Procesarea Imaginilor

Curs 7:

Convolutie. Transformata Fourier

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Convolutie

Notiuni preliminarii Imagine continua := functie continua de 2 variabile independente

Exemple: u(x, y); v(x, y); f(x, y)

Functie 2D separabila:

f(x,y)=f1(x)f2(y)

Imagine discreta (digitala) := secventa 2D (bidmensionala) de numere reale/intregi

Exemple: um,n ; v(m, n)

Notatii:

i,j,k,l,m,n indici intregi folositi in matrice, vectori:

1j

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Convolutie

Functia Dirac (impuls unitar Aria = 1)

Functia Dirac 2D (continua):

d(x,y) =d(x)d(y)

Functia discreta Kronecker:

d(m,n) =d(m)d(n)

Sistem

H

Sistem liniar (principiul superpozitiei liniare):

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Convolutie

Iesirea unui sistem liniar

Unde: Este raspunsul la impuls al sistemului iesirea sistemului H in punctul/locatia (m,n) atunci cand la intrare se aplica functia impuls unitar (Kronecker) la locatia (m,n).

H

Dc. y(m,n), x(m,n) >0 (ex: imagini), atunci h s.n. Points Spread Function (PSF)

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Convolutie

Regiune suport a raspunsului la impuls := cea mai mica regiune din planul (m,n) cu proprietatea ca in afara acestei regiuni raspunsul la impuls (h) este nul.

Sisteme FIR sisteme a caror raspuns la impuls au o regiune suport finita

Sisteme IIR sisteme a caror raspuns la impuls au o regiune suport infinita

Sistem invariant la translatie := sistem pt. care o translatie a intrarii va determina o translatie corespunzatoare a iesirii

forma raspunsului la impuls nu se schimba odata cu deplasarea impulsului in planul (m,n)

Convolutia intarii cu raspunsul la impuls:

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Convolutie

Convolutie 1D

Convolutie 2D

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Operaia de convoluie n domeniul spaial

Operaia de convoluie implic folosirea unei mti/nucleu de convoluie H (de obicei de form simetric de dimensiune wxw, cu w=2k+1) care se aplic peste imaginea surs:

SD IHI

( , ) ( , ) ( , ) , 0... 1, 0... 1k k

D Si k j k

I x y H i j I x i y j x Height y Width

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Convolutie Exemplu: filtru pt. detectia muchiilor verticale (calculeaza derivata imaginii pe directia x)

ImgSrc (i,j) ImgDst(i,j) =GX*ImgSrc (i,j)

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Notiuni preliminarii

Functie periodica serii Fourier (suma sin si cos de frecvente diferite)

Functie integrala (arie) finita trasnformata Fourier (inetgrala / suma de sin si cos)

Transformata Fourier (1D)

)sin()cos( xjxe jx

Transformata Fourier inversa (1D)

Se poate reface functia initiala plecand de la functia transformata !!!

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Transformata Fourier (2D):

Transformata Fourier discreta - DFT (1D):

u = 0, 1, 2, .. M-1 (domeniu de frecvente)

x = 0, 1, 2, .. M-1

F(0)= .. , F(1)= .. , , F(M-1)= .. (componente de frecvanta)

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Caracteruistici ale DFT

F(u) exprimata in coordonate polare:

Magnitudine (spectru) image enhancement

Faza (spectru de faza / unghi de faza)

Spectru de putere (densitate spectrala)

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Transformata Fourier discreta DFT (2D)

u = 0, 1, 2, .. M-1, v = 0, 1, 2, .. N-1 (coordonate/variabile de frecventa)

x = 0, 1, 2, .. M-1, y = 0, 1, 2, .. N-1 (coordonate/variabile spatiale)

Spectru:

Faza:

Spectru de putere:

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Shift-area transformatei Fourier:

M/2

N/2 F(0,0)

Componenta continua a spectrului:

intensitatea medie a imaginii

Pt. Imagini (f R):

Relatiile dintre esantioanele spatiale si frecventiale ale imaginii:

Spectru Fourier F(u,v) centrat

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Exemple de reprezentare a DFT

log(1+|F(u,v)|)

log(1+|F(u,v)|) F(u,v)

Pentru marirea contrastului zonelor intunecate din spectru se foloseste transformata log (pt. vizualizare mai buna)

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier

Original log(|F(u,v)|) f(u,v)

Transformata directa

Transformata inversa

f(u,v)=0 (se ignora faza) |A(u,v)|=0 (se ignora amplitudinea)

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Filtrarea imaginilor in domeniul frecventelor

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Filtrarea imaginilor in domeniul frecventelor exemple:

Negativul imaginii:

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Filtrarea imaginilor in domeniul frecventelor exemple:

FTJ

FTS

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Filtrarea imaginilor in domeniul frecventelor exemple:

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Filtrarea imaginilor in domeniul frecventelor exemple: FTJ

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Filtrarea imaginilor in domeniul frecventelor exemple: FTJ ideal

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Transformata Fourier [1]

Filtrarea imaginilor in domeniul frecventelor exemple: FTJ Gausian

Technical University of Cluj Napoca

Computer Science Department IMAGE PROCESSING

Referinte

[1] R.C.Gonzales, R.E.Woods, "Digital Image Processing, 2-nd Edition", Prentice Hall, 2002, pp 147-177