Linear Systems 2: Fourier transform, filtering, convolution theorem - - PowerPoint PPT Presentation

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COMP 546

Lecture 17

Linear Systems 2:

Fourier transform, filtering, convolution theorem

• Tues. March 20, 2018

Recall last lecture

• convolution
• special behavior of sines and cosines
• complex numbers and Euler’s formula

Today

• Fourier transform
• convolution theorem
• filtering

Key idea from linear algebra:

• rthonormal basis vectors

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• 1

1 1 = 𝑣 𝑤 𝑦 𝑧

1 2

1 1

• 1 1

= 𝑣 𝑤 𝑦 𝑧

Example: 1 2 𝑣 𝑤 𝑦 𝑧

Fourier transform uses

• rthogonal basis vectors

1 1 1 -1 = 𝐽 0 𝐽(1) 𝐽 0 𝐽(1)

Example: 𝐽 0 𝐽 1 𝐽 0 𝐽 1

1 1 1 -1 = 𝐽 0 𝐽(1) 𝐽 0 𝐽(1)

1 2

(1D) Fourier analysis

Fourier transform

map N-dimensional delta function (impulse function) basis to an N-dimensional sinusoid function basis

Inverse Fourier transform

𝐽 0 : : 𝐽(𝑂 − 1)

𝐆−1 𝐆

𝐽 0 : : 𝐽(𝑂 − 1)

𝐽 𝑙 =

𝑦=0 𝑂−1

cos 2𝜌 𝑂 𝑙𝑦 − 𝑗 sin 2𝜌 𝑂 𝑙𝑦 𝐽 𝑦

Fourier Transform

𝐽 𝑙 = 𝐆 𝐽 𝑦

𝑓 −𝑗 2 𝜌

𝑂 𝑙 𝑦

Fourier transform

𝐽 𝑙 = 𝐆 𝐽 𝑦

Define 𝑂 x 𝑂 Fourier transform matrix

Claim: (see lecture notes for proof)

where

𝐆𝑙,𝑦 ≡ 𝑓 −𝑗 2 𝜌

𝑂 𝑙𝑦

𝐆−1 = 1 𝑂 𝐆

𝐆𝑙,𝑦 ≡ 𝑓 𝑗 2𝜌

𝑂 𝑙𝑦

𝐽 𝑙 = 𝐽 𝑙

𝑓 𝑗 𝜚(𝑙)

amplitude phase spectrum spectrum

Convolution Theorem

𝑮 𝐽 𝑦 ∗ ℎ 𝑦 = 𝑮 𝐽 𝑦 𝑮 ℎ 𝑦 = 𝐽(𝑙) ℎ(𝑙)

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Let 𝐽(𝑦) and ℎ(𝑦) be defined on 𝑦 ∈ 0, 1, … , 𝑂 − 1 . See lecture notes for proof.

Convolution Theorem

𝑮 𝐽 𝑦 ∗ ℎ 𝑦 = 𝑮 𝐽 𝑦 𝑮 ℎ 𝑦 = 𝐽(𝑙) ℎ(𝑙)

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Let 𝐽(𝑦) and ℎ(𝑦) be defined on 𝑦 ∈ 0, 1, … , 𝑂 − 1 .

= 𝐽 𝑙 ℎ(𝑙)

𝑓 −𝑗 𝜚𝐽(𝑙) 𝑓 −𝑗 𝜚ℎ(𝑙)

Convolving an image 𝐽 𝑦 with a filter ℎ 𝑦 changes the amplitude and phase of each frequency component.

Concept of filtering (by size) filters

Linear Filtering (by frequency “band”)

= + + + +

Ideal Low Pass Filter ℎ 𝑦

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𝑂 𝑙 𝑂 2 ℎ 𝑙

Only consider up to N/2 because of the conjugacy property (coming soon).

Ideal High Pass Filter ℎ 𝑦

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𝑂 𝑙 𝑂 2 ℎ 𝑙

Ideal bandpass filter

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𝑂 𝑙1 𝑂 2 𝑙2

Bandwidth ≡ 𝑙2 − 𝑙1 Bandwidth (octaves) ≡ 𝑚𝑝𝑕2(𝑙2) − 𝑚𝑝𝑕2(𝑙1)

ℎ 𝑙

Non-Ideal Filters

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Bandwidth of Non-Ideal Bandpass Filter

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Why are we defining the low/band/high pass filters according to their properties

• n frequencies k in 0, .. N/2 only ?

Conjugacy Property of Fourier transform

Let ℎ 𝑦 be a real valued function. Then, for any integer 𝑙,

ℎ 𝑙 = ℎ 𝑂 − 𝑙 .

Proof: see the lecture notes. 𝑂 𝑙 𝑂 − 𝑙 𝑂 2

For any positive or negative integer 𝑛,

Periodicity Property of Fourier transform ℎ 𝑙 = ℎ 𝑙 + 𝑛𝑂 .

𝑂 𝑙 𝑙 + 𝑂 𝑙 − 𝑂

etc

For any positive or negative integer 𝑛, Proof: Use this:

Periodicity Property of Fourier transform ℎ 𝑙 = ℎ 𝑙 + 𝑛𝑂 .

𝑓 −𝑗 2 𝜌

𝑂 (𝑙+𝑛𝑂) 𝑦

= 𝑓 −𝑗 2 𝜌

𝑂 𝑙𝑦 𝑓 −𝑗 2 𝜌 𝑂 𝑛𝑂𝑦

1

𝑦=0 𝑂−1

𝑓 −𝑗 2 𝜌

𝑂 𝑙 𝑦

𝐽 𝑦 𝐽 𝑙 = The Fourier transform is well defined for any 𝑙 (not just in 0, …, 𝑂 − 1. )

𝑦 =−𝑂 2 𝑂 2−1

𝑓 −𝑗 2 𝜌

𝑂 𝑙 𝑦

𝑔 𝑦 𝑔 𝑙 = The Fourier transform is well defined for any range of 𝑂 consecutive values of 𝑦. e.g. Essentially we are treating 𝑔(𝑦) as periodic. 𝑓 −𝑗 2 𝜌

𝑂 (𝑦+𝑛𝑂) 𝑙

= 𝑓 −𝑗 2 𝜌

𝑂 𝑙𝑦 𝑓 −𝑗 2 𝜌 𝑂 𝑙𝑛𝑂

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Example 1 = ?

Example 1

Examples 2:

Local Difference:

𝐽 𝑦 ∗ 𝐸(𝑦) ≡ 1 2 𝐽 𝑦 + 1 − 1 2 𝐽 𝑦 − 1

− 1 2 , 𝑦 = 1

𝐸 𝑦 ≡

1 2 ,

0,

• therwise

𝑦 = −1

− 1 2 , 𝑦 = 1

𝐸 𝑦 ≡

1 2 ,

0,

• therwise

𝑦 = −1

𝑦=0 𝑂−1

𝑓 −𝑗 2 𝜌

𝑂 𝑙 𝑦

𝐸 𝑦 𝐸 𝑙 = = ?

(Done on blackboard. See lecture notes)

Example 3:

Local Average:

𝐽 𝑦 ∗ 𝐶 𝑦 ≡ 1 4 𝐽 𝑦 + 1 + 1 2 𝐽 𝑦 + 1 4 𝐽 𝑦 − 1

1 4 , 𝑦 = −1, 1

𝐶 𝑦 ≡

1 2 ,

0,

• therwise

𝑦 = 0

1 4 , 𝑦 = −1, 1

𝐶 𝑦 ≡

1 2 ,

0,

• therwise

𝑦 = 0

𝑦=0 𝑂−1

𝑓 −𝑗 2 𝜌

𝑂 𝑙 𝑦

𝐶 𝑦 𝐶 𝑙 = = ?

(Sketched on blackboard. See lecture notes)

Stopped here

(will finish next class)

Example 4:

𝑓 𝑗 2 𝜌

𝑂 𝑙0 𝑦 =

𝑂 𝜀 (𝑙 − 𝑙0)

𝐆

𝑂 𝑙0 𝑂 2 ℎ 𝑙 (Done on blackboard. See lecture notes.)

Example 5 & 6: cosine and sine

cos 2𝜌 𝑂 𝑙0𝑦 = ?

𝐆

sin 2𝜌 𝑂 𝑙0𝑦 = ?

𝐆

Example 5 & 6: cosine and sine

cos 2𝜌 𝑂 𝑙0𝑦 = 𝑂 2 𝜀 𝑙 − 𝑙0 + 𝜀(𝑙 + 𝑙0 )

𝐆

sin 2𝜌 𝑂 𝑙0𝑦 = 𝑂 2𝑗 𝜀 𝑙 − 𝑙0 − 𝜀(𝑙 + 𝑙0 )

𝐆

Use Euler’s formula and Example 4. Also, recall the conjugacy property.

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𝐻 𝑦, 𝜏 = 1 2𝜌 𝜏 𝑓− 1

2 𝑦 𝜏

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Example 7: Gaussian What is its Fourier transform?

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𝐻 𝑦, 𝜏 = 1 2𝜌 𝜏 𝑓− 1

2 𝑦 𝜏

2

with equality in the limit as distance between samples goes to 0 and N goes to infinity, i.e. continuous Fourier transform.

Note the inverse relationship

𝑮 𝐻 𝑦, 𝜏 ≈ 𝑓−1

2 2 𝜌 𝜏 𝑙 𝑂

2

Example 7: Gaussian

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Example 8: cosine Gabor

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cos 2𝜌 𝑂 𝑙0𝑦 = 𝑂 2 𝜀 𝑙 − 𝑙0 + 𝜀(𝑙 + 𝑙0 )

𝐆 𝐆 𝐻 𝑦, 𝜏 ≈ 𝑓−1

2 2 𝜌 𝜏 𝑙 𝑂

2

𝐆 𝑑𝑝𝑡𝐻𝑏𝑐𝑝𝑠 𝑦, 𝜏 = 𝐆 { cos

2𝜌 𝑂 𝑙0𝑦 𝐻 𝑦, 𝜏 }

= ?

Convolution Theorem (version 2)

𝑮 𝐽 𝑦 ℎ 𝑦 =

1 𝑂 𝑮 𝐽 𝑦 ∗ 𝑮 ℎ 𝑦

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Proof: see Appendix in lecture notes

Example 8: cosine Gabor

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cos 2𝜌 𝑂 𝑙0𝑦 = 𝑂 2 𝜀 𝑙 − 𝑙0 + 𝜀(𝑙 + 𝑙0 )

𝐆 𝐆 𝐻 𝑦, 𝜏 ≈ 𝑓

−1 2 2 𝜌 𝜏 𝑙 𝑂

2

𝐆 𝑑𝑝𝑡𝐻𝑏𝑐𝑝𝑠 𝑦, 𝜏 ≈ 𝑂 2 ( 𝑓

−1 2 2 𝜌 𝜏 (𝑙− 𝑙0) 𝑂

2

+ 𝑓

−1 2 2 𝜌 𝜏 (𝑙+ 𝑙0) 𝑂

2

) See lecture notes for proof, and formula for sine Gabor.

Example: cosine Gabor

[ADDED: April 12]

N = 128, k0 = 20, sigma = 5