Kronecker Delta Function # 1 for m = 0,n = 0. Kronecker Delta Function

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Bioegieerig 280A Priciples of Biomedical Imagig Fall Quarter 202 Sigals ad Fourier Trasforms Sigals ad Images Discrete-time/space sigałimage: cotiuous valued fuctio with a discrete time/space ide,

1 Bioegieerig 280A Priciples of Biomedical Imagig Fall Quarter 202 Sigals ad Fourier Trasforms Sigals ad Images Discrete-time/space sigałimage: cotiuous valued fuctio with a discrete time/space ide, deoted as s[] for D, s[m,] for 2D, etc. Cotiuous-time/space sigałimage: cotiuous valued fuctio with a cotiuous time/space ide, deoted as s(t or s( for D, s(,y for 2D, etc. m t y Kroecker Delta Fuctio # δ[] = $ for = 0 % 0 otherwise δ[] Kroecker Delta Fuctio # for m = 0, = 0 δ[m,] = $ % 0 otherwise δ[m,] δ[m-2,] 0 δ[-2] δ[m,-2] δ[m-2,-2] 0

2 Discrete Sigal Epasio g[] = g[k]δ[ k] g[m,] = k= k= l= g[k,l]δ[m k, l] δ[] g[] 0 -δ[-] 0.5δ[-2] 0 D Sigal Decompositio { 2,0,2,0 } = 2 {,0,0,0} + 0 {0,,0,0} + 2 {0,0,,0} + 0 {0,0,0,} { 2,0,2,0 } = a {,,,} + b {,0,,0} + c {0,,0, } + d {,,, } { 2,0,2,0 } = {,,,} + 0 {,0,,0} + 0 {0,,0, } + {,,, } D Sigal Decompositio { 2, 2,2, 2 } = 2 {,0,0,0} 2 {0,,0,0} + 2 {0,0,,0} 2 {0,0,0,} { 2, 2,2, 2 } = 0 {,,,} + 0 {,0,,0} + 0 {0,,0, } + 2 {,,, } 2D Sigal a 0 0 b a b = + c d c 0 0 d 2

Image Decompositio 0 c d 0 c d + 0 0 0 0 = + a b a 0 0 0 0 b + 0 0 Basis Fuctios /2 /2 Coefficiets = /2 -/2 Sum g[m,]

Image Compressio Eskimo Words for Sow tlapa tlacrigit kayi tlapat kli akli tlamo tlatim tlaslo tlapiti kripya tliyel

i large wet flakes sow that falls i small flakes sow that falls slowly sow that falls quickly sow that has melted ad

sold to Germa tourists sow sold to America tourists sow sold to Japaese tourists sow mied with the soud of old rock ad

3 Image Decompositio 0 c d 0 c d = + a b a b Basis Fuctios /2 /2 Coefficiets = /2 -/2 Sum g[m,] = aδ[m,] + bδ[m, ] + cδ[m,] + dδ[m, ] /2 /2 /2 -/2 k= 0 l= 0 = g[k,l] δ[m k, l] /2 /2 -/2 -/2 /2 -/2 -/2 /2 Object Image Compressio Eskimo Words for Sow tlapa tlacrigit kayi tlapat kli akli tlamo tlatim tlaslo tlapiti kripya tliyel tliyeli powder sow sow that is crusted o the surface driftig sow still sow remembered sow forgotte sow sow that falls i large wet flakes sow that falls i small flakes sow that falls slowly sow that falls quickly sow that has melted ad refroze sow that has bee marked by wolves sow that has bee marked by Eskimos tlalma tlalam tlaip tla-a-a depptla sow sold to Germa tourists sow sold to America tourists sow sold to Japaese tourists sow mied with the soud of old rock ad roll from a portable radio a small sowball, preserved i Lucite, that had bee hadled by Johy Depp 3

D Fourier Trasform The Fourier Trasform KPBS KIFM Fourier Trasform Fourier Trasform (FT G( f = g(te j 2πft Iverse Fourier Trasform dt = F{ g(t } g(t = G( f e j 2πft

4 D Fourier Trasform The Fourier Trasform KPBS KIFM Fourier Trasform Fourier Trasform (FT G( f = g(te j 2πft Iverse Fourier Trasform dt = F{ g(t } g(t = G( f e j 2πft df = F { G( f } KIOZ j = Comple Numbers Comple Numbers z = 2 +j j 2 =? (3 + 2 j(3 2 j =? j 2 = (3 + 2 j(3 2 j = 9 4 j 2 =3 z = = 5 $ θ = ta & ' = 30 degrees % 2( 4

5 e jθ Euler s Formula = cosθ + j siθ z = + jy = z e jθ = D Fourier Trasform ( g(ep j2πk d = g(cos(2πk d j g( si(2πk d The part of g( that "looks" like cos(2πk The part of g( that "looks" like si(2πk k k Uits Temporal Coordiates, e.g. t i secods, f i cycles/secod G( f = g(te j 2πft dt Fourier Trasform g(t = G( f e j 2πft df Iverse Fourier Trasform Spatial Coordiates, e.g. i cm, k is spatial frequecy i cycles/cm = g(e j 2πk d Fourier Trasform g( = e j 2πk dk Iverse Fourier Trasform 2D Fourier Trasform Fourier Trasform = F[ g(,y ] = g(, y Iverse Fourier Trasform g(, y = ( dk dk y e j 2π k +k y y ( ddy e j 2π k +k y y 5

6 Plae Waves e j 2 π (k +ky y = cos(2π (k + k y y + j si(2π (k + ky y k2 + ky2 /ky /k cos(2πk cos(2πkyy Figure 2.5 from Price ad Lik cos(2πk +2πkyy Plae Waves B BC BD = AB = AC C /k Image space ΔABC ~ ΔBDC AC AB = BC BD A θ /ky D θ k-space k ky + k 2 k y2 = k-space k 2 + k y2 ky y $k ' θ = arcta& y % k ( Fourier Trasform k 6

7 Eamples Eamples 7

8 Eamples Eamples Eamples Eercise 8

Eercise Eercise Dirac Delta Fuctio Notatio : δ( - D Dirac Delta Fuctio δ(, y or 2 δ(, y - 2D Dirac Delta Fuctio δ(, y,z or 3 δ(,y,z - 3D Dirac Delta Fuctio δ( r - N Dimesioal Dirac Delta Fuctio D

9 Eercise Eercise Dirac Delta Fuctio Notatio : δ( - D Dirac Delta Fuctio δ(, y or 2 δ(, y - 2D Dirac Delta Fuctio δ(, y,z or 3 δ(,y,z - 3D Dirac Delta Fuctio δ( r - N Dimesioal Dirac Delta Fuctio D Dirac Delta Fuctio δ( = 0 whe 0 ad δ(d = Ca iterpret the itegral as a limit of the itegral of a ordiary fuctio that is shrikig i width ad growig i height, while maitaiig a costat area. For eample, we ca use a shrikig rectagle fuctio such that δ(d = lim τ Π( /τd. τ 0 τ 0 -/2 /2 9

10 2D Dirac Delta Fuctio δ(, y = 0 whe 2 + y 2 0 ad δ(, yddy = where we ca cosider the limit of the itegral of a ordiary 2D fuctio that is shrikig i width but icreasig i height while maitaiig costat area. δ(,yddy = lim τ 2 Π /τ,y /τ τ 0 ( ddy. Useful fact : δ(, y = δ(δ(y Geeralized Fuctios Dirac delta fuctios are ot ordiary fuctios that are defied by their value at each poit. Istead, they are geeralized fuctios that are defied by what they do udereath a itegral. The most importat property of the Dirac delta is the siftig property δ( 0 g(d = g( 0 where g( is a smooth fuctio. This siftig property ca be uderstood by cosiderig the limitig case lim τ Π /τ τ 0 ( g(d = g( 0 g( τ 0 0 Area = (height(width= (g( 0 / τ τ = g( 0 Represetatio of D Fuctio From the siftig property, we ca write a D fuctio as g( = g(ξδ( ξdξ. To gai ituitio, cosider the approimatio g( = g(δ Δ Π * Δ -, / Δ. + Δ. Represetatio of 2D Fuctio Similarly, we ca write a 2D fuctio as g(, y = g(ξ,ηδ( ξ, y ηdξdη. To gai ituitio, cosider the approimatio g(, y g(δ,mδy Δ Π + - Δ. 0 =, Δ / Δy Π + y mδy. - 0 ΔΔy. m=, Δy / g( 0

11 Rectagle Fuctio Π( = 0 >/2 $ % & /2 -/2 /2 Also called rect( y /2 Π(, y = Π(Π(y -/2 /2 -/2 Computig Trasforms F(δ( = δ(e j 2πk d = F(δ( 0 = δ( 0 e j 2πk d = e j 2πk 0 / 2 F( Π( = e j 2πk / 2 d = e jπk e jπk j2πk = si(πk πk = sic(k Computig Trasforms F( = e j 2πk d =??? Defie h k ( = e j 2πk ( G( k h( k dk = G k d ad see what it does uder a itegral. e j 2πk ddk = G( k e j 2πk dk d = g( d = G(0 Therefore, F( = e j 2πk d = δ(k Liearity F ag(, y + bh(, y Basic Properties [ ] = a + bh(k Scalig [ ] = ab G k $ F g(a,by Shift " a, k % ' # b & F[ g( a,y b ] = e j 2π (k a +k y b Modulatio [ ] = a b j 2π (a +yb F g(, ye

12 Liearity The Fourier Trasform is liear. F{ ag( + bh( } = a + bh(k F[ ag(,y + bh(,y ] = a + bh(k Similarly, Computig Trasforms F e j 2πk 0 { } = δ(k k 0 F{ cos2πk 0 } = 2 ( δ(k k + δ(k + k 0 0 F{ si2πk 0 } = 2 j ( δ(k k δ(k + k 0 0 Eamples Eamples g(, y =+ e j 2πa = δ k ( + δ(k + aδ(k y g(, y =+ e j 2πay = δ(k + δ(k δ(k y a g(, y = cos(2π(a + by = 2 δ(k aδ(k y b + 2 δ(k + aδ(k y + b 2

13 Eamples Scalig Theorem F{ g(a } = a G " k % $ ' # a & [ ] = ab G k $ F g(a,by " a, k % ' # b & = δ(k + g(, y =??? δ(k + cδ(k y + δ(k δ(k y d + 2 δ(k aδ(k y b + 2 δ(k + aδ(k y + b g, y Separable Fuctios ( is said to be a separable fuctio if it ca be ( = g X ( g Y ( y writte as g,y Eample g(, y = Π(Π(y = sic(k sic(k y Eample (sic/rect The Fourier Trasform is the separable as well. = g(, y ( ddy e j 2π k +k y y = g X ( e j 2πk d g Y ( y e j 2πkyy dy = G X (k G Y (k y Eample g(, y = Π(Π(y = sic(k sic(k y y /2 -/2 /2 -/2 3

14 Eample (sic/rect Eamples g(, y = δ(, y = δ(δ(y = g(, y = δ( = δ(k y!!! Duality Note the similarity betwee these two trasforms { } = δ(k a F e j 2πa F{ δ( a } = e j 2πk a Applicatio of Duality F{ sic( } = siπ π e j 2πk d =?? These are specific cases of duality F{ G( } = g( k Recall that F{ Π( } = sic( k. Therefore from duality, F{ sic( } = Π( k = Π(k 4

Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2004 Lecture 2 Linear Systems. Topics

Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2004 Lecture 2 Linear Systems. Topics Bioegieerig 280A Priciples of Biomedical Imagig Fall Quarter 2004 Lecture 2 Liear Systems Topics 1. Liearity 2. Impulse Respose ad Delta fuctios 3. Superpositio Itegral 4. Shift Ivariace 5. 1D ad 2D covolutio