Introduction to the Discrete Fourier Transform

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1 Introduction to the Discrete ourier Transform Lucas J. van Vliet TNW: aculty of Applied Sciences IST: Imaging Science and technology PH: Linear Shift Invariant System A discrete image can be decomposed into a weighted field of equi-spaced impulses. + f( ) = f( n) δ( n) n= LSI system f( ) g( ) h( ) impulse at position n amplitude at position n + linear aδ ( ) ah( ) shift inv. δ( n ) h( n) superposition f( ) = f( n ) δ( n ) g( ) = f( n) h( n) n= δ ( ) h( ) h() = impulse response or Point Spread unction (PS) + n= f( ) h( ) Image g is the result of a convolution between image f and h Discrete ourier Transform

Convolution revisited Convolution: Replace the central piel by a weighted sum of the grayvalues inside an nn neighborhood.

Gauss σ=1 1 1 4 1 1 1 0-1 0-1 0-1 Gauss σ=4 0 1 0 1 4 1 0 1 0 Discrete ourier Transform 3 Eigenfunction of LSI systems Eigenfunctions of LSI

jω ϕ ( ) = e = cos ω + jsinω LSI system + ϕ ω ( ) h( ) Kϕω ( ) = ϕ( n) h( n) ω ϕ ω = ω1( ) = + j ϕ ω = ω ( ) = + j ϕ = ( ) = + j ω ω3 n= j e

2 Convolution revisited Convolution: Replace the central piel by a weighted sum of the grayvalues inside an nn neighborhood. Impulse response h() is the filter. Gauss σ= Gauss σ= Discrete ourier Transform 3 Eigenfunction of LSI systems Eigenfunctions of LSI systems are comple eponentials ϕ(). jω ϕ ( ) = e = cos ω + jsinω LSI system + ϕ ω ( ) h( ) Kϕω ( ) = ϕ( n) h( n) ω ϕ ω = ω1( ) = + j ϕ ω = ω ( ) = + j ϕ = ( ) = + j ω ω3 n= j e ω + + jωn jω jωm g( ) = e h( n) = e h( m) e n= n = = e jω H( ω ) H( ω ) An LSI system multiplies each eigenfunction ep(jω) by its corresponding eigenvalue H(ω). The ourier transform decomposes an image into a weighted set of comple eponentials of varying frequency ω: The ourier spectrum The system function H(ω) is the ourier Transform of h() Discrete ourier Transform 4

3 Convolution property A convolution between an image f() and an impulse response h() in space, corresponds to a multiplication of the ourier spectra (ω) and H(ω) in the ourier domain. LSI system + f( ) h( ) f( ) h( ) = f( n ) h( n ) + + { f( ) h( ) } = f( n) h( n) e = n= + + n= jωn jωm = f( n) e h( m) e n = = ( ω) = ( ω) H( ω) H( ω) jω Discrete ourier Transform 5 Gaussian derivatives How to compute a derivative in digital space? f [, y] f[, y] =? Introduce (Gaussian) scale ( σ ) ( 0 [ ] ) ( σ f, y = f [, y] g ) [, y] ( σ= 5 ) ( σ= 3 [ ] ) ( σ= 4 f, y = f [, y] g ) [, y] Derivative at scale σ is computed by convolution of the discrete image f[,y] with discrete Gaussian derivative ( σ ) ( 0 [ ] ) ( σ f, y = f [, y] g ) [, y] Note that the Gaussian function is known analytically, compute the derivative(s), and then sample to produce a discrete filter. The sampling of the Gaussian should be high enough, i.e. σ > 0.9 piels Discrete ourier Transform 6

ourier filters: Gaussian f f*g f*g f*g g g g G G G G G G Discrete ourier Transform 7 Gaussian derivative filters In continuous space: the derivative operator corresponds to multiplication of the

4 ourier filters: Gaussian f f*g f*g f*g g g g G G G G G G Discrete ourier Transform 7 Gaussian derivative filters In continuous space: the derivative operator corresponds to multiplication of the ourier spectrum with jω f ω ( ) ( ) d f ( ) j ω ( ω) d d ( ) ( ) f ω ω d In discrete space: combine the derivative operator with Gaussian smoothing σ=1 σ= σ=4 σ=1 σ= σ=4 ( σ) g ( ) f( ) G( ω) ( ω) d g ( σ) ( ) f ( ) j ω G ( ω) ( ω) d d ( ) ( ) ( ) ( ) ( ) g σ f ω G ω ω d Discrete ourier Transform 8

5 ourier: Gaussian derivative f f*g f*g f*g g g g Im{G } Im{G } Im{G } G G G Discrete ourier Transform 9 ourier: Laplace & sharpening f f*(g +g yy ) f*(g +g yy ) g +g yy g +g yy Re{G +G yy } f f*(g +g yy ) f f*(g +g yy ) Re{G +G yy } (G +G yy ) (G +G yy ) Discrete ourier Transform 10

6 Discrete ourier Transform Each image can be decomposed in weighted sum of comple eponentials (sines and cosines) of frequency f and angle φ. (or two frequency components u and image size NN N 1N 1 π j ( u+ vy ) N 1 g(, y) = G( u, e N u= 0v= 0 N 1N 1 π j ( u+ vy ) N 1 G( u, = g(, y) e N u= 0v= 0 or real-valued images: Ev { g(, y) } Re { G( u, } Od { g(, y) } j Im { G( u, } Discrete ourier Transform 11 ourier spectrum (u, is the comple amplitude of the eigenfunction ep(j (π/n)(u+vy)) Note that ep(j (π/n)(u+vy)) = cos((π/n)(u+vy)) + j sin((π/n)(u+vy)) Standard display is the logarithm of the magnitude: log( (u,v ) 0 0 f(,y) 10 log ( ( u, ) column N-1 column row row u N-1 y v (u, = (0,0) Discrete ourier Transform 1

7 Getting used to ourier () An image is a weighted sum of cos (even) and sin (odd) images. ourier domain with comple amplitude: a+jb a jb a+jb Discrete ourier Transform 13 Eigenfunctions: Even & Odd Real & even π 1 + cos u 0 = N π π j u 0 j ( u0) e N + e N 1+ Real & odd π sin u 0 = N π π j u 0 j ( u0) e N e N j Real & even δ( uv, ) + δ( u u0, + 1 δ( u + u0, 1 1 N Imag & odd 1 N 1 δ( u u0, j 1 + δ( u + u, 0 Discrete ourier Transform 14

Orientation & frequency Discrete ourier Transform 15

Tunneling Microscopy Atomic structure of graphite shows

(-½N,-½N) (c,r) = (½N,½N) (u, = (0,0) 0 row u N-1 y v

8 Orientation & frequency Discrete ourier Transform 15 Getting used to ourier (1) Graphite surface by Scanning Tunneling Microscopy Atomic structure of graphite shows a heagonal surface 0 column N-1 (c,r) = (0,0) (u, = (-½N,-½N) (c,r) = (½N,½N) (u, = (0,0) 0 row u N-1 y v (c,r) = (N-1,N-1) (u, = (½N-1, ½N-1) Discrete ourier Transform 16

9 Superposition ourier spectrum = = Discrete ourier Transform 17 ourier transforms magnitude phase Discrete ourier Transform 18

10 Magnitude & phase ( u, 1? j ( uv, ) ( u, e = ( u, 1? Discrete ourier Transform 19 Local variance filter: power Recipe: local variance filter (filter size = n) 1. Compute the local mean (blurring filter of size n). Subtract the local mean. 3. Compute the square of each piel value 4. Suppress the double response by local averaging (blurring filter of size n) Local variance is a measure for the local squaredcontrast. (zero = gray) step 1 step step 3 step 4 Discrete ourier Transform 0

11 Scaling: local vs global Problem: Choosing the proper scale is an important, but tedious task. Scale too small: Local characteristics are missed which yields an incomplete data description. Scale too large: Confusion (miing) of adjacent objects, lack of localization, and blindness for detail. Solution: Multi-scale analysis. Analyze the image as function of scale: from fine detail to course image-filling objects. π ( u π y ) ( v N N ) σ + + g(, y 1 ) = e σ G( u, = e πσ Discrete ourier Transform 1 Multi-Scale Series of images of increasing scale: Scale-space Sample the scales logarithmically using filters of size = base scale yields n scales per octave base,,,..., 1 n { } Input image Scale space scale 0 scale 1 scale scale 3 scale 4 scale 5 Gaussian scale-space difference scale-space var (scale 1) var (scale ) var (scale 3) var (scale 4) var (scale 5) local variance scale-space Discrete ourier Transform

12 Scale-spaces Morphological scale-space: Use openings (closings) Gaussian scale-space: Use Gaussian filters Increasing scales ourier domain with footprints of Gaussian filters of increasing scale ilter size is inversely proportional to footprint in ourier domain Discrete ourier Transform 3 Chirp eample Gaussian scale-space derivative scale-space local variance scale-space Discrete ourier Transform 4

13 Gaussian derivatives ( 0 ( 1 f ) [, y ] f ) [, y] ( 5 f ) [, y ] ( 1 f ) [, y ] ( 5 f ) [, y ] ( 0 f ) [, y ] =? Discrete ourier Transform 5 Sampling i = = Discrete ourier Transform 6

14 Interpolation Zero-order hold irst-order hold B-spline Discrete ourier Transform 7 Periodic images Periodic image yields ourier spectrum with impulses Discrete ourier Transform 8

Frequency-Domain C/S of LTI Systems

Frequency-Domain C/S of LTI Systems x(n) LTI y(n) LTI: Linear Time-Invariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the