Wavelet-based Denoising of Cardiac PET geogreen/ Wavelet-based Denoising of

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  • Wavelet-based Denoising

    of

    Cardiac PET Data

    by

    Geoffrey C. Green

    A thesis submitted to

    The Faculty of Graduate Studies and Research

    In partial fulfillment of the requirements for the degree of

    Master of Applied Science

    in Electrical Engineering

    Ottawa-Carleton Institute for Electrical and Computer Engineering

    Department of Systems and Computer Engineering

    Carleton University

    Ottawa, Ontario, Canada

    January 2005

    c©Geoffrey C. Green, 2005

  • The undersigned hereby recommend to

    the Faculty of Graduate Studies and Research

    acceptance of the thesis

    Wavelet-based Denoising

    of

    Cardiac PET Data

    Submitted by

    Geoffrey C. Green,

    in partial fulfillment of the requirements

    for the degree of Master of Applied Science

    –––––––––––––––––––

    Dr. Aysegul Cuhadar, Thesis Supervisor

    –––––––––––––––––––

    Dr. Robert deKemp, Thesis Supervisor

    –––––––––––––––––––

    Chair, Department of Systems and Computer Engineering

    Carleton University

    January 2005

    ii

  • To my late father, who taught me the importance of education.

    To my mother, whose strength and positive attitude toward life inspires me.

    And to Katherine and Timothy, my new family.

    iii

  • Acknowledgments

    I would like to thank my thesis supervisor, Dr. Aysegul Cuhadar, for giving me the

    chance to explore a research career in biomedical engineering. Her continual support,

    technical insight, and dedication to this research has made this opportunity most

    enjoyable.

    I would also like to thank Dr. Rob deKemp at the Ottawa Heart Institute for

    being so accessible and helpful, for the guidance he provided during this research

    project, and for taking the time to explain the practical and clinical aspects of PET

    imaging.

    iv

  • Abstract

    This thesis focuses on denoising of positron emission tomography (PET) data.

    Cardiac PET scans generated using a rubidium-82 radiotracer are a convenient, non–

    invasive method of diagnosing heart disease, but suffer from a high degree of noise.

    Denoising methods based on the wavelet transform are capable of outperforming

    existing clinical methods due to their ability to better preserve detail while simul-

    taneously suppressing noise at multiple scales. We investigate the applicability of

    recently developed wavelet denoising methods to cardiac PET data. A comprehen-

    sive set of experiments is performed, in which combinations of these techniques are

    applied to the different decomposition levels of wavelet coefficients. By doing so, we

    determine the relevant importance of each (and the domain in which it is applied)

    to the overall quality of the denoised result. With this information, we propose PET

    denoising protocols that substantially improve image quality (for static studies) and

    lead to better measures of myocardial perfusion (for dynamic studies).

    v

  • Contents

    1 Introduction 1

    1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.3 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2 Background — PET 7

    2.1 Physics of Positron Emission . . . . . . . . . . . . . . . . . . . . . . . 7

    2.2 Description of PET Scanner and Counts . . . . . . . . . . . . . . . . 9

    2.3 PET Measurements and Sinograms . . . . . . . . . . . . . . . . . . . 11

    2.4 Image Reconstruction from Projections . . . . . . . . . . . . . . . . . 13

    2.4.1 Filtered Backprojection (FBP) . . . . . . . . . . . . . . . . . . 17

    2.4.2 Iterative Reconstruction . . . . . . . . . . . . . . . . . . . . . 18

    2.5 Sources of Noise in PET Images . . . . . . . . . . . . . . . . . . . . . 20

    3 Background — PET in Cardiology 22

    3.1 Cardiac Anatomy and Function . . . . . . . . . . . . . . . . . . . . . 22

    3.2 Representations of the Heart used in PET . . . . . . . . . . . . . . . 24

    3.3 Quantitative Dynamic PET Studies . . . . . . . . . . . . . . . . . . . 27

    3.4 Tracer Kinetics and Compartmental Modeling . . . . . . . . . . . . . 30

    vi

  • 4 Image Denoising 34

    4.1 Classical Techniques for Image Denoising . . . . . . . . . . . . . . . . 34

    4.1.1 Noise Model and Properties . . . . . . . . . . . . . . . . . . . 35

    4.1.2 Low Pass Filtering . . . . . . . . . . . . . . . . . . . . . . . . 36

    4.1.3 Order Statistic Filters . . . . . . . . . . . . . . . . . . . . . . 39

    4.1.4 Wiener Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    4.2 Wavelet-based Image Denoising . . . . . . . . . . . . . . . . . . . . . 41

    4.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    4.2.2 Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . 43

    4.3 Wavelet-based Noise Removal . . . . . . . . . . . . . . . . . . . . . . 44

    4.4 Discrete Dyadic Wavelet Transform . . . . . . . . . . . . . . . . . . . 50

    4.4.1 One-dimensional Discrete Dyadic Wavelet Transform (DDWT) 52

    4.4.2 Multi-dimensional Discrete Dyadic Wavelet Transform . . . . 54

    5 Wavelet–based Denoising of Cardiac PET Data 58

    5.1 Description of Input Data . . . . . . . . . . . . . . . . . . . . . . . . 59

    5.1.1 Phantom PET Data . . . . . . . . . . . . . . . . . . . . . . . 59

    5.1.2 Clinical PET Data . . . . . . . . . . . . . . . . . . . . . . . . 61

    5.2 Noise Properties of PET Data . . . . . . . . . . . . . . . . . . . . . . 62

    5.2.1 Noise Level Estimation . . . . . . . . . . . . . . . . . . . . . . 65

    5.2.2 Noise Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 66

    5.3 Implementation Details for Discrete Dyadic Wavelet Transform . . . . 67

    5.3.1 One–dimensional DDWT . . . . . . . . . . . . . . . . . . . . . 67

    5.3.2 D–dimensional DDWT . . . . . . . . . . . . . . . . . . . . . . 69

    5.4 Description of Denoising Protocols . . . . . . . . . . . . . . . . . . . 73

    5.4.1 Development Environment . . . . . . . . . . . . . . . . . . . . 73

    5.4.2 General Denoising Procedure and Notation Used . . . . . . . . 73

    5.4.3 Details of Denoising Protocols . . . . . . . . . . . . . . . . . . 75

    vii

  • 5.4.4 Signal Extension at Boundaries . . . . . . . . . . . . . . . . . 80

    5.5 Figures of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    5.5.1 Phantom Data . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    5.5.2 Clinical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    5.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    5.6.1 Phantom Data . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    5.6.2 Clinical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

    5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

    6 Summary and Future Work 109

    6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

    6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    A Background — Wavelet Theory 122

    A.1 Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . . . 122

    A.2 Discrete Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . 124

    A.2.1 Signal Expansions as a Series . . . . . . . . . . . . . . . . . . 124

    A.2.2 Multiresolution Theory . . . . . . . . . . . . . . . . . . . . . . 126

    A.2.3 Representation of Signals in terms of MRA Functions . . . . . 131

    A.2.4 Fast Wavelet Transform (FWT) . . . . . . . . . . . . . . . . . 132

    viii

  • List of Tables

    3.1 Frame times and durations for dynamic 82Rb PET study . . . . . . . 28

    5.1 Description of Clinical PET Data . . . . . . . . . . . . . . . . . . . . 61

    5.2 Filter coefficients for implementation of spline-based DDWT . . . . . 71

    5.3 Denoising protocols used in this thesis . . . . . . . . . . . . . . . . . 76

    5.4 Classification of denoising protocols based on level of smoothing . . . 101

    5.5 Comparison of quantitative model outputs . . . . . . . . . . . . . . . 107

    ix

  • List of Figures

    1.1 Noisy 82Rb PET scan of myocardium . . . . . . . . . . . . . . . . . . 2

    2.1 Positron emission and annilihation . . . . . . . . . . . . . . . . . . . 8

    2.2 ECAT ART PET scanner with coordinate system . . . . . . . . . . . 10

    2.3 Types of PET counts . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    2.4 Line of response and sinogram . . . . . . . . . . . . . . . . . . . . . . 12

    2.5 Radiotracer distribution and “thin-strip” approximation . . . . . . . 14

    2.6 Illustration of the Fourier Slice Theorem . . . . . . . . . . . . . . . . 15

    2.7 Noisy FBP and OSEM reconstructions . . . . . . . . . . . . . . . . . 19

    3.1 Diagram of heart and left ventricle . . . . . . . . . . . . . . . . . . . 24

    3.2