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UCSD BENG 280A - Linear Systems

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1Thomas Liu, BE280A, UCSD, Fall 2004Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2004Lecture 2Linear SystemsThomas Liu, BE280A, UCSD, Fall 2004Topics1. Linearity2. Impulse Response and Delta functions3. Superposition Integral4. Shift Invariance5. 1D and 2D convolution6. Examples.2Thomas Liu, BE280A, UCSD, Fall 2004Signals 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, BE280A, UCSD, Fall 2004Linearity (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, BE280A, UCSD, Fall 2004Linearity (Scaling)I1(x,y)R(I)K1(x,y)a1I1(x,y)R(I)a1K1(x,y)Thomas Liu, BE280A, UCSD, Fall 2004LinearityA 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, BE280A, UCSD, Fall 2004ExampleAre 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, BE280A, UCSD, Fall 2004Rectangle 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, BE280A, UCSD, Fall 2004Kronecker Delta Function€ δ[n] =1 for n = 00 otherwise   nδ[n]nδ[n-2]00Thomas Liu, BE280A, UCSD, Fall 2004Kronecker 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, BE280A, UCSD, Fall 2004Discrete 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, BE280A, UCSD, Fall 2004Dirac 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, BE280A, UCSD, Fall 20041D 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, BE280A, UCSD, Fall 20042D 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, BE280A, UCSD, Fall 2004Generalized 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, BE280A, UCSD, Fall 2004Working with Dirac Delta Functions€ What is δ(ax - b)? What is dδ(x)/dx? How do we define generalized functions?There are two main approaches :1) Look at the limit of an integral with sequences.2) Consider the behavior of the function when integrated with a nice test function. Two generalized functions δ1(t) and δ2(t) are equivalent in the distributional sense when δ1(t)φ-∞∞∫(t)dt =δ2(t)φ-∞∞∫(t)dt Example : δ(ax) = ??9Thomas Liu, BE280A, UCSD, Fall 2004Representation 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)Thomas Liu, BE280A, UCSD, Fall 2004Representation 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= −∞∞∑.10Thomas Liu, BE280A, UCSD, Fall 2004Intuition: 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 ImageOriginalImageThomas Liu, BE280A, UCSD, Fall 2004Impulse 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,η11Thomas Liu, BE280A, UCSD, Fall 2004Superposition 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)-∞∞∫-∞∞∫dξdηThomas Liu, BE280A, UCSD, Fall 2004Pinhole Magnification Exampleηηabba-€ In this example, an impulse at ξ,η( ) will yield an impulseat Mξ, Mη( ) where M = −b/ a. Thus, h x2, y2;ξ,η( )= Lδx1−ξ, y1−η( )[ ]=δ(x2− Mξ, y2− Mη).12Thomas Liu, BE280A, UCSD, Fall 2004Pinhole Magnification Example€ I(x2, y2) = g(ξ,η)h(x2, y2;ξ,η)-∞∞∫-∞∞∫dξdη= C g(ξ,η)δ(x2− Mξ, y2− Mη)-∞∞∫-∞∞∫dξdηI(x2,y2)g(x1,y1)Thomas Liu, BE280A, UCSD, Fall 2004Space Invariance€ If a system is


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