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Computer Vision Spring 2012 15-385,-685 Instructor: S. Narasimhan Wean Hall 5409 T-R 10:30am – 11:50am

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Frequency domain analysis and Fourier Transform Lecture #4

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How to Represent Signals? Option 1: Taylor series represents any function using polynomials. Polynomials are not the best - unstable and not very physically meaningful. Easier to talk about “signals” in terms of its “frequencies” (how fast/often signals change, etc).

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Jean Baptiste Joseph Fourier (1768-1830) Had crazy idea (1807): Any periodic function can be rewritten as a weighted sum of Sines and Cosines of different frequencies. Don’t believe it? –Neither did Lagrange, Laplace, Poisson and other big wigs –Not translated into English until 1878! But it’s true! –called Fourier Series –Possibly the greatest tool used in Engineering

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A Sum of Sinusoids Our building block: Add enough of them to get any signal f(x) you want! How many degrees of freedom? What does each control? Which one encodes the coarse vs. fine structure of the signal?

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Fourier Transform We want to understand the frequency of our signal. So, let’s reparametrize the signal by instead of x: f(x) F( ) Fourier Transform F( ) f(x) Inverse Fourier Transform For every from 0 to inf, F( ) holds the amplitude A and phase of the corresponding sine –How can F hold both? Complex number trick!

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Time and Frequency example : g(t) = sin(2p i f t) + (1/3)sin(2pi (3f) t)

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Time and Frequency = + example : g(t) = sin(2p i f t) + (1/3)sin(2pi (3f) t)

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Frequency Spectra example : g(t) = sin(2p i f t) + (1/3)sin(2pi (3f) t) = +

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Frequency Spectra Usually, frequency is more interesting than the phase

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= + = Frequency Spectra

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= + =

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= + =

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= + =

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= + =

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=

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FT: Just a change of basis...... * = M * f(x) = F( )

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IFT: Just a change of basis...... * = M -1 * F( ) = f(x)

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Fourier Transform – more formally Arbitrary functionSingle Analytic Expression Spatial Domain (x)Frequency Domain (u) Represent the signal as an infinite weighted sum of an infinite number of sinusoids (Frequency Spectrum F(u)) Note: Inverse Fourier Transform (IFT)

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Also, defined as: Note: Inverse Fourier Transform (IFT) Fourier Transform

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Fourier Transform Pairs (I) angular frequency ( ) Note that these are derived using

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angular frequency ( ) Note that these are derived using Fourier Transform Pairs (I)

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Fourier Transform and Convolution Let Then Convolution in spatial domain Multiplication in frequency domain

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Fourier Transform and Convolution Spatial Domain (x)Frequency Domain (u) So, we can find g(x) by Fourier transform FT IFT

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Properties of Fourier Transform Spatial Domain (x)Frequency Domain (u) Linearity Scaling Shifting Symmetry Conjugation Convolution Differentiation frequency ( ) Note that these are derived using

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Properties of Fourier Transform

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Example use: Smoothing/Blurring We want a smoothed function of f(x) H(u) attenuates high frequencies in F(u) (Low-pass Filter)! Then Let us use a Gaussian kernel

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Does not look anything like what we have seen Magnitude of the FT Image Processing in the Fourier Domain

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Does not look anything like what we have seen Magnitude of the FT

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Convolution is Multiplication in Fourier Domain * f(x,y)f(x,y) h(x,y)h(x,y) g(x,y)g(x,y) |F(s x,s y )| |H(s x,s y )| |G(s x,s y )|

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Low-pass Filtering Let the low frequencies pass and eliminating the high frequencies. Generates image with overall shading, but not much detail

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High-pass Filtering Lets through the high frequencies (the detail), but eliminates the low frequencies (the overall shape). It acts like an edge enhancer.

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Boosting High Frequencies

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Most information at low frequencies!

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Fun with Fourier Spectra

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Next Class Image resampling and image pyramids Horn, Chapter 6

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