys- tems . This includes not only technical devices, but natural processes like attenuation and reverberation of speech in transmission through air as well. Systems have signals as input and as output. Inside the system, the properties of the signal are changed or signals are related to each other. We describe the processing of a signal using a system with the operator H{·} that is applied to the function f . A graphical representation of a system is shown in Fig. 2.1. An important subtype is the linear shift-invariant system . Linear shift- invariant systems are characterized by the two important properties of lin- earity and shift-invariance (cf. Geek Box 2.1 and 2.2). Another property important for the practical realization of linear shift- invariant systems is causality. A causal system does not react to the input 16 2 System Theory Geek Box 2.1: Linear Systems The linearity property of a system means that linear combinations of inputs can be represented as the same linear combination of the processed inputs H{ af ( t ) } = a H{ f ( t ) } (2.1) H{ f ( t ) + g ( t ) } = H{ f ( t ) } + H{ g ( t ) } , (2.2) with constant a and arbitrary signals f and g . The linearity property greatly simplifies the mathematical and practical treatment, as the behavior of the system can be studied on basic signals. The behav- ior on more complex signals can be inferred directly if they can be represented as a superposition of the basic signals. Geek Box 2.2: Shift-Invariant Systems Shift-invariance denotes the characteristic of a system that its re- sponse is independent of shifts of the independent variable of the signal. Mathematically, this is described as g 1 ( t ) = H{ f ( t ) } (2.3) g 2 ( t ) = H{ f ( t − τ ) } (2.4) g 1 ( t − τ ) = g 2 ( t ) , (2.5) for the shift τ . This means that shifting the signal by τ followed by processing with the system is identical