Analysis and design of univariate subdivision schemes:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2010
|
| Series: | Geometry and computing
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Literaturverz.: S. 207 - 212 |
| Physical Description: | XIV, 215 S. graph. Darst. 25 cm |
| ISBN: | 9783642136474 |
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IMAGE 1
TABLE OF CONTENTS
NOTATION
PART I. PREPENDICES
1. FUNCTIONS AND CURVES 5
1.1 SUMMARY 6
2. DIFFERENCES 7
2.1 FIRST DIFFERENCES 7
2.2 HIGHER DIFFERENCES 7
2.3 DIFFERENCES OF POLYNOMIALS 8
2.4 DIVIDED DIFFERENCES 8
2.5 SUMMARY 10
3. B-SPLINES 11
3.1 DEFINITION 11
3.2 DERIVATIVE PROPERTIES 12
3.3 CONSTRUCTION 13
3.4 REFINEMENT 14
3.5 SUMMARY IG
4. EIGENFACTORISATION 17
4.1 DEFINITION 17
4.2 UNIQUENESS 17
4.3 PROPERTIES OF EIGENVECTORS 19
4.3.1 W H AT HAPPENS WHEN A GENERAL VECTOR IS MULTIPLIED BY A MATRIX? 19
4.3.2 W H AT HAPPENS WHEN A GENERAL VECTOR IS MULTIPLIED BY A MATRIX
REPEATEDLY? 20
4.4 CALCULATING EIGENCOMPONENTS 20
4.5 THE EFFECT OF NON-ZERO OFF-DIAGONAL ELEMENTS 22
4.5.1 JORDAN BLOCKS 22
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1002540348
DIGITALISIERT DURCH
IMAGE 2
VIII TABLE OF CONTENTS
4.5.2 EFFECT OF A JORDAN BLOCK ON THE MULTIPLICATION OF A (GENERALISED)
EIGENVECTOR BY THE MATRIX 22
4.5.3 COMPLEX EIGENVALUES 24
4.6 SUMMARY 24
5. ENCLOSURES 25
5.1 DEFINITION 25
5.2 EXAMPLES OF ENCLOSURES 25
5.3 REPRESENTATIONS 27
5.4 SUMMARY 28
6. HOLDER CONTINUITY 29
6.1 CONTINUITY 29
6.2 DERIVATIVES 29
6.3 HOLDER CONTINUITY 30
6.4 SUMMARY 32
7. MATRIX NORMS 33
7.1 VECTOR NORMS 33
7.2 MATRIX NORMS 34
7.2.1 PROOFS OF MATRIX NORM PROPERTIES 34
7.2.2 EVALUATING MATRIX NORMS 35
7.3 SUMMARY 36
8. JOINT SPECTRAL RADIUS 37
8.1 SUMMARY 38
9. RADIX NOTATION 39
9.1 SUMMARY 40
10. Z-TRANSFBRMS 41
10.1 THE Z-TRANSFORM 41
10.2 WHY ZI 42
10.3 WHAT SORT OF OBJECT IS Z? 42
10.4 SOME SPECIAL SEQUENCES 44
10.5 NORMALISATION 45
10.6 SUMMARY 45
PART II. DRAMATIS PERSONAE
11. AN INTRODUCTION TO SOME REGULARLY-APPEARING CHARACTERS . 49 11.1
POLYGONS 49
11.2 LABELLING AND PARAMETRISATION 51
IMAGE 3
TABLE OF CONTENTS IX
11.3 PRIMAL AND DUAL SCHEMES 52
11.4 TERNARY SCHEMES AND HIGHER ARITIES 52
11.5 INTERPOLATORY SCHEMES 54
11.6 RANGE 54
11.7 REPRESENTATIONS OF SUBDIVISION SCHEMES 55
11.8 EXERCISES 58
11.9 SUMMARY 58
PART III. ANALYSES
12. SUPPORT 63
12.1 THE BASIS FUNCTION 63
12.2 SUPPORT WIDTH 65
12.2.1 PRIMAL BINARY SCHEMES 65
12.2.2 DUAL BINARY SCHEMES 67
12.2.3 TERNARY SCHEMES 67
12.2.4 'NEITHER' TERNARY SCHEMES 68
12.2.5 HIGHER ARITIES 68
12.3 FACTS WHICH WILL BE RELEVANT TO OTHER ANALYSES 68
12.4 THE MATRICES OF POWERS OF A SCHEME 71
12.5 PRACTICAL SUPPORT 71
12.6 EXERCISES 71
12.7 SUMMARY 72
13. ENCLOSURE 73
13.1 POSITIVITY 73
13.2 IF THE BASIS FUNCTIONS ARE SOMEWHERE NEGATIVE 74
13.3 EXERCISES 76
13.4 SUMMARY 76
14. CONTINUITY 1 - AT SUPPORT ENDS 77
14.1 DERIVATIVE CONTINUITY OF THE BASIS FUNCTION AT ITS ENDS . . . 77
14.1.1 WHY DOES THE LIMIT CURVE CONVERGE JUST BECAUSE THE CONTROL POINTS
DO? 79
14.1.2 HOW DO YOU KNOW SOMETHING NASTY DOESN'T HAPPEN AT PLACES IN
BETWEEN THE EXTREME CONTROL POINTS EXAMINED? 79
14.2 EXERCISES 80
14.3 SUMMARY 80
15. CONTINUITY 2 - EIGENANALYSIS 81
15.1 CONTINUITY AT MARK POINTS BY EIGENANALYSIS 81
15.1.1 ODD-EVEN PARTITIONING 83
IMAGE 4
X TABLE OF CONTENTS
15.1.2 USING BLOCK STRUCTURE 84
15.1.3 INTERPRETATION 84
15.2 A MOTIVATION QUESTION 88
15.3 DUAL SCHEMES 88
15.4 HIGHER ARITIES 88
15.5 PIECEWISE POLYNOMIAL SCHEMES 91
15.6 WHAT MARK POINTS CAN BE MADE ? 91
15.7 EXERCISES 92
15.8 SUMMARY 93
16. CONTINUITY 3 - DIFFERENCE SCHEMES 95
16.1 LOWER BOUNDS BY DIFFERENCE SCHEMES 95
16.1.1 CONTINUITY BY DIFFERENCE SCHEMES 95
16.2 CONTINUITY OF DERIVATIVES BY DIVIDED DIFFERENCE SCHEMES . . 97
16.3 DUAL SCHEMES 97
16.4 HIGHER ARITIES 98
16.5 TIGHTENING THE LOWER BOUND 98
16.6 A PROCEDURE FOR DETERMINING BOUNDS ON HOLDER CONTINUITY 100 16.7
EXERCISES 101
16.8 SUMMARY 101
17. CONTINUITY 4 - DIFFERENCE EIGENANALYSIS 103
17.1 EFFICIENT COMPUTATION OF THE EIGENCOMPONENTS 103
17.1.1 THE KERNEL 103
17.1.2 EIGENVECTORS 104
17.2 EXAMPLES 105
17.2.1 CUBIC B-SPLINE 105
17.2.2 FOUR POINT 105
17.3 DUAL SCHEMES 106
17.4 HIGHER ARITY SCHEMES 106
17.5 A SPECIAL CASE 106
17.6 EXERCISES 107
17.7 SUMMARY 107
18. CONTINUITY 5 - THE JOINT SPECTRAL RADIUS 109
18.1 THE JOINT SPECTRAL RADIUS APPROACH 109
18.2 THE CONTINUITY ARGUMENT I LL
18.3 A PROCEDURE FOR DETERMINING HOLDER CONTINUITY 112
18.4 SUMMARY 113
19. WHAT CONVERGES ? 115
19.1 A MORE APPROPRIATE DESCRIPTION 115
19.2 A MORE APPROPRIATE DEFINITION 116
19.3 EXAMPLE 116
IMAGE 5
TABLE OF CONTENTS XI
19.4 SUMMARY 118
20. REPRODUCTION OF POLYNOMIALS 119
20.1 GENERATION DEGREE 120
20.1.1 HIGHER ARITIES 120
20.1.2 LOCAL POLYNOMIAL STRUCTURE 120
20.1.3 HIGHER ARITIES 121
20.2 INTERPOLATING DEGREE 122
20.2.1 PRIMAL BINARY SCHEMES 122
20.2.2 DUAL BINARY SCHEMES 123
20.2.3 HIGHER ARITIES 123
20.3 REPRODUCTION DEGREE 123
20.4 EXERCISES 123
20.5 SUMMARY 124
21. ARTIFACTS 125
21.1 ARTIFACTS 125
21.2 FACTORISING THE SUBDIVISION MATRIX 127
21.2.1 EFFECT OF THE SAMPLING MATRIX 127
21.2.2 EFFECT OF THE SMOOTHING MATRIX 128
21.2.3 EFFECT OF THE KERNEL 129
21.3 EFFECT ON THE LIMIT CURVE 129
21.4 DUAL SCHEMES 130
21.5 EXAMPLES 130
21.6 HIGHER ARITIES 131
21.7 EXERCISES 131
21.8 SUMMARY 132
22. NORMALISATION OF SCHEMES 133
22.1 QUASI-B-SPLINES 133
22.2 SIMILAR EFFECTS 135
22.3 SUMMARY 136
23. SUMMARY OF ANALYSIS RESULTS 137
23.1 EXERCISES 137
23.2 SUMMARY 137
PART IV. DESIGN
24. THE DESIGN SPACE 141
24.1 BINARY SCHEMES 141
24.2 HIGHER ARITIES 142
24.3 EXERCISES 142
IMAGE 6
XII TABLE OF CONTENTS
24.4 SUMMARY 142
25. LINEAR SUBSPACES OF THE DESIGN SPACE 143
25.1 SUPPORT 143
25.2 GENERATION DEGREE 144
25.3 INTERPOLATION DEGREE 144
25.3.1 PRIMAL BINARY SCHEMES 144
25.3.2 NON-PRIMAL SCHEMES 145
25.3.3 HIGHER ARITIES 146
25.4 EXERCISES 146
25.5 SUMMARY 146
26. NON-LINEAR CONDITIONS 147
26.1 DERIVATIVE CONTINUITY 147
26.2 POSITIVITY 151
26.2.1 ANOTHER SUFFICIENT CONDITION 151
26.2.2 REALISTIC CONDITIONS 151
26.3 ARTIFACTS 152
26.3.1 ZERO ARTIFACT AT A GIVEN SPATIAL FREQUENCY 152
26.4 EXERCISES 153
26.5 SUMMARY 153
27. NON-STATIONARY SCHEMES 155
27.1 EXAMPLES 155
27.1.1 THE UP-FUNCTION 155
27.1.2 VARIANTS ON UP 156
27.1.3 STEP UP 157
27.1.4 DOWN 157
27.1.5 CIRCLE-PRESERVING SCHEMES 157
27.2 ANALYSES OF NON-STATIONARY SCHEMES 158
27.2.1 SUPPORT 158
27.2.2 REPRODUCTION DEGREE 158
27.2.3 CONTINUITY 159
27.2.4 POSITIVITY 159
27.2.5 ARTIFACTS 159
27.3 STEP-INDEPENDENCE 159
27.4 EXERCISES 160
27.5 SUMMARY 160
28. GEOMETRY SENSITIVE SCHEMES 161
28.1 SPAN CRITERIA 161
28.2 VERTEX CRITERIA 162
28.3 GEOMETRIC DUALITY 163
28.4 SUMMARY 163
IMAGE 7
TABLE OF CONTENTS XIII
PART V. IMPLEMENTATION
29. MAKING POLYGONS 167
29.1 PULL 167
29.2 PUSH 167
29.3 MULTI-STAGE 168
29.4 GOING DIRECT 168
29.5 GOING DIRECT TO LIMIT POINTS 168
29.6 SUMMARY 169
30. RENDERING 171
30.1 POLYGON RENDERING 171
30.2 B-SPLINE RENDERING 171
30.3 HERMITE RENDERING 172
30.4 WHAT ABOUT NON-STATIONARY SCHEMES ? 172
30.5 SUMMARY 172
31. INTERROGATION 173
31.1 EVALUATION AT GIVEN ABSCISSA VALUES 173
31.2 EVALUATION AT THE INTERSECTION WITH A GIVEN PLANE 173
31.3 EVALUATION OF A POINT NEAR A GIVEN POINT 174
31.4 SUMMARY 174
32. END CONDITIONS 175
32.1 END CONDITIONS 175
32.2 HOW MUCH SHORTER? 176
32.3 HOW DO YOU WANT THE LIMIT CURVE TO BE RELATED TO THE POLYGON? 176
32.4 REQUIREMENTS FOR APPROXIMATING SCHEMES 177
32.5 REQUIREMENTS FOR INTERPOLATING SCHEMES 177
32.6 HOW TO IMPLEMENT END-CONDITIONS 178
32.6.1 MODIFYING THE MATRICES FOR APPROXIMATING SCHEMES . 179 32.6.2
MODIFIED INITIAL POLYGON FOR INTERPOLATING SCHEMES . . 180 32.7 SUMMARY
180
33. MODIFYING THE ORIGINAL POLYGON 181
33.1 MAKING A POLYGON TO INTERPOLATE GIVEN POINTS 181
33.2 SUMMARY 182
PART VI. APPENDICES
1 PROOFS 185
IMAGE 8
XIV TABLE OF CONTENTS
1 THE KERNEL OF THE SQUARE OF A SCHEME IS A FACTOR OF THE SQUARE OF THE
KERNEL 185
2 THE KERNEL OF THE SQUARE OF A SCHEME IS THE SQUARE OF THE KERNEL 186
3 CONTRIBUTION TO THE JOINT SPECTRAL RADIUS FROM A SHARED EIGENVECTOR
186
4 CONTRIBUTION TO THE JOINT SPECTRAL RADIUS FROM NESTED INVARIANT
SUBSPACES 187
2 HISTORICAL NOTES 189
3 SOLUTIONS TO EXERCISES 193
1 DRAMATIS PERSONAE 193
2 SUPPORT 195
3 ENCLOSURES 195
4 CONTINUITY 1 - AT SUPPORT ENDS 196
5 CONTINUITY 2 - EIGENANALYSIS 196
6 CONTINUITY 3 - DIFFERENCE SCHEMES 199
7 CONTINUITY 4 - DIFFERENCE EIGENANALYSIS 199
8 REPRODUCTION OF POLYNOMIALS 200
9 ARTIFACTS 200
10 THE DESIGN SPACE 201
11 LINEAR SUBSPACES OF THE DESIGN SPACE 202
12 NON-LINEAR CONDITIONS 202
13 NON-STATIONARY SCHEMES 203
4 CODA 205
1 FOURIER DECAY ANALYSIS 205
2 FOURIER ENERGY 205
3 LINKS BETWEEN THE ARTIFACTS AND APPROXIMATION ORDER . . . 206 4
END-CONDITIONS FOR SCHEMES WITH HIGHER QUASI- INTERPOLATION DEGREE 206
5 NON-UNIFORM THEORY TO ENCOMPASS ENDCONDITIONS 206
BIBLIOGRAPHY 207
INDEX 213 |
| any_adam_object | 1 |
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| id | DE-604.BV041104222 |
| illustrated | Illustrated |
| indexdate | 2024-08-03T00:45:03Z |
| institution | BVB |
| isbn | 9783642136474 |
| language | English |
| lccn | 2010934300 |
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| owner | DE-83 DE-11 DE-188 |
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| physical | XIV, 215 S. graph. Darst. 25 cm |
| publishDate | 2010 |
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| publisher | Springer |
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| series | Geometry and computing |
| series2 | Geometry and computing |
| spelling | Sabin, Malcolm Verfasser aut Analysis and design of univariate subdivision schemes Malcolm Sabin Berlin [u.a.] Springer 2010 XIV, 215 S. graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Geometry and computing 6 Literaturverz.: S. 207 - 212 Unterteilungsalgorithmus (DE-588)4753239-7 gnd rswk-swf Kurve (DE-588)4033824-1 gnd rswk-swf Geometrische Modellierung (DE-588)4156717-1 gnd rswk-swf Kurve (DE-588)4033824-1 s Unterteilungsalgorithmus (DE-588)4753239-7 s Geometrische Modellierung (DE-588)4156717-1 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-13648-1 Geometry and computing 6 (DE-604)BV022959018 6 application/pdf http://d-nb.info/1002540348/04 Inhaltsverzeichnis text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3476793&prov=M&dok_var=1&dok_ext=htm Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026080552&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sabin, Malcolm Analysis and design of univariate subdivision schemes Geometry and computing Unterteilungsalgorithmus (DE-588)4753239-7 gnd Kurve (DE-588)4033824-1 gnd Geometrische Modellierung (DE-588)4156717-1 gnd |
| subject_GND | (DE-588)4753239-7 (DE-588)4033824-1 (DE-588)4156717-1 |
| title | Analysis and design of univariate subdivision schemes |
| title_auth | Analysis and design of univariate subdivision schemes |
| title_exact_search | Analysis and design of univariate subdivision schemes |
| title_full | Analysis and design of univariate subdivision schemes Malcolm Sabin |
| title_fullStr | Analysis and design of univariate subdivision schemes Malcolm Sabin |
| title_full_unstemmed | Analysis and design of univariate subdivision schemes Malcolm Sabin |
| title_short | Analysis and design of univariate subdivision schemes |
| title_sort | analysis and design of univariate subdivision schemes |
| topic | Unterteilungsalgorithmus (DE-588)4753239-7 gnd Kurve (DE-588)4033824-1 gnd Geometrische Modellierung (DE-588)4156717-1 gnd |
| topic_facet | Unterteilungsalgorithmus Kurve Geometrische Modellierung |
| url | http://d-nb.info/1002540348/04 http://deposit.dnb.de/cgi-bin/dokserv?id=3476793&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026080552&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV022959018 |
| work_keys_str_mv | AT sabinmalcolm analysisanddesignofunivariatesubdivisionschemes |
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