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Rank-Based Filter with a Frequency

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Rank-Based Filter with a Frequency MSc in Advanced Methods of Department of Computer Science Project Report 2008 Computer Science How effective is a rank-based filter with a frequency- and orientation- selective response? George Azzopardi August 2008 To my dear wife Charmaine Page i Acknowledgements I would like to express my gratitude to all those who gave me the possibility to complete this study. First of all, I would like to thank my supervisor, Dr. Fabrizio Smeraldi, for helping me in sorting out my ideas and for his constant kind assistance throughout this study. His incredibly broad knowledge-base and his systematic approach helped me to understand the complexities of such a study. I must say, that I have been honoured to share his outstanding experience in the field of Pattern Recognition and also his excellent way in approaching complex problems. I would like to thank the Malta Government Scholarship Scheme Board for awarding me a scholarship in order to pursue a postgraduate Masters degree with Queen Mary College. Furthermore, I would also like to express my gratitude to the management of Bank of Valletta for granting me a one year study leave from my place of work for making this experience possible. I am also grateful to Mr. Charles Theuma and Dr. Kenneth P. Camilleri who encouraged me to take this experience and for being present throughout the year. I also appreciate the proof reading performed by a special Maltese friend Trevor Spiteri. Special thanks go to my parents and parent’s-in-law for their kind support and concern which was of immense help throughout this academic year. Lastly, but not less, I would like to express my appreciation to my wife Charmaine for her encouragement and strong support throughout this year. Her love and presence were the ingredients needed to keep me focused on the studies especially during the last months. Page ii Abstract The pioneering study about the functions of the primary visual cortex by Hubel and Wiesel [38], motivated scientists to design machine vision models which tentatively simulate the same functions. In visual perception, there are two kinds of stimuli, usually referred to as first-order stimuli, characterized by a difference in luminance (the object tends to be lighter or darker than the background) and second-order stimuli that are characterized by difference in texture (the object and background share the same luminance but differ in texture). The literature shows that the acceptable models for first- and second-order stimuli are 2-D Gabor linear filters and the linear-nonlinear-linear (LNL) approach respectively. However, this is still not entirely satisfactory because first- and second-order stimuli are processed through different channels, which require an a priori knowledge of the nature of the image; this is not plausible either from the perceptual or from the application point of view. The existing orientation-selective, nonparametric features, called ranklets , which are based on the computation of Wilcoxon rank-sum test statistics [64] has motivated us to investigate the applicability of other rank statistics which would be suitable for both kinds of stimuli. This study proposes an innovative rank-based filter, with orientation- and frequency- selective response. The filter is based on an approximation of a 2-D Gabor filter and on a combination of the Wilcoxon (location) and the Siegel-Tukey (dispersion) non- parametric statistics. The promising results obtained from experiments on perceptual stimuli show that the proposed filter is sensitive to both first- and second-order stimuli. Page iii Table of Contents Acknowledgements...................................................................................................... ii Abstract....................................................................................................................... iii Table of Contents ....................................................................................................... iv List of Figures ........................................................................................................ vi List of Tables.......................................................................................................... vi List of Abbreviations................................................................................................. vii Chapter 1 Introduction ............................................................................................ 1 1.1 Vision ................................................................................................................... 2 1.2 Visual Pathway..................................................................................................... 2 1.2.1 Primary Visual Cortex (V1) ..................................................................... 3 1.2.2 Other Extra-Striate Regions ..................................................................... 4 1.3 Visual Texture...................................................................................................... 5 1.3.1 First-Order versus Second-Order Stimuli ................................................ 6 1.4 Motivation for the Investigation........................................................................... 7 1.5 Aims of the Project............................................................................................... 7 1.6 Changes to Original Proposal............................................................................... 7 1.7 Project Plan .......................................................................................................... 7 1.8 Organisation of the Dissertation........................................................................... 8 Chapter 2 Perceptual Models .................................................................................. 9 2.1 Introduction ........................................................................................................ 10 2.2 First-Order Stimuli............................................................................................. 10 2.3 Second-Order Stimuli......................................................................................... 12 Chapter 3 Analytical Tools .................................................................................... 16 3.1 Introduction ........................................................................................................ 17 3.2 Linear Filters ...................................................................................................... 17 3.2.1 Fourier Transform .................................................................................. 17 3.2.2 Continuous Fourier Transform............................................................... 18 3.2.3 Discrete Fourier Transform (DFT)......................................................... 18 3.2.4 Filtering in the Spatial Domain .............................................................. 19 3.2.5 Filtering in the Frequency Domain ........................................................ 19 3.2.6 Complex 2-D Gabor Linear Filter.......................................................... 20 3.3 Rank-Based Filters ............................................................................................. 21 3.3.1 Non-Parametric Statistics....................................................................... 21 3.3.2 Mann-Whitney-Wilcoxon (MWW) Test Statistics ................................ 22 3.3.3 Rank-Based Features.............................................................................. 23 Chapter 4 Hypothesis ............................................................................................. 25 4.1 Research Gap...................................................................................................... 26 4.2 Hypothesis.......................................................................................................... 26 Chapter 5 Methodology.......................................................................................... 27 5.1 Introduction ........................................................................................................ 28 5.2 Filter Design....................................................................................................... 28 5.2.1 Approx 1: Naïve Approximation of Complex 2-D Gabor Filter............ 28 5.2.2 Approx 2: Adaptive Approximation of Complex 2-D Gabor Filter....... 29 5.2.3 Approx 3: Weighted Gabor Coefficients ............................................... 30 Page iv 5.3 Non-Parametric Statistics: Location and Dispersion Tests................................ 31 5.3.1 Ansari-Bradley Dispersion Test............................................................. 31 5.3.2 Siegel-Tukey Dispersion Test ................................................................ 32 5.3.3 Combined Location and Dispersion Test Statistics ............................... 32 5.3.3.1 Innovative Combination of Wilcoxon-Siegel-Tukey (WST) Test 33 5.3.4 Treatment of Ties ................................................................................... 33 5.3.5 Ranklet Calculation................................................................................ 34 5.4 Rank-Based Filtering in the Spatial Domain...................................................... 36 5.5 Conclusion.......................................................................................................... 37 Chapter 6 Implementation and Results...............................................................
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