1 Fourier Transformation Fourier Transformasjon Fourier Transformasjon f(x) F(u)
Fourier Transform and Applications By Njegos Nincic Fourier.
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Transcript of Fourier Transform and Applications By Njegos Nincic Fourier.
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Fourier Transform and Applications
By Njegos Nincic
Fourier
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Overview
Transforms Mathematical Introduction
Fourier Transform Time-Space Domain and Frequency Domain Discret Fourier Transform
Fast Fourier Transform Applications
Summary References
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Transforms
Transform: In mathematics, a function that results when a
given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits. (Britannica)
Can be thought of as a substitution
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Transforms
Example of a substitution: Original equation: x + 4x² – 8 = 0 Familiar form: ax² + bx + c = 0 Let: y = x² Solve for y x = ±√y
4
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Transforms
Transforms are used in mathematics to solve differential equations: Original equation: Apply Laplace Transform:
Take inverse Transform: y = Lˉ¹(y)
y'' 9y 15e 2t
s 2 L y 9 L y 1 5 s 2
Ly 15
s2s29
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Fourier Transform
Property of transforms: They convert a function from one domain to
another with no loss of information
Fourier Transform:
converts a function from the time (or spatial) domain to the frequency domain
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Time Domain and Frequency Domain
Time Domain: Tells us how properties (air pressure in a sound function, for
example) change over time:
Amplitude = 100 Frequency = number of cycles in one second = 200 Hz
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Time Domain and Frequency Domain
Frequency domain: Tells us how properties (amplitudes) change over
frequencies:
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Time Domain and Frequency Domain
Example: Human ears do not hear wave-like oscilations,
but constant tone
Often it is easier to work in the frequency domain
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Time Domain and Frequency Domain
In 1807, Jean Baptiste Joseph Fourier showed that any periodic signal could be represented by a series of sinusoidal functions
In picture: the composition of the first two functions gives the bottom one
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Time Domain and Frequency Domain
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Fourier Transform
Because of the
property:
Fourier Transform takes us to the frequency domain:
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Discrete Fourier Transform
In practice, we often deal with discrete functions (digital signals, for example)
Discrete version of the Fourier Transform is much more useful in computer science:
O(n²) time complexity
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Fast Fourier Transform
Many techniques introduced that reduce computing time to O(n log n)
Most popular one: radix-2 decimation-in-time (DIT) FFT Cooley-Tukey algorithm:
(Divide and conquer)
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Applications
In image processing: Instead of time domain: spatial domain (normal
image space) frequency domain: space in which each image
value at image position F represents the amount that the intensity values in image I vary over a specific distance related to F
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Applications: Frequency Domain In Images
If there is value 20 at the point that represents the frequency 0.1 (or 1 period every 10 pixels). This means that in the corresponding spatial domain image I the intensity values vary from dark to light and back to dark over a distance of 10 pixels, and that the contrast between the lightest and darkest is 40 gray levels
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Applications: Frequency Domain In Images
Spatial frequency of an image refers to the rate at which the pixel intensities change
In picture on right: High frequences:
Near center Low frequences:
Corners
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Applications: Image Filtering
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Other Applications of the DFT
Signal analysis Sound filtering Data compression Partial differential equations Multiplication of large integers
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Summary
Transforms: Useful in mathematics (solving DE)
Fourier Transform: Lets us easily switch between time-space domain
and frequency domain so applicable in many other areas
Easy to pick out frequencies Many applications
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References
Concepts and the frequency domain http://www.spd.eee.strath.ac.uk/~interact/fourier/concepts.html
THE FREQUENCY DOMAIN Introduction http://www.netnam.vn/unescocourse/computervision/91.htm
JPNM Physics Fourier Transform http://www.med.harvard.edu/JPNM/physics/didactics/improc/intro/fourier2.html
Introduction to the Frequency Domain http://zone.ni.com/devzone/conceptd.nsf/webmain/
F814BEB1A040CDC68625684600508C88 Fourier Transform Filtering Techniques
http://www.olympusmicro.com/primer/java/digitalimaging/processing/fouriertransform/index.html
Fourier Transform (Efunda) http://www.efunda.com/math/fourier_transform/
Integral Transforms http://www.britannica.com/ebc/article?tocId=9368037&query=transform&ct=