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An Adaptive Chirplet Transform ADAPTIVE CHIRPLET TRANSFORM FOR THE ANALYSIS OF VISUAL EVOKED POTENTIALS by Jie Cui A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Institute of Biomaterials and Biomedical Engineering University of Toronto Copyright © 2006 by Jie Cui Abstract Adaptive Chirplet Transform for the Analysis of Visual Evoked Potentials Doctor of Philosophy, 2006 Jie Cui Institute of Biomaterials and Biomedical Engineering, University of Toronto Visual evoked potentials (VEPs) are electrical signals measured on the surface of the scalp in response to rapid and repetitive visual stimuli. These signals possess complex time-frequency structures and are difficult to characterize with conventional methods. In this work, we propose a new approach based on the adaptive chirplet transform (ACT) to represent a complete VEP response from the transient to the steady-state portion. Our implementation of the ACT involves both a windowed and a non-windowed approach. The non-windowed ACT employs a coarse-refinement algorithm to estimate multiple chirplets under low signal-to-noise ratio condition. The method decomposes VEPs into chirplet basis functions with four adjustable parameters (i.e., time-spread, chirp rate, time-center and frequency-center). We show how these parameters can be used to separate the transient from the steady-state portions of the response, and that as few as three chirplets are required to represent a complete VEP signal. In the windowed ACT method, the signal is partitioned into equal-length non-overlapping segments before estimating one chirplet from each segment. The concept of the optimal window length with reference to the windowed ACT method is proposed and calculations are made in terms of the signal characteristics. Both the windowed and non-windowed methods reveal a similar pattern of the VEP response – that a short transient VEP precedes the steady-state VEP. It is shown, however, that the computational time of the windowed method is significantly lower. Finally, we demonstrate that the adaptive chirplet spectrogram (ACS) offers a clearer visualization of the time-frequency structure of the VEP signal than the conventional spectrogram, because the ACS avoids cross-term interference in the time-frequency plane. Possible applications of VEP chirplet analysis to on-line signal classification and the technical limitations of the ACT approach are also discussed. ii Acknowledgements “Therefore, it does not matter whether “是故无贵无贱，无长无少，道之 a person is high or low in position, 所存，师之所存也。” young or old in age. Where there is the doctrine, there is my teacher.” —《师说》• 韩愈 —— Han Yu (768-824) I would like to express my gratitude to my supervisors Dr. Willy Wong and Dr. Hans Kunov for their guidance and support, and to my supervisory committee members, Dr. Milos R. Popovic and Dr. Steve Mann, for their continuing interest and suggestions in this work. I would also like to acknowledge the external examiners, Dr. Terrence Picton, Dr. Kenneth H. Norwich and Dr. William MacKay, from the University of Toronto, and the external appraiser, Dr. Rangaraj M. Rangayyan, from the University of Calgary. Of equal importance has been the camaraderie of the associates and students here at Sensory Communication Laboratory of the Institute of Biomaterials & Biomedical Engineering. I would like to thank Alberto Behar, Dr. Hilmi Dajani, Dr. Dave Purcell, Dr. Elad Sagi, Dr. Taha Jaffer, Ewen MacDonald, Kevin Cannons, Jason Lee, Graham Greenland, Gerry Fung, Jan Rubak, Fei Fan, and Hafiz Noordin for their friendship as well as teamwork. Finally, I wish to thank and dedicate this thesis to the members of my family. Most treasured of all is my wife, Dinghui Wang, whose love and support kept me inspired. Words alone cannot express how grateful I am to my parents for their moral support and encouragement when I am on this never-ending road of pursuing the truth. Richard Jie Cui Toronto iii Contents Abstract ................................................................................................................ii Acknowledgements...............................................................................................iii Contents...............................................................................................................iv List of Tables ......................................................................................................vii List of Figures....................................................................................................viii List of Abbreviations ............................................................................................x Mathematical Notations......................................................................................xii Chapter 1 Introduction ....................................................................................1 1.1 Motivation ..............................................................................................1 1.2 Current Status of EP Signal Processing..................................................5 1.3 Challenges............................................................................................. 10 1.4 Objective and Hypothesis ..................................................................... 12 1.5 Why Adopt Time-Frequency Analysis? ................................................ 14 1.6 Thesis Outline....................................................................................... 18 Chapter 2 MPLEM – An Adaptive Chirplet Transform ................................ 19 2.1 Introduction to the Gaussian Chirplet Transform ................................ 19 2.1.1 An overview of time-frequency analysis............................................. 20 2.1.1.1 The need for time-frequency analysis ......................................... 20 2.1.1.2 Logon and the short-time Fourier transform.............................. 23 2.1.1.3 “Wavelet” and the wavelet transform........................................ 26 iv 2.1.2 Chirplet and the Gaussian chirplet transform ................................... 29 2.2 The Adaptive Chirplet Transform (ACT) ............................................ 32 2.2.1 The MP algorithm............................................................................. 34 2.2.2 The LEM algorithm .......................................................................... 42 2.2.3 MPLEM algorithm of the ACT......................................................... 44 2.2.3.1 Signal model and CRLBs of chirplet estimates .......................... 46 2.2.3.2 Numerical simulation ................................................................. 50 2.2.4 Measures for stopping criterion and compactness.............................. 54 2.3 The Windowed Adaptive Chirplet Transform ...................................... 55 2.3.1 Computational method...................................................................... 57 2.3.2 Discussion.......................................................................................... 57 2.4 Summary .............................................................................................. 60 Chapter 3 Application of the Non-Windowed ACT ....................................... 61 3.1 Experimental Method ........................................................................... 61 3.1.1 Subjects............................................................................................. 62 3.1.2 Visual stimulus.................................................................................. 62 3.1.3 Apparatus and VEP recording .......................................................... 64 3.2 VEP Data Processing Results............................................................... 64 3.2.1 Number of chirplets in the dictionary ............................................... 64 3.2.2 Chirplet estimation ........................................................................... 66 3.2.3 Visualization...................................................................................... 68 3.3 Discussion ............................................................................................. 70 3.3.1 Model validation ............................................................................... 71 3.3.2 Compactness comparison................................................................... 76 3.3.3 Visualization effect............................................................................ 79 3.3.4 Separation of tVEP and ssVEP......................................................... 81 v 3.4 Conclusions........................................................................................... 83 Chapter 4 Application of the Windowed ACT .............................................. 86 4.1 Introduction.......................................................................................... 86 4.2 Optimal Window Length ...................................................................... 89 4.3 VEP Data Processing Results............................................................... 93 4.3.1 Number of chirplets in dictionary...................................................... 93 4.3.2 Chirplet estimation ........................................................................... 93 4.3.3 Visualization...................................................................................... 94 4.3.4 Statistical information....................................................................... 96 4.4 Discussion
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