signal and image processing a short introsaad/onbi/sj_fourier.pdf · signal and image processing a...
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Signal and Image processing A short intro
Saad Jbabdi, FMRIB Centre, Oxford
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When I say signal processing…
• Fourier series, Fourier transform, FFT
• A bit of image processing (but related to Fourier)
• A bit on image reconstruction (also related to Fourier)
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Fourier Analysis what’s it all about?
• Sines and cosines are building blocks
source: wikipedia
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Why?• Frequency content
time
freq
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Why?• Cleaning
Fourier
inverse
spectrum
cut spectrum
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Why?
• Frequency content
• Cleaning (filtering)
• Computational efficiency
• Also very useful in abstract maths (beyond signal processing)
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Fourier Series
y = sin 2πt( )
mag
nitu
de
frequency1
1
time/length
ampl
itude
• Spectrum of the sinusoid:
Signal domain Fourier domain
The ‘fundamental’ frequency
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Frequency - 2D
• Define sinusoid going in x and y directions.
x frequency = 2 y frequency = 0
x frequency = 0 y frequency = 2
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Fourier SeriesApproximate periodic signals with sines and cosines
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Fourier Series
time/length
ampl
itude
mag
nitu
de
frequency
1
Signal Frequency Spectrum
• Keep going: y t( ) = an sin nf × 2πt( ) + bn cos nf × 2πt( )n∑
n∑
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Fourier Analysis what’s it all about?
• Sines and cosines are building blocks
fundamental frequency
harmonic
harmonic
harmonic
magnitudedetails increasing
with frequency
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Fourier TransformApproximate non-periodic signals with sines and cosines
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Fourier Series --> Transforms
• Fourier Series good for periodic signals, what do we do if they are not? – Classic example: the top hat function
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Fourier Series --> Transforms
time/length
ampl
itude
mag
nitu
de
frequency
• Try making it periodic over a certain period:
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Fourier Series --> Transforms
time/length
ampl
itude
mag
nitu
de
frequency
• Make the spacing larger, so that there is a larger space in which only the top hat appears:
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Fourier Series --> Transforms
time/length
ampl
itude
mag
nitu
de
frequency
• In the limit (T = infinity) we get a smooth spectrum:
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y t( ) = A f( )e j2π f tdf−∞
∞
∫
Fourier Transform
• Fourier series was a sum at specific frequencies:
• Fourier transform is a sum over all frequencies:
– Note: this formula is usually called the inverse FT.
y t( ) = an sin nf × 2πt( ) + bn cos nf × 2πt( )n∑
n∑
Negative frequencies
Frequency
Sine/Cosine (compact notation)
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Fourier Transform
• To find the frequency spectrum we need to do an integral:
Basically: at frequency f, how much of sin(f x 2πt) do I need?
• In practice we get computers do deal with this using the Fast Fourier Transform (FFT).
A f( ) = y t( )e− j2π f tdt−∞
∞
∫
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Fourier transform
Fast Fourier Transform
Mathematical operation
Algorithm
Continuous
Discrete
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Fourier Transform
• Examples:
FFT Result‘White’ noise
Ideally flat - all frequencies
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Fourier Transform - 1D
• Examples:
FFT ResultPiano note
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Fourier transform
Powerspectrum
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Power spectrum lacks phase information
Powerspectrum
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A f( ) = y t( )e− j2π f tdt−∞
∞
∫signal
Fourier transform
complex number
complex number
Complex numbers easier to manipulate than sine and cosine functions
Information on phase and magnitude
mag
phase
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Magnitude, PhaseWhat? Where?
FFT Same magnitudeDifferent phases
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Fourier Transform - 2D
• 2D signals, i.e. images have 2D Fourier transforms • We now have x and y frequencies:
Low frequencies
High x frequencies
High y frequencies
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Fourier Transform - 2D
• Sinusoid in x direction
Image 2D spectrum
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Fourier Transform - 2D
• Sinusoid in x plus sinusoid in y direction
Image 2D spectrum
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Fourier Transform - 2D
• Single frequency in all directions
Image 2D spectrum
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Fourier Transform - 2D
• Examples:
• Spectrum is ‘bright’ in the centre.
• Detail involves high frequency.
• Spectrum is symmetric.
http://www.cs.unm.edu/~brayer/vision/fourier.html
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Image 1 Image 2
What kind of image would we get if we mix the phase and magnitude from these?
Courtesy of JM Brady
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A: Phaseor
B: Magnitude
A: Magnitudeor
B: Phase
Courtesy of JM Brady
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B: Magnitude B: Phase
Courtesy of JM Brady
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Phase Magnitude
Courtesy of JM Brady
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Magnitude Phase
What?
Where?
FFT Same magnitudeDifferent phases
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Filtering
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Filtering - 1D
• In Fourier signals are a mixture of different frequency components.
• Often we want some components and not others.
Signal Spectrum
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Filtering - 1D
• I want to get rid of high frequency noise component. • Low Pass filter - throw away high frequencies
Signal Spectrum
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Filtering - 1D
• I want to get rid of low frequency nosie/drift. • High Pass filter - throw away low frequencies.
Signal Spectrum
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• I want to get rid of that annoying component at [pick a frequency] (e.g. mains noise).
• Band Stop filter - let everything through except...
Filtering - 1D
Signal Spectrum
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• I want to get rid of all this other stuff that is not my signal (not always this simple!)
• Band Pass filter - get rid of everything but...
Filtering - 1D
Signal Spectrum
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Filtering - 2D
• Same principles as in 1D.
• Low Pass • Remove high
frequencies. • Loose detail.
http://www.cs.unm.edu/~brayer/vision/fourier.html
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Sampling, aliasing—> details in the practical
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real space Fourier space
T 1/T
T 1/T
Sampling in real space is like repeating in Fourier space and vice versa
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Time-frequency analysis
—> details in the practical
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Time-frequency
• Three different signals - same frequency spectrum.
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Time-frequency
• Need a ‘dynamic’ time-frequency representation.
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Image reconstruction—> details in the practical
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Radon transform (X-ray CT)
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Magnetic Resonance Imaging
Object of interest Measurement (Fourier space)
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www.fmrib.ox.ac.uk/~saad/GP/fourier.html