Statistical Geometry Processing Winter Semester 2011/2012 A Very - - PowerPoint PPT Presentation

A Very Short Primer on Signal Theory

Statistical Geometry Processing

Winter Semester 2011/2012

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Topics

Topics

• Fourier transform
• Theorems
• Analysis of regularly sampled signals
• Irregular sampling

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Fourier Basis

Fourier Basis

• Function space: {𝑔: ℝ → ℝ, 𝑔 sufficiently smooth}
• Fourier basis can represent

– Functions of finite variation – Lipchitz-smooth functions

• Basis: sine waves of different frequency and phase:
• Real basis:

{sin 2𝜌𝜕𝑦 , cos 2𝜌𝜕𝑦 𝜕 ∈ ℝ

• Complex variant:

{𝑓−2𝜌𝑗𝜕𝑦 𝜕 ∈ ℝ

(Euler‘s formula: 𝑓𝑗𝑦 = cos 𝑦 + 𝑗 sin 𝑦 )

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Fourier Transform

Fourier Basis properties:

• Fourier basis: {𝑓−𝑗2𝜌𝜕𝑦 𝜕 ∈ ℝ
• Orthogonal basis
• Projection via scalar products  Fourier transform
• Fourier transform: (f: ℝ → ℂ) → F: ℝ → ℂ

𝐺(𝜕) = 𝑔 𝑦 𝑓−2𝜌𝑗𝑦𝜕𝑒𝑦

∞ −∞

• Inverse Fourier transform: F: ℝ → ℂ → (f: ℝ → ℂ)

𝑔(𝜕) = 𝐺 𝑦 𝑓2𝜌𝑗𝑦𝜕𝑒𝑦

∞ −∞

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Fourier Transform

Interpreting the result:

• Transforming a real function f: ℝ → ℝ
• Result: F 𝜕 : ℝ → ℂ
• 𝜕 are frequencies (real)
• Real input 𝑔:

Symmetric F −𝜕 = F 𝜕

• Output are complex numbers

– Magnitude: “power spectrum”

(frequency content)

– Phase: phase spectrum

(encodes shifts) 𝜕 = 𝑓−𝑗𝑦 𝜕 ∡𝜕 Im Re

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Important Functions

Some important Fourier-transform pairs

• Box function:

𝑔 𝑦 = box 𝑦 → 𝐺 𝜕 = sin 𝜕 𝜕 ≔ sinc 𝜕

• Gaussian:

𝑔 𝑦 = 𝑓−𝑏𝑦2 → 𝐺 𝜕 = 𝜌 𝑏 ⋅ 𝑓− 𝜌𝜕 2

𝑏

box(x) sinc(𝜕)

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Higher Dimensional FT

Multi-dimensional Fourier Basis:

• Functions f: ℝ𝑒 → ℂ
• 2D Fourier basis:

𝑔(𝑦, 𝑧) represented as combination of {𝑓−𝑗2𝜌𝜕𝑦𝑦 ⋅ 𝑓−𝑗2𝜌𝜕𝑧𝑧 𝜕𝑦, 𝜕𝑧 ∈ ℝ

• In general: all combinations of 1D functions

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Convolution

Convolution:

• Weighted average of functions
• Definition:

Example:



  

   dx t x g x f t g t f ) ( ) ( ) ( ) (

t g f





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Theorems

Fourier transform is an isometry:

• 𝑔, 𝑕 = 𝐺, 𝐻
• In particular 𝑔

= 𝐺

Convolution theorem:

• 𝐺𝑈 𝑔⨂𝑕 = 𝐺 ⋅ G
• Fourier Transform converts convolution into

multiplication

• All other cases as well:

𝐺𝑈−1 𝑔 ⋅ 𝑕 = 𝐺⨂G, 𝐺𝑈 𝑔 ⋅ 𝑕 = 𝐺⨂G, 𝐺𝑈−1 𝐺 ⋅ 𝐻 = 𝐺⨂G

• Fourier basis diagonalizes shift-invariant linear operators

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Sampling a Signal

Given:

• Signal 𝑔: ℝ → ℝ
• Store digitally:
• Sample regularly … 𝑔 0.3 , 𝑔 0.4 , 𝑔 0.5 …
• Question: what information is lost?

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Sampling

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Regular Sampling

Results: Sampling

• Band-limited signals can be represented exactly
• Sampling with frequency 𝜉𝑡:

Highest frequency in Fourier spectrum ≤ 𝜉𝑡/2

• Higher frequencies alias
• Aliasing artifacts (low-frequency patterns)
• Cannot be removed after sampling (loss of information)

band-limited aliasing

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Regular Sampling

Result: Reconstruction

• When reconstructing from discrete samples
• Use band-limited basis functions
• Highest frequency in Fourier spectrum ≤ 𝜉𝑡/2
• Otherwise: Reconstruction aliasing

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Regular Sampling

Reconstruction Filters

• Optimal filter: sinc

(no frequencies discarded)

• However:
• Ringing artifacts in spatial domain
• Not useful for images (better for audio)
• Compromise
• Gaussian filter

(most frequently used)

• There exist better ones,

such as Mitchell-Netravalli, Lancos, etc...

2D sinc 2D Gaussian

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Irregular Sampling

Irregular Sampling

• No comparable formal theory
• However: similar idea
• Band-limited by “sampling frequency”
• Sampling frequency = mean sample spacing

– Not as clearly defined as in regular grids – May vary locally (adaptive sampling)

• Aliasing
• Random sampling creates noise as aliasing artifacts
• Evenly distributed sample concentrate noise in higher frequency

bands in comparison to purely random sampling

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Consequences for our applications

When designing bases for function spaces

• Use band-limited functions
• Typical scenario:
• Regular grid with spacing 𝜏
• Grid points 𝐡𝑗
• Use functions: exp − 𝐲−𝐡𝑗 2

𝜏2

• Irregular sampling:
• Same idea
• Use estimated sample spacing instead of grid width
• Set 𝜏 to average sample spacing to neighbors