h [.,.] h [.,.] f [.,.] f [.,.] 0 0 0 0 0 0 0 0 0 0 - PowerPoint PPT Presentation

Review - Point Operations BBM 413 Fundamentals of • Smallest possible neighborhood is of size 1x1 Image Processing • Process each point independently of the others • Output image g depends only on the value of f at a single point (x,y) • Transformation function T remaps the sample’s value: Erkut Erdem s = T(r) Dept. of Computer Engineering Hacettepe University where – r is the value at the point in question – s is the new value in the processed result Frequency Domain – T is a intensity transformation function Techniques – Part1 1 1 1 1 1 1 × × × × Review – Spatial Filtering Review – Spatial Filtering g [ , ] g [ , ] 1 1 1 1 1 1 1 1 1 1 1 1 h [.,.] h [.,.] f [.,.] f [.,.] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 90 0 90 90 90 0 0 0 0 0 0 0 0 90 90 0 0 90 90 90 90 90 90 0 0 0 0 0 0 0 90 0 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 0 0 0 0 0 0 0 0 0 0 0 90 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = å = å + + + + h [ m , n ] g [ k , l ] f [ m k , n l ] h [ m , n ] g [ k , l ] f [ m k , n l ] k , l k , l Slide credit: S. Seitz Slide credit: S. Seitz

1 1 1 1 1 1 × × × × Review – Spatial Filtering g [ , ] Review – Spatial Filtering g [ , ] 1 1 1 1 1 1 1 1 1 1 1 1 h [.,.] h [.,.] f [.,.] f [.,.] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 20 0 10 20 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 0 0 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 0 0 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = å = å + + + + h [ m , n ] g [ k , l ] f [ m k , n l ] h [ m , n ] g [ k , l ] f [ m k , n l ] k , l k , l Slide credit: S. Seitz Slide credit: S. Seitz 1 1 1 1 1 1 × × × × Review – Spatial Filtering Review – Spatial Filtering g [ , ] g [ , ] 1 1 1 1 1 1 1 1 1 1 1 1 h [.,.] h [.,.] f [.,.] f [.,.] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 20 30 30 0 0 0 0 0 0 0 0 0 0 0 10 20 30 30 30 20 10 0 0 0 90 90 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 20 40 60 60 60 40 20 0 0 0 90 90 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 30 60 90 90 90 60 30 0 0 0 90 90 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 30 50 80 80 90 60 30 0 0 0 90 0 90 90 90 0 0 0 0 0 90 0 90 90 90 0 0 0 30 50 80 80 90 60 30 0 0 0 90 90 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 20 30 50 50 60 40 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 20 30 30 30 30 20 10 0 0 90 0 0 0 0 0 0 0 0 0 90 0 0 0 0 0 0 0 10 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = å = å + + + + h [ m , n ] g [ k , l ] f [ m k , n l ] h [ m , n ] g [ k , l ] f [ m k , n l ] k , l k , l Slide credit: S. Seitz Slide credit: S. Seitz

Review – Spatial Filtering Review – Spatial Filtering Fill in the blanks: A Filtering Operator a) _ = D * B b) A = _ * _ c) F = D * _ B d) _ = D * D E 1 0 -1 G 2 0 -2 C F 1 0 -1 Sobel D H I Slide credit: J. Hays Slide credit: D. Hoiem Today Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts? • Frequency domain techniques • Images in terms of frequency Gaussian Box filter • Fourier Series • Convolution Theorem Slide credit: D. Hoiem

Why does a lower resolution image still make sense How is it that a 4MP image can be compressed to a to us? What do we lose? few hundred KB without a noticeable change? Slide credit: D. Hoiem Slide credit: J. Hays Image: http://www.flickr.com/photos/igorms/136916757/ Jean Baptiste Joseph Fourier (1768-1830) Answer to these questions? had crazy idea (1807): • Thinking images in terms of frequency. Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies. • Treat images as infinite-size, continuous periodic ...& functions. )& ...& =& =& ...& ...& Slide credit: A. Efros

Jean Baptiste Joseph Fourier (1768-1830) Jean Baptiste Joseph Fourier (1768-1830) ...the manner in which the author arrives at these had crazy idea (1807): had crazy idea (1807): equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired Any univariate function can be Any univariate function can be on the score of generality and even rigour . rewritten as a weighted sum of rewritten as a weighted sum of sines and cosines of different sines and cosines of different frequencies. frequencies. • Don’t believe it? • Don’t believe it? – Neither did Lagrange, – Neither did Lagrange, Laplace Laplace, Poisson and Laplace, Poisson and other big wigs other big wigs – Not translated into – Not translated into English until 1878! English until 1878! • But it’s (mostly) true! Lagrange Legendre – called Fourier Series – there are some subtle restrictions Slide credit: A. Efros A sum of sines Frequency Spectra Our building block: • example : g ( t ) = sin( 2 π f t ) + ( 1/3 )sin( 2 π ( 3f ) t ) w x + f ) A sin( Add enough of them to get any signal f(x) you want! = + Slide credit: A. Efros Slide credit: A. Efros

Frequency Spectra Frequency Spectra = + = Slide credit: A. Efros Slide credit: A. Efros Frequency Spectra Frequency Spectra = = + + = = Slide credit: A. Efros Slide credit: A. Efros

Frequency Spectra Frequency Spectra = = + + = = Slide credit: A. Efros Slide credit: A. Efros Frequency Spectra Frequency Spectra ¥ 1 sin(2 å p = A kt ) k = k 1 Slide credit: A. Efros Image credit: Lucas V. Barbosa

Example: Music Other signals • We think of music in terms of frequencies at different • We can also think of all kinds of other signals the same way magnitudes. xkcd.com Slide credit: J. Hays Slide credit: D . Hoeim Fourier Transform Fourier Transform • Fourier transform stores the magnitude and phase at each We want to understand the frequency w of our signal. So, let ’ s frequency reparametrize the signal by w instead of x : – Magnitude encodes how much signal there is at a particular frequency Fourier – Phase encodes spatial information (indirectly) f( f(x) F( F(w) – For mathematical convenience, this is often notated in terms of real and Transform complex numbers For every w from 0 to inf, F(w) holds the amplitude A and w x + f ) A sin( w I ( ) phase f of the corresponding sine f = - = ± w 2 + w 2 tan 1 A R ( ) I ( ) Amplitude: Phase: • How can F hold both? Complex number trick! w R ( ) w = w + w F ( ) R ( ) iI ( ) w I ( ) - = ± w + w f = tan 1 2 2 A R ( ) I ( ) w R ( ) We can always go back: Inverse Fourier F(w) F( f(x) f( Transform Slide credit: A. Efros