Noise reduction by wavelet thresholding:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2001
|
| Series: | Lecture notes in statistics
161 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 191 S. Ill., graph. Darst. |
| ISBN: | 0387952446 |
Staff View
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| 100 | 1 | |a Jansen, Maarten |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Noise reduction by wavelet thresholding |c Maarten Jansen |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2001 | |
| 300 | |a XIX, 191 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 650 | 4 | |a Traitement du signal - Techniques numériques - Méthodes statistiques | |
| 650 | 4 | |a Active noise and vibration control | |
| 650 | 4 | |a Electronic noise |x Automatic control | |
| 650 | 4 | |a Signal processing |x Digital techniques |x Statistical methods | |
| 650 | 4 | |a Wavelets (Mathematics) | |
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MAARTEN JANSEN NOISE REDUCTION BY WAVELET THRESHOLDING SPRINGER CONTENTS
PREFACE V NOTATIONS AND ABBREVIATIONS XIII LIST OF FIGURES XVII LIST OF
TABLES XXI 1. INTRODUCTION AND OVERVIEW 1 1.1 OUTLINE AND SITUATION 1
1.2 NOTIONS AND NOTATIONS 3 1.2.1 MATHEMATICAL PRELIMINARIES 3 1.2.2
FOURIER ANALYSIS AND DIGITAL SIGNALS 5 1.2.3 A NOTE ON IMAGES 6 1.2.4
GENERALITY 7 2. WAVELETS AND WAVELET THRESHOLDING : 9 2.1- EXPLOITING
SAMPLE CORRELATIONS 9 2.1.1 THE INPUT PROBLEM: SPARSITY 9 2.1.2 BASIS
FUNCTIONS AND MULTIRESOLUTION 11 2.1.3 THE DILATION EQUATION \ 15 2.1.4
(FAST) WAVELET TRANSFORMS AND FILTER BANKS 16 2.1.5 LOCALITY 19 2.1.6
VANISHING MOMENTS 21 2.1.7 TWO-DIMENSIONAL WAVELET TRANSFORMS 23 2.2
CONTINUOUS WAVELET TRANSFORM 25 2.3 NON-DECIMATED WAVELET TRANSFORMS AND
FRAMES 26 2.4 WAVELET PACKETS 29 2.5 SMOOTH LOCAL TRIGONOMETRIC BASES 30
2.6 LIFTING AND SECOND GENERATION WAVELETS *. -. 30 2.6.1 THE IDEA
BEHIND LIFTING 30 2.6.2 SUBDIVISION 33 2.6.3 THE INTEGER WAVELET
TRANSFORM 33 2.6.4 NON-EQUIDISTANT DATA 33 VIII CONTENTS 2.7 NOISE
REDUCTION BY THRESHOLDING WAVELET COEFFICIENTS 35 2.7.1 NOISE MODEL AND
DEFINITIONS 35 2.7.2 THE WAVELET TRANSFORM OF A SIGNAL WITH NOISE 36
2.7.3 WAVELET THRESHOLDING, MOTIVATION 36 2.7.4 HARD- AND
SOFT-THRESHOLDING, SHRINKING .38 2.7.5 THRESHOLD ASSESSMENT 39 2.7.6
THRESHOLDING AS NON-LINEAR SMOOTHING 39 2.8 OTHER COEFFICIENT SELECTION
PRINCIPLES 41 2.9 BASIS SELECTION METHODS 42 2.10 WAVELETS IN OTHER
DOMAINS OF APPLICATION 43 2.11 SUMMARY AND CONCLUDING REMARKS 44 3. THE
MINIMUM MEAN SQUARED ERROR THRESHOLD 47 3.1 MEAN SQUARE ERROR AND RISK
FUNCTION 48 3.1.1 DEFINITIONS 48 3.1.2 VARIANCE AND BIAS 49 3.2 THE RISK
CONTRIBUTION OF EACH COEFFICIENT (GAUSSIAN NOISE) 50 3.3 THE ASYMPTOTIC
BEHAVIOR OF THE MINIMUM RISK THRESHOLD FOR PIECE- WISE POLYNOMIALS 53
3.3.1 MOTIVATION 53 3.3.2 ASYMPTOTIC EQUIVALENCE 54 3.3.3 THE ASYMPTOTIC
BEHAVIOR 55 3.3.4 AN EXAMPLE 57 3.3.5 WHY DOES THE THRESHOLD DEPEND ON
THE NUMBER OF DATA POINTS? 58 3.4 UNIVERSAL THRESHOLD 60 3.4.1 ORACLE
MIMICKING 60 3.4.2 MINIMAX PROPERTIES 61 3.4.3 ADAPTIVITY, OPTIMALITY
WITHIN FUNCTION CLASSES 61 3.4.4 SMOOTHNESS 62 3.4.5 PROBABILISTIC UPPER
BOUND 62 3.4.6 UNIVERSAL THRESHOLD IN PRACTICE : 63 3.5 FALSE DISCOVERY
RATE (FDR) I 63 3.6 BEYOND THE PIECEWISE POLYNOMIAL CASE 67 3.6.1 FOR
WHICH COEFFICIENTS IS A GIVEN THRESHOLD TOO LARGE/SMALL? . 67 3.6.2
INTERMEDIATE RESULTS FOR THE RISK IN ONE COEFFICIENT 70 3.6.3 ;
PIECEWISE SMOOTH FUNCTIONS 72 3.7 FUNCTION SPACES 75 3.7.1 LIPSCHITZ
REGULARITY 75 3.7.2 BESOV SPACES 76 3.7.3 I V BALLS 78 3.8 CONCLUSION 79
CONTENTS IX ESTIMATING THE MINIMUM MSE THRESHOLD 81 4.1 SURE, A FIRST
ESTIMATOR FOR THE MSE 82 4.1.1 THE EFFECT OF THE THRESHOLD OPERATION 82
4.1.2 COUNTING THE NUMBER OF COEFFICIENTS BELOW THE THRESHOLD . 83
4.1.3 SURE IS ADAPTIVE 84 4.2 'ORDINARY' CROSS VALIDATION 85 4.3
GENERALIZED CROSS VALIDATION 87 4.3.1 DEFINITION 87 4.3.2 THE LINK
BETWEEN GCV AND SURE 87 4.3.3 ASYMPTOTIC BEHAVIOR 88 4.4 GCV FOR A
FINITE NUMBER OF DATA 92 4.4.1 THE MINIMIZATION PROCEDURE 93 4.4.2
CONVEXITY AND CONTINUITY 94 4.4.3 BEHAVIOR FOR LARGE THRESHOLDS AND
PROBLEMS NEAR THE ORIGIN . 94 4.4.4 GCV IN ABSENCE OF SIGNAL AND IN
ABSENCE OF NOISE 96 4.4.5 ABSOLUTE AND RELATIVE ERROR 97 4.4.6 WHICH IS
BETTER: GCV OR UNIVERSAL? 98 4.5 CONCLUDING REMARKS 99 THRESHOLDING AND
GCV APPLICABILITY IN MORE REALISTIC SITUATIONS . 101 5.1 SCALE
DEPENDENT THRESHOLDING 102 5.1.1 CORRELATED NOISE 102 5.1.2 LEVEL
DEPENDENT THRESHOLD ESTIMATION BY GCV 105 5.1.3 NON-ORTHOGONAL
TRANSFORMS 107 5.1.4 SCALE-ADAPTIVITY 107 5.2 NON-DECIMATED WAVELET
TRANSFORMS 107 5.3 INTERSCALE DEPENDENCIES 109 5.3.1 TREE-STRUCTURED
THRESHOLDING , 109 '5.3.2 AN EXAMPLE ILL 5.3.3 INTERSCALE CORRELATIONS
115 5.4 INTRASCALE DEPENDENCIES 115 5.5 HARD- AND SOFT-THRESHOLDING
REVISITED \ 118 5.5.1 RISK FOR HARD AND SOFT-THRESHOLDING \ 118 5.5.2
ORIGIN AND CHARACTER OF THE ERROR FOR HARD AND SOFT THRESH- OLDING 118
5.5.3 ESTIMATION OF THE OPTIMAL HARD THRESHOLD 121 5.6 TEST EXAMPLES AND
COMPARISON OF DIFFERENT METHODS 122 5.6.1 ORTHOGONAL TRANSFORM, WHITE
NOISE . 122 5.6.2 BIORTHOGONAL TRANSFORM, COLORED NOISE 123 5.7
INTEGER WAVELET TRANSFORMS 129 5.8 NON-GAUSSIAN NOISE 132 X CONTENTS 6.
BAYESIAN CORRECTION WITH GEOMETRICAL PRIORS FOR IMAGE NOISE REDUCTION
139 6.1 AN APPROXIMATION THEORETIC POINT OF VIEW 139 6.1.1 STEP FUNCTION
APPROXIMATION IN ONE DIMENSION 139 6.1.2 APPROXIMATIONS IN TWO
DIMENSIONS 141 6.1.3 SMOOTHNESS SPACES 143 6.1.4 OTHER BASIS FUNCTIONS
144 6.2 THE BAYESIAN APPROACH 144 6.2.1 MOTIVATION AND OBJECTIVES 144
6.2.2 PLUGGING THE THRESHOLD PROCEDURE INTO A FULLY RANDOM MODEL 145
6.2.3 THRESHOLD MASK IMAGES 146 6.2.4 BINARY IMAGE ENHANCEMENT METHODS
148 6.2.5 BAYESIAN CLASSIFICATION 149 6.3 PRIOR AND CONDITIONAL MODEL
150 6.3.1 THE PRIOR MODEL 150 6.3.2 THE CONDITIONAL MODEL 152 6.4 THE
BAYESIAN ALGORITHM 153 6.4.1 POSTERIOR PROBABILITIES 153 6.4.2
STOCHASTIC SAMPLING 154 6.5 PARAMETER ESTIMATION 155 6.5.1 PARAMETERS OF
THE CONDITIONAL MODEL 155 6.5.2 FULL BAYES OR EMPIRICAL BAYES 156 6.6
THE ALGORITHM AND ITS RESULTS 157 6.6.1 ALGORITHM OVERVIEW 157 6.6.2
RESULTS AND DISCUSSION 157 6.6.3 RELATED METHODS 158 6.6.4 POSSIBLE
EXTENSIONS T 159 6.7 ' SUMMARY AND CONCLUSIONS 160 7. SMOOTHING
NON-EQUIDISTANTLY SPACED DATA USING SECOND GENERATION WAVELETS AND
THRESHOLDING ' 161 7.1 THRESHOLDING SECOND GENERATION COEFFICIENTS 162
7.1.1 THE MODEL AND PROCEDURE 162 7.1.2 THRESHOLD SELECTION 162 7.1.3
TWO EXAMPLES 163 7.2 THE BIAS 166 7.2.1 THE PROBLEM 166 7.2.2 CONDITION
OF THE WAVELET TRANSFORM 166 7.2.3 WHERE DOES THE BAD CONDITION COME
FROM? 167 7.3 HOW TO DEAL WITH THE BIAS? 169 7.3.1 COMPUTING THE IMPACT
OF A THRESHOLD 169 7.3.2 HIDDEN COMPONENTS AND CORRELATION BETWEEN
COEFFICIENTS . 170 7.3.3 STABILIZING MODIFICATIONS 172 7.3.4 STARTING
FROM A FIRST-GENERATION SOLUTION 172 CONTENTS XI 7.3.5 THE PROPOSED
ALGORITHM 173 7.3.6 RESULTS AND DISCUSSION 175 BIBLIOGRAPHY 177 INDEX
188 SOFTWARE 193 |
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| isbn | 0387952446 |
| language | English |
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| series2 | Lecture notes in statistics |
| spelling | Jansen, Maarten Verfasser aut Noise reduction by wavelet thresholding Maarten Jansen New York [u.a.] Springer 2001 XIX, 191 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in statistics 161 Bruit électronique - Commande automatique Ondelettes Traitement du signal - Techniques numériques - Méthodes statistiques Active noise and vibration control Electronic noise Automatic control Signal processing Digital techniques Statistical methods Wavelets (Mathematics) Signalverarbeitung (DE-588)4054947-1 gnd rswk-swf Geräuschminderung (DE-588)4129292-3 gnd rswk-swf Bildverarbeitung (DE-588)4006684-8 gnd rswk-swf Wavelet (DE-588)4215427-3 gnd rswk-swf Signalverarbeitung (DE-588)4054947-1 s Geräuschminderung (DE-588)4129292-3 s Wavelet (DE-588)4215427-3 s DE-604 Bildverarbeitung (DE-588)4006684-8 s Lecture notes in statistics 161 (DE-604)BV002447846 161 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009371655&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jansen, Maarten Noise reduction by wavelet thresholding Lecture notes in statistics Bruit électronique - Commande automatique Ondelettes Traitement du signal - Techniques numériques - Méthodes statistiques Active noise and vibration control Electronic noise Automatic control Signal processing Digital techniques Statistical methods Wavelets (Mathematics) Signalverarbeitung (DE-588)4054947-1 gnd Geräuschminderung (DE-588)4129292-3 gnd Bildverarbeitung (DE-588)4006684-8 gnd Wavelet (DE-588)4215427-3 gnd |
| subject_GND | (DE-588)4054947-1 (DE-588)4129292-3 (DE-588)4006684-8 (DE-588)4215427-3 |
| title | Noise reduction by wavelet thresholding |
| title_auth | Noise reduction by wavelet thresholding |
| title_exact_search | Noise reduction by wavelet thresholding |
| title_full | Noise reduction by wavelet thresholding Maarten Jansen |
| title_fullStr | Noise reduction by wavelet thresholding Maarten Jansen |
| title_full_unstemmed | Noise reduction by wavelet thresholding Maarten Jansen |
| title_short | Noise reduction by wavelet thresholding |
| title_sort | noise reduction by wavelet thresholding |
| topic | Bruit électronique - Commande automatique Ondelettes Traitement du signal - Techniques numériques - Méthodes statistiques Active noise and vibration control Electronic noise Automatic control Signal processing Digital techniques Statistical methods Wavelets (Mathematics) Signalverarbeitung (DE-588)4054947-1 gnd Geräuschminderung (DE-588)4129292-3 gnd Bildverarbeitung (DE-588)4006684-8 gnd Wavelet (DE-588)4215427-3 gnd |
| topic_facet | Bruit électronique - Commande automatique Ondelettes Traitement du signal - Techniques numériques - Méthodes statistiques Active noise and vibration control Electronic noise Automatic control Signal processing Digital techniques Statistical methods Wavelets (Mathematics) Signalverarbeitung Geräuschminderung Bildverarbeitung Wavelet |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009371655&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002447846 |
| work_keys_str_mv | AT jansenmaarten noisereductionbywaveletthresholding |