Discrete Fourier Transform (DFT) in 2D Signals - Prof. Alregib, Assignments of Digital Image Processing

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ECE 6258 | ECE 4803 | BMED 8813

Digital Image Processing

Fall 2023

DFT

Image Transform: Outline

• DFT (Discrete Fourier Transform)
• DCT (Discrete Cosine Transform)
• Unitary Transform
• KLT (Karhunen-Loève Transform)
• Directional Transforms
• Autoencoders

Transforms are Everywhere

• e.g. Image Coding

Introduction

• We studied the Discrete Time Fourier Transform (DTFT), 𝐹(𝜇, 𝜈) but in

practice computing the Fourier Transform is not possible

• It’s still continuous in frequency domain!!
• Computers are digital!
• Thus, the DTFT is only an analytical tool
• In this part of the course, we study DFT
• DTFT: discrete time & continuous frequency
• DFT: discrete time & discrete frequency

Review:

DTFT pair

𝐹 𝜇, 𝜈 = ෍

𝑚=−∞

∞

෍

𝑛=−∞

∞

𝑓 𝑚, 𝑛 𝑒

−𝑗2𝜋(𝑚𝜇+𝑛𝜈)

𝑓 𝑚, 𝑛 = න

−

1

2

1

2

න

−

1

2

1

2

𝐹(𝜇, 𝜈) 𝑒

𝑗2𝜋(𝑚𝜇+𝑛𝜈) 𝑑𝜇𝑑𝜈

DFT Outline

• Definition of DFT
• Properties of DFT
• Linear Convolution vs. Circular Convolution
• Computing DFT for real images
• DFT in matrix form
• Plotting DFT

2 - D Discrete Fourier Transform

• The DFT is obtained by sampling the DTFT (on a rectangular lattice)
  • continuous frequency  discrete frequency
• The DFT transfers the image into the frequency domain and then

performs various operations in that domain

• The DFT is energy compactor
• It helps with analyzing various processes
• It is derived for a periodic sequence

Discrete Fourier Series (DFS)

• DFS: discrete time & discrete frequency  We can compute DFS
• The 2 - D DFS gives us a way to write

𝑓 𝑚, 𝑛 as a superposition of

harmonically related complex sinusoids:

𝑘= 0

𝑀− 1

𝑙= 1

𝑁− 1

𝐹 𝑘, 𝑙 exp 𝑗

Fourier Series Coefficient

Periodic in both k , l and m , n

Note: we don’t limit the range of m & n

Discrete Fourier Series (DFS)

• The coefficients,

𝐹 𝑘, 𝑙 , can be computed from

𝑓[𝑚, 𝑛] using:

𝑚= 0

𝑀− 1

𝑛= 0

𝑁− 1

𝑓 𝑚, 𝑛 exp −𝑗

Note: we don’t limit the range of k & l

Definition of DFT

• Recall:

𝑚= 0

𝑀− 1

𝑛= 0

𝑁− 1

−𝑗2𝜋(𝜇𝑚+𝜈𝑛)

thus, we can write:

• So far we know:
  • DFT = finite region version of DFS
    • The 2-D DFT is a Fourier Series representation for the periodic extension of 𝑓[𝑚, 𝑛]
  • 𝐹[𝑘, 𝑙] = 𝐹

𝑘

𝑀

,

𝑙

𝑁

• The 2-D DFT consists of samples of the Fourier Transform

𝜇 →

𝑘

𝑀

, 𝜈 →

𝑙

𝑁

DFT Outline

• Definition of DFT
• Properties of DFT
• Linear Convolution vs. Circular Convolution
• Computing DFT for real images
• DFT in matrix form
• Plotting DFT

Linearity

• When 𝑓[𝑚, 𝑛] and 𝑔[𝑚, 𝑛] have the support on the same 𝑀 × 𝑁

region

• Note: If the two images don’t have support on the same region, we

perform zero padding

• Circular Shift

0 𝑀

0 𝑁

𝑀

𝑚 0

𝑘

𝑁

𝑛 0

𝑙

𝐹[𝑘, 𝑙]

• Example：

Circular Shift

(( )) mod ( *int( / )) N

q  q N  q  N q N

𝑊 𝑁

= 𝑒

−𝑗

2𝜋

𝑁

Symmetry

• Similarly, if we have a sequence that has Hermitian symmetric and

asymmetric portions, then we can find the following:

𝑠

∗

𝑀 − 𝑚

𝑀

𝑁

𝑎

∗

𝑀 − 𝑚

𝑀

𝑁

Symmetry

• Combining all four equation lead us to:

𝑠

∗

[𝑘, 𝑙] = Re 𝐹[𝑘, 𝑙]

𝑎

∗

[𝑘, 𝑙] = 𝑗 Im 𝐹[𝑘, 𝑙]

OR

Hermitian Symmetric Portion ↔ Real Part of DFT

Hermitian Asymmetric Portion ↔ 𝑗 × imaginary part of DFT