Organization and Introduction - TUM · Motivation Organizational Things An Overview updated...

Organization andIntroduction

Michael Moeller

Motivation

Organizational Things

An Overview

updated 14.04.2015

Chapter 0Organization and IntroductionNonlinear Multiscale Methods for Image and Signal AnalysisSS 2015

Michael MoellerComputer Vision

Department of Computer ScienceTU Munchen


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

What do you hear?

images/sound.wav


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

Let’s look at the sound signal ...

0 0.05 0.1 0.15 0.2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Soundwave of a guitar string

Time (sec)

Am

plitu

de


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

Much better: DCT coefficients!

0 200 400 600 800 1000 12000

5

10

15DCT coefficients of sound signal

Frequency (Hz)

Mag

nitu

de


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

What do you see?


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

What do you see?


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

Shouldn’t there be a ’spectral’ representation with three peaks?

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500Spectrum of a TV decomposition

Time (ISS flow)

Am

plitu

de


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation


0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000


Time (ISS flow)

Am

plitu

de


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation


0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000


Time (ISS flow)

Am

plitu

de


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

What could spectral representations be good for?

0 20 40 60 80 100 120 140 160 180 200−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x

ampl

itude

Input signal


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation


0 20 40 60 80 100 120 140 160 180 200

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

frequency

mag

nitu

de

DCT of input signal


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation


0 20 40 60 80 100 120 140 160 180 200

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

frequency

mag

nitu

de

DCT of input signal


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation


0 20 40 60 80 100 120 140 160 180 200

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

frequency

mag

nitu

de

DCT of input signal


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation


0 20 40 60 80 100 120 140 160 180 200−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x

ampl

itude

Filtered signal


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation


0 20 40 60 80 100 120 140 160 180 200−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x

ampl

itude

Input signal


Michael Moeller

Motivation


An Overview

updated 14.04.2015

MotivationLooks boring for cosine signals?


Michael Moeller

Motivation


An Overview

updated 14.04.2015



Michael Moeller

Motivation


An Overview

updated 14.04.2015



Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

Looks boring for cosine signals?


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

Looks boring for cosine signals?


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

GUI


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Motivation

How does this work?


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Variational Approach

u(t) = arg minu

12‖u − f‖2

2 + t J(u)

φ(t) = t ∂ttu(t), ψ(t) = φ(1/t)1t2

Forward Flow

∂tu(t) = −p(t), p(t) ∈ ∂J(u(t))

φ(t) = t ∂ttu(t), ψ(t) = φ(1/t)1t2

Inverse Scale Space Flow

∂tp(t) = f − v(t), p(t) ∈ ∂J(v(t))

ψ(t) = ∂tv(t), φ(t) = ψ(1/t)1t2

Filtering

ufiltered =

∫ ∞

0ω(t)ψ(t) dt


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Organizational Stuff


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Requirements, or “is this something for me?”

Necessary

• Interest in mathematical theory• Image processing and convex analysis• Numerics (Matlab)

Nice to know

• Optimization• Partial differential equations


Michael Moeller

Motivation


An Overview

updated 14.04.2015


For those who heard Variational Methods for Computer Vision

• It will become more theoretical and more mathematicallychallenging.

• We will prove theorems on the board.• You’ll have an advantage in terms of possible applications

and their numerical implementation.

For those who heard Ill-posed Problems

• We will need very little functional analysis – everything willbe in Rn.

• You’ll have an advantage in terms of the theoreticalmathematical concepts.

• I highly recommend learning how to implement thediscussed approaches. It is useful and fun!


Michael Moeller

Motivation


An Overview

updated 14.04.2015


For those who heard Variational Methods for Computer Vision

• It will become more theoretical and more mathematicallychallenging.

• We will prove theorems on the board.• You’ll have an advantage in terms of possible applications

and their numerical implementation.

For those who heard Ill-posed Problems

• We will need very little functional analysis – everything willbe in Rn.

• You’ll have an advantage in terms of the theoreticalmathematical concepts.

• I highly recommend learning how to implement thediscussed approaches. It is useful and fun!


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Formalities

Exercises

• There are no exercises for this lecture.• Occasionally: theory or programming problems.• Solution one week later in the lecture.• The more we discuss in the lecture, the more interesting

the course will be! Please don’t be shy to say something!• We’ll use OnlineTed to make the lecture more interactive.


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Formalities

Examination

• Depending on the number of attendees, the final exam willbe either oral or written.

• ECTS credits: 4


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Formalities

Miscellaneous

• My office: 02.09.061• Office hours: Tuesday 4–5pm• Lecture: Starts at 2:15pm. Short break in between.• Course website: https://vision.in.tum.de/teaching/ss2015/multiscale_methods

https://vision.in.tum.de/teaching/ss2015/multiscale_methods

https://vision.in.tum.de/teaching/ss2015/multiscale_methods


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Overview


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Introduction

Step 1: Make sure we are all on the same page!

Basics of convex analysis:• Convex extended real valued functions in Rn

• Minimization problems (existence, optimality condition)• Duality, Saddle point problems

Goal: Everyone knows all necessary tools to follow the lecture!


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Introduction

Step 2: Make sure everyone can try out what we are doing!

An optimization method for non-smooth convex minimization• Praxis oriented – focus on a Matlab implementation.• Idea rather than convergence analysis.

Goal: Everyone can try out him-/herself what we are doing!


Michael Moeller

Motivation


An Overview

updated 14.04.2015

Multiscale decomposition

Step 3: Analyzing multiscale methods!

3.1 Classical theory: How does linear filtering work?

• Transform signal to different representation.• Filter coefficients.• Transform back.

→ Analyze behavior via eigendecomposition!


Michael Moeller

Motivation


An Overview

updated 14.04.2015



3.2 A nonlinear singular vector analysis

• Is there any analogy to singular vectors for nonlinearregularization methods?

• What properties do nonlinear singular vectors have?


Michael Moeller

Motivation


An Overview

updated 14.04.2015



3.3 Nonlinear variational methods

• Define a spectral decomposition for one-homogeneousregularizations.


Michael Moeller

Motivation


An Overview

updated 14.04.2015



3.4 Nonlinear scale space flows

• Analyze behavior of scale space flows.• Define a spectral decomposition for one-homogeneous

regularizations.


Michael Moeller

Motivation


An Overview

updated 14.04.2015



3.5 Nonlinear inverse scale space flows

• Analyze behavior of inverse scale space flows• Define a spectral decomposition for one-homogeneous

regularizations.


Michael Moeller

Motivation


An Overview

updated 14.04.2015



3.6 Properties of spectral decomposition methods

• Under which conditions do the spectral decompositionapproaches yield a discrete spectrum?

• Under which conditions are the three spectraldecomposition approaches equivalent?

• Under which conditions do we obtain a nonlineareigenvalue decomposition?

Date post:	11-Aug-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Organization and Introduction - TUM · Motivation Organizational Things An Overview updated...

Documents