Mastering the Discrete Fourier Transform in One, Two or Several Dimensions: Pitfalls and Artifacts

388 Pages·2013·35.21 MB·English

by  Isaac Amidror (auth.)

Computational Imaging and Vision 43
Isaac Amidror
Mastering
the Discrete Fourier
Transform in One,
Two or Several
Dimensions
Pitfalls and Artifacts
Mastering the Discrete Fourier Transform in One,
Two or Several Dimensions: Pitfalls and Artifacts
Computational Imaging and Vision
ManagingEditor
MAXVIERGEVER
UtrechtUniversity,Utrecht,TheNetherlands
SeriesEditors
GUNILLABORGEFORS,CentreforImageAnalysis,SLU,Uppsala,Sweden
DANIELCREMERS,TechnischeUniversitätMünchen,München,Germany
RACHIDDERICHE,INRIA,SophiaAntipolis,France
KATSUSHIIKEUCHI,TokyoUniversity,Tokyo,Japan
REINHARDKLETTE,UniversityofAuckland,Auckland,NewZealand
ALESLEONARDIS,ViCoS,UniversityofLjubljana,Ljubljana,Slovenia
STANZ.LI,CASIA,Beijing&CIOTC,Wuxi,China
DIMITRISN.METAXAS,RutgersUniversity,NewBrunswick,NJ,USA
HEINZ-OTTOPEITGEN,CeVis,Bremen,Germany
JOHNK.TSOTSOS,YorkUniversity,Toronto,Canada
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Volume43
Forfurthervolumes:
http://www.springer.com/series/5754
Isaac Amidror
Mastering the Discrete Fourier
Transform in One, Two or
Several Dimensions
Pitfalls and Artifacts
Isaac Amidror
School of Computer and Communication Sciences
Peripheral Systems Laboratory
Ecole Polytechnique Fédérale de Lausanne
Lausanne, Switzerland

ISSN1381-6446
ISBN 978-1-4471-5166-1 ISBN 978-1-4471-5167-8 (eBook)
DOI 10.1007/978-1-4471-5167-8
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To my parents
An expert is a man who has made all the mistakes,
which can be made, in a very narrow field.
Niels Bohr [Mackay91 p. 35]
Front cover image: Aliasing due to insufficient sampling rate
may give unexpected shapes both in the signal domain and
in the spectral domain. See Fig. 5.17(k).
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The discrete Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 A brief historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The scope of the present book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Overview of the following chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 About the graphic presentation of sampled signals and discrete data . . . . . . 7
1.5.1 Graphic presentations in the 1D case . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.2 Graphic presentations in the 2D case . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.3 Graphic presentations in the MD case . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 About the exercises and the internet site . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2. Background and basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The continuous Fourier transform: definitions and notations . . . . . . . . . . . . 15
2.3 The discrete Fourier transform: definitions and notations . . . . . . . . . . . . . . . 17
2.4 Rules for deriving new Fourier transforms from already known ones . . . . . 21
2.4.1 Rules for the 1D continuous Fourier transform . . . . . . . . . . . . . . . . 22
2.4.2 Rules for the 2D continuous Fourier transform . . . . . . . . . . . . . . . . 23
2.4.3 Rules for the MD continuous Fourier transform . . . . . . . . . . . . . . . 25
2.4.4 Rules for the 1D discrete Fourier transform . . . . . . . . . . . . . . . . . . 26
2.4.5 Rules for the 2D discrete Fourier transform . . . . . . . . . . . . . . . . . . 28
2.4.6 Rules for the MD discrete Fourier transform . . . . . . . . . . . . . . . . . 29
2.5 Graphical development of the DFT — a three-stage process . . . . . . . . . . . . 31
2.6 DFT as an approximation to the continuous Fourier transform . . . . . . . . . . 36
2.7 The use of DFT in the case of periodic or almost-periodic functions . . . . . . 41
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3. Data reorganizations for the DFT and the IDFT . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Reorganization of the output data of the DFT . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Reorganization of the input data of the DFT . . . . . . . . . . . . . . . . . . . . . . . . . 47
vii
viii Contents
3.4 Data reorganizations in the case of IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4. True units along the axes when plotting the DFT . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 True units for the input array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 True units for the output array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1 The particular case of periodic functions . . . . . . . . . . . . . . . . . . . . . 74
4.4 True units for the DFT element values (heights along the vertical axis) . . . . 78
4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5. Issues related to aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Aliasing in the one dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Examples of aliasing in the one dimensional case . . . . . . . . . . . . . . . . . . . . . 95
5.4 Aliasing in two or more dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5 Examples of aliasing in the multidimensional case . . . . . . . . . . . . . . . . . . . . 114
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.7 Signal-domain aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6. Issues related to leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2 Leakage in the one dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.3 Examples of leakage in the one dimensional case . . . . . . . . . . . . . . . . . . . . . 148
6.4 Errors due to signal-domain truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.5 Spectral impulses that fall between output array elements . . . . . . . . . . . . . . . 162
6.6 Leakage in two or more dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.6.1 The case of 2D 1-fold periodic functions . . . . . . . . . . . . . . . . . . . . 170
6.6.2 The case of 2D 2-fold periodic functions . . . . . . . . . . . . . . . . . . . . 172
6.6.3 The general MD case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.7 Signal-domain leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7. Issues related to resolution and range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.2 The choice of the array size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Contents ix
7.3 The choice of the sampling interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.4 The choice of the sampling range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.5 The choice of the frequency step and of the frequency range . . . . . . . . . . . . 189
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8. Miscellaneous issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.2 Representation of discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.3 Phase related issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.4 Symmetry related issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.5 Jaggies on sharp edges as aliasing or reconstruction phenomena . . . . . . . . . 210
8.6 Sub-Nyquist artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.7 Displaying considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8.8 Numeric precision considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Appendices
A. Impulses in the continuous and discrete worlds . . . . . . . . . . . . . . . . . . . . . . . 247
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
A.2 Continuous-world impulses vs. discrete-world impulses . . . . . . . . . . . . . . . 247
A.3 Impulses in the spectral domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
A.4 Impulses in the signal domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
B. Data extensions and their effects on the DFT results . . . . . . . . . . . . . . . . . . . 265
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
B.2 Method 1: Extending the input data by adding new values beyond the
original range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

B.3 Method 2: Extending the input data by denser sampling within the
original range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

B.4 Method 3: Extending the input data by adding zeroes after each value
(zero packing) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

B.5 Method 4: Extending the input data by replicating it . . . . . . . . . . . . . . . . . . 269
B.6 Method 5: Extending the input data by replicating each of its elements . . . . 269
B.7 Method 6: Extending the input data by adding zeroes beyond its original
range (zero padding) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

B.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272