3/20/2012 2 Periodicity of the DFTcnikou/Courses/Digital_Image_Processing/... · 2012. 3. 20. · 3/20/2012 4 19 The DFT and Image Processing To filter an image in the frequency domain:

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Filtering in the Frequency Domain(Application)

Digital Image Processing

University of Ioannina - Department of Computer Science

( pp )

Christophoros [email protected]

2 Periodicity of the DFT

C. Nikou – Digital Image Processing (E12)

• The range of frequencies of the signal is between [-M/2, M/2].• The DFT covers two back-to-back half periods of the signal

as it covers [0, M-1].• For display and computation purposes it is convenient to shift

the DFT and have a complete period in [0, M-1].

3 Periodicity of the DFT (cont...)

• From DFT properties: 02 ( / )0[ ] ( )j N n Mf n e F k Nπ ⇔ −

Letting N0=M/2:And F(0) is now located at M/2.

[ ]( 1) ( / 2)nf n F k M− ⇔ −


4 Periodicity of the DFT (cont...)

• In two dimensions:

and F(0,0) is now located at (M/2, N/2).

[ , ]( 1) ( / 2, / 2)m nf m n F k M l N+− ⇔ − −


5 DFT & Images

The DFT of a two dimensional image can be visualised by showing the spectrum of the image component frequencies

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DFT

6 DFT & Images

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7 DFT & Images

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8 DFT & Images (cont…)

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DFT


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Scanning electron microscope image of an integrated circuit

magnified ~2500 times

Fourier spectrum of the image


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Although the images differ by a simple geometrictransformation no intuitive information may beextracted from their phases regarding their relation.


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15 DFT synopsis

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16 DFT synopsis (cont.)

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19 The DFT and Image Processing

To filter an image in the frequency domain:1. Compute F(u,v) the DFT of the image2. Multiply F(u,v) by a filter function H(u,v)3. Compute the inverse DFT of the result

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20 Some Basic Frequency Domain Filters

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The DFT is centered after multiplication of the image by (-1)m+n

21Some Basic Frequency Domain Filters

(cont.)

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High Pass Filter

22 The importance of zero padding

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• The image and the DFT are considered to be periodic.• The vertical edges of the middle image are not blurred

if no padding is applied. Why?

23The importance of zero padding

(cont.)

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24 Spatial zero-padding and filters

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•The filter is defined in the frequency domain.•A naïve approach:

Compute the inverse DFT of the filter


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25Spatial zero-padding and filters

(cont.)

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(cont.)

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• Spatial truncation of the filter results in ringing effects.


(cont.)

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components and simultaneously perform zero-padding to avoid aliasing.

• A decision on which limitation to accept is i d


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preferable than ringing.• Another solution is to choose filters attenuating gradually instead of ideal filters.

28 Steps of filtering in the DFT domain

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29Steps of filtering in the DFT domain

(cont.)

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-1 0 1


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-2 0 2

-1 0 1

• Apply a 3x3 Sobel filter to the 600x600 image in the frequency domain.


(cont.)

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2. Place the filter to the center of the 602x602 padded array.

3 Multiply the filter by (-1)m+n to place the


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4. Compute the DFT of the filter.5. Multiply the DFT by (-1)m+n to place it to

the center of the array.6. Perform filtering in the frequency domain.

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(cont.)

Imag

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filter periodically and compute the DFT.0 20 1

-2-1


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0 1 -1

-1 0 1-2 0 2-1 0 1

32The importance of zero padding

(cont...)

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33 Smoothing Frequency Domain Filters

• Smoothing is achieved in the frequency domain by dropping out the high frequency components

• The basic model for filtering is:


G(u,v) = H(u,v)F(u,v)• where F(u,v) is the Fourier transform of the

image being filtered and H(u,v) is the filter transform function

• Low pass filters – only pass the low frequencies, drop the high ones.

34 Ideal Low Pass Filter

Simply cut off all high frequency components that are a specified distance D0 from the origin of the transform.

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Changing the distance changes the behaviour of the filter.Im

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35 Ideal Low Pass Filter (cont…)

The transfer function for the ideal low pass filter can be given as:

⎩⎨⎧ ≤

= 0

)(if0),( if 1

),(DDDvuD

vuH


where D(u,v) is given as:

⎩⎨ > 0),( if 0

),(DvuD

2/122 ])2/()2/[(),( NvMuvuD −+−=


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An image, its Fourier spectrum and a series of ideal low pass filters of radius 5, 15, 30, 80 and 230 superimposed on top of it.

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37 Ideal Lowpass Filters (cont...)

• ILPF in the spatial domain is a sincfunction that has to be truncated and produces ringing effects.

• The main lobe is responsible for blurring Imag

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and the side lobes are responsible for ringing.

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Originalimage

ILPF of radius 5

ILPF of radi s 30ILPF of radius 15


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ILPF of radius 230ILPF of radius 80

ILPF of radius 15

39 Butterworth Lowpass Filters

• The transfer function of a Butterworth lowpass filter of order n with cutoff frequency at distance D0 from the origin is defined as:

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1


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nDvuDvuH 2

0 ]/),([11),(

+=

40 Butterworth Lowpass Filters (cont...)

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41 Butterworth Lowpass Filter (cont…)

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Original image BLPF n=2, D0=5

BLPF n=2, D0=30BLPF n=2, D0=15


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BLPF n=2, D0=230BLPF n=2, D0=80

Less ringing than ILPFdue to smoothertransition

42 Gaussian Lowpass Filters

• The transfer function of a Gaussian lowpass filter is defined as:

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2 2/),(),( DvuDevuH −=


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43 Gaussian Lowpass Filters (cont…)

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Gaussian D0=30Gaussian D =15


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Gaussian D0=230Gaussian D0=85

Gaussian D0=15

Less ringing thanBLPF but also lesssmoothing

44 Lowpass Filters Compared

ILPF D0=15 BLPF n=2, D0=15

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Gaussian D0=15

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45 Lowpass Filtering Examples

A low pass Gaussian filter is used to connect broken text

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46 Lowpass Filtering Examples

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47 Lowpass Filtering Examples (cont…)

• Different lowpass Gaussian filters used to remove blemishes in a photograph.

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48 Lowpass Filtering Examples (cont…)

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49 Sharpening in the Frequency Domain

• Edges and fine detail in images are associated with high frequency components

• High pass filters – only pass the high


frequencies, drop the low ones• High pass frequencies are precisely the

reverse of low pass filters, so:Hhp(u, v) = 1 – Hlp(u, v)

50 Ideal High Pass Filters

• The ideal high pass filter is given by:

⎩⎨⎧

>≤

=0

0

),( if 1),( if 0

),(DvuDDvuD

vuH

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• D0 is the cut off distance as before.

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51 Ideal High Pass Filters (cont…)

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IHPF D0 = 15 IHPF D0 = 30 IHPF D0 = 80

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52 Butterworth High Pass Filters

• The Butterworth high pass filter is given as:

nvuDDvuH 2

0 )],(/[11),(

+=

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• n is the order and D0 is the cut off distance as before.

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53 Butterworth High Pass Filters (cont…)

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BHPF n=2, D0 =15

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BHPF n=2, D0 =30 BHPF n=2, D0 =80

54 Gaussian High Pass Filters

• The Gaussian high pass filter is given as:

D i th t ff di t b f

20

2 2/),(1),( DvuDevuH −−=

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• D0 is the cut off distance as before.

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55 Gaussian High Pass Filters (cont…)

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Gaussian HPF n=2, D0 =15



56 Highpass Filter Comparison

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IHPF D0 = 15

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BHPF n=2, D0 =15

57 Highpass Filtering Example

Orig

inal

imag

e

Highpass filtering reIm

age

Pro

cess

ing

(200

2)


esult

Hig

h fre

quen

cy

emph

asis

resu

lt After histogram

equalisation

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58 Laplacian In The Frequency Domain

Lapl

acia

n in

the

requ

ency

dom

ain

(not

cen

tere

d)

2-D im

age of Laplacin the frequency

domain

(not centered)

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fr

cian

Inve

rse

DFT

of

Lapl

acia

n in

the

frequ

ency

dom

ain

Zoomed section of the image on

the left compared to spatial filterIm

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59 Frequency Domain Laplacian Example

Original image

Laplacian filtered image


Laplacian image scaled

Enhanced image

60 Band-pass and Band-stop Filters


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61 Band-Pass Filters (cont...)


62 Band-Pass Filters (cont...)


63 Fast Fourier Transform

•The reason that Fourier based techniques have become so popular is the development of the Fast Fourier Transform (FFT) algorithm.


• It allows the Fourier transform to be carried out in a reasonable amount of time.

•Reduces the complexity from O(N4) to O(N2logN2).

64Frequency Domain Filtering & Spatial

Domain Filtering• Similar jobs can be done in the spatial and

frequency domains.• Filtering in the spatial domain can be

easier to understand.


• Filtering in the frequency domain can be much faster – especially for large images.

Documents

3/20/2012 2 Periodicity of the DFTcnikou/Courses/Digital_Image_Processing/... · 2012. 3. 20. · 3/20/2012 4 19 The DFT and Image Processing To filter an image in the frequency domain: