Discrete tomography with two directions
Dalen, B.E. van
Citation
Dalen, B. E. van. (2011, September 20). Discrete tomography with two directions. Retrieved from https://hdl.handle.net/1887/17845
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Discrete tomography with two directions
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus prof.mr. P.F. van der Heijden,
volgens besluit van het College voor Promoties te verdedigen op dinsdag 20 september 2011
klokke 15:00 uur
door
Birgit Ellen van Dalen
geboren te ’s-Gravenhage in 1984
Samenstelling van de promotiecommissie
Promotoren prof.dr. R. Tijdeman
prof.dr. K.J. Batenburg (Centrum Wiskunde & Informatica, Universiteit Antwerpen) Overige leden
prof.dr. S.J. Edixhoven prof.dr. H.W. Lenstra, Jr.
prof.dr. A. Schrijver (Centrum Wiskunde & Informatica) prof.dr. P. Stevenhagen
Discrete tomography with two directions
Birgit van Dalen
ISBN/EAN 9789461081803 Birgit van Dalen, Leiden, 2011c bevandalen@gmail.com
Typeset using LATEX
Printed by Gildeprint Drukkerijen, Enschede Cover design by Ad van den Broek
Contents
1 Introduction 1
1.1 Discrete tomography . . . 1
1.2 Applications . . . 2
1.3 Two directions . . . 3
1.4 Stability . . . 4
1.5 Difference between reconstructions . . . 5
1.6 Boundary length . . . 7
1.7 Shape of binary images . . . 9
1.8 Overview . . . 9
2 Stability results for uniquely determined sets 11 2.1 Introduction . . . 11
2.2 Notation and statement of the problems . . . 12
2.3 Staircases . . . 14
2.4 A new bound for the disjoint case . . . 17
2.5 Two bounds for general α . . . 19
vi
2.6 Generalisation to unequal sizes . . . 29
3 Upper bounds for the difference between reconstructions 31 3.1 Introduction . . . 31
3.2 Notation . . . 32
3.3 Some lemmas . . . 33
3.4 Uniquely determined neighbours . . . 35
3.5 Sets with equal line sums . . . 38
3.6 Sets with different line sums . . . 41
3.7 Concluding remarks . . . 46
4 A lower bound on the largest possible difference 47 4.1 Introduction . . . 47
4.2 Definitions and notation . . . 48
4.3 Main result . . . 49
4.4 Proof . . . 50
4.5 Example . . . 53
5 Minimal boundary length of a reconstruction 57 5.1 Introduction . . . 57
5.2 Definitions and notation . . . 59
5.3 The main theorem . . . 60
5.4 Some examples and a corollary . . . 63
5.5 An extension . . . 67
vii
6 Reconstructions with small boundary 81
6.1 Introduction . . . 81
6.2 Definitions and notation . . . 82
6.3 The construction . . . 83
6.4 Boundary length of the constructed solution . . . 90
6.5 Examples . . . 94
6.6 Generalising the results for arbitrary c1 and r1 . . . 97
7 Boundary and shape of binary images 99 7.1 Introduction . . . 99
7.2 Definitions and notation . . . 100
7.3 Largest connected component . . . 101
7.4 Balls of ones in the image . . . 108
Bibliography 115 Samenvatting 119 1 Binaire plaatjes en Japanse puzzels . . . 119
2 Onoplosbare puzzels . . . 121
3 Saaie puzzels . . . 124
4 Puzzels met meerdere oplossingen . . . 125
5 Rand . . . 128
Curriculum Vitae 131
viii
