Image and Multidimensional Signal Processing

Transcription

1 Image and Mltidimensional Signal Processing Professor William Hoff Dept of Electrical Engineering &Compter Science

2 Forier Transform Part : D discrete transforms 2

3 Overview This topic is abot Representing images in the freqency domain Understanding the operation of filters in this domain Topics: Forier transform definition and properties The 2D discrete Forier transform Convoltion theorem mltiplication in the freqency domain is eqivalent to convoltion in the spatial domain Filters and applications smoothing, sharpening, enhancement 3

4 Forier Invented Forier analysis to solve the heat eqation Original application was to nderstand the flow of heat when boring a cannon barrel Jean Baptiste Joseph Forier ( ) Main idea Any periodic fnction can be expressed as a sm of sines and cosines of different freqencies, each mltiplied by a different coefficient This is called a Forier series Other ideas Given enogh terms, the fnction can be represented exactly The fnction can be recovered from the freqency domain representation First five terms of the Forier series representing a sawtooth fnction (from 4

5 Complex Nmbers A Forier series is a sm of sines and cosines We can simplify its representation by sing complex nmbers Let j be the imaginary nmber j A complex nmber is defined as C R ji In polar coordinates C C cosq j sinq Eler s formla jq e cosq j sin q, so C C e jq imag I Complex conjgate * C R ji q R C real Matlab fnctions complex, conj, abs, angle Forier series f t c e j2n / T ( ) n n 5

6 Implse fnction The implse fnction is helpfl to nderstanding Forier transforms It is defined as if t 0 () t 0 if t 0 where we constrain it sch that ( t) dt Discrete eqivalents: if x 0 ( x) 0 if x 0 x x ( x) f ( x) ( x x ) f ( x ) 0 0 Sifting property f ( t) ( t) dt f (0), also f ( t) ( t t ) dt f ( t ) 0 0 6

7 Forier Transform The D Forier transform (continos) F( ) f ( x) e j2x Inverse Forier transform f ( x) F( ) e j2x dx d x: spatial domain : freqency domain We often write F F f x F ( ) F( ) ( ) f ( x) f(x) and F() are called Forier transform pairs f ( x) F( ) 7

8 Example Find the Forier transform of the box or rectangle fnction Soltion F( ) A j2 A f ( t) e j2t j2t W / 2 e jtw e j 2 W / 2 Ae W / 2 A e j2 sin W AW W j2t dt jtw W / 2 e e dt jtw jtw This is known as the sinc fnction sin m sinc( m) m 8

9 Discrete Forier Transform Discrete Forier transform and inverse M j2 x/ M ( ) ( ) for 0,,..., F f x e M x0 f ( x) M M 0 F( ) e j2x/ M for x 0,,, M We consider f(x) and F() to be periodic in the discrete domain -M What is F(0)? M 2M 9

10 A Note on Units Units Spatial nits (x): length (e.g., x has nits of meters) Freqency (): / length (e.g., has nits of meters - ) We have M samples of f(x) taken at integer spacing: 0,,, M- We can consider the samples to be taken at intervals of Dx: f(0), f(dx), f(2dx),, f( (M-)Dx) 0.8 Similarly F() is sampled at: F(0), F(D), F(2D),, F( (M-) D) The longest possible wavelength is MDx => corresponds to min freq of D = /(MDx)

11 D discrete Forier transform x = 0:M-; = 0:M-; Example - Matlab % Create an inpt series f(),,f(m) % : for n=:m F(n) = 0; for m = :M F(n) = F(n) + f(m) * exp(-j*2*pi*(n)*x(m)/m); end end Example of the rectangle fnction Inpt series: f = zeros(,m); f(:4) =.0; f(m-2:m) =.0; Otpt series: F = Colmns throgh i i i Colmns 5 throgh i i i i Colmns 9 throgh i i i i Colmns 3 throgh i i i i Colmns 7 throgh Department i of Electrical Engineering i and Compter Science i i

12 Fast Forier Transform Redces cost from O(M 2 ) to O(M log M) Recrsively divides problem into 2, and comptes transform of each Reqires #samples to be a power of 2 (if not, can pad with zeros) F(µ) = F even (µ) + F odd (µ)*e -j2pi µ /2K for 0 µ < K F(µ) = F even (µ) F odd (µ)*e -j2pi µ /2K for K µ < 2K Matlab s fft fnction 2

13 Important Forier Transform Pairs Rectangle Sin A rectangle transforms to a sinc fnction rect( t) sinc( ) Cos A cosine transforms to two implses cos(2 0x) ( 0) ( 0) 2 A sine transforms to two (imaginary) implses sin(2 0x) j ( 0) ( 0) 2 Gassian A Gassian transforms to a Gassian e x e /

14 Important Forier Transform Pairs Implse ( x) j2x j2x F( ) f ( x) e dx ( x) e dx j20 0 e e Shifted implse ( x a) e j2 a j2 ax e a ( ) 4

15 Important Forier Transform Pairs Periodic implse train (comb) n n ( x ndx) ( ) Dx Dx n Dx x Proof Let By definition of Forier series, we can write as where s ( ) ( ) Dx x x ndx n sd x( x) cne n 2 n j x Dx 2 n Dx/2 j x Dx cn s ( ) x/2 Dx x e dx D x D D /Dx Since integral covers only the single implse at the origin, 2 n Dx/2 j x 0 Dx cn ( x) e dx e Dx Dx/2 Dx Dx 5

16 Periodic implse train (contined) Proof (contined) The Forier series expansion of the plse train is ths sd x( x) e D x n 2 n j x Dx Now we take Forier transform 2n 2 n j x F j x Dx Dx sdx ( x) F e e Dx n Dx F n By the translation property j2 ax e a This becomes ( ) Dx n n Dx 6

17 Convoltion Theorem Convoltion in one domain is eqivalent to mltiplication in the other domain h( x)* f ( x) H ( ) F( ) convoltion point-by-point mltiplication h( x) f ( x) H ( )* F( ) point-by-point mltiplication convoltion Similarly for correlation, except that we have complex conjgate h( x) h * f ( x) H ( ) F( ) ( x) f ( x) H ( ) F( ) * As we will see a little later, this is very sefl for implementing large filters 7

18 Convoltion in D Proof h( x) f ( x) h( t ) f ( x t ) dt Forier transform of the convoltion F t t t t t t j t j t h t f t h f t d e dt h f t e dt d 2 2 ( ) ( ) ( ) ( ) ( ) ( ) The term in brackets is the Forier transform of f(t-t) From the translation property, we know that So F F F j2 f ( t t ) F( ) e t j j t h t f t h F e d F h e d 2 t 2 ( ) ( ) ( ) ( ) ( ) ( ) h( t) f ( t) H( ) F( ) t t t t 8

19 Understanding Sampling and Aliasing We often sbsample a signal When we originally digitize it When we shrink it We can reconstrct the signal exactly from the samples if the samples are dense enogh The sampling theorem says that the sampling rate mst be more than twice the maximm freqency of the inpt signal (this is the Nyqist rate ) If the sampling rate is lower, we can get errors in the reconstrcted signal (called aliasing ) 9

20 Sampling Sampling of a continos fnction f(t) can be modeled by mltiplying it with an implse train s ( t) ( t ndt ) f ( t) f ( t) s ( t) DT n n DT f ( t) ( t ndt ) 20

21 Sampling Recall that mltiplication in the spatial domain convoltion in the freqency domain We know that the Forier transform of a comb fnction is n S( ) DT n DT So f ( t) sd T ( t) F( ) S( ) F( ) S( ) F( t ) S( t ) dt n F( t ) t dt T D n DT n n F( t ) t dt F T D n DT DT n DT Copies of F() at intervals of /DT 2

22 Forier transform of the sampling fnction Sampling S() 2Dx /Dx /Dx 2Dx Forier transform of the original fnction F() Convolving the two reslts in periodic copies of F() S()*F() 2Dx /Dx /Dx 2Dx /2Dx /2Dx 22

23 Reconstrction We can reconstrct the original continos fnction from the samples We jst need to eliminate the extra copies by mltiplying by an ideal low pass filter F s () = S()*F() 2Dx /Dx /Dx 2Dx H() Ideal low pass filter 2Dx /Dx /2Dx /2Dx /Dx 2Dx H() F s () Mltiplication - gets back the original F() 2Dx /Dx /Dx 2Dx 23

24 Reconstrction Recall that mltiplication in freqency domain is eqivalent to convoltion in the spatial domain F s () /2Dx /2Dx H() /2Dx /2Dx H() F s () /2Dx /2Dx 24

25 Forier transform of the sampling fnction Under Sampling S() 2Dx /Dx /Dx 2Dx Forier transform of the original fnction F() If freqency range of original fnction exceeds ±/2Dx, we have aliasing /2Dx /2Dx S()*F() 2Dx /Dx /2Dx /2Dx /Dx 2Dx 25

26 Under Sampling S()*F() 2Dx /Dx /2Dx /2Dx /Dx 2Dx Ideal low pass filter 2Dx /Dx /Dx 2Dx H() F s () Mltiplication 2Dx /Dx /2Dx /2Dx /Dx 2Dx 26

27 Example of aliasing 27

28 Smmary / Qestions Any periodic fnction can be expressed as a sm of sines and cosines of different freqencies, each mltiplied by a different coefficient. The Forier transform expresses this concisely sing complex nmbers. What is the implse fnction? the convoltion theorem? aliasing? 28