Filtering in the Frequency Domain

Chapter 4

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

Chapter Objectives

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.

History

The Big Idea

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

Fourier Series◦ 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

Background

Euler’s formulae jθ = cosθ + j sinθ

The Frequency Domain

1D Discrete Fourier Transform 2D Discrete Fourier Transform

Discrete Fourier Transform

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

Basic properties of frequency domain

Filtering Images in theFrequency Domain

Filtering Images in theFrequency Domain

Filter or filter transfer function◦ Suppresses certain frequencies in transform

while leaving others unchanged

Fourier transform of the output imageG(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)

Filtering Images in theFrequency Domain

Filtering Scheme

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

Filters in the Frequency Domain