Multidimensional signal recognition, invariant to affine ... - DiVA

Multidimensional signal recognition, invariant to ane transformation and time-shift, using canonical correlation Thesis project done at Computer Vision Laboratory Linkoping University Sweden by

Bjorn Johansson

Reg. nr: LiTH-ISY-EX-1825

Supervisor: Magnus Borga Examiner: Hans Knutsson Linkoping, October 6, 1997

Contents 1 Introduction 1.1 1.2 1.3 1.4

Acknowledgements . Notation . . . . . . . Previous knowledge . Outlines . . . . . . .

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2 Canonical correlation

2.1 De nition and a short explanation . . . . . . . . . . . . . . . . . . 2.2 The gradient-search algorithm . . . . . . . . . . . . . . . . . . . . .

3 What this thesis is about 4 LP- ltering signals

4.1 Experiments . . . . . . . . . . . . . . . . . . 4.1.1 One-dimensional signals . . . . . . . 4.1.2 Multidimensional signals . . . . . . . 4.2 An ounce of theory #1 . . . . . . . . . . . . 4.2.1 Multidimensional frequencies . . . . 4.2.2 Time shift and linear transformation 4.3 Discussion . . . . . . . . . . . . . . . . . . .

5 LP- ltering canonical correlation

5.1 An idea . . . . . . . . . . . . . . . . . . . 5.2 Experiments . . . . . . . . . . . . . . . . . 5.3 An ounce of theory #2 . . . . . . . . . . . 5.3.1 Covariance and signal frequencies . 5.3.2 Filtered canonical correlation . . . 5.3.3 One step further . . . . . . . . . . 5.4 Discussion . . . . . . . . . . . . . . . . . .

6 Signal norm

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4 4 4 5 5

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6.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2 An ounce of theory #3 . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2

7 Round-up

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A Proof of theorem 4.1

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7.1 Summary and comparisment . . . . . . . . . . . . . . . . . . . . . 38 7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3

Chapter 1

Introduction 1.1 Acknowledgements I would like to thank my supervisor licentiate Magnus Borga and my examiner associate professor Hans Knutsson for many inspiring discussions and ideas that has moved the work forward (I got no help from literature - could not nd any concerning this topic). I would also like to thank Magnus Borga for all constructive criticism in proof-reading my thesis. Thank you also to the rest of the sta at the Computer Vision laboratory for helping me with practical things during the work. Thanks also to Stefan Eriksson for taking on the job as opponent. Finally I would like to thank the word processing program LATEX and the mathematical software MATLAB for being cooperative (most of the time).

1.2 Notation Italics (x) are used for scalars. Lowercase letters in boldface (x) are used for vectors. Uppercase letters in boldface (X) are used for matrices. Transpose is denoted by a 'T ' (e.g. xT ). Adjungation (transpose + conjugation) is denoted by a '' (e.g. x ). '^' over a vector indicates unit length (e.g. jv^ j = 1.)