The Radon transform and medical imaging:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia, Pa.
Society for Industrial & Applied Mathematics
2014
|
| Series: | CBMS-NSF Regional Conference series in applied mathematics
85 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XVI, 240 S. Ill. |
| ISBN: | 9781611973280 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
List of Figures
xiii
Preface
xv
I Introduction
1
1
Why Use Mathematics in Medical Imaging?
3
1.1
Types of
medicai
imaging
........................ 3
1.2
Tomography. Computed tomography (CT). Inverse problems
3
1.3
Applications
................................. 4
1.4
General types of CT
............................ 5
1.5
What kind of mathematics is involved?
............... 5
2
A Brief and Incomplete History of CT
7
3
Some Major CT Modalities and Their Features to Watch for
11
3.1 Tomographie
modalities
.........................11
3.2
What features one should pay attention to?
.............12
3.3
PDE classification
.............................13
4
Organization of the Book
15
II Traditional Computed Tomography Techniques and Integral Ge¬
ometry
19
5
Standard CT and X-ray and Radon Transforms
21
5.1
X-ray projection imaging: X-ray pictures and old tomography
21
5.2
X-ray CT
................................... 22
5.3
Beer s law and the X-ray /Radon transforms
............ 23
5.4
Some notation and conventions
.................... 28
5.5
Properties of X-ray
(—
Radon) transform in two dimensions
. 30
5.6
Backprojection
............................... 34
5.7
Inversion formulas
............................. 36
5.8
Stability
ofinversion
........................... 42
5.9
Fourier series and Cormack inversion formulas
.......... 44
5.10
Range conditions for the Radon transform
............. 45
5.11
Support (or hole ) theorem
...................... 48
5.12
Chapter s final remarks and conclusions
............... 48
6
Emission Tomography
53
6.1
Radiative transfer (transport) equation
(RTE)
........... 53
6.2
Attenuated X-ray transform and SPECT (single photon emis¬
sion computed tomography)
...................... 54
6.3
PET (positron emission tomography)
................ 59
6.4
Chapter s final remarks and conclusions
............... 60
7
Artifacts, Incomplete Data,
Microlocal
Analysis, and Such
65
7.1
Some common artifacts
......................... 65
7.2
Microlocal
detection of singularities:
Wavefront
sets of dis¬
tributions
.................................. 68
7.3
Detection of singularities, stability, and incomplete data prob¬
lems
...................................... 70
7.4
Local (high frequency) tomography
—
Sharpening singularities
72
7.5
Chapter s final remarks and conclusions
............... 73
8
More about
3D
Radon and X-ray Transforms
77
8.1 3D
Radon transform
........................... 77
8.2
X-ray transform: Overdetermined parametrization and John s
equation
................................... 79
8.3
Projectivization,
Ar-operator,
and such
................ 80
8.4
Fully
3D
X-ray CT
............................ 80
9
A Brief Overview of Numerical Methods
83
9.1
Analytic techniques
............................ 83
9.2
Algebraic reconstruction techniques (ART)
............ 84
9.3
Optimization techniques
......................... 84
9.4
Statistical techniques
........................... 85
9.5
Parametrix
+
ART
............................ 85
9.6
Some
MATLAB®
sources
........................ 86
9.7
Numerical techniques for inverse problems
............. 86
10
MRI,
EIT,
ОТ,
Elastography,
UT
87
10.1
MRI
(magnetic resonance imaging)
..................
S7
10.2
EIT (electrical impedance tomography)
............... 88
10.3
Optical tomography
(ОТ)
........................ 90
10.4
Elastography
................................ 91
10.5
Ultrasound imaging and tomography
(UT)
............. 91
Ill
Hybrid
(Coupled Physics) Imaging Techniques
93
11
Thermo-, Photo-, and Optoacoustic Tomography (TAT/PAT/OAT)
97
11.1
Mathematical model
............................98
11.2
Analysis of TAT: Acoustically homogeneous medium case
. . . 105
11.3
Analysis of TAT: Acoustically inhomogeneous media
......116
11.4
Reconstruction methods in TAT/PAT
................121
11.5
Chapter s final remarks and conclusions
...............137
12
Ultrasound Modulation in EIT and
ОТ
141
12.1
AET (acousto-electric tomography)
..................142
12.2
Synthetic focusing
.............................144
12.3
Ultrasound modulated optical tomography (UMOT)
......147
12.4
Chapter s final remarks and conclusions
...............149
13
Inverse Problems with Interior Information
151
13.1
The QPAT example worked out (well, sketched)
.........154
13.2
Other examples
...............................155
13.3
Generic linearized uniqueness
.....................156
IV Appendices
157
A Notation
159
A.I Sets, vectors, etc
...............................159
A.2 Multi-index notation and derivatives
.................159
A.3 Some useful functions
...........................160
A.4 Radon transform
..............................161
A.5 Some linear algebra
............................162
В
Brief Notes on the Fourier Transform and Harmonic Analysis
165
B.I Harmonic analysis
.............................165
B.2 Fourier series expansions
........................
Î67
B.3 Properties of Fourier series expansions
...............168
B.4 Smoothness of
ƒ(%)<=>
decay of Fourier coefficients/^
.....169
B.5 Relations with shifts and derivatives
.................170
B.6 Product-convolution relations
.....................171
B.7 Convolution onR
............................171
B.8 Fourier transform (FT)
..........................172
B.9 Properties of the FT
............................173
B.10 FT of some common functions
....................174
B.ll The Paley-Wiener theorem
.......................174
B.12 Smoothness and decay of the FT
....................175
B.13 Smoothing
..................................175
B.14 Sobolev spaces
...............................176
8.15
Sampling
...................................177
8.
16
Mellin transform
..............................179
В.
17
Wavefront
sets of distributions
.....................180
С
Geometric Rays, Trapping, Energy Decay
181
C.I Geometric rays
...............................181
C.2 Trapping
...................................182
C.3 Local energy decay
............................182
D
Some Classes of Linear Operators and Operator Functions
183
D.I Some notation concerning linear operators
.............183
D.2
Fredholm
and semi-Fredholm operators.
Fredholm
index.
. . . 183
D.3 Schatten-von Neumann classes
.....................185
D.4 Regularized determinants
........................187
D.5 Analytic
Fredholm
operator functions
................188
E
Recommended Books and Surveys
191
Bibliography
193
Index
237
This book surveys the main mathematical ideas and techniques behind
some well-established imaging modalities such as X-ray CT and emission
tomography, as well as a variety of newly developing coupled-physics or
hybrid techniques, including thermoacoustic tomography.
The Radon Transform and Medical Imaging
•
emphasizes mathematical techniques and ideas arising across the
spectrum of medical imaging modalities, and
•
explains important concepts concerning inversion, stability, incomplete
data effects, the role of interior information, and other issues critical
to all medical imaging methods.
For nonexperts, the author provides appendices that cover background
information on notation, Fourier analysis, geometric rays, and linear operators.
The vast bibliography, with over
800
entries, directs readers to a wide array
of additional information sources on medical imaging for further study.
Graduate students and researchers in mathematics, physics, and engineering
who are interested in the mathematics behind medical imaging will find this
book useful
|
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| isbn | 9781611973280 |
| language | English |
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| spelling | Kuchment, Peter 1949- Verfasser (DE-588)1046398547 aut The Radon transform and medical imaging Peter Kuchment Philadelphia, Pa. Society for Industrial & Applied Mathematics 2014 XVI, 240 S. Ill. txt rdacontent n rdamedia nc rdacarrier CBMS-NSF Regional Conference series in applied mathematics 85 Mathematik (DE-588)4037944-9 gnd rswk-swf Bildgebendes Verfahren (DE-588)4006617-4 gnd rswk-swf Bildgebendes Verfahren (DE-588)4006617-4 s Mathematik (DE-588)4037944-9 s DE-604 CBMS-NSF Regional Conference series in applied mathematics 85 (DE-604)BV000016696 85 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027187747&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027187747&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Kuchment, Peter 1949- The Radon transform and medical imaging CBMS-NSF Regional Conference series in applied mathematics Mathematik (DE-588)4037944-9 gnd Bildgebendes Verfahren (DE-588)4006617-4 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4006617-4 |
| title | The Radon transform and medical imaging |
| title_auth | The Radon transform and medical imaging |
| title_exact_search | The Radon transform and medical imaging |
| title_full | The Radon transform and medical imaging Peter Kuchment |
| title_fullStr | The Radon transform and medical imaging Peter Kuchment |
| title_full_unstemmed | The Radon transform and medical imaging Peter Kuchment |
| title_short | The Radon transform and medical imaging |
| title_sort | the radon transform and medical imaging |
| topic | Mathematik (DE-588)4037944-9 gnd Bildgebendes Verfahren (DE-588)4006617-4 gnd |
| topic_facet | Mathematik Bildgebendes Verfahren |
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