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Frequency Domain Processing

Frequency domain processing�Homomorphic filtering

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Recap: Fig 4.1

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An example: Fig 4.2

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Basic Scheme

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An image and its DFT: Fig 4.4

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filtering

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Ideal Low Pass Filter

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An image: Fig 4.11

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LPF at different cut-off frequencies

Notice the�ringing�in b-f.

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Butterworth LPF

LPF : 1H(u,v)

D0 D(u,v)

Butterworth filter:1

H (u,v)

D(u,v)/D0

G u v H u v F u v

H u vD u v DD u v D

D u v u v

( , ) ( , ) ( , )

( , )( , )( , )

( , )

=

=≤>

RST= +

10

0

0

2 2

ifif

(Circularly symmetric)

H u vD u v D n( , )

[ ( , ) / ]=

+1

1 02

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�Ringing�: Fig 4.13

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Butterworth filter: Fig 4.14

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LPF example again

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Spatial representation of BWF

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Gaussian LPF

2 2( , ) exp( ( , ) / 2 )H u v D u v σ= −

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Filtering using GLPF

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Another example

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Frequency domain processingHPF

HPF :

Butterworth filter:

H u vD u v DD u v D

( , )( , )( , )

=≤>

RST01

0

0

ifif

H u vD D u v n( , )

[ / ( , )]=

+1

1 02

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HPF: Fig 4.22

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Spatial representation of HPF

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Ideal HPF

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BWF

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GHPF

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Homomorphic filtering

Consider f (x,y) = i (x,y) . r (x,y)

Illumination Reflectance

Now So cannot operate on individual components directlyLet

Let

Let

ℑ ≠ ℑ

= = +ℑ = ℑ + ℑ

= + = = += ℑ + ℑ

′ = ℑ ′ = ℑ

− −

− −

{ ( , )} { . }

( , ) ln ( , ) ln ( , ) ln ( , ){ ( , )} {ln } {ln }( , ) ; ( , )( , ) { } { }

( , ) { } ; ( , ) { }

f x y i r

z x y f x y i x y r x yz x y i r

Z u v I R S u v HZ HI HRs x y HI HR

i x y HI r x y HR

1 1

1 1

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Homomorphic filtering (contd.)

In practice: i slowly varying => LFr fast varying => HF

f(x,y)

ln i

ln

LPF

HPFln r

α<1

ρ>1

X expg(x,y)

∴ = ′ + ′== ′ + ′=

s x y i rg x y s x y

i ri x y r x yo o

( , )( , ) exp ( ( , ) )

exp exp( , ) ( , )

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Homomorphic filtering (cont.)

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An example