1Thomas Liu, BE280B, UCSD, Spring 2005Bioengineering 280BPrinciples of Biomedical ImagingSpring Quarter 2005Lecture 1Linear SystemsThomas Liu, BE280B, UCSD, Spring 2005Topics1. Linearity2. Impulse Response and Delta functions3. Superposition Integral4. Shift Invariance5. 1D and 2D convolution6. Signal Representations2Thomas Liu, BE280B, UCSD, Spring 2005Signals and ImagesDiscrete-time/space signal/image: continuous valuedfunction with a discrete time/space index, denoted ass[n] for 1D, s[m,n] for 2D , etc.Continuous-time/space signal/image: continuousvalued function with a continuous time/space index,denoted as s(t) or s(x) for 1D, s(x,y) for 2D, etc.nt mnyxxThomas Liu, BE280B, UCSD, Spring 2005Linearity (Addition)I1(x,y)R(I)K1(x,y)I2(x,y)R(I)K2(x,y)I1(x,y)+ I2(x,y)R(I)K1(x,y) +K2(x,y)3Thomas Liu, BE280B, UCSD, Spring 2005Linearity (Scaling)I1(x,y)R(I)K1(x,y)a1I1(x,y)R(I)a1K1(x,y)Thomas Liu, BE280B, UCSD, Spring 2005LinearityA system R is linear if for two inputs I1(x,y) and I2(x,y) with outputsR(I1(x,y))=K1(x,y) and R(I2(x,y))=K2(x,y)the response to the weighted sum of inputs is theweighted sum of outputs:R(a1I1(x,y)+ a2I 2(x,y))=a1K1(x,y)+ a2K2(x,y)4Thomas Liu, BE280B, UCSD, Spring 2005ExampleAre these linear systems? g(x,y) g(x,y)+10+10g(x,y) 10g(x,y)X10g(x,y)Move upBy 1Move rightBy 1g(x-1,y-1)Thomas Liu, BE280B, UCSD, Spring 2005Rectangle Function€ Π(x) =0 x > 1/21 x ≤1/2 -1/2 1/21-1/2 1/21/2x-1/2xyAlso called rect(x)€ Π(x, y) = Π(x)Π (y)5Thomas Liu, BE280B, UCSD, Spring 2005Kronecker Delta Function€ δ[n] =1 for n = 00 otherwise nδ[n]nδ[n-2]00Thomas Liu, BE280B, UCSD, Spring 2005Kronecker Delta Function€ δ[m,n] =1 for m = 0,n = 00 otherwise δ[m,n] δ[m-2,n]δ[m,n-2]δ[m-2,n-2]6Thomas Liu, BE280B, UCSD, Spring 2005Discrete Signal Expansion€ g[n] = g[k]δ[n − k]k=−∞∞∑g[m,n] =l= −∞∞∑g[k,l]δ[m − k, n − l]k=−∞∞∑nδ[n]n1.5δ[n-2]0n-δ[n-1]00ng[n]nThomas Liu, BE280B, UCSD, Spring 2005Dirac Delta Function € Notation : δ(x) - 1D Dirac Delta Functionδ(x, y) or 2δ(x, y) - 2D Dirac Delta Functionδ(x, y,z) or 3δ(x, y,z) - 3D Dirac Delta Functionδ(r r ) - N Dimensional Dirac Delta Function7Thomas Liu, BE280B, UCSD, Spring 20051D Dirac Delta Function€ δ(x) = 0 when x ≠ 0 and δ(x)dx =1−∞∞∫Can interpret the integral as a limit of the integral of an ordinary function that is shrinking in width and growing in height, while maintaining aconstant area. For example, we can use a shrinking rectangle function such that δ(x)dx =−∞∞∫limτ→0τ−1Π(x /τ)dx−∞∞∫.-1/2 1/21xτ→0Thomas Liu, BE280B, UCSD, Spring 20052D Dirac Delta Function€ δ(x, y) = 0 when x2+ y2≠ 0 and δ(x, y)dxdy = 1−∞∞∫−∞∞∫where we can consider the limit of the integral of an ordinary 2D functionthat is shrinking in width but increasing in height while maintaining constant area.δ(x, y)dxdy =−∞∞∫−∞∞∫limτ→0τ−2Π x /τ, y /τ( )dxdy−∞∞∫−∞∞∫.Useful fact : δ(x, y) =δ(x)δ(y)τ→08Thomas Liu, BE280B, UCSD, Spring 2005Generalized Functions€ Dirac delta functions are not ordinary functions that are defined by theirvalue at each point. Instead, they are generalized functions that are defined by what they do underneath an integral. The most important property of the Dirac delta is the sifting propertyδ(x − x0)g(x)dx = g(x0−∞∞∫) where g(x) is a smooth function. This siftingproperty can be understood by considering the limiting case limτ→0τ−1Π x /τ( )g(x)dx = g(x0−∞∞∫)g(x)Area = (height)(width)= (g(x0)/ τ) τ = g(x0)x0Thomas Liu, BE280B, UCSD, Spring 2005Representation of 1D Function€ From the sifting property, we can write a 1D function as g(x) = g(ξ)δ(x −ξ)dξ.−∞∞∫ To gain intuition, consider the approximationg(x) ≈ g(nΔx)1ΔxΠx − nΔxΔx n=−∞∞∑Δx.g(x)9Thomas Liu, BE280B, UCSD, Spring 2005Representation of 2D Function€ Similarly, we can write a 2D function as g(x, y) = g(ξ,η)δ(x −ξ, y −η)dξdη.−∞∞∫−∞∞∫ To gain intuition, consider the approximationg(x, y) ≈ g(nΔx,mΔy)1ΔxΠx − nΔxΔx n=−∞∞∑1ΔyΠy − mΔyΔy ΔxΔym=−∞∞∑.Thomas Liu, BE280B, UCSD, Spring 2005Intuition: the impulse response is the response ofa system to an input of infinitesimal width andunit area.Impulse ResponseSince any input can be thought of as theweighted sum of impulses, a linear system ischaracterized by its impulse response(s).Blurred ImageOriginalImage10Thomas Liu, BE280B, UCSD, Spring 2005Impulse Response€ The impulse response characterizes the response of a system over all space to a Dirac delta impulse function at a certain location. h(x2;ξ) = Lδx1−ξ( )[ ] 1D Impulse Response h(x2, y2;ξ,η) = Lδx1−ξ, y1−η( )[ ] 2D Impulse Responsex1y1x2y2€ h(x2, y2;ξ,η)€ Impulse at ξ,ηThomas Liu, BE280B, UCSD, Spring 2005Superposition Integral€ What is the response to an arbitrary function g(x1,y1)? Write g(x1,y1) = g(ξ,η)δ(x1-∞∞∫-∞∞∫−ξ, y1−η)dξdη.The response is given by I(x2, y2) = L g1(x1,y1)[ ] = L g(ξ,η)δ(x1-∞∞∫-∞∞∫−ξ, y1−η)dξdη[ ] = g(ξ,η)Lδ(x1−ξ, y1−η)[ ]-∞∞∫-∞∞∫dξdη = g(ξ,η)h(x2, y2;ξ,η)-∞∞∫-∞∞∫dξdη11Thomas Liu, BE280B, UCSD, Spring 2005Space Invariance€ If a system is space invariant, the impulse response depends onlyon the difference between the output coordinates and the position ofthe impulse and is given by h(x2, y2;ξ,η) = h x2−ξ, y2−η( ) Thomas Liu, BE280B, UCSD, Spring 20052D Convolution€ I(x2, y2) = g(ξ,η)h(x2, y2;ξ,η)-∞∞∫-∞∞∫dξdη= g(ξ,η)h(x2−ξ, y2−η)-∞∞∫-∞∞∫dξdη= g(x2, y2) **h(x2, y2)For a space invariant linear system, the superposition integralbecomes a convolution integral.where ** denotes 2D convolution. This will sometimes beabbreviated as *, e.g. I(x2, y2)= g(x2, y2)*h(x2, y2).12Thomas Liu, BE280B, UCSD, Spring 20051D Convolution€ I(x) = g(ξ)h(x;ξ)dξ-∞∞∫= g(ξ)h(x −ξ)-∞∞∫dξ= g(x) ∗ h(x)For completeness, here is the 1D version.Useful fact: € g(x) ∗δ(x − Δ) = g(ξ)δ(x − Δ −ξ)-∞∞∫dξ= g(x − Δ)Thomas Liu, BE280B, UCSD, Spring 20052D
View Full Document
Unlocking...