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

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1 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 6. Examples. 1

2 Sigals ad Images Discrete-time/space sigal/image: cotiuous valued fuctio with a discrete time/space idex, deoted as s[] for 1D, s[m,] for 2D, etc. Cotiuous-time/space sigal/image: cotiuous valued fuctio with a cotiuous time/space idex, deoted as s(t) or s(x) for 1D, s(x,y) for 2D, etc. m t y x x Liearity (Additio) I 1 (x,y) R(I) K 1 (x,y) I 2 (x,y) R(I) K 2 (x,y) I 1 (x,y)+ I 2 (x,y) R(I) K 1 (x,y) +K 2 (x,y) 2

3 Liearity (Scalig) I 1 (x,y) R(I) K 1 (x,y) a 1 I 1 (x,y) R(I) a 1 K 1 (x,y) Liearity A system R is liear if for two iputs I 1 (x,y) ad I 2 (x,y) with outputs R(I 1 (x,y))=k 1 (x,y) ad R(I 2 (x,y))=k 2 (x,y) the respose to the weighted sum of iputs is the weighted sum of outputs: R(a 1 I 1 (x,y)+ a 2 I 2 (x,y))=a 1 K 1 (x,y)+ a 2 K 2 (x,y) 3

4 Example Are these liear systems? g(x,y) + g(x,y)+10 g(x,y) X 10g(x,y) g(x,y) Move up By 1 Move right By 1 g(x-1,y-1) Rectagle Fuctio Π(x) = 0 x >1/2 1 1 x 1/2-1/2 1/2 x Also called rect(x) y 1/2 Π(x, y) = Π(x)Π(y) -1/2 1/2-1/2 x 4

5 Kroecker Delta Fuctio δ[] = 1 for = 0 0 otherwise δ[] 0 0 δ[-2] Kroecker Delta Fuctio 1 for m = 0, = 0 δ[m,] = 0 otherwise δ[m,] δ[m-2,] δ[m,-2] δ[m-2,-2] 5

6 Discrete Sigal Expasio g[] = g[k]δ[ k] g[m,] = g[] k= k= l= g[k,l]δ[m k, l] δ[] 0 -δ[-1] 0 1.5δ[-2] 0 Dirac Delta Fuctio Notatio : δ(x) - 1D Dirac Delta Fuctio δ(x, y) or 2 δ(x, y) - 2D Dirac Delta Fuctio δ(x, y,z) or 3 δ(x, y,z) - 3D Dirac Delta Fuctio δ( r ) - N Dimesioal Dirac Delta Fuctio 6

7 1D Dirac Delta Fuctio δ(x) = 0 whe x 0 ad δ(x)dx =1 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 example, we ca use a shrikig rectagle fuctio such that δ(x)dx = lim τ 1 Π(x /τ)dx. τ 0 1 τ 0-1/2 1/2 x 2D Dirac Delta Fuctio δ(x, y) = 0 whe x 2 + y 2 0 ad δ(x,y)dxdy =1 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(x, y)dxdy = lim τ 2 Π x /τ,y /τ τ 0 ( )dxdy. Useful fact : δ(x, y) = δ(x)δ(y) τ 0 7

8 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(x x 0 )g(x)dx = g(x 0 ) where g(x) is a smooth fuctio. This siftig property ca be uderstood by cosiderig the limitig case lim τ 1 Π x /τ τ 0 ( )g(x)dx = g(x 0 ) g(x) Area = (height)(width)= (g(x 0 )/ τ) τ = g(x 0 ) x 0 Workig with Dirac Delta Fuctios What is δ(ax - b)? What is dδ(x)/dx? How do we defie geeralized fuctios? There are two mai approaches : 1) Look at the limit of a itegral with sequeces. 2) Cosider the behavior of the fuctio whe itegrated with a ice test fuctio. Two geeralized fuctios δ 1 (t) ad δ 2 (t) are equivalet i the distributioal sese whe δ 1 (t)φ(t)dt = δ 2 (t)φ(t)dt - - Example : δ(ax) =?? 8

9 Represetatio of 1D Fuctio From the siftig property, we ca write a 1D fuctio as g(x) = g(ξ)δ(x ξ)dξ. To gai ituitio, cosider the approximatio g(x) = g(δx) 1 Δx Π x Δx Δx. Δx g(x) Represetatio of 2D Fuctio Similarly, we ca write a 2D fuctio as g(x, y) = g(ξ,η)δ(x ξ,y η)dξdη. To gai ituitio, cosider the approximatio g(x, y) g(δx,mδy) 1 Δx Π x Δx 1 = Δx Δy Π y mδy ΔxΔy. m= Δy 9

10 Impulse Respose Ituitio: the impulse respose is the respose of a system to a iput of ifiitesimal width ad uit area. Origial Image Blurred Image Sice ay iput ca be thought of as the weighted sum of impulses, a liear system is characterized by its impulse respose(s). Impulse Respose The impulse respose characterizes the respose of a system over all space to a Dirac delta impulse fuctio at a certai locatio. h(x 2 ;ξ) = L[ δ( x 1 ξ) ] 1D Impulse Respose [ ( )] 2D Impulse Respose h(x 2, y 2 ;ξ,η) = L δ x 1 ξ, y 1 η y 1 Impulse at ξ,η y 2 h(x2 ;ξ,η) x 1 x 2 10

11 Superpositio Itegral What is the respose to a arbitrary fuctio g(x 1,y 1 )? Write g(x 1,y 1 ) = g(ξ,η)δ(x 1 ξ, y 1 η)dξdη. - - The respose is give by [ ] I(x 2 ) = L g 1 (x 1,y 1 ) [ - - ] = L g(ξ,η)δ(x 1 ξ,y 1 η)dξdη [ ] = g(ξ,η)l δ(x 1 ξ, y 1 η) dξdη - - = g(ξ,η)h(x 2 ;ξ,η) dξdη - - Pihole Magificatio Example - b a η η a b I this example, a impulse at ξ,η ( ) where M = b /a. ( ) = L δ( x 1 ξ,y 1 η) at Mξ, Mη Thus, h x 2, y 2 ;ξ,η ( ) will yield a impulse [ ] = δ(x 2 Mξ, y 2 Mη). 11

12 Pihole Magificatio Example I(x 2 ) = - - g(ξ,η)h(x 2, y 2 ;ξ,η) dξdη = C g(ξ,η)δ(x 2 Mξ, y 2 Mη) dξdη - - I(x 2, y 2 ) g(x 1, y 1 ) Space Ivariace If a system is space ivariat, the impulse respose depeds oly o the differece betwee the output coordiates ad the positio of ( ) the impulse ad is give by h(x 2, y 2 ;ξ,η) = h x 2 ξ η 12

13 Pihole Magificatio Example - b a η η a b h( x 2 ;ξ,η) = Cδ(x 2 Mξ Mη). Is this system space ivariat? Pihole Magificatio Example, the pihole system space ivariat. 13

14 I(x 2 ) = 2D Covolutio For a space ivariat liear system, the superpositio itegral becomes a covolutio itegral. - - g(ξ,η)h(x 2 ;ξ,η) dξdη = g(ξ,η)h(x 2 ξ, y 2 η) dξdη - - = g(x 2 ) **h(x 2 ) where ** deotes 2D covolutio. This will sometimes be abbreviated as *, e.g. I(x 2, y 2 )= g(x 2, y 2 )*h(x 2, y 2 ). I(x) = 1D Covolutio For completeess, here is the 1D versio. - g(ξ)h(x;ξ)dξ = g(ξ)h(x ξ) dξ = g(x) h(x) Useful fact: - g(x) δ(x Δ) = - g(ξ)δ(x Δ ξ) dξ = g(x Δ) 14

15 1D Covolutio Review g(x) h(x) = - g(ξ)h(x ξ)dξ Basic Rule: Flip oe fuctio, slide it past the other fuctio, ad itegrate as you go. g(x)=rect(x) h(x)=rect(x-1/2) -1/2 1/2 1 1D Covolutio Review h(-1/2-ξ) g(ξ) I(x) h(-ξ) h(1/2-ξ) -1/2 1/2 3/2 h(3/2-ξ) 15

16 2D Covolutio Example g(x)= δ(x+1/2,y) + δ(x,y) y h(x)=rect(x,y) y x -1/2 12 x I(x,y)=g(x)**h(x,y) x 2D Covolutio Example 16

17 Summary 1. The respose to a liear system ca be characterized by a spatially varyig impulse respose ad the applicatio of the superpositio itegral. 2. A shift ivariat liear system ca be characterized by its impulse respose ad the applicatio of a covolutio itegral. 3. Dirac delta fuctios are geeralized fuctios. Pihole Magificatio Example I(x 2 ) = - - g(ξ,η)h(x 2 ;ξ,η) dξdη = g(ξ,η)δ(x 2 Mξ Mη) dξdη - - after substitutig ξ = Mξ ad η = Mη, we obtai = 1 g( ξ / M, M η / M)δ(x ξ,y η ) d ξ d η = 1 M g(x 2 2 / M / M) δ ( x 2 ) = 1 M 2 g(x 2 / M / M) 17