1TT Liu, BE280A, UCSD Fall 2008Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2008X-Rays Lecture 2TT Liu, BE280A, UCSD Fall 2008Topics• Review of Signal Expansions• Linearity• Superposition• ConvolutionTT Liu, BE280A, UCSD Fall 2008Kronecker Delta Function! "[n] =1 for n = 00 otherwise# $ % nδ[n]nδ[n-2]00TT Liu, BE280A, UCSD Fall 2008Kronecker Delta Function! "[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]2TT Liu, BE280A, UCSD Fall 2008Discrete Signal Expansion! g[n] = g[k]"[n # k]k=#$$%g[m,n] =l= #$$%g[k,l]"[m # k,n # l]k=#$$%nδ[n]n1.5δ[n-2]0n-δ[n-1]00ng[n]nTT Liu, BE280A, UCSD Fall 20082D Signala bdc00d0=+++a0000b00000cTT Liu, BE280A, UCSD Fall 2008Image Decomposition! g[m,n] = a"[m,n] + b"[m,n #1] + c"[m #1,n] + d"[m #1,n #1]= g[k, l]l= 01$k= 01$"[m # k, n # l]c dba0010=+++cdab100001000001TT Liu, BE280A, UCSD Fall 2008Dirac Delta Function ! Notation : "(x) - 1D Dirac Delta Function"(x, y) or 2"(x, y) - 2D Dirac Delta Function"(x, y,z) or 3"(x, y,z) - 3D Dirac Delta Function"(r r ) - N Dimensional Dirac Delta Function3TT Liu, BE280A, UCSD Fall 2008Rectangle Function! "(x) =0 x > 1/21 x # 1/2$ % & -1/2 1/21-1/2 1/21/2x-1/2xyAlso called rect(x)! "(x, y) = " ( x)"(y)TT Liu, BE280A, UCSD Fall 20081D Dirac Delta Function! "(x) = 0 when x # 0 and "(x)dx =1$%%&Can interpret the integral as a limit of the integral of an ordinary function that is shrinking in width and growing in height, while maintaining aconstant area. For example, we can use a shrinking rectangle function such that "(x)dx =$%%&lim'(0'$1)(x /')dx$%%&.-1/2 1/21xτ→0TT Liu, BE280A, UCSD Fall 20082D Dirac Delta Function! "(x, y) = 0 when x2+ y2# 0 and "(x, y)dxdy =1$%%&$%%&where we can consider the limit of the integral of an ordinary 2D functionthat is shrinking in width but increasing in height while maintaining constant area."(x, y)dxdy =$%%&$%%&lim'(0'$2) x /', y /'( )dxdy$%%&$%%&.Useful fact : "(x, y) ="(x)"(y)τ→0TT Liu, BE280A, UCSD Fall 2008Generalized Functions! Dirac delta functions are not ordinary functions that are defined by theirvalue at each point. Instead, they are generalized functions that are defined by what they do underneath an integral. The most important property of the Dirac delta is the sifting property"(x # x0)g(x)dx = g(x0#$$%) where g(x) is a smooth function. This siftingproperty can be understood by considering the limiting case lim&'0( x /&( )g(x)dx = g(x0#$$%)g(x)Area = (height)(width)= (g(x0)/ τ) τ = g(x0)x04TT Liu, BE280A, UCSD Fall 2008Representation of 1D Function! From the sifting property, we can write a 1D function as g(x) = g(")#(x $")d".$%%& To gain intuition, consider the approximationg(x) ' g(n(x)1(x)x $ n(x(x* + , - . / n=$%%0(x.g(x)TT Liu, BE280A, UCSD Fall 2008Representation of 2D Function! Similarly, we can write a 2D function as g(x, y) = g(",#)$(x %", y %#)d"d#.%&&'%&&' To gain intuition, consider the approximationg(x, y) ( g(n)x,m)y)1)x*x % n)x)x+ , - . / 0 n=%&&11)y*y % m)y)y+ , - . / 0 )x)ym=%&&1.TT Liu, BE280A, UCSD Fall 2008Intuition: 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 ImageOriginalImageTT Liu, BE280A, UCSD Fall 2008Bushberg et al 20015TT Liu, BE280A, UCSD Fall 2008Full Width Half Maximum (FWHM) is a measure of resolution. Prince and Link 2005TT Liu, BE280A, UCSD Fall 2008Impulse 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;") = L#x1$"( )[ ] 1D Impulse Response h( x2, y2;",%) = L#x1$", y1$%( )[ ] 2D Impulse Responsex1y1x2y2! h(x2, y2;",#)! Impulse at ",#TT Liu, BE280A, UCSD Fall 2008Pinhole Magnification Exampleηηabba-! In this example, an impulse at ",#( ) will yield an impulseat m",m#( ) where m = $b /a. Thus, h x2, y2;",#( )= L%x1$", y1$#( )[ ]=%(x2$ m", y2$ m#). y1y2TT Liu, BE280A, UCSD Fall 2008Linearity (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)6TT Liu, BE280A, UCSD Fall 2008Linearity (Scaling)I1(x,y)R(I)K1(x,y)a1I1(x,y)R(I)a1K1(x,y)TT Liu, BE280A, UCSD Fall 2008LinearityA 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 2008ExampleAre 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 2008Superposition! g[m] = g[0]"[m] + g[1]"[m #1] + g[2]"[m # 2]h[m',k] = L["[m # k]]! = g[0]h[m',0] + g[1]h[m',1] + g[2]h[m',2]= g[k]h[m ', k]k= 02"! 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][ ]7TT Liu, BE280A, UCSD Fall 2008Superposition 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#TT Liu, BE280A, UCSD Fall 2008Pinhole 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 2008Space 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 2008Pinhole Magnification Exampleηηabba-! h x2, y2;",#( )= C$(x2% m", y2% m#) . Is this system space invariant?8TT Liu, BE280A, UCSD Fall 2008Pinhole Magnification Example____, the pinhole system ____ space invariant.TT Liu, BE280A, UCSD Fall 2008Convolution! g[m] = g[0]"[m] + g[1]"[m #1] + g[2]"[m # 2]h[m',k] = L["[m # k]] = h[$ 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 20081D Convolution! I(x) =
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