1. Filtering in the Frequency Domain
Chapter 4

2. Chapter Objectives
This chapter is concerned primarily with establishing a foundation for the Fourier transform and how it is used in basic image filtering.
Fourier Transformation History
The big Idea
Background

3. History Jean Baptiste Joseph Fourier
Fourier was born in Auxerre,France in 1768
Most famous for his work:
“LaThéorie Analitique de la Chaleur”
published in 1822.
Translated into English in 1878: “The Analytic Theory of Heat”
Nobody paid much attention when the work was first published.
One of the most important mathematical theories in modern engineering.

4. The Big Idea

5. The Big Idea
Any function that periodically repeats itself can be expressed as a sum of sines and cosines of different frequencies each multiplied by a different coefficient – a Fourier series

6. Background Fourier Series Fourier Transform
Any periodically repeated function can be expressed of the sum of sines/cosines of different frequencies, each multiplied by a different coefficient
Fourier Transform
Finite curves can be expressed as the integral of sines/cosines multiplied by a weighing function
wildly used in signal processing field
Fourier Series/Transform can be reconstructed completely via an inverse process with no loss of information

7. The Frequency Domain
Euler’s formula
e jθ = cosθ + j sinθ

8. Discrete Fourier Transform
1D Discrete Fourier Transform
2D Discrete Fourier Transform

9. Basic properties of frequency domain
Each term of F(u, v) contains all values of f(x, y) modified by the values of exponential terms
Not easy to make direct association between specific components of an image and its transform
Frequency is related to rate of change
Frequencies in Fourier transform can be associated with patterns of intensity variations in image
Slowest varying frequency component (u = v = 0) corresponds to average gray level of image
As you move away from origin of transform, low frequencies correspond to slowly varying components of image
Farther away from origin, you have higher frequencies corresponding to faster gray level changes

10. Filtering Images in the Frequency Domain

11. Filtering Images in the Frequency Domain

12. Filtering Images in the Frequency Domain
Filter or filter transfer function
Suppresses certain frequencies in transform while leaving others unchanged
Fourier transform of the output image
G(u, v) = H(u, v) · F(u, v)
Multiplication of H and F is only between corresponding elements
G(0, 0) = H(0, 0) · F(0, 0)

13. Filtering Scheme

14. Filters in the Frequency Domain
Image Smoothing Using Frequency Domain Filters:
Ideal Lowpass Filters
Butterworth Lowpass Filters
Gaussian Lowpass Filters.
Image sharpening Using Frequency Domain Filters:
Ideal Highpass Filters
Butterworth Highpass Filters
Gaussian Highpass Filters.
The Laplacian in the Frequency Domain
Unsharp Masking, Highboost Filtering, and High- Frequency-Emphasis Filtering