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Medical Image Processing: A Comprehensive Introduction, Lecture notes of Engineering

FAST - National University of Computer and Emerging Sciences (NUCES)Engineering

This comprehensive book provides a thorough introduction to the field of medical image processing, covering a wide range of imaging modalities and techniques. It delves into the physics behind various medical imaging technologies, including x-ray, ct, mri, ultrasound, and more. Key aspects of medical image processing, such as image file formats, image reconstruction, image segmentation, image registration, and 3d visualization. With over 70 matlab scripts illustrating the algorithms, the book offers a hands-on, practical approach to learning medical image processing. It is an invaluable resource for students, researchers, and professionals in the fields of medical physics, biomedical engineering, and computer science who want to gain a deep understanding of the fundamentals of this important field.

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ISBN: 978-1-4665-5557-0
9 781466 555570
90000
Biomedical Engineering
“I mention the book every time I meet someone who asks what the best way is to get
into this field.”
—Professor Leo Joskowicz, Head of Computer-Assisted Surgery and Medical Image
Processing Laboratory, The Hebrew University of Jerusalem, Israel
“… exceptionally useful, practical, and approachable.”
—Professor Gabor Fichtinger, Queen’s University, Kingston, Ontario, Canada
A widely used, classroom-tested text, Applied Medical Image Processing: A Basic Course
delivers an ideal introduction to image processing in medicine, emphasizing the clinical
relevance and special requirements of the field. Avoiding excessive mathematical formalisms,
the book presents key principles by implementing algorithms from scratch and using simple
MATLAB®/Octave scripts with image data and illustrations on an accompanying CD-ROM
and companion website. Organized as a complete textbook, it provides an overview of the
physics of medical image processing and discusses image formats and data storage, intensity
transforms, filtering of images and applications of the Fourier transform, three-dimensional
spatial transforms, volume rendering, image registration, and tomographic reconstruction.
This Second Edition of the bestseller:
• Contains two brand-new chapters on clinical applications and image-guided
therapy
• Devotes more attention to the subject of color space
• Includes additional examples from radiology, internal medicine, surgery, and
radiation therapy
• Incorporates freely available programs in the public domain (e.g., GIMP, 3DSlicer,
and ImageJ) when applicable
Beneficial to students of medical physics, biomedical engineering, computer science, applied
mathematics, and related fields, as well as medical physicists, radiographers, radiologists,
and other professionals, Applied Medical Image Processing: A Basic Course, Second Edition
is fully updated and expanded to ensure a perfect blend of theory and practice.
Applied Medical Image Processing
A Basic Course
SECOND
EDITION
Birkfellner
Applied
Medical
Image
Processing
SECOND EDITION
Wolfgang Birkfellner
6000 Broken Sound Parkway, NW
Suite 300, Boca Raton, FL 33487
711 Third Avenue
New York, NY 10017
2 Park Square, Milton Park
Abingdon, Oxon OX14 4RN, UK
an informa business
www.crcpress.com
ww w.crcpress.com
K15380
K15380 mech rev.indd 1 1/7/14 12:48 PM
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Download Medical Image Processing: A Comprehensive Introduction and more Lecture notes Engineering in PDF only on Docsity!

A Basic Course

Applied

Medical

Image

Processing

SECOND EDITION

Wolfgang Birkfellner

Applied

Medical

Image

Processing

SECOND EDITION

Applied

Medical

Image

Processing

A Basic Course

SECOND EDITION

Wolfgang Birkfellner

Center for Medical Physics and Biomedical Engineering
Medical University of Vienna
Vienna, Austria

With contributions by Michael Figl and Johann Hummel Center for Medical Physics and Biomedical Engineering Medical University of Vienna Vienna, Austria

Ziv Yaniv and Özgür Güler Sheikh Zayad Institute for Pediatric Surgical Innovation Children’s National Medical Center Washington, District of Columbia, USA

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

Taylor & Francis Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487- © 2014 by Taylor & Francis Group, LLC Taylor & Francis is an Informa business No claim to original U.S. Government works Version Date: 20140114 International Standard Book Number-13: 978-1-4665-5559-4 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents-in-Brief

List of Figures, xiii

List of Tables, xxi

Foreword, xxiii

Preface to the First Edition, xxv

Preface to the Second Edition, xxvii

User Guide, xxix

Acknowledgments, xxxi

Chapter 1 ^ A Few Basics of Medical Image Sources 1

Chapter 2 ^ Image Processing in Clinical Practice 45

Chapter 3 ^ Image Representation 59

Chapter 4 ^ Operations in Intensity Space 91

Chapter 5 ^ Filtering and Transformations 115

Chapter 6 ^ Segmentation 177

Chapter 7 ^ Spatial Transforms 215

Chapter 8 ^ Rendering and Surface Models 251

Chapter 9 ^ Registration 297

vii

Contents

List of Figures, xiii

List of Tables, xxi

Foreword, xxiii

Preface to the First Edition, xxv

Preface to the Second Edition, xxvii

User Guide, xxix

Acknowledgments, xxxi

Chapter 1 ^ A Few Basics of Medical Image Sources 1 Johann Hummel

1.1 RADIOLOGY 2
1.2 THE ELECTROMAGNETIC SPECTRUM 2
1.3 BASIC X-RAY PHYSICS 3
1.4 ATTENUATION AND IMAGING 7
1.5 COMPUTED TOMOGRAPHY 10
1.6 MAGNETIC RESONANCE TOMOGRAPHY 17
1.7 ULTRASOUND 28
1.8 NUCLEAR MEDICINE AND MOLECULAR IMAGING 33
1.9 OTHER IMAGING TECHNIQUES 36
1.10 RADIATION PROTECTION AND DOSIMETRY 39
1.11 SUMMARY AND FURTHER REFERENCES 43

Chapter 2 ^ Image Processing in Clinical Practice 45 Wolfgang Birkfellner

2.1 APPLICATION EXAMPLES 45
2.2 IMAGE DATABASES 45
2.3 INTENSITY OPERATIONS 46
2.4 FILTER OPERATIONS 47
2.5 SEGMENTATION 48
2.6 SPATIAL TRANSFORMS 49

ix

x  Contents

Wolfgang Birkfellner

xii  Contents

Özgür Güler and Ziv Yaniv

  • Chapter 10  CT Reconstruction
  • Chapter 11  A Tutorial on Image-Guided Therapy
  • Chapter 12  A Selection of MATLAB ©R Commands
  • Glossary,
  • List of MATLAB sample scripts,
  • Epilogue,
  • Index,
  • 2.7 RENDERING AND SURFACE MODELS
  • 2.8 REGISTRATION
  • 2.9 CT RECONSTRUCTION
  • 2.10 SUMMARY
  • Chapter 3  Image Representation
  • 3.1 PIXELS AND VOXELS Wolfgang Birkfellner
  • 3.2 GRAY SCALE AND COLOR REPRESENTATION
  • 3.3 IMAGE FILE FORMATS
  • 3.4 DICOM
  • 3.5 OTHER FORMATS – ANALYZE 7.5, NIFTI AND INTERFILE
  • 3.6 IMAGE QUALITY AND THE SIGNAL-TO-NOISE RATIO
  • 3.7 PRACTICAL LESSONS
  • 3.8 SUMMARY AND FURTHER REFERENCES
  • Chapter 4  Operations in Intensity Space
  • DYNAMIC RANGE 4.1 THE INTENSITY TRANSFORM FUNCTION AND THE
  • 4.2 WINDOWING
  • 4.3 HISTOGRAMS AND HISTOGRAM OPERATIONS
  • 4.4 DITHERING AND DEPTH
  • 4.5 PRACTICAL LESSONS
  • 4.6 SUMMARY AND FURTHER REFERENCES
  • Chapter 5  Filtering and Transformations
  • 5.1 THE FILTERING OPERATION Wolfgang Birkfellner
  • 5.2 THE FOURIER TRANSFORM
  • 5.3 OTHER TRANSFORMS
  • 5.4 PRACTICAL LESSONS
  • 5.5 SUMMARY AND FURTHER REFERENCES
  • Chapter 6  Segmentation
  • 6.1 THE SEGMENTATION PROBLEM Wolfgang Birkfellner
  • 6.2 ROI DEFINITION AND CENTROIDS
  • 6.3 THRESHOLDING
  • 6.4 REGION GROWING
  • 6.5 MORE SOPHISTICATED SEGMENTATION METHODS
  • 6.6 MORPHOLOGICAL OPERATIONS
  • 6.7 EVALUATION OF SEGMENTATION RESULTS Contents  xi
  • 6.8 PRACTICAL LESSONS
  • 6.9 SUMMARY AND FURTHER REFERENCES
  • Chapter 7  Spatial Transforms
  • 7.1 DISCRETIZATION – RESOLUTION AND ARTIFACTS Wolfgang Birkfellner
  • 7.2 INTERPOLATION AND VOLUME REGULARIZATION
  • 7.3 TRANSLATION AND ROTATION
  • 7.4 REFORMATTING
  • 7.5 TRACKING AND IMAGE-GUIDED THERAPY
  • 7.6 PRACTICAL LESSONS
  • 7.7 SUMMARY AND FURTHER REFERENCES
  • Chapter 8  Rendering and Surface Models
  • 8.1 VISUALIZATION Wolfgang Birkfellner
  • VIEWPOINT 8.2 ORTHOGONAL AND PERSPECTIVE PROJECTION, AND THE
  • 8.3 RAYCASTING
  • 8.4 SURFACE–BASED RENDERING
  • 8.5 PRACTICAL LESSONS
  • 8.6 SUMMARY AND FURTHER REFERENCES
  • Chapter 9  Registration
  • 9.1 FUSING INFORMATION Wolfgang Birkfellner
  • 9.2 REGISTRATION PARADIGMS
  • 9.3 MERIT FUNCTIONS
  • 9.4 OPTIMIZATION STRATEGIES
  • 9.5 SOME GENERAL COMMENTS
  • 9.6 CAMERA CALIBRATION
  • 9.7 REGISTRATION TO PHYSICAL SPACE
  • 9.8 EVALUATION OF REGISTRATION RESULTS
  • 9.9 PRACTICAL LESSONS
  • 9.10 SUMMARY AND FURTHER REFERENCES
  • Chapter 10  CT Reconstruction
  • 10.1 INTRODUCTION Michael Figl
  • 10.2 RADON TRANSFORM
  • 10.3 ALGEBRAIC RECONSTRUCTION
  • FILTERING 10.4 SOME REMARKS ON FOURIER TRANSFORM AND
  • 10.5 FILTERED BACKPROJECTION
  • 10.6 PRACTICAL LESSONS
  • 10.7 SUMMARY AND FURTHER REFERENCES
  • Chapter 11  A Tutorial on Image-Guided Therapy
  • IMAGE-GUIDED THERAPY 11.1 A HANDS-ON APPROACH TO CAMERA CALIBRATION AND
  • 11.2 TRANSFORMATIONS
  • 11.3 CAMERA CALIBRATION
  • 11.4 IMAGE-GUIDED THERAPY, INTRODUCTION
  • 11.5 IMAGE-GUIDED THERAPY, NAVIGATION SYSTEM
  • 11.6 IMAGE-GUIDED THERAPY, THEORY IN PRACTICE
  • 11.7 SUMMARY AND FURTHER REFERENCES
  • Chapter 12  A Selection of MATLAB ©R Commands
  • 12.1 CONTROL STRUCTURES AND OPERATORS
  • 12.2 I/O AND DATA STRUCTURES
  • 12.3 MATHEMATICAL FUNCTIONS
  • 12.4 FURTHER REFERENCES
  • Glossary,
  • List of MATLAB sample scripts,
  • Epilogue,
  • Index,
  • 1.1 A sketch of a typical x-ray spectrum. List of Figures
  • 1.2 The principle of an x-ray tube.
  • 1.3 An image of an x-ray tube.
  • 1.4 An illustration of the Compton- and photoeffect as well as pair production.
  • 1.5 The principle of computed tomography.
  • 1.6 A simple sketch of a sinogram.
  • 1.7 A photograph of the first commercially available CT.
  • 1.8 The principle of the four CT-scanner generations.
  • 1.9 A multislice CT scanner.
  • 1.10 An illustration of the amount of image data produced by multislice CT.
  • 1.11 A linear accelerator for radiation oncology with an imaging unit attached.
  • 1.12 A cone beam CT for dental radiology.
  • 1.13 Metal artifacts in CT.
  • 1.14 A sketch of precession motion.
  • 1.15 An interventional MR machine.
  • 1.16 Differences in T 1 and T 2 MR images.
  • 1.17 Effects of insufficient field homogeneity in MR.
  • 1.18 Effects of dental fillings in MR.
  • 1.19 A modern 7 Tesla MR machine.
  • 1.20 Two B-mode ultrasound transducer.
  • 1.21 Speckle in ultrasound images.
  • 1.22 A 3D ultrasound transducer.
  • 1.23 The principle of a γ-camera.
  • 1.24 A ring artifact in SPECT-reconstruction.
  • 1.25 A dual-head gamma camera.
  • 1.26 A modern PET-CT machine.
  • 1.27 An OCT image of basalioma.
  • 1.28 Effects of ionizing radiation on a video surveillance camera.
  • 2.1 Screenshot of a DICOM database patient selection dialog.
  • 2.2 Various views of a CT image with different intensity windowing.
  • 2.3 Sobel filtering applied to planning CT data.
  • 2.4 An application of intensity thresholding.
  • 2.5 An application of atlas based segmentation. xiv  List of Figures
  • 2.6 Interpolation of image resolution in a CT image.
  • 2.7 An example of reformatting.
  • 2.8 DRR - rendering for beam verification in radiotherapy.
  • therapy. 2.9 Surface rendering of organs-at-risk and planned target volumes in radio-
  • 2.10 3D visualization of reformatted slices and surface renderings.
  • 2.11 Rigid registration of CT and CBCT volume data.
  • 2.12 Deformable registration of CT and CBCT volume data.
  • 2.13 Registration of 3D CT and 2D x-ray data.
  • 3.1 A CT slice showing the pixel structure of images.
  • 3.2 A CT slice and its representation as a surface.
  • 3.3 An illustration of physical vs. isotropic voxel spacing in a CT volume.
  • 3.4 An early example of color photography.
  • 3.5 A CT slice, saved in PGM format.
  • 3.6 JPG image representation.
  • 3.7 Header of a DICOM file.
  • 3.8 Another DICOM header.
  • 3.9 An Analyze 7.5 header, viewed in a hex-editor.
  • 3.10 Image data used for Example 3.7.1.
  • 3.11 A DSA image, the result of Example 3.7.1.
  • 3.12 A file properties dialog showing the size of an image file.
  • 3.13 A MATLAB screenshot.
  • 3.14 The output from Example 3.7.2.
  • 3.15 The output from Example 3.7.2 saved as PGM.
  • 3.16 The "Open File" Dialog of ImageJ.
  • 3.17 A DICOM file as displayed by ImageJ.
  • 3.18 A screenshot of 3DSlicer.
  • 3.19 Importing patient data into the 3DSlicer database.
  • 3.20 A screenshot of images from a DICOM volume imported to 3DSlicer.
  • 3.21 Output from Example 3.7.4.
  • 3.22 Eight x-ray images acquired with different dose.
  • 3.23 Selection of a signal free area in an x-ray image for SNR evaluation.
  • 4.1 Effects of different gray scale representations.
  • 4.2 A response curve.
  • 4.3 Sigmoid curves as a model for a contrast transfer function.
  • 4.4 A clinical example illustrating the importance of windowing.
  • 4.5 A screenshot of the effects of windowing in 3DSlicer.
  • 4.6 The shape of an intensity transfer function for windowing.
  • 4.7 Effects of windowing a 12 bit image.
  • 4.8 Sample intensity histograms. List of Figures  xv
  • 4.9 The output from LinearIntensityTransform_4.m.
  • 4.10 Three analog photographs taken with different color filters.
  • 4.11 The analog B&W photographs from Example 4.5.2 after inversion.
  • 4.12 The result of Example 4.5.2, a color composite from B&W photographs.
  • using GIMP. 4.13 A screenshot of intensity transfer function manipulation on color images
  • 4.14 The output from LogExample_4.m.
  • 4.15 Non-linear intensity transforms using a Sigmoid-function.
  • 4.16 The output from Example 4.16.
  • 4.17 A screenshot from ImageJ, showing its contrast enhancement functionality.
  • 4.18 Contrast enhancement in ImageJ.
  • 4.19 A well-behaved image from Example 4.16.
  • 4.20 An image with a cluttered histogram from Example 4.16.
  • 4.21 Linear intensity transforms in ImageJ.
  • 4.22 Intensity windowing in ImageJ.
  • ing system. 5.1 The spherical aberration, an example for aberrations in a suboptimal imag-
  • 5.2 The spot diagram of a spherical mirror.
  • 5.3 Effects of smoothing an image in the spatial domain.
  • 5.4 An example of the sharpening operation.
  • 5.5 An example of apparent changes in brightness due to convolution.
  • 5.6 Numerical differentiation examples of simple functions.
  • 5.7 The effect of simple differentiating kernels.
  • 5.8 An image after differentiation.
  • 5.9 Illustration of four- and eight-connectedness.
  • 5.10 Effects of median filtering.
  • 5.11 The effects of anisotropic diffusion filtering.
  • 5.12 Representation of a vector in Cartesian coordinates.
  • a given vector. 5.13 An illustration on how a change in unit vectors affects the components of
  • 5.14 Characteristic features of a plane wave.
  • 5.15 Representation of numbers in the complex plane.
  • 5.16 The principle of the Fourier transform of images with finite size.
  • 5.17 A wraparound artifact in MRI.
  • 5.18 Motion artifacts in MRI.
  • 5.19 The power spectrum of an image.
  • 5.20 M13, the globular cluster in Hercules.
  • 5.21 The modulation transfer function of a spherical mirror.
  • 5.22 Principle of the Hesse normal form.
  • 5.23 A simple and classical example for the distance transform.
  • 5.24 Output from Example 5.4.1, a lowpass-filtering operation on an image. xvi  List of Figures
  • 5.25 Screenshot of ImageJ for basic filtering operations.
  • 5.26 Basic filtering operations in ImageJ.
  • 5.27 An image after Sobel filtering.
  • 5.28 Output from Example 5.4.5 showing the effects of median filtering.
  • 5.29 A simple example of frequency filtering.
  • 5.30 The spectra of a C major chord played on different instruments.
  • 5.31 Frequency filtering effects on a rectangle.
  • 5.32 A simple 2D image with defined frequencies.
  • 5.33 The result of directional filtering on a 2D image.
  • 5.34 An illustrative example for the convolution operation.
  • 5.35 Numerical differentiation in k-space.
  • 5.36 Frequency filtering on an image.
  • 5.37 High-pass filtering of an image.
  • 5.38 A typical PSF derived from a MTF.
  • 5.39 A sharp Gaussian PSF and the associated MTF.
  • 5.40 Effects of convolution in k-space.
  • 5.41 Images of a point source from a γ camera at various resolutions.
  • 5.42 The MTF of an Anger camera with a resolution of 64 x 64 pixels.
  • 5.43 The MTF of an Anger camera with a resolution of 1024 x 1024 pixels.
  • 5.44 A binary sample image used for a Hough-transform.
  • 5.45 An illustration of the basic principles of the Hough transform.
  • 5.46 A representation in Hough-space.
  • 5.47 An image after Hough-transform.
  • 5.48 A second sample image for the Hough transform.
  • 5.4.10. 5.49 The result of applying a distance transformation to the outcome of Example
  • 6.1 A segmentation example on a CT slice of the human heart.
  • 6.2 Examples of thresholding as a segmentation method.
  • 6.3 Segmentation using region growing – an example using GIMP.
  • 6.4 Region growing applied to a CT-scan of the heart.
  • 6.5 Two CT slices of the aorta in different orientations.
  • 6.6 An example of a segmentation algorithm from ITKSnap.
  • 6.7 Progress of a level-set segmentation algorithm.
  • 6.8 Final segmentation results from ITKSnap.
  • 6.9 A simple example of erosion and dilation.
  • 6.10 An example for the Hausdorff distance applied to simple graphs.
  • 6.11 An illustration of the asymmetric nature of the Hausdorff distance.
  • 6.12 ROI example images from a scintigraphic camera.
  • 6.13 A sample image from Example 6.8.3, a simple region growing algorithm.
  • 6.14 A medical example for segmentation using region growing. List of Figures  xvii
  • 6.15 Another medical example for segmentation using region growing.
  • 6.16 A binary phantom for 3D-segmentation.
  • 6.17 The result of a simple active contours example.
  • 6.18 Effects of improving the active contours example from Example 6.8.5.
  • gorithm in Example 6.8.5. 6.19 The effect of a further refinement of the active contours segmentation al-
  • 6.20 An illustration for an active contour segmentation example.
  • 6.21 The result from Example 6.8.6, an implementation of erosion.
  • 6.22 An x-ray of a pelvis during brachytherapy.
  • 6.23 Various segmentation efforts on an x-ray image using simple thresholding.
  • 6.24 Several preprocessing steps prior to segmentation in Example 6.8.8.1.
  • 6.25 The output from Example 6.8.8.1.
  • 6.26 Segmentation of structures in Example 6.8.8.1 using region growing.
  • 6.27 Preprocessing steps for Example 6.8.8.2.
  • 6.28 The enhanced CT-slice for segmentation from Example 6.8.8.2.
  • 6.29 The result of Example 6.8.8.2.
  • 6.30 A manual segmentation effort on an MR slice.
  • 7.1 An interpolation example on a PET image.
  • 7.2 A grid for bilinear intensity interpolation.
  • 7.3 A grid for trilinear intensity interpolation.
  • 7.4 Various types of interpolation.
  • 7.5 An illustration of non-commutative spatial transforms.
  • 7.6 The principle of the PCA transform.
  • 7.7 An example of curved reformatting from dentistry.
  • 7.8 An optical tracking system.
  • 7.9 An electromagnetic tracking system.
  • 7.10 A passive tool for an optical tracking system.
  • 7.11 An image-guided surgery system.
  • 7.12 A result from Example 7.6.1 – the rotation of 2D images.
  • 7.13 Effects of changing the center of rotation in Example 7.6.1.
  • 7.14 The result of NNInterpolation_7.m.
  • 7.15 The output of the BiLinearInterpolation_7.m script.
  • 7.16 A demonstration of the PCA on binary images.
  • 7.17 The result of Example 7.6.3, the principal axes transform.
  • 7.18 A few results of reformatting a 3D cone.
  • in Example 7.6.5. 7.19 Three sample slices in original axial orientation from the CT-dataset used
  • 7.20 A set of orthogonally reformatted slices from the pigBig.img CT dataset.
  • Big_7.m. 7.21 An oblique reformatting of the volume generated by ThreeDConvolution-
  • 8.1 The principle of raycasting without perspective. xviii  List of Figures
  • 8.2 The principle of raycasting with perspective.
  • 8.3 An example of maximum intensity projection.
  • 8.4 An example of summed voxel rendering.
  • 8.5 An example of volume compositing.
  • 8.6 A transfer function for volume compositing.
  • 8.7 An example of depth shading.
  • 8.8 An example of surface shading.
  • 8.9 The principle of computing gradients for surface shading.
  • 8.10 A comparison of rendered voxel surfaces and triangulated models.
  • 8.11 A screenshot from a software for simulating long bone fractures.
  • 8.12 The basic setup of the marching squares algorithm.
  • 8.13 Two out of 15 possible configurations of the marching cubes algorithm.
  • 8.14 The principle of Gouraud-shading.
  • 8.15 A virtual endoscopy of the spinal canal.
  • 8.16 The effects of perspective projection on a simple geometric shape.
  • 8.17 Output from Example 8.5.2, a simple raycasting algorithm.
  • 8.18 Output from Example 8.5.2.
  • 8.19 A summed voxel rendering from Example 8.5.2.
  • 8.20 Output from Example 8.5.3, a perspective splat rendering.
  • 8.21 A perspective DRR.
  • 8.22 A color-coded volume rendering.
  • 8.23 A depth shading generated by a splat rendering algorithm.
  • 8.24 A depth shading following the inverse square law.
  • 8.25 A surface plot of an intermediate result from Example 8.5.6.
  • 8.26 The result from Example 8.5.6, a surface rendering algorithm.
  • 8.27 More results from Example 8.5.6.
  • 8.28 Screenshot for downloading sample data for 3DSlicer.
  • 8.29 Menu settings for rendering using 3DSlicer.
  • 8.30 A volume rendering using 3DSlicer.
  • 8.31 Another example of volume rendering using 3DSlicer.
  • 8.32 A result from the Triangulation_8.m example, rendered in Paraview.
  • 8.33 A better surface model, also rendered in Paraview.
  • 8.34 Modified depth shadings for improved surface rendering in Example 8.5.9.
  • 8.35 Surface rendering with additional shading effects.
  • 9.1 A sample image from PET-CT.
  • 9.2 An example of unregistered CT and MR slices.
  • 9.3 A screenshot of the registration tool in AnalyzeAVW.
  • tration tool. 9.4 The volume transformation matrix as derived by the AnalyzeAVW regis-
  • 9.5 The principle of comparing gray values in registration merit functions. List of Figures  xix
  • 9.6 An intramodal example of joint histograms.
  • 9.7 Histograms for two random variables.
  • 9.8 A surface plot of joint histograms.
  • 9.9 An intermodal example of joint histograms.
  • 9.10 A second intermodal example of joint histograms.
  • 9.11 A plot of Shannon’s entropy from a registration example.
  • 9.12 A plot of the mutual information merit function.
  • 9.13 An example of PET-MR registration.
  • 9.14 An edge representation of co-registered slices.
  • 9.15 The distance transform of an MR image from Example 9.9.4.
  • 9.16 The resulting merit function plots from Example 9.9.4.
  • 9.17 A simple illustration of the Nelder-Mead algorithm.
  • 9.18 An illustration of the camera calibration problem.
  • using fiducial markers. 9.19 The Vogele-Bale-Hohner mouthpiece, a device for non-invasive registration
  • 9.20 Three images of a spine reference-dataset for 2D/3D image registration.
  • function during image rotation. 9.21 The output of Example 9.9.1, a computation of the cross-correlation merit
  • 9.22 Two sample outputs from modified versions of More2DRegistration_9.m.
  • 9.23 Three sample images used in the practical lessons.
  • 9.24 Two simple functions to be optimized in Example 9.9.5.
  • 9.25 The result of a local optimization in dependence of the starting value.
  • 9.26 The result of a local optimization using a different implementation.
  • an inverse rect-function. 9.27 The result of a local optimization in dependence of the starting value for
  • 10.1 Principle of a parallel beam CT.
  • 10.2 Normal form of a line.
  • 10.3 Simple Radon transform examples.
  • 10.4 Image discretization.
  • 10.5 Rotation of an image after discretization.
  • 10.6 Radon transform of the Shepp and Logan phantom.
  • 10.7 Rotation by nearest neighbor interpolation.
  • 10.8 Shape of a system matrix.
  • 10.9 Orthogonal projection to a subspace.
  • 10.10 Kaczmarz algorithm.
  • 10.11 Minimal distance of a point to a plane.
  • 10.12 Algebraic reconstructed Shepp and Logan phantom.
  • 10.13 Filtering using the Fourier transform.
  • 10.14 Projection slice theorem.
  • 10.15 Backprojection.