Chap 4-2. Frequency domain processing

Jen-Chang Liu, 2006

Extend to 2-D DFT from 1-D

2-D DFT: 1-D DFT in horizontal then vertical

1

0

1

0

)(2),(

1),(

N

y

M

x

yM

vx

M

uj

eyxfMN

vuF

1

0

1

0

)(2),(),(

M

u

N

v

yM

vx

M

ujevuFyxf

Complex Quantities to Real Quantities

Useful representation2/122 )],(),([),( vuIvuRvuF

]),(

),([tan),( 1

vuR

vuIvu

magnitude

phase

),(),(),(),( 222vuIvuRvuFvuP

Power spectrum

Real part

2d DFT basis functions

127,...,2,1,0,127,...,2,1,0,),()

128

1

128

0(2

yxeyxeyxj

1

0

1

0

)(2),(),(

M

u

N

v

yM

vx

M

ujevuFyxf

iDFT:

將影像用 )(2 yN

vx

M

uje

合成，其中(u, v) 代表頻率

DFT

More DFT basis (real part)

(u,v)=(0,2)

(0,30)

u

v

(0,63)

(1,1)

(1,30)

(30,30)

(1,0)

Example: reconstruction from DFT coefficients

1

0

1

0

)(2),(),(

M

u

N

v

yM

vx

M

ujevuFyxf

…Zigzag scan

Example: reconstruction from DFT coefficients

http://www.ncnu.edu.tw/~jcliu/course/dip2005/lenaidft.m

Notes on showingDFT

Lena 256x256

F=fft2(I);imshow(abs(F), [])

F(1,1)=7761921 F(1,127)=334.79+10i

imshow(log(abs(F)), [])

Log transformations

s = c log(1+r)

Compress the dynamic range of images with large variation in pixel values

M

N

M/2

N/2

0

Periodicity and conjugate symmetry property of 2-D DFT

Outline

Frequency domain operations Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering

mask coefficients

underlying neighborhood

X (product) output

Convolution – 2-D case

2d convolution 旋積

Masking operation

1

0

1

0

),(),(1

),(),(),(M

m

N

n

nymxhnmfMN

yxhyxfyxg

1

0

1

0

),(),(1

),(M

m

N

n

ynxmhnmfMN

yxg

Convolution theorem

),(),(),(),( vuHvuFyxhyxf

f:image

Fouriertransform

F

h: filter or mask

Fouriertransform

H

Timedomain

Frequencydomain

convolution

multiplication

Filtering in the frequency domain

fftshift

Outline

Frequency Domain Operations Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering

Smoothing frequency-domain filters

Design issue G(u,v)=F(u,v) H(u,v) Remove high freq. component (details,

noise, …) Ideal low-pass filter Butterworth filter Gaussian filter

More smoothin the edge ofcut-off frequency

Ideal low-pass filter

Sharp cut-off frequency

0

0

),( if 0

),( if 1),(

DvuD

DvuDvuH

where D(u,v) is the distance to the center freq.

2/122 ])2/()2/[(),( NvMuvuD

Ideal low-pass filter (cont.)

Cut-off freq.

Ideal low-pass filter (con.t)

ILPF can not be realized in electronic components, but can be implemented in a computer

Decision of cut-off freq.? Measure the percentage of image power

within the low freq.

freqoffcutvu

TPvuP ),(

/)],([100

1

0

1

0

),(M

u

N

vT vuPPTotal image power

ILPF: distribution of image power

original Freq.

99.5

98

96.4

94.692

original =92D0=5

=94.6D0=15

=96.4D0=30

=98D0=80

=99.5D0=230

Ideal low-passfiltering

Ringingeffect

Effects of ideal low-pass filtering

Blurring and ringing

ILPF: Freq.

F-1

blurring

ringing

ILPF: spatial

Effects of ideal low-pass filtering (cont.)

spatial

impulse

ILPF

spatial

Butterworth low-pass filters

nDvuDvuH

20 ]/),([1

1),(

H=0.5 whenD(u,v)=D0

Order of butterworth filters

n=1 n=2 n=5 n=20

Spatial domain filter of butterworth filters

Ringing likeIdeal LPF

Butterworth filtersOrder = 2

original D0=5

D0=15 D0=30

D0=80 D0=230

Gaussian low-pass filters

20

2 2/),(),( DvuDevuH Variance orcut-off freq.

D(u,v)=D0

H = 0.607

Gaussian smoothing

original D0=5

D0=15 D0=30

D0=80 D0=230

Practical applications: 1

444x508 GLPF, D0=80

斷點

Practical applications: 2

GLPF, D0=100

GLPF, D0=80

1028x732

Practical applications: 3

588x600 GLPF, D0=30 GLPF, D0=10

Scan line

Outline

Frequency Domain Operations Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering

Sharpening frequency-domain filters

Image details corresponds to high-frequency Sharpening: high-pass filters Hhp(u,v)=1-Hlp(u,v)

Ideal high-pass filters Butterworth high-pass filters Gaussian high-pass filters Difference filters

Laplacian filters

Ideal HPF

Butterworth HPF

Gaussian HPF

Spatial-domain HPF

ideal Butterworth Gaussian

negative

Ideal high-pass filters

0

0

),( if 1

),( if 0),(

DvuD

DvuDvuH

D0=15 D0=30 D0=80ringing

original

Butterworth high-pass filters

nvuDDvuH

20 )],(/[1

1),(

n=2, D0=15 D0=30 D0=80

Gaussian high-pass filters

20

2 2/),(1),( DvuDevuH

D0=15 D0=30 D0=80

Laplacian frequency-domain filters

Spatial-domain Laplacian (2nd derivative)

Fourier transform

2

2

2

22

y

f

x

ff

)()()(

uFjux

xf nn

n

),()(

),()(),()(),(),(

22

222

2

2

2

vuFvu

vuFjvvuFjuy

yxf

x

yxf

Laplacian frequency-domain filters

2

2

2

22

y

f

x

ff

Inputf(x,y)

Laplacian

F(u,v)F

F-(u2+v2)F(u,v)

-(u2+v2)

The Laplacian filter in the frequency domain isH(u,v) = -(u2+v2)

0

frequency

spatial

H(u,v) = -(u2+v2)

original Laplacian

ScaledLaplacian

original+Laplacian

Outline

Frequency Domain Operations Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering

Image Formation Model

Illumination source

scene

reflection

eye

Homomorphic filtering

Image formation model f(x,y)=i(x,y) r(x,y)

illumination: reflectance:

Slow spatial variations vary abruptly, particularlyat the junctions of dissimilarobjects

Homomorphic filtering Product term

Log of product f(x,y)=i(x,y) r(x,y)=> ln f(x,y)=ln i(x,y)+ ln r(x,y)

)},({)},({)},(),({),( yxryxiyxryxiyxf

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

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

yxryxi

yxryxiyxf

Separation of signal source:

Homomorphic filtering approach

ln i(x,y)

ln r(x,y)

illumination

reflection

filtering

How to identify the illumination and reflection?

Illumination -> low frequency Reflection -> high frequency

Radius fromthe origin

Example filter: sharpening

illumination reflection

Homomophic filtering: example

original Homomorphic filtering