The mathematics of medical imaging: a beginner's guide
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Springer
2010
|
| Schriftenreihe: | Springer undergraduate texts in mathematics and technology (SUMAT)
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 141 S. Ill., graph. Darst. |
| ISBN: | 9780387927114 9780387927121 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804140667889254400 |
|---|---|
| adam_text | Titel: The mathematics of medical imaging
Autor: Feeman, Timothy G.
Jahr: 2010
Contents
Preface.
XI
1 X-rays........................................................... 1
1.1 Introduction.................................................. 1
1.2 X-ray behavior and Beer s law................................... 3
1.3 Lines in the plane ............................................. 7
1.4 Exercises .................................................... 9
2 The Radon Transform............................................. 11
2.1 Definition.................................................... 11
2.2 Examples.................................................... 12
2.3 Linearity..................................................... 14
2.4 Phantoms.................................................... 15
2.5 The domain of Si ............................................. 17
2.6 Exercises .................................................... 18
3 Back Projection................................................... 19
3.1 Definition and properties....................................... 19
3.2 Examples.................................................... 20
3.3 Exercises .................................................... 23
4 Complex Numbers................................................ 25
4.1 The complex number system.................................... 25
4.2 The complex exponential function ............................... 27
4.3 Wave functions ............................................... 29
4.4 Exercises .................................................... 30
vu
Contents
The Fourier Transform............................................ 33
5.1 Definition and examples........................................ 33
5.2 Properties and applications...................................... 34
5.3 Heaviside and Dirac 8 ......................................... 38
5.4 Inversion of the Fourier transform................................ 40
5.5 Multivariable forms............................................ 43
5.6 Exercises .................................................... 45
Two Big Theorems................................................ 47
6.1 The central slice theorem....................................... 47
6.2 Filtered back projection........................................ 48
6.3 The Hilbert transform.......................................... 50
6.4 Exercises .................................................... 51
Filters and Convolution............................................ 53
7.1 Introduction.................................................. 53
7.2 Convolution.................................................. 55
7.3 Filter resolution............................................... 58
7.4 Convolution and the Fourier transform............................ 58
7.5 The Rayleigh-Plancherel theorem................................ 60
7.6 Convolution in 2-dimensional space.............................. 62
7.7 Convolution, SB, and Si ........................................ 62
7.8 Low-pass filters............................................... 63
7.9 Exercises .................................................... 66
Discrete Image Reconstruction ..................................... 67
8.1 Introduction.................................................. 67
8.2 Sampling .................................................... 68
8.3 Discrete low-pass filters........................................ 70
8.4 Discrete Radon transform....................................... 73
8.5 Discrete functions and convolution............................... 74
8.6 Discrete Fourier transform...................................... 77
8.7 Discrete back projection........................................ 82
8.8 Interpolation.................................................. 82
8.9 Discrete image reconstruction................................... 86
8.10 Matrix forms................................................. 91
8.11 FFT?the fast Fourier transform................................. 92
8.12 Fan beam geometry............................................ 97
8.13 Exercises .................................................... 98
Contents ix
9 Algebraic Reconstruction Techniques ...............................101
9.1 Introduction..................................................101
9.2 Least squares approximation....................................103
9.3 Kaczmarz s method............................................107
9.4 ART in medical imaging .......................................110
9.5 Variations of Kaczmarz s method................................Ill
9.6 ART or the Fourier transform?...................................112
9.7 Exercises ....................................................113
10 MRI?An Overview...............................................115
10.1 Introduction..................................................115
10.2 Basics.......................................................116
10.3 The Bloch equation............................................118
10.4 The RF field..................................................119
10.5 RF pulse sequences; T and T2...................................122
10.6 Gradients and slice selection....................................124
10.7 The imaging equation..........................................127
10.8 Exercises ....................................................128
Appendix A Integrability ............................................129
A. 1 Improper integrals.............................................129
A.2 Iterated improper integrals......................................131
A.3 L1 and L2....................................................132
A.4 Summability..................................................133
Appendix B Topics for Further Study.................................135
References............................................................137
Index................................................................139
|
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| callnumber-first | R - Medicine |
| callnumber-label | RC78 |
| callnumber-raw | RC78.7.D53 |
| callnumber-search | RC78.7.D53 |
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| callnumber-subject | RC - Internal Medicine |
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| dewey-search | 616.0754 |
| dewey-sort | 3616.0754 |
| dewey-tens | 610 - Medicine and health |
| discipline | Mathematik Medizin |
| format | Book |
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| spelling | Feeman, Timothy G. Verfasser aut The mathematics of medical imaging a beginner's guide Timothy Feeman New York, NY [u.a.] Springer 2010 XII, 141 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer undergraduate texts in mathematics and technology (SUMAT) Mathematik Diagnostic imaging Mathematics Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Bildgebendes Verfahren (DE-588)4006617-4 gnd rswk-swf Medizin (DE-588)4038243-6 gnd rswk-swf Computertomografie (DE-588)4113240-3 gnd rswk-swf Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Bildgebendes Verfahren (DE-588)4006617-4 s Mathematisches Modell (DE-588)4114528-8 s Medizin (DE-588)4038243-6 s DE-604 Computertomografie (DE-588)4113240-3 s Mathematische Methode (DE-588)4155620-3 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018610170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Feeman, Timothy G. The mathematics of medical imaging a beginner's guide Mathematik Diagnostic imaging Mathematics Mathematisches Modell (DE-588)4114528-8 gnd Bildgebendes Verfahren (DE-588)4006617-4 gnd Medizin (DE-588)4038243-6 gnd Computertomografie (DE-588)4113240-3 gnd Mathematische Methode (DE-588)4155620-3 gnd |
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| title | The mathematics of medical imaging a beginner's guide |
| title_auth | The mathematics of medical imaging a beginner's guide |
| title_exact_search | The mathematics of medical imaging a beginner's guide |
| title_full | The mathematics of medical imaging a beginner's guide Timothy Feeman |
| title_fullStr | The mathematics of medical imaging a beginner's guide Timothy Feeman |
| title_full_unstemmed | The mathematics of medical imaging a beginner's guide Timothy Feeman |
| title_short | The mathematics of medical imaging |
| title_sort | the mathematics of medical imaging a beginner s guide |
| title_sub | a beginner's guide |
| topic | Mathematik Diagnostic imaging Mathematics Mathematisches Modell (DE-588)4114528-8 gnd Bildgebendes Verfahren (DE-588)4006617-4 gnd Medizin (DE-588)4038243-6 gnd Computertomografie (DE-588)4113240-3 gnd Mathematische Methode (DE-588)4155620-3 gnd |
| topic_facet | Mathematik Diagnostic imaging Mathematics Mathematisches Modell Bildgebendes Verfahren Medizin Computertomografie Mathematische Methode |
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