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Filtering in the Frequency Domain. Chapter 4 . 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. History. - PowerPoint PPT Presentation
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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