Optical Measurement of Wide Bandwidth Ultrasound Fields by Todd Alan Pitts A thesis submitted in ...

180 Pages·2000·3.4 MB·English

Optical Measurement of Wide Bandwidth Ultrasound Fields
by
Todd Alan Pitts
A thesis submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Biomedical Sciences
(cid:5)
Biomedical Imaging
in the
MAYO GRADUATE SCHOOL
Examining committee:
Dr. James F. Greenleaf, Chair
Dr. Erik L. Ritman
Dr. Hal Ottesen
Dr. Armando Manduca
Dr. James Leger
Dr. Ahmed Tew(cid:12)k
1998
Optical Measurement of Wide Bandwidth Ultrasound Fields
Copyright 1998
by
Todd Alan Pitts
Acknowledgements
I am very grateful to my advisor, Dr. James F. Greenleaf, for his support and insight. I greatly
appreciate the assistance of Randall R. Kinnick and Tom Kinter in the experimental and computer
administrativeaspectsoftheresearch. Tothefellowstudentswhoendurednumerousconversationson
topics relatedto this research,I o(cid:11)er my thanks as well. I amparticularly gratefulfor the unflagging
patience and encouragement of my wife Emily and our four children through the many years of
graduate school.
Optical Measurement of Wide Bandwidth Ultrasound Fields iii
To my dear wife Emily,
and our children:
Matthew, Christopher, Erin, and Emily
Optical Measurement of Wide Bandwidth Ultrasound Fields iv
Contents
1 INTRODUCTION 3
1.1 Optical Measurement of Acoustic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Signal Phase Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 PHYSICAL MODEL 10
2.1 Experiment Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Optical Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Scalar Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.4 Modal Coupling via Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.5 Fresnel Model for Scalar Di(cid:11)raction . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Refractive Index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 The Acousto-Optic E(cid:11)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 The Piezooptic Coe(cid:14)cient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Optical Scattering Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Temporal and Spatial Bandwidth Relationships . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Acoustic PropagationModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Propagationof Spatial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7.1 Single Measurement in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7.2 Measurement at Multiple Distinct Time Points . . . . . . . . . . . . . . . . . . 27
2.8 Propagationof Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.10 Calculation of Particle Velocity From Pressure Field . . . . . . . . . . . . . . . . . . . 29
3 SIGNAL PROCESSING 31
3.1 Video Signal Digitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 BackgroundCorrection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Basic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Ring or Ridge Artifact Suppression . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Signal Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Signal Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 Phase Retrieval via Generalized Projections . . . . . . . . . . . . . . . . . . . . 42
3.4 The Fourier Slice Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 NUMERICAL SIMULATION 50
Optical Measurement of Wide Bandwidth Ultrasound Fields v
CONTENTS
5 CALIBRATION AND PREPROCESSING 60
5.1 System Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1.1 CCD Camera Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1.2 Digitization of Optical Intensity Data . . . . . . . . . . . . . . . . . . . . . . . 60
5.1.3 Imaging Plane Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Measurement Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 Superposition of Refractive Index Perturbations . . . . . . . . . . . . . . . . . 64
5.2.2 Amplitude Scaling of Refractive Index Perturbation . . . . . . . . . . . . . . . 64
5.3 BackgroundCorrection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.1 Simple Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.2 Tomographic Reconstruction: Circular Aperture . . . . . . . . . . . . . . . . . 66
6 TOMOGRAPHIC RECONSTRUCTION 77
6.1 Annular Aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Semi-Circular Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 Wide Bandwidth Double Slit: Narrow Spacing . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Wide Bandwidth Double Slit: Wide Spacing . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5 Tissue Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7 HYDROPHONE COMPARISON 102
8 PROPAGATION, VELOCITY, AND ENERGY FLOW 110
8.1 Complicated Field Pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.2 Wide Bandwidth Double Slit: Narrow Spacing . . . . . . . . . . . . . . . . . . . . . . 112
8.3 Wide Bandwidth Double Slit: Wide Spacing . . . . . . . . . . . . . . . . . . . . . . . . 114
8.4 Semi-Circular Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9 SUMMARY 123
A EXPERIMENTAL DATA 127
B PROPAGATION OF THE RADON TRANSFORM 143
C MEASUREMENT SYSTEM MODEL 145
D FOURIER OPTICS OPERATOR NOTATION 149
D.1 De(cid:12)nition of Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
D.2 Operator Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Optical Measurement of Wide Bandwidth Ultrasound Fields vi
List of Figures
2.1 Experiment Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Weak Scattering Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 De(cid:12)nition of Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Measurement Data Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Optical Imaging Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Image Background Removal and Normalization Flow Diagram. . . . . . . . . . . . . . 34
3.4 Fourier Space Location of BackgroundCorruption . . . . . . . . . . . . . . . . . . . . 35
3.5 Filtered and Un(cid:12)ltered Intensity Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 Experiment Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7 Examples of Convex and Non-Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.8 Convergence Traps and Tunnels in the Method of Generalized Projections . . . . . . . 46
3.9 Fourier Slice Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.10 Nullspace of the Sampling Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Rendered Experiment Scene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Simulation: Optical Phase Phantom and Intensity Measurement Comparison . . . . . 53
4.3 Simulation: Optical Phase Phantom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Simulation: Optical Intensity Measurements . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Simulation: Optical Phase Retrieval Versus Iteration Count . . . . . . . . . . . . . . . 56
4.6 Simulation: Optical Phase Retrieval and Summed Distance Error . . . . . . . . . . . . 57
4.7 Simulation: Radon Transform From Phase Retrieved Data . . . . . . . . . . . . . . . . 58
4.8 Simulation: Reconstruction of Three-Dimensional Optical Phase Phantom . . . . . . . 59
5.1 Charge Coupled Device (CCD) Array Integration Linearity . . . . . . . . . . . . . . . 61
5.2 Charge Coupled Device (CCD) Array Dark(cid:12)eld Image . . . . . . . . . . . . . . . . . . 61
5.3 Least Squares Fit of Optical Imaging System Parameters . . . . . . . . . . . . . . . . 63
5.4 Measurement Linearity: Superposition of Refractive Index Perturbations . . . . . . . . 65
5.5 Measurement Linearity: Amplitude Scaling of Refractive Index Perturbation . . . . . 67
5.6 Measurement Linearity: Amplitude Scaling of Refractive Index Perturbation . . . . . 68
5.7 BackgroundCorrection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.8 Simple Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.9 E(cid:11)ect of Wavelet Filter Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.10 Various Di(cid:11)raction Plane Views of Optical Intensity . . . . . . . . . . . . . . . . . . . 73
5.11 Tomographic Views of Retrieved Phase: Circular Aperture. . . . . . . . . . . . . . . . 74
5.12 Tomographic Reconstruction: Circular Aperture . . . . . . . . . . . . . . . . . . . . . 75
5.13 Tomographic Reconstruction: Un(cid:12)ltered Circular Aperture . . . . . . . . . . . . . . . 76
Optical Measurement of Wide Bandwidth Ultrasound Fields vii
LIST OF FIGURES
6.1 Annular Aperture: Di(cid:11)raction Plane Optical Intensity . . . . . . . . . . . . . . . . . . 78
6.2 Annular Aperture: Tomographic Phase Retrieved Views . . . . . . . . . . . . . . . . . 79
6.3 Annular Aperture: Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.4 Annular Aperture: Tomographic Reconstruction . . . . . . . . . . . . . . . . . . . . . 81
6.5 Annular Aperture: Tomographic Reconstruction . . . . . . . . . . . . . . . . . . . . . 82
6.6 Annular Aperture: Reconstructed Surface Plot . . . . . . . . . . . . . . . . . . . . . . 83
6.7 Annular Aperture: Transverse Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.8 Semi-Circular Aperture: Tomographic Reconstruction . . . . . . . . . . . . . . . . . . 86
6.9 Semi-Circular Aperture: Tomographic Reconstruction . . . . . . . . . . . . . . . . . . 87
6.10 Semi-Circular Aperture: Transverse Sections . . . . . . . . . . . . . . . . . . . . . . . 88
6.11 Double Slit Aperture (Narrow): Tomographic Reconstruction . . . . . . . . . . . . . . 90
6.12 Double Slit Aperture (Narrow): Tomographic Reconstruction . . . . . . . . . . . . . . 91
6.13 Double Slit Aperture (Narrow): Transverse Sections . . . . . . . . . . . . . . . . . . . 92
6.14 Double Slit Aperture (Wide): Tomographic Reconstruction . . . . . . . . . . . . . . . 94
6.15 Double Slit Aperture (Wide): Tomographic Reconstruction . . . . . . . . . . . . . . . 95
6.16 Double Slit Aperture (Wide): Transverse Sections . . . . . . . . . . . . . . . . . . . . 96
6.17 Circular Aperture: Tomographic Reconstruction . . . . . . . . . . . . . . . . . . . . . 98
6.18 Aberrated Aperture: TomographicReconstruction . . . . . . . . . . . . . . . . . . . . 99
6.19 Circular Aperture: Tomographic Reconstruction . . . . . . . . . . . . . . . . . . . . . 100
6.20 Aberrated Aperture: TomographicReconstruction . . . . . . . . . . . . . . . . . . . . 101
7.1 Hydrophone: Pressure Time Series, Elements 30 and 34 of Linear Array . . . . . . . . 104
7.2 Hydrophone: Pressure Time Series, Elements 31 and 34 of Linear Array . . . . . . . . 105
7.3 Hydrophone: Pressure Time Series, Elements 32 and 34 of Linear Array . . . . . . . . 106
7.4 Hydrophone Versus Optical Measurement: Reference Images . . . . . . . . . . . . . . 107
7.5 Hydrophone Versus Optical Measurement: Line Plot Comparisons . . . . . . . . . . . 108
7.6 Hydrophone Versus Optical Measurement: Line Plot Comparisons . . . . . . . . . . . 109
8.1 PropagationTime Sequence: Original Projection of Complicated Field . . . . . . . . . 111
8.2 PropagationTime Sequence: Complicated Field . . . . . . . . . . . . . . . . . . . . . . 112
8.3 PropagationTime Sequence: Original Projection of Double Slit Field (Narrow) . . . . 113
8.4 PropagationTime Sequence: Double Slit Field (Narrow) . . . . . . . . . . . . . . . . . 114
8.5 Particle Velocity and Power: Original Projection of Double Slit Field (Wide) . . . . . 115
8.6 PropagationTime Sequence: Double Slit Field (Wide) . . . . . . . . . . . . . . . . . . 116
8.7 Poynting Vector: Overlay on Projection of Double Slit Field (Wide) . . . . . . . . . . 117
8.8 Particle Velocity: Overlay on Projection of Double Slit Field (Wide) . . . . . . . . . . 117
8.9 Poynting Vector: Overlay on Projection of Double Slit Field (Wide) . . . . . . . . . . 118
8.10 Particle Velocity: Overlay on Projection of Double Slit Field (Wide) . . . . . . . . . . 118
8.11 Poynting Vector: Overlay on Projection of Double Slit Field (Wide) . . . . . . . . . . 119
8.12 Particle Velocity: Overlay on Projection of Double Slit Field (Wide) . . . . . . . . . . 119
8.13 PropagationTime Sequence: Original Projection of Semi-Circular Field . . . . . . . . 120
8.14 PropagationTime Sequence: Semi-Circular Field . . . . . . . . . . . . . . . . . . . . . 121
8.15 Poynting Vector: Overlay on Projection of Semi-Circular Field . . . . . . . . . . . . . 122
8.16 Particle Velocity: Overlay on Projection of Semi-Circular Field . . . . . . . . . . . . . 122
A.1 Experiment Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.2 D-Shaped Aperture: Optical Intensity and Recovered Phase . . . . . . . . . . . . . . . 128
A.3 D-Shaped Aperture: Tomographic Phase Retrieved Views . . . . . . . . . . . . . . . . 129
A.4 D-Shaped Aperture: Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.5 Double Slit Aperture (Narrow): Di(cid:11)raction Plane Optical Intensity . . . . . . . . . . . 131
A.6 Double Slit Aperture (Narrow): Tomographic Phase Retrieved Views . . . . . . . . . . 132
A.7 Double Slit Aperture (Narrow): Radon Transform . . . . . . . . . . . . . . . . . . . . 133
A.8 Double Slit Aperture (Wide): Di(cid:11)raction Plane Optical Intensity . . . . . . . . . . . . 134
A.9 Double Slit Aperture (Wide): Tomographic Phase Retrieved Views . . . . . . . . . . . 135
A.10Double Slit Aperture (Wide): Radon Transform. . . . . . . . . . . . . . . . . . . . . . 136
Optical Measurement of Wide Bandwidth Ultrasound Fields viii
LIST OF FIGURES
A.11Circular Aperture: Di(cid:11)raction Plane Optical Intensity . . . . . . . . . . . . . . . . . . 137
A.12Aberrated Aperture: Di(cid:11)raction Plane Optical Intensity . . . . . . . . . . . . . . . . . 138
A.13Circular Aperture: Tomographic Phase Retrieved Views . . . . . . . . . . . . . . . . . 139
A.14Aberrated Aperture: Tomographic Phase Retrieved Views . . . . . . . . . . . . . . . . 140
A.15Circular Aperture: Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.16Aberrated Aperture: Radon Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C.1 Ideal Two Lens Imaging System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Optical Measurement of Wide Bandwidth Ultrasound Fields ix
List of Tables
5.1 Video Signal Analog to Digital ConversionData . . . . . . . . . . . . . . . . . . . . . 62
5.2 Data for Parametric Least Squares Fit of Optical System . . . . . . . . . . . . . . . . 64
Optical Measurement of Wide Bandwidth Ultrasound Fields x