Þ,1.o2 Stochastic Process¡ng for Enhancement of Artificial lnsect Vision by Gregory P. Harmer B.Sc. (Applied Maths & Computer Science), Adelaide University, Australia, 1996 B.E. (First Class Honours), Adelaide University, Australia, 1997 Thesis submitted for the degree of Doctor of Philosophy tn Department of Electrical and Electronic Engineering, Faculty of Engineering, Computer and Mathematical Sciences Adelaide University, Australia November, 2001 CBME @ 2001 ADELAIDE UNIVERSITY Gregory P. Harmer AUSTRALIA Cøto ftr Bion¡dlc¡l r"ñæino Àtll¡irlD U¡¡wity All Rights Reserved Contents Heading Page Contents t¡t Abstract tx Statement of Originality x¡ Acknowledgments xilt Publications xv List of Figures xvii List of Tables xx¡t¡ Chapter 1. lntroduction and Motivation 1 1.L Thesis Layout 2 L.2 Introduction 3 1.3 Background 4 L.3.1 Motion Detection . 6 1..4 Motivation 7 1.4|1, Millimetre Wave Antenna Array 9 L.5 StochasticProcessing 12 L.5.1 BrownianRatchets 12 1,.5.2 Stochastic Resonance 13 1,.6 Current State of Knowledge 1.4 L.7 Original Contributions 15 Chapter 2. Brownian Ratchets t7 2.1, Brownian Motion L8 Page iii Contents 2.2 Rectifying Brownian Motion . 19 Laws 2.2.L of Thermodynamics . 20 2.2.2 Ratchet and Pawl Device 2L 2.2.3 Maxwell's Demon 24 2.2.4 Utilising Thermal Fluctuations 27 2.3 Brownian tchets 28 2.3.7 FlashingRatchets 30 A 2.3.2 Very Brief Review 31 2.3.3 Examples of Ratchets . 31 2.4 Summary 35 Chapter Parrondo's 3. Games 37 Introduction 3.1 38 Construction 3.2 and Characteristics 39 3.2.7 Construction of the Games 39 3.2.2 Playing the Games 47 3.2.3 Faimess 41 3.2.4 Distributions and Behaviour 45 3.2.5 MixingStrategies 47 3.2.6 Explanation of the Games 49 3.2.7 Observations 52 3.3 Analysis of the Games 54 Discrete-Time 3.3.1 Markov-Chains (DTMCs) 54 3.3.2 Modelling the Games as DTMCs 57 3.3.3 Constraints of Parrondo's Games . 63 3.3.4 Rate of Winning 67 3.4 Entropy 69 3.4.1, Entropy and Probability Space 71, 3.4.2 Entropy Paradox 73 3.5 History Dependent Games 74 3.5.1 Construction 74 Page iv Contents 3.5.2 Results and Distributions 75 3.5.3 Analysis of the Games 76 3.5.4 Probability Space 81 3.6 Other Phenomena . 83 3.6.1 Co-operative Parrondo's Games 83 3.6.2 Fractal Properties of Parrondo's Games 83 3.6.3 Parrondo's Games are Ubiquitous 86 3.6.4 Related Works 88 3.7 Summary . 88 Chapter 4. Stochastic Resonance 91 4.L Introduction 92 4.1.1. The Basics 93 4.1.2 Examples of SR 98 4.2 Systems Demonstrating SR 1,09 4.2.1, ThresholdDevices 110 4.2.2 Neuron Models L11 4.2.3 BistableDynamics 115 4.2.4 Other Nonlinear Systems r20 4.3 Noise 120 4.3.1 \Atrhite - Gaussian 120 4.3.2 Coloured -Ornstein-Uhlenbeck r21. 4.3.3 Generation of Random Numbers t22 4.3.4 Simulating Stochastic Systems 124 4.4 Quantifying SR 124 4.4.1. Signal-to-Noise Ratio L25 4.4.2 Residence Time Distribution 127 4.4.3 Correlation Coefficients 128 4.4.4 Average Mutual Information t32 4.4.5 Channel Capacity . 139 4.5 Noise Related Phenomena 1.44 4.5.1, Noise Linearisation . 1,44 4.5.2 Phase Space . 145 4.6 Summary 1.47 Page v Contents Chapter 5. Motion Detection 149 5.1 Introducti . 150 5.1.1 Local antl Widefield Configuration . 151 5.t.2 Classification of Detection Schemes . 752 5.1.3 Criteria for a Bidirectional Detector . 153 5.2 Biological Processing Schemes . 754 5.2.1, Reichardt's Delay and Compare Detector . L54 5.2.2 Barlow and Levick's Inhibitory Mechanism . t60 5.2.3 Directionally selective Local Inhibitory Motion Detector . 165 5.2.4 Horridge's Têmplate Model . t67 5.2.s Gradient Schemes . 172 Rudimentary 5.3 Noise Analysis . 174 Summary 5.4 . 174 Chapter processing 6. Stochastic Motion L77 6.1, Schemes Investigated . L78 6.2 Architecture of Detectors . 178 6.2.1, Template Model Configurations 779 Effectiveness 6.3 of Additive Noise 18L 6.3.1 Individual Widefield Characteristics t82 6.3.2 Parallel Network of Detectors . L87 6.3.3 Detector Linearity L87 parameter 6.3.4 Suboptimal Settings 188 6.3.5 DistributedThresholdsettings 192 6.3.6 Oversampling 193 6.4 Limitations of SR in Motion Detection 194 6.5 Summary 795 Chapter 7. Conclusions and Directions 197 7.1, Summary 198 7.2 Future Directions 200 7.3 Closing Comments 202 Page vi Contents Appendix A. Parrondo's Games Analysis 203 4.1 Constraints for the Generalised modulo M Game . 203 A.2 Calculating the Equilibrium Distribution 207 4.3 Rate of Winning . 209 A.4 Quasi-Birth-and-Death Analysis 210 Appendix B. SR Calculations 2r3 8.1 Random Number Generators 2L3 8.2 Numerically Integrating SDEs 215 8.3 AMI Expressions for Gaussian Signals and Noise . 277 8.3.1 Preliminary Calculations 217 8.3.2 Simplifying the average mutual information 220 8.4 Shannon's Channel Capacity 221. Appendix C. Lowpass & Highpass Filters 223 Bibliography 229 Glossary 247 lndex 249 Resume 253 Page vii Page viii Abstract taditionally, noise has been an enemy of engineers - its unpredictability limiting the performance of many types of systems. It places an upper limit on the data transfer rate of communications systems, plays havoc with sensors and is difficult to avoid. The traditional approach to improve performance is to minimise the effects of noise andbuild redundancies into a system. Flowever, within the last 10-20 years, new fields have emerged that use noise to enhance a system's performance. Two such fields that we will examine are stochastic resonance and Brownian ratchets. This thesis explores these noise enhancing phenomena with the intent of applying them to motion detection schemes. We consider the classical Reichardt detector, an inhibitory scheme based on a shunting inhibition neurons, and the Horridge template model. All the schemes have a biological background; this is to take advantage of the robust and simple algorithms produced by many years of evolution. The first half of this thesis looks at Brownian ratchets - devices that use noise to gen- erate directed motion of Brownian particles. The concept of Brownian ratchets dates back to the discovery of Brownian motion, where a number of schemes were proposed to harness the random fluctuations. One of the proposed rectification devices, referred to as the "ratchet and pawl machine," showed that when an external energy source is available, thermal noise is able to generate directed motion. It was this device that eventually inspired the development of Brownian ratchets. Many types of Brownian ratchets have since been constructed and the theory devel- oped. Based on this theory, we expound a set of mathematical games that mimic the behaviour of the Brownian ratchet. Called "Parrondo's games," th"y are two gambling games that have the counter-intuitive property of losing when played individually, but when ptayed in a random order produce a winning expectation. The games are ex- plained heuristically and via a mathematical analysis, which show this result is indeed the case. Though they do not have any direct link to utilising noise for motion detec- tion, the concept was an instructive example of using noise or randomness to enhance the performance of a system and led us to look more closely at stochastic resonance. The second half of this thesis deals with stochastic resonance and its application to mo- tion detection. Stochastic resonance (SR) is the phenomenon where the performance Page ix Abstract of a nonlinear system is optimised with the addition of noise. The noise improves the coherence between the input and output signals. We consider a variety of non- linear systems and explain the clifferent measures, including the channel capacity of SR. Different types of SR are also described, in particular suprathreshold SR, which has potential for the application of motion detection. Flowever, whatever the measure or type of SR used, a system operating optimally carurot benefit from the addition of noise. The motion detection schemes used are simple, biologically inspired algorithms that specifically detect edges. By using only local information, the detector can be im- plemented as a smart sensor. We present a quantitative noise analysis of the three schemes, and employ the framework used in SR. The results show that for an optimally configured scheme, noise does not help. Flowever, by utilising the network structure of suprathreshold SR, the performance is improved. By constraining the threshold in one of the schemes, the performance of the detector is improved by adding noise. In many real systems, there are trade-offs, and unavoidable intemal noise can cause suboptimal operation of a system. It is in these cases where stochastic resonance be- comes beneficial. We know that biological sensory neurons are very noisy and utilise SR, which they have achieved after millions of years of evolution. Thus, not all noise is bad. Page x Page xii Acknowledgments Many people have made undertaking this Ph.D. an enjoyable and memorable experi- ence. This includes fellow sfudents, colleagues and friends. Firstly, I would like to extend my thanks to my supervisor, Dr Derek Abbott, who has supported my work throughout my Ph.D. He has been particularly helpful in bringing to light ideas from a range of diverse fields that I may not have otherwise been aware of.