Fourier Analysis Without Tears15-463: Computational PhotographyAlexei Efros, CMU, Fall 2006Somewhere in Cinque Terre, May 2005Capturing what’s importantx1x0x1x01stprincipal component2ndprincipal componentFast vs. slow changesA nice set of basisThis change of basis has a special name…Teases away fast vs. slow changes in the image.Jean Baptiste Joseph Fourier (1768-1830)had crazy idea (1807):Any periodic function can be rewritten as a weighted sum of sinesand 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 SeriesA sum of sinesOur 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?)+φωxAsin(Fourier TransformWe want to understand the frequency ω of our signal. So, let’s reparametrize the signal by ωinstead of x:)+φωxAsin(f(x)F(ω)Fourier TransformF(ω)f(x)Inverse Fourier TransformFor every ωfrom 0 to inf, F(ω) holds the amplitude A and phase φ of the corresponding sine • How can F hold both? Complex number trick!)()()(ωωωiIRF+=22)()(ωωIRA +±=)()(tan1ωωφRI−=We can always go back:Time and Frequencyexample : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)Time and Frequencyexample : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)= +Frequency Spectraexample : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)= +Frequency SpectraUsually, frequency is more interesting than the phase= += Frequency Spectra= += Frequency Spectra= += Frequency Spectra= += Frequency Spectra= += Frequency Spectra= 11sin(2 )kAktkπ∞=∑Frequency SpectraFrequency SpectraFT: Just a change of basis...*=M * f(x) = F(ω)IFT: Just a change of basis...*=M-1* F(ω) = f(x)Finally: Scary MathFinally: Scary Math…not really scary:is hiding our old friend:So it’s just our signal f(x) times sine at frequency ω)sin()cos( xixexiωωω+=⎟⎟⎠⎞⎜⎜⎝⎛ = +±=)+=+−QPQPΑxAxQxP122tansin()sin()cos(φφ)+φωxAsin(phase can be encodedby sin/cos pairExtension to 2Din Matlab, check out: imagesc(log(abs(fftshift(fft2(im)))));2D FFT transformMan-made SceneCan change spectrum, then reconstructMost information in at low frequencies!CampbellCampbell--Robson contrast sensitivity curveRobson contrast sensitivity curveWe don’t resolve high frequencies too well…… let’s use this to compress images… JPEG!Lossy Image Compression (JPEG)Block-based Discrete Cosine Transform (DCT)Using DCT in JPEG A variant of discrete Fourier transform• Real numbers• Fast implementationBlock size• small block–faster – correlation exists between neighboring pixels• large block– better compression in smooth regionsUsing DCT in JPEG The first coefficient B(0,0) is the DC component, the average intensityThe top-left coeffs represent low frequencies, the bottom right – high frequenciesImage compression using DCTDCT enables image compression by concentrating most image information in the low frequenciesLoose unimportant image info (high frequencies) by cutting B(u,v) at bottom right The decoder computes the inverse DCT – IDCT •Quantization Table3 5 7 9 11 13 15 175 7 9 11 13 15 17 197 9 11 13 15 17 19 219 11 13 15 17 19 21 2311 13 15 17 19 21 23 2513 15 17 19 21 23 25 2715 17 19 21 23 25 27 2917 19 21 23 25 27 29 31JPEG compression comparison89k
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