Week 3: Images in Frequency Domain

By: Eslam Adel

email: eslam.a.mahmoud@eng1.cu.edu.eg

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### 2D Discrete Fourier Transform

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For Image I(x,y) DFT is I(u,v) where

<img style="width:100%" src="../../images/DFT.png">
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### Basis Functions

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<img style="width:60%" src="../../images/Basis_functions.png">

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<img style="scale:100%" src="../../images/Original_Image.png">

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#### Example Cont'd

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<img style="scale:100%" src="../../images/Fourier_Spectrum_of_the_image.png">

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Consider Image f(x,y) Shown

find

F(0,0), F(0,1) and F(1,0)

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Complex ( Magnitude and Phase)

Dynamic Range Compression 
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<img style="scale:80%" src="../../images/DRC.png">

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### Properties of FT Cont'd

Fourier transform of the image is symmetric
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Application MRI Half Fourier Imaging
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<img style="scale:80%" src="../../images/HFI.png">
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Where redundancy comes from ??

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DFT is periodic

Discretization implies periodicity
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No one to one corresponding
 $$I(u,v) = T(I)(x,y)$$ not $$T(I(x,y))$$ 
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#### Inverse Fourier Transform
<img style="width:100%" src="../../images/IFT.png">

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#### Fourier Filtering
<img style="width:100%" src="../../images/Fourier_Filtering.png">

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#### Color Image Processing
<img style="width:100%" src="../../images/block_diagram.png">

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Point Processor (Each individual Point)

Example : Thresholding

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### Point Operator Cont'd
Negative Image
<img style="width:80%" src="../../images/negatve.png">

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Enhance Contrast of the image

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Consider a 64 × 64 image with 7 gray levels( 0 1 2 3 4 5 6 ).

The histogram of this image is given by:

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<img style="width:30%" src="../../images/histEx.png">

Apply Histogram Equalization
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