EE 7730

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EE 7730

• 2D Discrete Fourier Transform (DFT)

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2D Discrete Fourier Transform

• 2D Fourier Transform

• 2D Discrete Fourier Transform (DFT)

2D DFT is a sampled version of 2D FT.
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2D Discrete Fourier Transform

• 2D Discrete Fourier Transform (DFT)

where and

• Inverse DFT

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2D Discrete Fourier Transform

• It is also possible to define DFT as follows

where and

• Inverse DFT

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2D Discrete Fourier Transform

• Or, as follows

where and

• Inverse DFT

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2D Discrete Fourier Transform
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2D Discrete Fourier Transform
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2D Discrete Fourier Transform
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2D Discrete Fourier Transform
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Periodicity

• M,N point DFT is periodic with period M,N

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Periodicity

• M,N point DFT is periodic with period M,N

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Convolution

• Be careful about the convolution property!

LengthPQ-1
LengthQ
LengthP
For the convolution property to hold, M must be
greater than or equal to PQ-1.
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Convolution

• Zero padding

4-point DFT (M4)
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DFT in MATLAB

• Let f be a 2D image with dimension M,N, then
  its 2D DFT can be computed as follows
• Df fft2(f,M,N)
• fft2 puts the zero-frequency component at the
  top-left corner.
• fftshift shifts the zero-frequency component to
  the center. (Useful for visualization.)
• Example
• f imread(saturn.tif) f double(f)
• Df fft2(f,size(f,1), size(f,2))
• figure imshow(log(abs(Df)), )
• Df2 fftshift(Df)
• figure imshow(log(abs(Df2)), )

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DFT in MATLAB
f
Df fft2(f)
After fftshift
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DFT in MATLAB

• Lets test convolution property
• f 1 1
• g 2 2 2
• Conv_f_g conv2(f,g) figure plot(Conv_f_g)
• Dfg fft (Conv_f_g,4) figure plot(abs(Dfg))
• Df1 fft (f,3)
• Dg1 fft (g,3)
• Dfg1 Df1.Dg1 figure plot(abs(Dfg1))
• Df2 fft (f,4)
• Dg2 fft (g,4)
• Dfg2 Df2.Dg2 figure plot(abs(Dfg2))
• Inv_Dfg2 ifft(Dfg2,4)
• figure plot(Inv_Dfg2)

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DFT in MATLAB

• Increasing the DFT size
• f 1 1
• g 2 2 2
• Df1 fft (f,4)
• Dg1 fft (g,4)
• Dfg1 Df1.Dg1 figure plot(abs(Dfg1))
• Df2 fft (f,20)
• Dg2 fft (g,20)
• Dfg2 Df2.Dg2 figure plot(abs(Dfg2))
• Df3 fft (f,100)
• Dg3 fft (g,100)
• Dfg3 Df3.Dg3 figure plot(abs(Dfg3))

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DFT in MATLAB

• Scale axis and use fftshift
• f 1 1
• g 2 2 2
• Df1 fft (f,100)
• Dg1 fft (g,100)
• Dfg1 Df1.Dg1
• t linspace(0,1,length(Dfg1))
• figure plot(t, abs(Dfg1))
• Dfg1_shifted fftshift(Dfg1)
• t2 linspace(-0.5, 0.5, length(Dfg1_shifted))
• figure plot(t, abs(Dfg1_shifted))

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Example
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Example
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DFT-Domain Filtering

• a imread(cameraman.tif')
• Da fft2(a)
• Da fftshift(Da)
• figure imshow(log(abs(Da)),)
• H zeros(256,256)
• H(128-2012820,128-2012820) 1
• figure imshow(H,)
• Db Da.H
• Db fftshift(Db)
• b real(ifft2(Db))
• figure imshow(b,)

H
Frequency domain
Spatial domain
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Low-Pass Filtering
81x81
61x61
121x121
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Low-Pass Filtering
h


DFT(h)
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High-Pass Filtering
h


DFT(h)
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High-Pass Filtering
High-pass filter
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Anti-Aliasing
aimread(barbara.tif)
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Anti-Aliasing
aimread(barbara.tif) bimresize(a,0.25) cimr
esize(b,4)
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Anti-Aliasing
aimread(barbara.tif) bimresize(a,0.25) cimr
esize(b,4) Hzeros(512,512) H(256-6425664,
256-6425664)1 Dafft2(a) Dafftshift(Da) Dd
Da.H Ddfftshift(Dd) dreal(ifft2(Dd))
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Noise Removal

• For natural images, the energy is concentrated
  mostly in the low-frequency components.

Einstein
DFT of Einstein
Profile along the red line
Signal vs Noise
Noise40rand(256,256)
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Noise Removal

• At high-frequencies, noise power is comparable to
  the signal power.

Signal vs Noise

• Low-pass filtering increases signal to noise
  ratio.

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Appendix
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Appendix Impulse Train

• The Fourier Transform of a comb function is

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Impulse Train (contd)

• The Fourier Transform of a comb function is

(Fourier Trans. of 1)
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Impulse Train (contd)

• Proof

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Appendix Downsampling

• Question What is the Fourier Transform of
  ?

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Downsampling

• Let

Using the multiplication property
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Downsampling
where
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Example
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Example
?
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Downsampling
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Example
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Example
No aliasing if