Computer Vision Spring 2012 15-385,-685 Instructor: S. Narasimhan Wean Hall 5409 T-R 10:30am – 11:50am Frequency domain analysis and Fourier Transform Lecture #4 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). 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 A Sum of Sinusoids • Our building block: Asin( x • 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? Fourier Transform • We want to understand the frequency of our signal. So, let’s reparametrize the signal by instead of x: Fourier Transform f(x) F() • For every from 0 to inf, F() holds the amplitude A and phase of the corresponding sine Asin( x – How can F hold both? Complex number trick! F ( ) R( ) iI ( ) A R( ) I ( ) 2 F() 2 Inverse Fourier Transform I ( ) tan R( ) 1 f(x) Time and Frequency • example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) Time and Frequency • example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) = + Frequency Spectra • example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) = + Frequency Spectra • Usually, frequency is more interesting than the phase Frequency Spectra = = + Frequency Spectra = = + Frequency Spectra = = + Frequency Spectra = = + Frequency Spectra = = + Frequency Spectra 1 = A sin(2 kt ) k 1 k Frequency Spectra FT: Just a change of basis M * f(x) = F() * . . . = IFT: Just a change of basis M-1 * F() = f(x) * . . . = Fourier Transform – more formally Represent the signal as an infinite weighted sum of an infinite number of sinusoids F u f x e i 2ux dx Note: e cos k i sin k ik i 1 Arbitrary function Single Analytic Expression Spatial Domain (x) Frequency Domain (u) (Frequency Spectrum F(u)) Inverse Fourier Transform (IFT) f x F u e i 2ux dx Fourier Transform • Also, defined as: F u f x eiuxdx Note: e cos k i sin k ik • Inverse Fourier Transform (IFT) 1 f x 2 F u eiuxdx i 1 Fourier Transform Pairs (I) Note that these are derived using angular frequency ( e iux ) Fourier Transform Pairs (I) Note that these are derived using angular frequency ( e iux ) Fourier Transform and Convolution Let g f h Then Gu g x e i 2ux dx f hx ei 2uxddx f e f e i 2u i 2u d d hx ei 2u x dx hx'e i 2ux' dx' F u H u Convolution in spatial domain Multiplication in frequency domain Fourier Transform and Convolution Spatial Domain (x) Frequency Domain (u) g f h g fh G FH G F H So, we can find g(x) by Fourier transform g IFT G f FT F h FT H Properties of Fourier Transform Spatial Domain (x) Frequency Domain (u) c1 f x c2 g x c1F u c2Gu Scaling f ax 1 u F a a Shifting f x x0 e i 2ux0 F u Symmetry F x f u Conjugation f x Convolution f x g x F u Differentiation d n f x dx n i 2u n F u Linearity F u Gu Note that these are derived using frequency ( e i 2ux )