1TT Liu, BE280A, UCSD Fall 2007Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2007X-Rays Lecture 3TT Liu, BE280A, UCSD Fall 2007Topics• Linearity• Superposition• ConvolutionTT Liu, BE280A, UCSD Fall 2007Linearity (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)TT Liu, BE280A, UCSD Fall 2007Linearity (Scaling)I1(x,y)R(I)K1(x,y)a1I1(x,y)R(I)a1K1(x,y)2TT Liu, BE280A, UCSD Fall 2007LinearityA 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)TT Liu, BE280A, UCSD Fall 2007ExampleAre 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)TT Liu, BE280A, UCSD Fall 2007Superposition! g[m] = g[0]"[m] + g[1]"[m #1] + g[2]"[m # 2]h[m',k] = L["[m # k]]y[m'] = L g[m][ ]= L g[0]"[m] + g[1]"[m #1] + g[2]"[m # 2][ ]= L g[0]"[m][ ]+ L g[1]"[m #1][ ]+ L g[2]"[m # 2][ ]= g[0]L"[m][ ]+ g[1]L"[m #1][ ]+ g[2]L"[m # 2][ ]= g[0]h[m',0] + g[1]h[m',1] + g[2]h[m',2]= g[k]h[m',k]k= 02$TT Liu, BE280A, UCSD Fall 2007Superposition Integral! What is the response to an arbitrary function g(x1,y1)? Write g(x1,y1) = g(",#)$(x1-%%&-%%&'", y1'#)d"d#.The response is given by I(x2, y2) = L g1(x1,y1)[ ] = L g(",#)$(x1-%%&-%%&'", y1'#)d"d#[ ] = g(",#)L$(x1'", y1'#)[ ]-%%&-%%&d"d# = g(",#)h(x2, y2;",#)-%%&-%%&d"d#3TT Liu, BE280A, UCSD Fall 2007Pinhole Magnification Example! I(x2, y2) = g(",#)h(x2, y2;",#)-$$%-$$%d"d#= C g(",#)&(x2' m", y2' m#)-$$%-$$%d"d#I(x2,y2)g(x1,y1)TT Liu, BE280A, UCSD Fall 2007Space 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;",#) = h x2$", y2$#( ) TT Liu, BE280A, UCSD Fall 2007Pinhole Magnification Exampleηηabba-! h x2, y2;",#( )= C$(x2% m", y2% m#) . Is this system space invariant? TT Liu, BE280A, UCSD Fall 2007Pinhole Magnification Example____, the pinhole system ____ space invariant.4TT Liu, BE280A, UCSD Fall 2007Convolution! g[m] = g[0]"[m] + g[1]"[m #1] + g[2]"[m # 2]h[m',k] = L["[m # k]] = h[m # k]y[m'] = L g[m][ ]= L g[0]"[m] + g[1]"[m #1] + g[2]"[m # 2][ ]= L g[0]"[m][ ]+ L g[1]"[m #1][ ]+ L g[2]"[m # 2][ ]= g[0]L"[m][ ]+ g[1]L"[m #1][ ]+ g[2]L"[m # 2][ ]= g[0]h[m'#0] + g[1]h[m'#1] + g[2]h[m'#2]= g[k]h[m'#k]k= 02$TT Liu, BE280A, UCSD Fall 20071D Convolution! I(x) = g(")h(x;")d"-##$= g(")h(x %")-##$d"= g(x) & h( x)Useful fact: ! g(x) "#(x $ %) = g(&)#(x $ % $&)-''(d&= g(x $ %)TT Liu, BE280A, UCSD Fall 20072D Convolution! I(x2, y2) = g(",#)h(x2, y2;",#)-$$%-$$%d"d#= g(",#)h(x2&", y2&#)-$$%-$$%d"d#= g(x2, y2) **h(x2, y2)For a space invariant linear system, the superpositionintegral becomes a convolution integral.where ** denotes 2D convolution. This will sometimes beabbreviated as *, e.g. I(x2, y2)= g(x2, y2)*h(x2, y2).TT Liu, BE280A, UCSD Fall 2007Rectangle Function! "(x) =0 x > 1/21 x #1/2$ % & -1/2 1/21-1/2 1/21/2x-1/2xyAlso called rect(x)! "(x, y) = "(x)"(y)5TT Liu, BE280A, UCSD Fall 20071D Convolution Examples-1/2 1/21x*-3/4x= ?-1/2 1/21x*-1/2 1/21x= ?1/4TT Liu, BE280A, UCSD Fall 20072D Convolution Example-1/2 1/2xyh(x)=rect(x,y)yxg(x)= δ(x+1/2,y) + δ(x,y)xI(x,y)=g(x)**h(x,y)TT Liu, BE280A, UCSD Fall 20072D Convolution ExampleTT Liu, BE280A, UCSD Fall 2007Pinhole Magnification Example! I( x2, y2) = s(",#)h(x2, y2;",#)-$$%-$$%d"d#= s(",#)&(x2' m", y2' m#)-$$%-$$%d"d#after substituting ( " = m" and ( # = m#, we obtain=1m2s(( " /m,( # /m)&(x2'( " , y2'( # )-$$%-$$%d( " d( # =1m2s(x2/m, y2/m) ))&x2, y2( )=1m2s(x2/m, y2/m)6TT Liu, BE280A, UCSD Fall 2007X-Ray Image Equation! I( x2, y2) = s(",#)h(x2, y2;",#)-$$%-$$%d"d#= s(",#)tx2& m"M,y2& m#M' ( ) * + , -$$%-$$%d"d#after substituting - " = m" and - # = m#, we obtain=1m2s(- " /m,- # /m)tx2&- " M,y2&- # M' ( ) * + , -$$%-$$%d- " d- # =1m2s(x2/m, y2/m) ..t x2/ M, y2/ M( )Note: we have ignored obliquity factors etc.TT Liu, BE280A, UCSD Fall 2007Summary1. The response to a linear system can becharacterized by a spatially varying impulseresponse and the application of the superpositionintegral.2. A shift invariant linear system can becharacterized by its impulse response and theapplication of a convolution
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