The Mathematics of Medical Imaging: A Beginner's Guide

205 Pages·2015·5.512 MB·English

by  Timothy G. Feeman

Springer Undergraduate Texts
in Mathematics and Technology
Timothy G. Feeman
The
Mathematics of
Medical Imaging
A Beginner’s Guide
Second Edition
Springer Undergraduate Texts
in Mathematics and Technology
SeriesEditors:
J.M.Borwein,Callaghan,NSW,Australia
H.Holden,Trondheim,Norway
V.Moll,NewOrleans,LA,USA
EditorialBoard:
L.Goldberg,Berkeley,CA,USA
A.Iske,Hamburg,Germany
P.E.T.Jorgensen,IowaCity,IA,USA
S.M.Robinson,Madison,WI,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/7438

Timothy G. Feeman
The Mathematics of Medical Imaging
A Beginner’s Guide
Second Edition
123
TimothyG.Feeman
DepartmentofMathematicsandStatistics
VillanovaUniversity
Villanova,PA,USA
ISSN1867-5506 ISSN1867-5514 (electronic)
SpringerUndergraduateTextsinMathematicsandTechnology
ISBN978-3-319-22664-4 ISBN978-3-319-22665-1 (eBook)
DOI10.1007/978-3-319-22665-1
LibraryofCongressControlNumber:2015949223
MathematicsSubjectClassification(2010):42A38,42B10,44A12,65T50,68U10,94A08,94A20
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Contents
Prefacetothesecondedition.............................................................. xi
Preface....................................................................................... xiii
1 X-rays.................................................................................. 1
1.1 Introduction..................................................................... 1
1.2 X-raybehavior andBeer’slaw ................................................ 3
1.3 Linesintheplane............................................................... 7
1.4 Exercises........................................................................ 10
2 TheRadonTransform ............................................................... 13
2.1 Definition ....................................................................... 13
2.2 Examples ....................................................................... 14
2.3 Someproperties ofR .......................................................... 18
2.4 Phantoms ....................................................................... 22
2.5 Designingphantoms ........................................................... 24
2.5.1 Plottinganelliptical region............................................ 25
2.5.2 TheRadontransform ofanellipse .................................... 27
2.5.3 Plottingasquareregion................................................ 30
2.5.4 TheRadontransform ofasquareregion.............................. 32
2.6 Thedomainof R............................................................... 34
2.7 Theattenuated Radontransform............................................... 34
2.8 Exercises........................................................................ 36
3 BackProjection....................................................................... 39
3.1 Definitionandproperties....................................................... 39
3.2 Examples ....................................................................... 41
3.3 Exercises........................................................................ 45
vii
viii Contents
4 ComplexNumbers.................................................................... 47
4.1 Thecomplexnumbersystem .................................................. 47
4.2 Thecomplexexponential function ............................................ 49
4.3 Wavefunctions ................................................................. 51
4.4 Exercises........................................................................ 52
5 TheFourierTransform .............................................................. 55
5.1 Definitionandexamples ....................................................... 55
5.2 Properties andapplications .................................................... 57
5.3 Heaviside andDiracı.......................................................... 61
5.4 Inversion oftheFouriertransform............................................. 63
5.5 Multivariable forms ............................................................ 66
5.6 Exercises........................................................................ 68
6 TwoBigTheorems.................................................................... 71
6.1 Thecentralslicetheorem ...................................................... 71
6.2 Filteredbackprojection........................................................ 72
6.3 TheHilberttransform .......................................................... 74
6.4 Exercises........................................................................ 75
7 FiltersandConvolution.............................................................. 77
7.1 Introduction..................................................................... 77
7.2 Convolution..................................................................... 80
7.2.1 Someproperties ofconvolution ....................................... 80
7.3 Filterresolution................................................................. 82
7.4 Convolution andtheFouriertransform........................................ 84
7.5 TheRayleigh–Plancherel theorem............................................. 86
7.6 Convolution in2-dimensional space .......................................... 87
7.7 Convolution, B,R,andı ...................................................... 88
7.8 Low-passfilters................................................................. 91
7.9 Exercises........................................................................ 94
8 DiscreteImageReconstruction...................................................... 97
8.1 Introduction..................................................................... 97
8.2 Sampling........................................................................ 98
8.3 Discretelow-passfilters........................................................ 100
8.4 DiscreteRadontransform...................................................... 103
8.5 Discretefunctions andconvolution............................................ 104
8.6 DiscreteFouriertransform..................................................... 107
8.7 Discretebackprojection ....................................................... 113
8.8 Interpolation .................................................................... 114
8.9 Discreteimagereconstruction ................................................. 121
8.10 Matrixforms.................................................................... 127
8.11 FFT—thefastFouriertransform ............................................. 129
Contents ix
8.12 Fanbeamgeometry ............................................................ 133
8.13 Exercises........................................................................ 134
9 AlgebraicReconstruction Techniques.............................................. 137
9.1 Introduction..................................................................... 137
9.2 Kaczmarz’smethod ............................................................ 139
9.2.1 Affinespaces ........................................................... 140
9.2.2 Kaczmarz’smethod.................................................... 141
9.2.3 VariationsofKaczmarz’smethod..................................... 145
9.3 Leastsquares approximation................................................... 147
9.4 Pseudoinverses andleastsquares.............................................. 149
9.5 Spectralfilteringandregularization ........................................... 152
9.6 ARTortheFouriertransform? ................................................ 159
9.7 Exercises........................................................................ 159
10 MRI—anoverview.................................................................. 163
10.1 Introduction..................................................................... 163
10.2 Basics ........................................................................... 164
10.3 TheBlochequation............................................................. 165
10.4 TheRFfield .................................................................... 167
10.5 RFpulsesequences; T andT ................................................ 169
1 2
10.6 Gradientsandsliceselection................................................... 171
10.7 Theimagingequation.......................................................... 174
10.8 Exercises........................................................................ 175
Appendix A Integrability ................................................................ 177
A.1 Improperintegrals.............................................................. 177
A.2 Iteratedimproperintegrals..................................................... 179
A.3 L1 andL2........................................................................ 181
A.4 Summability.................................................................... 182
Appendix B Matrices, transposes,andfactorization ................................. 183
B.1 Transposeofamatrix .......................................................... 183
B.2 Eigenvalue decomposition ..................................................... 186
B.3 Singularvaluedecomposition ................................................. 187
Appendix C Topicsforfurtherstudy................................................... 191
Bibliography................................................................................ 193
Index......................................................................................... 195