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Reconstruction of Undersampled Signals and Alignment in The RECONSTRUCTION OF UNDERSAMPLED SIGNALS AND ALIGNMENT IN THE FREQUENCY DOMAIN by David M. Simpson June, 1988 A thesis submitted for the degree of Doctor of Philosophy. Department of Electrical Engineering Imperial College London, SW7 ABSTRACT This thesis describes a technique for the digital reconstruction of one and two dimensional signals from a series of undersampled versions. To do the necessary alignment of the signals, a novel technique of shift estimation is developed, based on the phase of the Fourier components. The effects of windowing, when finite length signals are processed, are also studied. The reconstruction technique- is based on the frequency domain, and requires that the signal is bandlimited and the total number of samples is adequate. The technique is evaluated in the presence of noise and with alignment errors. The relative shift between the sampled signals is found to sub-sample resolution using the phase of consecutive harmonics. Particular attention is paid to the effect of aliasing on the delay estimate. The use of data windows for processing finite length signals is investigated for both the signal alignment and reconstruction methods. The usefulness and reliability of the complete reconstruction algorithm is evaluated on simulated one and two dimensional signals and a number of real images. It is concluded that the shift estimation technique developed provides an efficient and useful new algorithm. The investi­ gation into windowing shows that the required window properties here are different to the usual ones in spectral estimation. The reconstruction technique was found to be sensitive to noise and errors in alignment but could recover detail lost in digitizing the signals. 2 CONTENTS Page Title Page 1 Abstract 2 Contents 3 List of Figures 8 List of Tables 14 Acknowledgement 16 Statement of Originality 17 1. Introduction 19 1.1. BACKGROUND TO THE TECHNIQUES PRESENTED 19 1.2. THEORETICAL BACKGROUND 34 1.3. SUMMARY 54 2. Signal Reconstruction 55 2.1. INTRODUCTION 55 2.2. LITERATURE REVIEW 59 2.2.1. Introduction 59 2.2.2. Reconstruction from a Single Aliased Signal 60 2.2.3. Reconstruction from Non-uniformly Sampled Signals 62 2.3. BL RECONSTRUCTION OF ONE DIMENSIONAL SIGNALS 71 2.3.1. First Order Aliasing 71 2.3.2. Higher Order Aliasing 76 2.4. EVALUATION OF THE RECONSTRUCTION TECHNIQUE 83 2.4.1. Introduction 83 2.4.2. Simulations with Noisy SIGNALS 87 3 2.4.3. Simulations with Errors in Delay Estimates 90 2.4.4. Simulations with Both Noisy Signals and Errors in Delay Estimates 93 2.4.5. Prediction of Noise in the Reconstruction due to Noisy Input Signals 95 2.4.6. Prediction of Distortion in the Reconstruction due to Errors in the Delay Estimates 100 2.4.7. Summary 103 2.5. BL RECONSTRUCTION OF TWO DIMENSIOANL SIGNALS 105 2.6. SUMMARY AND CONCLUSION 113 3. Signal Alignment 117 3.1. INTRODUCTION 117 3.2. REVIEW OF SIGNAL ALIGNMENT TECHNIQUES 122 3.2.1. Introduction 122 3.2.2. Techniques in the Discrete Space or Time Domain 122 3.2.3. Cross-correlation Methods 125 3.2.4. Techniques Based on the Phase of the Fourier Transform 129 3.2.5. Discussion of Delay Estimation from Phase 134 3.3. THE PCF ESTIMATOR FOR ONE DIMENSIONAL SIGNALS WITHOUT ALIASING 141 3.3.1. Introduction 141 3.3.2. Phase Unwrapping 143 3.3.3. Estimation of the Gradient 148 3.3.4. Evaluation of the PCF Estimator 153 3.3.5. Summary 161 3.4. THE EFFECT OF ALIASING ON THE DELAY ESTIMATE 162 4 3.5. MODIFIED PCF ESTIMATOR FOR ONE DIMENSIONAL ALIASED DATA 168 3.6. THE PCF ESTIMATE IN TWO DIMENSIONS 174 3.7 . SUMMARY AND CONCLUSIONS 179 4. Windowing 182 4.1. INTRODUCTION 182 4.1.1. Windowing in Delay Estimation 182 4.1.2. Literature Review 185 4.2. WINDOWING AND THE ESTIMATION OF SIGNAL PHASE AND PHASE DIFFERENCE 195 4.2.1. Introduction 195 4.2.2. Windowing and the Estimation of Phase 196 4.2.3. Windowing and the Phase Difference of Delayed Signals 204 4.3. OPTIMAL WINDOWS FOR DELAYED SIGNALS 218 4.3.1. Introduction 218 4.3.2. The Least Mean Square Error Criterion 219 4.3.3. The Least Mean Square Error Criterion Applied to Tukey and Trapezium Windows 224 4.4. SUMMARY AND CONCLUSIONS 230 5. Evaluation of Alignment and Reconstruction Techniques 231 5.1. INTRODUCTION 231 5.2. ONE DIMENSIONAL SIGNALS 234 5.2.1. Introduction 234 5.2.2.. PCF Delay Estimates 235 5.2.3.. BL Reconstruction 251 5 TWO DIMENSIONAL SIGNALS 261 Int roduct ion 261 Evaluation of Two Dimensional PCF Alignment 268 BL Reconstruction of Correctly Aligned Signals 279 BL Reconstruction with PCF Shift Estimates 283 SUMMARY 294 Summary, Conclusions and Suggestions for Future Work 298 References 305 Appendices Markov 1 Signals 315 Listing of Markov 1 Generator 318 Listing of LU-factorization for the Solution of Simultaneous Equations 319 Listing of the BL Reconstruction Algorithm for n-th Order Aliasing 325 Listing of the Two Dimensional BL reconstruction Algorithm 330 Power in BL Reconstructed Signals 337 Distortion in BL Reconstructed Signals due to Inaccurate Delay Estimates 339 Variance in the Phase of Signals with Added Noise 341 Sequential Minimum Variance Estimator 344 Variance in Predicted Phase Difference 346 Listing of the One Dimensional PCF Alignment Algorithm 348 6 3.5. Listing of the Two Dimensional PCF Alignment Algorithm 352 7 LIST OF FIGURES Reconstruction of undersampled signals 20 Windowing a shifted image 22 A digital X-ray imaging system 27 A complex number in the complex plane 36 Sampling and aliasing in the time and frequency domain 40 A periodic signal and a circularly delayed version 44 A signal with a data window 45 Examples of Markov 1 signals 51 Power spectra of Markov 1 signals 52 Composing functions for a recurring group of two samples 64 A Time-Division-Multiplex frame 67 Output signal-to-noise ratio for BL recon­ struction and cubic spline interpolation with noisy input signals (40dB) 88 Output signal-to-noise ratio for BL recon­ struction and cubic spline interpolation for noisy input signals (20dB) 89 Output signal-to-noise ratios for BL recon­ struction and cubic spline interpolation with inaccurate delay estimates (p = 0.0) 90 Output signal-to-noise ratios for BL recon- -8- struction and cubic spline interpolation with inaccurate delay estimates (p = 0.9) 91 2.7 Output signal-to-noise ratios for BL recon­ struction with inaccurate delay estimates 92 2.8 Output signal-to-noise ratios for cubic spline interpolation with inaccurate delay estimates 93 2.9 Output signal-to-noise ratios for BL recon­ struction with inaccurate delay estimates and noisy data 94 2.10 Predicted output signal-to-noise ratios and experimental results with noisy input signals 99 2.11 Predicted output signal-to-noise ratios and experimental results with inaccurate delay estimates 102 2.12 Aliasing in two dimensional spectra 105 3.1 Sampled, delayed signals and the superimposed samples 118 3.2 Unwrapped phase difference 131 3.3 Phase of the cross-correlation function, averaged in the frequency domain 137 3.4 Errors in phase difference due to frequency domain averaging 138 3.5 Errors in phase difference with noisy input data and frequency domain averaging 139 3.6 Mean PCF and PX delay estimates for noisy Markov 1 data 154 3.7 Standard deviation of PCF and PX delay -9- estimates for noisy (20dB) Markov 1 data 155 Standard deviation of PCF and PX delay estimates for noisy (40dB) Markov 1 data 156 Standard deviation of PX delay estimates for noise free and noisy (20dB) Markov 1 signals 157 Standard deviation of PCF and PX delay estimates for one Markov 1 signal and added noise 159 Mean and standard deviation of PCF and PX delay estimates for noise free and aliased Markov 1 data 165 Phasor diagram for aliased and delayed signals 165 Mean PCF delay estimates with a range of frequency dependent weightings for aliased Markov 1 data 169 Standard deviation of PCF delay estimates with a range of frequency dependent weightings for aliased Markov 1 data 170 Mean PCF and PX delay estimates for aliased Markov 1 data 172 Standard deviation of PCF and PX delay estimates for aliased Markov 1 data 173 PCF motion estimation in two dimensional signals 176 Data windows in delayed signals 183 Tapered data windows with a range of rise times 186 Amplitude spectra of trapezium windows 187 -10- Amplitude spectra of Tukey windows 188 The transform of the Hanning window as the sum of sine functions 190 The phase of a windowed cosine function 197 The phase of a windowed cosine function with DC offset 201 Phase difference for delayed, windowed signals consisting of a single harmonic 206 Phase difference for delayed, windowed signals consisting of a single harmonic and added DC offset 208 Phasor diagram showing error in phase difference as a result of windowing 211 Mean error in phase difference due to windowing, as a function of frequency 214 Standard deviation in phase difference as a function of frequency 216 Windowing with wraparound 223 The mean square error in trapezium windows 226 The normalized mean square error for trapezium and Tukey windows 227 Minimum error values for Tukey and trapezium windows 228 Mean and standard deviation of PCF2 delay estimates for undersample Markov 1 signals 236 PCF delay estimates for undersampled Markov 1 signals (integer delay) 238 -11- PCF1 and PCF2 delay estimates for undersampled Markov 1 signals (integer delay) 242 PCF delay estimatest for undersampled Markov 1 signals 245 PCF1 and PCF2 delay estimates for undersampled Markov 1 signals 248 SNR1 and SNR2 for BL reconstructed Markov 1 signals 253 SNR1 of BL reconstructed Markov 1 signals using correct delay 254
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