EECE253_06_FourierTransform

EECE253_06_FourierTransform - EECE\CS 253 Image Processing...

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Unformatted text preview: EECE\CS 253 Image Processing Lecture Notes: The 1&2-Dimensional Fourier Transforms Lecture Notes Richard Alan Peters II Department of Electrical Engineering and Computer Science Fall Semester 2011 This work is licensed under the Creative Commons Attribution-Noncommercial 2.5 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/2.5/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA. Signal: A measurable phenomenon that changes over time or throughout space. image sound code 01101000101101110110010110001 2011-09-27 1999-2011 by Richard Alan Peters II 2 Signals: Space-Time vs. FrequencyDomain Representation Space/time representation: a graph of the measurements with respect to a point in time and/or positions in space. Fact: signals undulate (otherwise they’d contain no information). Frequency-domain representation: an exact description of a signal in terms of its undulations. 2011-09-27 1999-2011 by Richard Alan Peters II 3 Origin of Sounds 2011-09-27 The mechanical vibrations of an object in an atmosphere. Vibrations: internal elastic motions of the material. The surface of the object undulates causing compressions and rarefactions in the air which propagate through the air away from the surface. An object vibrates with different modes. A mode is a vibratory pattern with a distinctive shape — part of the object surface moves out while another part moves in — a standing wave. 1999-2011 by Richard Alan Peters II 4 Vibratory Modes / Standing Waves: Examples displacement from rest position internal pressure string modes pipe modes 2011-09-27 Note that the modes are all sinusoids. 1999-2011 by Richard Alan Peters II Note that the negatives of these also will occur 5 Sound Waves: Emerge from the superposition of the modes. 1999-2011 by Richard Alan Peters II pipe sound string sound 2011-09-27 6 Sound Waves: The vibratory modes add up to one complex motion that pushes the air around the vibrating object 1999-2011 by Richard Alan Peters II Odd-order harmonics pipe sound 2011-09-27 string sound Even-order harmonics Emerge from the superposition of the modes. 7 Fact: Any Real Signal has a Frequency-Domain Representation Odd-order harmonics sq (t ) = é 2p ù 1 sin ê (2n + 1) t ú å 2n +1 êë l úû n=-¥ ¥ The modes shown (blue) sum to the rippling square wave (black). As the number of modes in the sum becomes large, it approaches a square wave (red). 2011-09-27 1999-2011 by Richard Alan Peters II 8 Frequency-Domain Representation Any periodic signal can be described by a sum of sinusoids. sq (t ) = é 2p ù 1 sin ê (2n + 1) t ú å 2n +1 êë l úû n=-¥ ¥ The sinusoids are called “basis functions”. The multipliers are called “Fourier coefficients”. 2011-09-27 1999-2011 by Richard Alan Peters II 9 Frequency-Domain Representation Any periodic signal can be described by a sum of sinusoids. sq (t ) = é 2p ù 1 sin ê (2n + 1) t ú å 2n +1 êë l úû n=-¥ ¥ The sinusoids are called “basis functions”. The multipliers are called “Fourier coefficients”. 2011-09-27 Basis functions 1999-2011 by Richard Alan Peters II 10 Frequency-Domain Representation Any periodic signal can be described by a sum of sinusoids. sq (t ) = é 2p ù 1 sin ê (2n + 1) t ú å 2n +1 êë l úû n=-¥ ¥ The sinusoids are called “basis functions”. The multipliers are called “Fourier coefficients”. 2011-09-27 The Fourier coefficients (of a square wave). 1999-2011 by Richard Alan Peters II 11 Example: Partial Sums of a Square Wave The limit of the given sequence of partial sums1 is exactly a square wave 1 sine 2 sines 4 sines the limit as n approaches infinity of the sum of n sines. 1 8 sines 2011-09-27 16 sines 1999-2011 by Richard Alan Peters II 32 sines 12 Anatomy of a Sinusoid f (t) 0 æ 2p ö ç f (t ) = A sin ç t - f÷ ÷ ÷ èl ø 2011-09-27 1999-2011 by Richard Alan Peters II 13 The Inner Product: a Measure of Similarity The similarity between functions f and g on the interval (-λ / 2, λ / 2) can be defined by l /2 f,g = ò f (t ) g * (t ) dt -l /2 where g * (t ) is the complex conjugate of g (t ). This number, called the inner product of f and g , can also be thought of as the amount of g in f or as the projection of f onto g . If f and g have the same energy, then their inner product is maximal if f = g . On the other hand if f , g = 0, then f and g have nothing in common. 2011-09-27 1999-2011 by Richard Alan Peters II 14 Inner Products a function, f pointwise product f(t)g(t) 1024 g is a component of f f (t ) g (t )dt ≈ 0.12 0 2011-09-27 1999-2011 by Richard Alan Peters II 15 Inner Products a function, f pointwise product f(t)h(t) 1024 h is a not a comp. of f f (t )h(t )dt ≈ 0 0 2011-09-27 1999-2011 by Richard Alan Peters II 16 Inner Product of a Periodic Function and a Sinusoid 3 different representations 2011-09-27 1999-2011 by Richard Alan Peters II 17 Inner Product of a Periodic Function and a Sinusoid real number results yield the amplitude of that sinusoid in the function. 2011-09-27 1999-2011 by Richard Alan Peters II 18 Inner Product of a Periodic Function and a Sinusoid Complex number result yields the amplitude and phase of that sinusoid in the function. 2011-09-27 1999-2011 by Richard Alan Peters II 19 The Fourier Series ¥ f (t ) = A0 + å n=1 is the decomposition of a λ-periodic signal into a sum of sinusoids. æ 2p n ö ÷ + Bn sin æ 2p n t ö ç ç An cos ç t÷ ÷ çl ÷ ÷ ÷ èl ø è ø periodic : $l Î such that f (t nl ) = f (t ) 2 An = l 2 Bn = l 2011-09-27 l /2 ó ô ô õ -l /2 l /2 ó ô ô õ -l /2 The representation of a function by its Fourier Series is the sum of sinusoidal “basis functions” multiplied by coefficients. é æ 2p n öù f (t ) êcos ç t - jn ÷ú dt for n ³ 0 ÷ ÷ú è øû êë ç l Fourier coefficients are generated by taking the inner product of the function with the basis. é æ 2p n öù f (t ) êsin ç t - jn ÷ú dt for n ³ 0 ÷ ÷ú è øû êë ç l The basis functions correspond to modes of vibration. 1999-2011 by Richard Alan Peters II 20 can also be written in terms of complex exponentials The Fourier Series f (t ) = ¥ åCe n n=-¥ = p i 2l n t + = ¥ å Cn e æ 2 pn ö ÷ ç +ç +fn ÷ ÷ çl è ø i t n=-¥ æ 2pn ö æ 2p n ö Cn cos ç t + fn ÷ + Cn sin ç t + fn ÷ ÷ ÷ å çl çl ÷ ÷ è ø è ø n=-¥ ¥ Cn = Cn e + ifn l /2 2 pn -i t 1 l dt = ò f (t ) e l -l /2 l /2 é æ 2p n ö æ 2p n ö 1 ÷ ÷ù ç = ò f (t ) ê cos ç t + fn ÷ + Cn sin ç t + fn ÷ú dt çl ÷ ÷ú ø è øû êë è l l -l /2 2011-09-27 i = -1 1999-2011 by Richard Alan Peters II Cn = Cn e+ifn eix = cos x + i sin x f (t + nl ) = f (t ) for all integers n 21 The Fourier Series Cont’d. on next page. Relationship between the real and the complex Fourier Series ¥ f (t ) = å [ An cos wnt + Bn sin wnt ], where wn = n=0 2pn l l /2 l /2 ù 2 ¥ éê = å ê ò f (h ) cos wn h d h cos wnt + ò f (h ) sin wn h d h sin wn t úú l n=0 êë-l /2 úû -l /2 l /2 2¥ = å ò f (h )[cos wn h cos wnt + f (h ) sin wn h sin wnt ] d h l n=0 -l /2 l /2 The sine-plus-cosine 1¥ = å ò f (h ) cos (wn h - wnt ) d h form results from the l n=0 -l /2 projection of f onto a cosine that is in phase with the current time. 2011-09-27 1999-2011 by Richard Alan Peters II 22 Relationship between the real and the complex Fourier Series (cont’d.) Claim: 0= ¥ å sin (w h - w t ). n n n=-¥ l /2 Therefore: ò -l /2 Thus: Cont’d. on next page. ¥ é ù ê f (h ) å sin (wn h - wnt )ú d h = 0. êë úû n=-¥ l /2 ù 1 ¥ éê -i å ê ò f (h ) sin (wn h - wn t ) d h úú = 0. l n=-¥ êë-l /2 úû Then add zero to the equation at the end of the previous page: l /2 l /2 ù ù 1 ¥ éê 1 ¥ éê ú -i å ú f (t ) = å ê ò f (h ) cos (wn h - wnt ) d h ú ê ò f (h ) sin (wn h - wnt ) d h ú. l n=-¥ êë-l /2 úû l n=-¥ êë-l /2 úû 2011-09-27 1999-2011 by Richard Alan Peters II 23 Relationship between the real and the complex Fourier Series (cont’d.) l /2 l /2 ù ù 1 ¥ éê 1 ¥ éê ú -i å ú f (t ) = å ê ò f (h ) cos (wn h - wnt ) d h ú ê ò f (h ) sin (wn h - wnt ) d h ú l n=-¥ êë-l /2 úû l n=-¥ êë-l /2 úû l /2 1¥ = å ò f (h )[cos wn (h - t ) - i sin wn (h - t )] d h l n=-¥ -l /2 l /2 1¥ = å ò f (h ) e-iwn (h-t )d h l n=-¥ -l /2 ¥ l /2 ¥ + Then some algebraic manipulations lead to the result. 2 pn 2 pn h -i +i t 1 f (h ) e l d h e l =å ò n=-¥ l -l /2 = åCe n n=-¥ 2011-09-27 p i 2l n t = ¥ å n=-¥ ifn Cn e e p i 2l n t + = ¥ å Cn e tf æ 2 pn ö ÷ +ç ç ÷ ç l + n÷ è ø i n=-¥ 1999-2011 by Richard Alan Peters II 24 Relationship between the real and the complex Fourier Series (cont’d.) l /2 l /2 ù ù 1 ¥ éê 1 ¥ éê ú -i å ú f (t ) = å ê ò f (h ) cos (wn h - wnt ) d h ú ê ò f (h ) sin (wn h - wnt ) d h ú l n=-¥ êë-l /2 úû l n=-¥ êë-l /2 úû l /2 1¥ = å ò f (h )[cos wn (h - t ) - i sin wn (h - t )] d h l n=-¥ -l /2 l /2 1¥ = å ò f (h ) e-iwn (h-t )d h l n=-¥ -l /2 ¥ l /2 ¥ + Then some algebraic manipulations lead to the result. 2 pn 2 pn h -i +i t 1 f (h ) e l d h e l =å ò n=-¥ l -l /2 = åCe n n=-¥ 2011-09-27 p i 2l n t = ¥ å n=-¥ ifn Cn e e p i 2l n t + = ¥ å Cn e tf æ 2 pn ö ÷ +ç ç ÷ ç l + n÷ è ø i n=-¥ 1999-2011 by Richard Alan Peters II 25 Why are Fourier Coefficients Complex Numbers? f (t ) = ¥ å Cn e +i 2 pn t l where Cn = Cn e +ifn . n=-¥ Cn represents the amplitude, A=|Cn|, and relative phase, φ , of that part of the original signal, f (t), that is a sinusoid of frequency ωn = n / λ. 2011-09-27 intensity 0 0 frequency, ω = 1/λ 1999-2011 by Richard Alan Peters II 26 What about real + imaginary? ℑ F (w ) = ( N2A co s ϕ )[d (w + N l ) + d (w - N l )] + i ( N2A sin ϕ )[- d (w + N l ) + d (w - N l )] The FS of a cosine is a pair of impulses with complex amplitudes 2011-09-27 1999-2011 by Richard Alan Peters II 27 The real and imaginary parts at the positive frequency, N/λ ... Real + Imaginary to Magnitude & Phase ← ← … form a magnitude, NA/2, and a phase, φ0. 2011-09-27 1999-2011 by Richard Alan Peters II 28 The real and imaginary parts at the negative frequency, -N/λ ... Real + Imaginary to Magnitude & Phase → → … form a magnitude, NA/2, and a phase, -φ0. 2011-09-27 1999-2011 by Richard Alan Peters II 29 Fourier Series of a Square Wave Time-domain signal Fourier magnitude Fourier phase 2011-09-27 1999-2011 by Richard Alan Peters II 30 The Fourier Transform is the decomposition of a nonperiodic signal into a continuous sum* of sinusoids. ¥ F ( w ) = F (w ) e iF(w ) = ò f (t ) e i 2 p w t dt -¥ ¥ = ò f (t )[cos ( 2pw t ) + i sin ( 2pw t )] dt -¥ ¥ ¥ f (t ) = ò F (w ) e -i 2 p w t -¥ d w = ò F (w ) e-i(2 p w t +F(w)) d w -¥ ¥ = ò F (w ) [cos ( 2pw t ) - i sin (2pw t )] d w -¥ ¥ = ò F (w ) [cos ( 2pw t +F (w )) - i sin (2pw t +F (w ))] d w -¥ 2011-09-27 1999-2011 by Richard Alan Peters II * i.e., an integral. 31 Mammals Use the FT in Hearing 2011-09-27 1999-2011 by Richard Alan Peters II 32 The Discrete Fourier Transform A discrete signal, {hk k = 0,1, 2, , N − 1 }, of finite length N can be repre - { } sented as a weighted sum of N sinusoids, e −i 2π k n /N n = 0,1, 2, , N − 1 through N −1 hk = ∑ H n e −i 2π k n /N n= 0 where the set, { H n n = 0,1, 2, , N − 1}, are the Fourier coefficients defined as the projection of the original signal onto sinusoid, n, given by : 1 N −1 Hn = ∑ hk e +i 2π k n /N N k=0 2011-09-27 1999-2011 by Richard Alan Peters II 33 The Two-Dimensional Fourier Transform Primary Uses of the FT in Image Processing: 2011-09-27 Explains why down-sampling can add distortion to an image and shows how to avoid it. Useful for certain types of noise reduction, deblurring, and other types of image restoration. For feature detection and enhancement, especially edge detection. 1999-2011 by Richard Alan Peters II 34 The Fourier Transform: Discussion The expressions ¥ F (w ) = ò f (t ) e-i 2 p w t dt = f (t ) , e+ i 2 p w t -¥ continuous signals defined over all real numbers and 1 Hn = N N -1 åhe -i 2 p k n / N k = hk , e +i 2 p k n /N n= 0 discrete signals with N terms or samples. for the Fourier coefficients are “inner products” which can be thought of as measures of the similarity between the functions f (t ) and e + i 2π ω t for t ∈ (− ∞, ∞ ) or between the sequences N -1 + i 2 p k n / N N -1 { hk } k =0 and { e } k =0 . 2011-09-27 1999-2011 by Richard Alan Peters II 35 The Fourier Transform: Discussion (cont’d.) In the context of inner products, the complex exponentials { e-i2pw t } and { e-i 2pk n/N w ÎÂ and w Î(-¥ , ¥) , -2, -1, 0,1, 2, } are called “orthogonal sets” since they have the property: ¥ e-i 2 p w 1 t , e-i 2p w 2 t = ò e-i 2 p w 1 t ⋅ e+ i 2 p w 2 t dt = { 0, ¥ , if w 1= w 2 if w 1¹ w 2 -¥ -i 2 p j n / N e -i 2 p k n / N ,e = å e-i 2 p j n /N ⋅e+i 2 p k n /N = { 0, N -1 c , if j = k if j ¹ k n= 0 , The function sets are called “orthogonal basis sets” They are called “basis sets” since for any function1, f (t), of a real variable there exists a complex-valued function F(w), and for any sequence1, hk , there exist complex numbers, Hn , such that ¥ f (t ) = ò F (w ) e -i 2 p w t -¥ 2011-09-27 N -1 d w and hk = å H n e-i 2p k n /N . n= 0 1999-2011 by Richard Alan Peters II 1 with finite energy. 36 The Fourier Transform: Discussion (cont’d.) Consider the 2-dimensional functions {e -i 2 p ( ux + vy ) u , v, x, y Î Â } and {e jm -i 2 p ( M + kn ) N } j , m Î 0, ..., M - 1, k , n Î 0,..., N- 1 These are, likewise, orthogonal: ¥¥ =ò e-i 2 p(u1x+v1 y) , e-i 2 p (u2 x+v2 y) ò e-i 2 p (u1x+v1 y) ⋅ e+ i 2 p (u2 x+v2 y) dxdy -¥ -¥ = æ j m k nö ÷ ç -i 2 p ç 1 + 1 ÷ ÷ çM è Nø e æ j m k n÷ ö ç -i 2 p ç 2 + 2 ÷ ÷ çM è Nø ,e M -1 N -1 m= 0 n= 0 { , ( jMm + kNn)⋅ e+i 2p( jMm + kNn) -i 2 p = åå e = 2011-09-27 { ¥ , if u1 =u2 and v1 =v2 0, otherwise 1 1 c , if j1 = j2 and k1 = k2 0, otherwise 1999-2011 by Richard Alan Peters II 2 2 . 37 The Fourier Transform: Discussion (cont’d.) Therefore {e - i 2 p (ux + vy ) u , v, x, y Î } and {e -i2p( jm M + kn N ) j, k , m , n, M Î } are orthogonal basis sets. This suggests that function f (x,y) defined on the real plane, and sequence {{ hmn }} for integers m and n have analogous Fourier representations, ¥¥ f ( x, y ) = ò ò M - 1N - 1 F (u, v) e+i 2 p (ux+vy ) dudv and hmn = å å H jk e æ jm kn ö +i 2 p ç + ÷ ÷ ç çM è ø N÷ j= 0 k = 0 -¥ -¥ . where the Fourier coefficients are given by ¥¥ F (u , v) = ò ò M -1 N -1 æ jm kn ö ÷ ç -i 2 p ç + ÷ ÷ çM è Nø f ( x, y) e-i 2 p ( ux+vy ) dxdy and H jk = å å hmn e m=0 n=0 -¥ -¥ . (True for finite energy functions f (x,y) and {{ hmn }}.) 2011-09-27 1999-2011 by Richard Alan Peters II 38 Continuous Fourier Transform The continuous Fourier transform assumes a continuous image exists in a finite region of an infinite plane. The BoingBoing Bloggers 2011-09-27 1999-2011 by Richard Alan Peters II 39 Discrete Fourier Transform The discrete Fourier transform assumes a digital image exists on a closed surface, a torus. The BoingBoing Bloggers 2011-09-27 1999-2011 by Richard Alan Peters II 40 Discrete Fourier Transform The discrete Fourier transform assumes a digital image exists on a closed surface, a torus. The BoingBoing Bloggers 2011-09-27 1999-2011 by Richard Alan Peters II 41 The 2D Fourier Transform of a Digital Image Let I(r,c) be a single-band (intensity) digital image with R rows and C columns. Then, I(r,c) has Fourier representation R-1 C -1 I ( r , c ) = å å I ( v ,u ) e æ vr uc ö +i 2 p ç + ÷ ç ÷ çR C ÷ è ø u =0 v=0 , where R-1 C -1 these complex exponentials are 2D sinusoids. æ vr uc ö ÷ -i 2 p ç + ÷ ç ç èR C ÷ ø 1 I (v,u ) = RC å å I(r ,c) e r =0 c=0 are the R x C Fourier coefficients. 2011-09-27 1999-2011 by Richard Alan Peters II 42 What are 2D sinusoids? To simplify the situation assume R = C = N. Then e ± i 2π ( vr R + uc C ) =e ±i 2π (vr + uc) N =e ±i 2πω ( r sin θ + c cos θ) N , where v = w sin θ, u = w cos θ, w = v 2 + u 2 , and Write Note: since images are indexed by row & col with r down and c to the right, θ is positive in the clockwise direction. N λ= , w Then by Euler’s relation, e 1 i 2 p l (r sin θ + c cos θ) v θ = tan -1 ( u ) . = cos [ 2lp (r sin θ + c cos θ)] i sin [ 2lp (r sin θ + c cos θ)]. Cont’d. on next page. 2011-09-27 1999-2011 by Richard Alan Peters II 43 What are 2D sinusoids? (cont’d.) Both the real part of this, { Re e 1 i 2 p l (r sin θ + c cos θ) } = + cos[ 2p l (r sin θ + c cos θ)] 2p l (r sin θ + c cos θ)] and the imaginary part, { Im e 1 i 2 p l (r sin θ + c cos θ) } = sin [ are sinusoidal “gratings” of unit amplitude, period λ and direction θ. Then w 2pw is the radian frequency, and the frequency, of the wavefront N N and λ = 2011-09-27 N is the wavelength in pixels in the wavefront direction. w 1999-2011 by Richard Alan Peters II 44 2D Sinusoids: I ( r , c) = { } é 2p ù A cos ê (r ⋅ sin θ + c ⋅ cos θ) + j ú + 1 êë l úû 2 ... are plane waves with grayscale amplitudes, periods in terms of lengths, ... θ orientation A φ = phase shift 2011-09-27 1999-2011 by Richard Alan Peters II 45 2D Sinusoids: 2011-09-27 ... specific orientations, and phase shifts. 1999-2011 by Richard Alan Peters II 46 The Fourier Transform of an Image c v v u u r I 2011-09-27 Re[F{I}] 1999-2011 by Richard Alan Peters II Im[F{I}] 47 Points on the Fourier Plane If R=C=N the point at column freq. u and row freq. v represents a sinusoid with freq. ω and orientation θ. If R ≠ C then ω = 1/λ where λ is the length of vector (C/u, R/v) and the wavefront orientation is θ = tan-1[(v/R)/(u/C)]. 2011-09-27 1999-2011 by Richard Alan Peters II 48 Points on the Fourier Plane (of a Digital Image) In the Fourier transform of an R×C digital image, positions u and v indicate the number of repetitions of the sinusoid in those directions. Therefore the wavelengths along the column and row axes are λu = C u and λ v = R v -v direction pixels, and the wavelength in the wavefront direction is of a digital image ( ) +( ) . C2 u λ wf = R2 v v , ω v = R , and ω wf = 1 ()() C2 + u R2 v cycles. (0,0) The wavefront direction is given by ω θ wf = tan -1 ( ω v ) = tan -1 ( v C ). uR u 2011-09-27 row freq. ──────── column freq. 1999-2011 by Richard Alan Peters II u direction u C -θ direction The frequency is the fraction of the sinusoid traversed over one pixel, ωu = More about this later (pp. 66-86). Note that the wave front direction = θ only if R=C. 49 Points on the Fourier Plane Note that θ is the wavefront direction only if R=C. y x This point represents this particular sinusoidal grating 2011-09-27 1999-2011 by Richard Alan Peters II 50 The Value of a Fourier Coefficient … … is a complex number with a real part and an imaginary part. If you represent that number as a magnitude, A, and a phase, φ, … ..these represent the amplitude and offset of the sinusoid with frequency ω and direction θ.* 2011-09-27 1999-2011 by Richard Alan Peters II *See p. 49. 51 The Value of a Fourier Coefficient The magnitude and phase representation makes more sense physically… …since the Fourier magnitude, A (ω,θ), at point (ω,θ) represents the amplitude of the sinusoid… and the phase, φ(ω,θ), represents the offset of the sinusoid relative to origin. 2011-09-27 1999-2011 by Richard Alan Peters II 52 The Fourier Coefficient at (u,v) So, the point (u,v) on the Fourier plane… …represents a sinusoidal grating of frequency ω and orientation θ.* The complex value, F(u,v), of the FT at point (u,v)… …represents the amplitude, A, and the phase offset, φ, of the sinusoid. 2011-09-27 1999-2011 by Richard Alan Peters II *See p. 49. 53 The Sinusoid from the Fourier Coeff. at (u,v) Note that the wave front direction = θ only if R=C. 2011-09-27 1999-2011 by Richard Alan Peters II 54 FT of an Image (Magnitude + Phase) I 2011-09-27 log{|F{I}|2+1} 1999-2011 by Richard Alan Peters II ∠[F{I}] 55 FT of an Image (Real + Imaginary) I 2011-09-27 Re[F{I}] 1999-2011 by Richard Alan Peters II Im[F{I}] 56 The Power Spectrum The power spectrum of a signal is the square of the magnitude of its Fourier Transform. For display, the log of the power spectrum is often used. 2 I (u ,v) = I (u ,v) I*(u ,v) = [ Re I (u ,v) + i Im I (u ,v)][ Re I (u ,v) - i Im I (u ,v)] 2 2 = [ Re I (u ,v)] + [ Im I (u ,v)] . At each location (u,v) it indicates the squared intensity of the frequency component with period l = 1 / u 2 + v 2 and orientation q = tan -1 (v / u ). For display in Matlab: PS = fftshift(2*log(abs(fft2(I))+1)); 2011-09-27 1999-2011 by Richard Alan Peters II 57 On the Computation of the Power Spectrum The power spectrum (PS) is defined by PS( I ) = F {I(u, v )} . We take the base-e logarithm of the PS in order to view it. Otherwise its dynamic range could be too large to see everything at once. We add 1 to it first so that the minimum value of the result is 0 rather than –infinity, which it would be if there were any zeros in the PS. Recall that log( f 2) = 2log( f ). Multiplying by 2 is not necessary if you are generating a PS for viewing, since you'll probably have to scale it into the range 0-255 anyway. It is much easier to see the structures in a Fourier plane if the origin is in the center. Therefore we usually perform an fftshift on the PS before it is displayed. >> PS = fftshift(log(abs(fft2(I))+1)); >> M = max(PS(:)); >> image(uint8(255*(PS/M))); 2 If the PS is being calculated for later computational use -- for example the autocorrelation of a function is the inverse FT of the PS of the function -- it should be calculated by >> PS = abs(fft2(I)).^2; 2011-09-27 1999-2011 by Richard Alan Peters II 58 The Uncertainty Relation space frequency FT If Δ x Δ y is the extent of the object in space and if Δ u Δ v is its extent in frequency then, Δ x Δ y ⋅ Δu Δ v ≥ space frequency FT 2011-09-27 1 16π 2 A small object in space has a large frequency extent and vice-versa. 1999-2011 by Richard Alan Peters II 59 The Uncertainty Relation → small extent ← IFT → small extent ← ← large extent → space IFT 2011-09-27 A symmetric pair of lines in the frequency domain becomes a sinusoidal line in the spatial domain. ← large extent → frequency → small extent ← Recall: a symmetric pair of impulses in the frequency domain becomes a sinusoid in the spatial domain. → small extent ← space ← large extent → frequency ← large extent → 1999-2011 by Richard Alan Peters II 60 The Fourier Transform of an Edge edge 2011-09-27 Power Spectrum 1999-2011 by Richard Alan Peters II Phase Spectrum 61 The Fourier Transform of a Bar bar 2011-09-27 Power Spectrum 1999-2011 by Richard Alan Peters II Phase Spectrum 62 Coordinate Origin of the FFT Center = (floor(R/2)+1, floor(C/2)+1) Even Odd Even Odd Image Origin Image Origin Weight Matrix Origin Weight Matrix Origin After FFT shift After FFT shift After IFFT shift After IFFT shift 2011-09-27 1999-2011 by Richard Alan Peters II 63 Matlab’s fftshift and ifftshift I = ifftshift(J): J = fftshift(I): origin origin from FFT2 or ifftshift after fftshift J ( R/2 +1, C/2 +1) → I (1,1) I (1,1) → J ( R/2 +1, C/2 +1) where x = floor(x) = the largest integer smaller than x. 2011-09-27 1999-2011 by Richard Alan Peters II 64 Matlab’s fftshift and ifftshift 5 6 4 8 J = fftshift(I): 9 7 1 3 5 6 7 2 3 4 I (1,1) → J ( R/2 +1, C/2 +1) 2 8 9 1 5 1 J ( R/2 +1, C/2 +1) → I (1,1) 2 3 4 5 8 4 8 9 7 3 1 6 7 6 2 I = ifftshift(J): 9 where x = floor(x) = the largest integer smaller than x. 2011-09-27 1999-2011 by Richard Alan Peters II 65 Points on the Fourier Plane (of a Digital Image) In the Fourier transform of an R×C digital image, positions u and v indicate the number of repetitions of the sinusoid in those directions. Therefore the wavelengths along the column and row axes are λu = C u and λ v = R v -v direction pixels, and the wavelength in the wavefront direction is of a digital image ( ) +( ) . C2 u λ wf = R2 v u C v , ω v = R , and ω wf = 1 ()() C2 + u R2 v cycles. (0,0) The wavefront direction is given by ω θ wf = tan -1 ( ω v ) = tan -1 ( v C ). uR u 2011-09-27 row freq. ──────── column freq. 1999-2011 by Richard Alan Peters II u direction ωu = -θ direction The frequency is the fraction of the sinusoid traversed over one pixel, Note that the wave front direction = θ only if R=C. 66 Coordinates and Directions in the Fourier Plane decreasing rows (-r,-c) (-r,+c) (-r,-c) θ>0 θ<0 θ<0 (+r,-c) (-r,+c) increasing cols (+r,+c) decreasing cols (+r,-c) θ>0 (+r,+c) increasing rows Since rows increase down and columns to the right, slopes and angles are opposite those of a right-handed coordinate system. 2011-09-27 1999-2011 by Richard Alan Peters II 67 Inverse FFTs of Impulses “horizontal” is the wavefront direction. fftshifted highest-possible-frequency horizontal sinusoid (C is even) 2011-09-27 1999-2011 by Richard Alan Peters II 68 Inverse FFTs of Impulses “vertical” is the wavefront direction. fftshifted highest-possible-frequency vertical sinusoid (R is even) 2011-09-27 1999-2011 by Richard Alan Peters II 69 Inverse FFTs of Impulses a checker-board pattern. fftshifted highest-possible-freq horizontal+vertical sinusoid (R & C even) 2011-09-27 1999-2011 by Richard Alan Peters II 70 Inverse FFTs of Impulses “horizontal” is the wavefront direction. fftshifted lowest-possible-frequency horizontal sinusoid 2011-09-27 1999-2011 by Richard Alan Peters II 71 Inverse FFTs of Impulses “vertical” is the wavefront direction. fftshifted lowest-possible-frequency vertical sinusoid 2011-09-27 1999-2011 by Richard Alan Peters II 72 Inverse FFTs of Impulses “negative diagonal” is the wavefront direction. fftshifted lowest-possible-frequency negative diagonal sinusoid 2011-09-27 1999-2011 by Richard Alan Peters II 73 Inverse FFTs of Impulses “positive diagonal” is the wavefront direction. fftshifted lowest-possible-frequency positive diagonal sinusoid 2011-09-27 1999-2011 by Richard Alan Peters II 74 Frequencies and Wavelengths in the Fourier Plane 512 columns 384 rows +u direction +v direction Note this … … and this. frequencies: (u,v) = (4,3); wavelengths: (λu, λv) = (128,128) How can that be? 2011-09-27 1999-2011 by Richard Alan Peters II 75 Frequencies and Wavelengths in the Fourier Plane λu = C / u 384 rows 512 columns u = # of complete cycles in the horizontal direction frequencies: (u,v) = (1,0); wavelength: λu= 512 2011-09-27 1999-2011 by Richard Alan Peters II 76 Frequencies and Wavelengths in the Fourier Plane v = # of complete cycles in the vertical direction λv = R / v 384 rows 512 columns frequencies: (u,v) = (0,1); wavelength: λv= 384 2011-09-27 1999-2011 by Richard Alan Peters II 77 Frequencies and Wavelengths in the Fourier Plane λu = C / u 384 rows 512 columns u = # of complete cycles in the horizontal direction frequencies: (u,v) = (2,0); wavelength: λu= 256 2011-09-27 1999-2011 by Richard Alan Peters II 78 Frequencies and Wavelengths in the Fourier Plane v = # of complete cycles in the vertical direction λv = R / v 384 rows 512 columns frequencies: (u,v) = (0,2); wavelength: λv= 192 2011-09-27 1999-2011 by Richard Alan Peters II 79 Frequencies and Wavelengths in the Fourier Plane λu = C / u 384 rows 512 columns u = # of complete cycles in the horizontal direction frequencies: (u,v) = (3,0); wavelength: λu= 170 ⅔ 2011-09-27 1999-2011 by Richard Alan Peters II 80 Frequencies and Wavelengths in the Fourier Plane v = # of complete cycles in the vertical direction λv = R / v 384 rows 512 columns frequencies: (u,v) = (0,3); wavelength: λv= 128 2011-09-27 1999-2011 by Richard Alan Peters II 81 In the Fourier plane of a square image, the orientation of the line through the point pair = the orientation of the wave front in the image. Not so for a non-square image. In the F plane the angle is -45˚ in this image it’s about -53˚ (yellow line). That’s because the fraction of R covered by one pixel is 4/3 the fraction of C covered by one pixel. Frequencies and Wavelengths in the Fourier Plane θ θ θ θ 384 rows 512 columns Also as a result, the wavelength is 213⅓. frequencies: (u,v) = (3,3); wavelengths: (λu, λv) = (170 ⅔,128) 2011-09-27 1999-2011 by Richard Alan Peters II 82 In general the slope of the wavefront direction in the image is given by (v/R) / (u/C). Therefore its angle is Frequencies and , Wavelengths in the Fourier Plane æ vC ö θ = tan ç ÷ -1 wf ç uR ÷ è÷ ø 512 columns λwf θ 384 rows θ θwf θwf and the wavelength is: frequencies: (u,v) = (3,3); wavelengths: 2011-09-27 1999-2011 by Richard Alan Peters II 2 2 æ Rö C (λwf, =v)æ= ö(170 ⅔,128) λu λ ç ÷ +ç ÷ , çu÷ çv÷ è÷ è÷ ø ø 83 Frequencies and Wavelengths in the Fourier Plane 384 rows 512 columns frequencies: (u,v) = (3,3); wavelengths: (λu, λv) = (170 ⅔,128) 2011-09-27 1999-2011 by Richard Alan Peters II 84 Frequencies and Wavelengths in the Fourier Plane 384 rows 512 columns frequencies: (u,v) = (4,3); wavelengths: (λu, λv) = (128,128) 2011-09-27 1999-2011 by Richard Alan Peters II 85 The ratio R/C = ¾ in this image. Therefore at frequency (4,3) the wave front angle is æ 3 ⋅ 512 ö æ 3⋅ 4 ö ÷ Wavelengths1 (1) = 45 ,Fourier ÷ ç ç θ wf = tan -1 ç and= tan -1 ç Frequencies⋅ 384 ÷ ÷ = tan in the ÷ ÷ è4 ø è 4 ⋅ 3ø Plane 512 columns θwf θwf 384 rows λwf and the wavelength is frequencies: (u,v) = (4,3); wavelengths: (λöu, λv) = (128,128) 2 2 æ öæ 512 ÷ 384 ÷ 2 λ wf = ç ÷ +ç ç 4 ø è 3 ÷ = 2 ⋅128 = 128 2, ç ÷ ÷ è ø 2011-09-27 1999-2011 by Richard Alan Peters II 86 Power Spectrum of an Image 2011-09-27 1999-2011 by Richard Alan Peters II 87 Relationship between Image and FT phase power spectrum power spectrum 2011-09-27 1999-2011 by Richard Alan Peters II phase 88 Features in the FT and in the Image Lines in the Power Spectrum are … … perpendicular to lines in the image. 2011-09-27 1999-2011 by Richard Alan Peters II 89 Fourier Magnitude and Phase I 2011-09-27 1999-2011 by Richard Alan Peters II 90 Fourier Magnitude log F {I} 2011-09-27 1999-2011 by Richard Alan Peters II 91 Fourier Phase F { I} 2011-09-27 1999-2011 by Richard Alan Peters II 92 Q: Which contains more visually relevant information; magnitude or phase? original image 2011-09-27 Fourier log magnitude 1999-2011 by Richard Alan Peters II Fourier phase 93 Magnitude Only Reconstruction 2011-09-27 1999-2011 by Richard Alan Peters II 94 Phase Only Reconstruction 2011-09-27 1999-2011 by Richard Alan Peters II 95 ...
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