Digital Image Processing Digital Image Processing [PDF]

Digital Image Processing

Chapter 4: Image Enhancement in the Frequency Domain

Background: Fourier Series

Fourier series: Any periodic signals can be viewed as weighted sum of sinusoidal signals with different frequencies Frequency D F Domain: i view frequency as an i d independent d t variable i bl (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Fourier Tr. and Frequency Domain

Fourier Tr.

Time, spatial Ti ti l Domain Signals

Frequency F Domain Signals

Inv Fourier Tr Tr.

1-D, Continuous case 

Fourier Tr.:

F (u ) 

 f ( x )e

 

Inv. Fourier Tr.:

f ( x) 

 j 2ux

 F ( u )e



j 2ux

dx

du

Fourier Tr. and Frequency Domain (cont.) 1 D Discrete case 1-D, Fourier Tr.: I Fourier Inv. F i Tr.: T

M 1

1 F (u )  M f ( x) 



f ( x )e  j 2ux / M

x 0

u = 0,…,M 0 M-11

M 1

j 2ux / M F ( u ) e 

x = 0,…,M-1 0 M1

u 0

F(u) can be written as

F (u )  R(u )  jI (u )

or

F (u )  F (u ) e  j ( u )

where

F (u )  R (u )  I (u ) 2

2

 I (u )   (u )  tan   (u )   R (u 1

Example of 1-D Fourier Transforms

Notice that the longer the time domain signal, The shorter its Fourier transform

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Relation Between x and u For a signal F i l f(x) f( ) with ith M points, i t let l t spatial ti l resolution l ti x be b space between samples in f(x) and let frequency resolution u be space between frequencies components in F(u), F(u) we have

1 u  Mx Example: E l for f a signal i l f(x) f( ) with i h sampling li period i d 0.5 0 5 sec, 100 point, i we will get frequency resolution equal to

1 u   0.02 Hz 100  0.5 This means that in F(u) ( ) we can distinguish g 2 frequencies q that are apart by 0.02 Hertz or more.

2-Dimensional Discrete Fourier Transform For an image of size MxN pixels 2 D DFT 2-D

1 F ( u, v )  MN

M 1 N 1



f ( x, y )e  j 2 ( ux / M  vy / N )

x 0 y 0

u = frequency in x direction, u = 0 ,…, M-1 v = frequency in y direction, v = 0 ,…, N-1

2-D IDFT

f ( x, y ) 

M 1 N 1

 F (u, v )e u 0 v 0

j 2 ( ux / M  vy / N )

x = 0 ,…, M-1 M1 y = 0 ,…, N-1

2-Dimensional Discrete Fourier Transform (cont.) F(u,v) can be written as  j ( u ,v ) or F ( u, v )  F ( u, v ) e F (u, v )  R(u, v )  jI (u, v )

where

F ( u, v )  R ( u, v )  I ( u, v ) 2

2

 I (u, v )   (u, v )  tan    R ( u, v )  1

For the purpose of viewing, we usually display only the Magnitude ag ude pa part oof F(u,v) (u,v)

2-D DFT Properties

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

2-D DFT Properties (cont.)

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

2-D DFT Properties (cont.)

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

2-D DFT Properties (cont.)

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Computational Advantage of FFT Compared to DFT

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Relation Between Spatial and Frequency Resolutions

1 u  M x M where

1 v  Ny N

x = spatial resolution in x direction y = spatial resolution in y direction

x and yy are ppixel width and height. g ) u = frequency resolution in x direction v = frequency resolution in y direction N,M = image width and height

How to Perform 2-D DFT by Using 1-D DFT

f(x,y)

1-D DFT by row

F(u,y) 11-D D DFT by column

F(u,v)

How to Perform 2-D DFT by Using 1-D DFT (cont.) Alternative method

f(x,y) 11-D D DFT by column

F(x,v)

1-D DFT by row

F(u,v)

Periodicity of 1-D DFT From DFT:

-N N

1 F (u )  M

0

M 1



f ( x )e  j 2ux / M

x 0

N

2N

We display only in this range

DFT repeats itself every N points (Period = N) but we usually display it for n = 0 ,…, N-1

Conventional Display for 1-D DFT F (u )

f(x)) f( DFT

N-1

0

N-1

0

Time Domain Signal g High frequency area L frequency Low f area

The graph F(u) is not easy to understand !

Conventional Display for DFT : FFT Shift F (u )

FFT Shift: Shift center of the graph F(u) to 0 to get better Display which is easier to understand.

0

F (u )

N1 N-1

High frequency area Low frequency area

-N/2

0

N/2-1

Periodicity of 2-D DFT 1 F (u, v )  MN

2-D DFT:

M 1 N 1

 j 2 ( ux / M  vy / N ) f ( x , y ) e  x 0 y 0

g(x,y) -M

0

For an iimage off size F i NxM N M pixels, its 2-D DFT repeats itself every N points in xdirection and every M points in yy-direction.

M

2M -N

0

N

2N

We display only in this range (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Conventional Display for 2-D DFT F(u,v) has low frequency areas at corners of the image g while high g frequency areas are at the center of the image which is inconvenient to interpret.

High frequency area Low frequency area

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

2-D FFT Shift : Better Display of 2-D DFT 2 D FFT Shift is a MATLAB function: Shift the zero frequency 2-D of F(u,v) to the center of an image.

2D FFTSHIFT

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

High frequency area

Low frequency area

2-D FFT Shift (cont.) : How it works -M

0 Display of 2D DFT Aft FFT Shift After M

2M -N

0

N

Original display of 2D DFT

2N

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Example of 2-D DFT

Notice that the longer the time domain signal, signal The shorter its Fourier transform (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Example of 2-D DFT

Notice that direction of an object in spatial image and Its Fourier transform are orthogonal to each other. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Example of 2-D DFT

2D DFT

Original image

2D FFT Shift

Example of 2-D DFT

2D DFT

Original image

2D FFT Shift

Basic Concept of Filtering in the Frequency Domain From Fourier Transform Property:

g ( x , y )  f ( x , y )  h ( x , y )  F ( u, v )  H ( u, v )  G ( u, v ) We cam perform filtering process by using

Multiplication M ltiplication in the frequency freq enc domain is easier than convolution in the spatial Domain. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Filtering in the Frequency Domain with FFT shift H(u,v) (User defined)

F(u,v)

FFT shift hif

X

2D FFT

f(x,y)

g(x,y)

2D IFFT

FFT shift

G(u,v)

In this case, F(u,v) and H(u,v) must have the same size and have the zero frequency at the center.

Multiplication in Freq. Domain = Circular Convolution

f( ) f(x)

DFT

F( ) F(u) G(u) = F(u)H(u)

h(x)

DFT

IDFT

g(x)

H(u)

Multiplication of DFT off 2 signals DFTs i l is equivalent to perform circular convolution in the spatial domain.

1

f( ) f(x)

0.5 0 0

20

40 0

60

80

120

1

h( ) h(x)

05 0.5

“Wrapp around” effect

100

0 0

20

40

60

80

100

120

80

100

120

40

g(x)

20 0

0

20

40

60

Multiplication in Freq. Domain = Circular Convolution

Original image

H(u,v) Gaussian Lowpass Filter with D0 = 5

Filtered image (obtained using circular convolution) Incorrect areas at image rims

Linear Convolution by using Circular Convolution and Zero Padding

f( ) f(x)

Zero padding

DFT

F( ) F(u) G(u) = F(u)H(u)

h(x)

Zero padding

DFT

H(u) IDFT

1 0.5

Concatenation

0 0

50

100

150

200

250

g(x)

1 0.5 0 0

50

100

150

200

250

Padding zeros Before DFT

40

Keep p only y this ppart 20 0

0

50

100

150

200

250

Linear Convolution by using Circular Convolution and Zero Padding

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Linear Convolution by using Circular Convolution and Zero Padding

Filtered image Zero padding area in the spatial Domain of the mask image (th ideal (the id l lowpass l filter) filt )

Only this area is kept.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Filtering in the Frequency Domain : Example In this hi example, l we set F(0,0) ( ) to zero which means that the zero frequency component is removed. removed

N t Zero Note: Z frequency f = average intensity of an image

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Filtering in the Frequency Domain : Example

Lowpass Filter

Highpass Filter (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Filtering in the Frequency Domain : Example (cont.)

Result of Sharpening Filter (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Filter Masks and Their Fourier Transforms

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Ideal Lowpass Filter Ideal LPF Filter Transfer function

1 H (u, v )   0

D (u, v )  D0 D (u, v )  D0

where D(u,v) ( , ) = Distance from ((u,v) , ) to the center of the mask.

(Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing Wood, Processing, 2nd Edition. Edition

Examples of Ideal Lowpass Filters

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

The smaller D0, the more high frequency components are removed.

Results of Ideal Lowpass Filters

Ringing effect can be obviously seen!

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

How ringing effect happens 1 H (u, v )   0

D (u, v )  D0 D (u, v )  D0 Surface Plot

1 0.8 0.6

Ideal Lowpass Filter with D0 = 5

0.4 0.2 0 20 20

0

Abrupt change in the amplitude

0 -20

-20 20

How ringing effect happens (cont.)

Surface Plot -3

x 10

15

Spatial S i l Response R off Ideal Id l Lowpass Filter with D0 = 5

10 5 0

20

Ripples that cause ringing effect

20

0

0 -20

-20

How ringing effect happens (cont.)

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Butterworth Lowpass Filter Transfer function H ( u, v ) 

1 2N 1  D (u, v ) / D0 

Where D0 = Cut off frequency, frequency N = filter order. order

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Results of Butterworth Lowpass Filters

There is less ringing effect compared to those of ideal lowpass filters! (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Spatial Masks of the Butterworth Lowpass Filters

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Some ripples can be seen.

Gaussian Lowpass Filter Transfer function H ( u, v )  e

 D 2 ( u ,v ) / 2 D0 2

Where D0 = spread factor.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Note: the Gaussian filter is the only filter that has no ripple and hence no ringing effect.

Gaussian Lowpass Filter (cont.) H ( u, v )  e

 D 2 ( u ,v ) / 2 D0 2

1 08 0.8

Gaussian lowpass filter with D0 = 5

0.6 0.4 0.2

20 20

0

0 -20

-20 0.03

Spatial respones of the Gaussian lowpass filter with D0 = 5

0.02

0 01 0.01

0 20 20

0

0 -20

Gaussian shape

-20

Results of Gaussian Lowpass Filters

No ringing effect!

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Application of Gaussian Lowpass Filters

Original image

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Better Looking

The GLPF can bbe used Th d tto remove jjagged d edges d and “repair” broken characters.

Application of Gaussian Lowpass Filters (cont.) R Remove wrinkles i kl

Original image

Softer-Looking

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Application of Gaussian Lowpass Filters (cont.)

Original image : The gulf of Mexico and Florida from NOAA satellite.

Filtered image (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Remove artifact lines: this is a simple but crude way to do it!

Highpass Filters

Hhp = 1 - Hlp

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Ideal Highpass Filters Ideal LPF Filter Transfer function

0 H (u, v )   1

D (u, v )  D0 D (u, v )  D0

where D(u,v) ( , ) = Distance from ((u,v) , ) to the center of the mask.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Butterworth Highpass Filters Transfer function H ( u, v ) 

1 2N 1  D0 / D (u, v )

Where D0 = Cut off frequency, frequency N = filter order. order

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Gaussian Highpass Filters Transfer function H ( u, v )  1  e

 D 2 ( u ,v ) / 2 D0 2

Where D0 = spread factor.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Gaussian Highpass Filters (cont.) H ( u, v )  1  e

1

 D 2 ( u ,v ) / 2 D0 2

0.8

Gaussian highpass filter with D0 = 5

0.6 0.4 0.2 0 60 40 20 10

20

30

40

50

60

3000 2000

Spatial respones of the Gaussian highpass filter with ith D0 = 5

1000 0 60 40 20 10

20

30

40

50

60

Spatial Responses of Highpass Filters

Ripples pp

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Results of Ideal Highpass Filters

Ringing effect can be obviously b i l seen!!

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Results of Butterworth Highpass Filters

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Results of Gaussian Highpass Filters

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Laplacian Filter in the Frequency Domain From Fourier Tr. Tr Property:

d n f ( x) n    ju F (u ) n dx

Th for Then f Laplacian L l i operator t 2 2 f f   2  f  2  2  u 2  v 2 F (u, v ) x y

We get

2   u 2  v 2 

Image g of –(u2+v2)

Surface plot

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Laplacian Filter in the Frequency Domain (cont.) Spatial response of –(u2+v2)

Cross section

Laplacian mask in Chapter 3

Sharpening Filtering in the Frequency Domain Spatial Domain

f hp ( x, y )  f ( x, y )  f lp ( x, y ) f hb ( x, y )  Af ( x, y )  f lp ( x, y ) f hb ( x, y )  ( A  1) f ( x, y )  f ( x, y )  f lp ( x, y ) f hb ( x, y )  ( A  1) f ( x, y )  f hp ( x, y ) Frequency Domain Filter

H hp (u, v )  1  H lp (u, v ) H hb (u, v )  ( A  1)  H hp (u, v )

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Sharpening Filtering in the Frequency Domain (cont.)

p

2 P

2 P

P  2 P

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Sharpening Filtering in the Frequency Domain (cont.) f hb ( x, y )  ( A  1) f ( x, y )  f hp ( x, y )

f

f hp  2 P

A=2

A = 2.7 27

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

High Frequency Emphasis Filtering H hfe hf ( u, v )  a  bH hp h ( u, v ) Butterworth highpass g pass filtered image

Original

High g freq. q emphasis p filtered image

After Hist Eq.

a = 0.5, b = 2 (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Homomorphic Filtering An image g can be expressed p as

f ( x, y )  i ( x, y ) r ( x, y ) i(x,y) = illumination component r(x,y) = reflectance component We need to suppress effect of illumination that cause image Intensity changed slowly slowly.

Homomorphic Filtering

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Homomorphic Filtering

More details in the room can be seen!

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Correlation Application: Object Detection

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.