BE280A 06 xct4 - University of California, San Diego · TT Liu, BE280A, UCSD Fall 2006 Macovski...
Transcript of BE280A 06 xct4 - University of California, San Diego · TT Liu, BE280A, UCSD Fall 2006 Macovski...
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TT Liu, BE280A, UCSD Fall 2006
Bioengineering 280APrinciples of Biomedical Imaging
Fall Quarter 2006X-Rays/CT Lecture 4
TT Liu, BE280A, UCSD Fall 2006
Topics
• Review Signal Expansions, Impulse Response andLinearity
• Superposition• Space Invariance• Convolution• Start Computed Tomography
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TT Liu, BE280A, UCSD Fall 2006
Discrete Signal Expansion
!
g[n] = g[k]"[n # k]k=#$
$
%
g[m,n] =l=#$
$
% g[k,l]"[m # k,n # l]k=#$
$
%
n
δ[n]
n
1.5δ[n-2]
0
n
-δ[n-1]
0
0
n
g[n]
n
TT Liu, BE280A, UCSD Fall 2006
Image Decomposition
!
g[m,n] = a"[m,n]+ b"[m,n #1]+ c"[m #1,n]+ d"[m #1,n #1]
= g[k, l]l= 0
1
$k= 0
1
$ "[m # k,n # l]
c d
ba 0 010
=
+
+
+
c d
a b
1 000
0 100
0 001
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TT Liu, BE280A, UCSD Fall 2006
Representation of 1D Function
!
From the sifting property, we can write a 1D function as
g(x) = g(")#(x $")d" .$%
%
& To gain intuition, consider the approximation
g(x) ' g(n(x)1
(x)
x $ n(x
(x
*
+ ,
-
. /
n=$%
%
0 (x.
g(x)
TT Liu, BE280A, UCSD Fall 2006
Representation of 2D Function
!
Similarly, we can write a 2D function as
g(x,y) = g(",#)$(x %",y %#)d"d#.%&
&
'%&
&
'
To gain intuition, consider the approximation
g(x,y) ( g(n)x,m)y)1
)x*
x % n)x
)x
+
, -
.
/ 0
n=%&
&
11
)y*
y %m)y
)y
+
, -
.
/ 0 )x)y
m=%&
&
1 .
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TT Liu, BE280A, UCSD Fall 2006
Impulse 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[m',k] = L #[m $ k][ ]
h(x2,y2;",%) = L # x1 $",y1 $%( )[ ] 2D Impulse Response
h[m',n';k, l] = L #[m $ k,n $ l][ ]
x1
y1
x2
y2
!
h(x2,y
2;",#)
!
Impulse at ",#
TT Liu, BE280A, UCSD Fall 2006
LinearityA system R is linear if for two inputs I1(x,y) and I2(x,y) with outputs
R(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)
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TT Liu, BE280A, UCSD Fall 2006
Superposition
!
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= 0
2
$
TT Liu, BE280A, UCSD Fall 2006
Superposition 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#
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TT Liu, BE280A, UCSD Fall 2006
Pinhole Magnification Example
η
η
a b
ba
-
!
In this example, an impulse at ",#( ) will yield an impulse
at m",m#( ) where m = $b /a.
Thus, h x2,y2;",#( ) = L % x1 $",y1 $#( )[ ] = %(x2 $m",y2 $m#).
TT Liu, BE280A, UCSD Fall 2006
Pinhole Magnification Example
!
I(x2,y
2) = g(",#)h(x
2,y
2;",#)
-$
$
%-$
$
% d"d#
= C g(",#)&(x2'm",y
2'm#)
-$
$
%-$
$
% d"d#
I(x2,y2)
g(x1,y1)
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TT Liu, BE280A, UCSD Fall 2006
Space Invariance
!
If a system is space invariant, the impulse response depends only
on the difference between the output coordinates and the position of
the impulse and is given by h(x2,y2;",#) = h x2 $",y2 $#( )
TT Liu, BE280A, UCSD Fall 2006
Pinhole Magnification Example
η
η
a b
ba
-
!
h x2,y2;",#( ) = C$(x2 %m",y2 %m#) .
Is this system space invariant?
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TT Liu, BE280A, UCSD Fall 2006
Pinhole Magnification Example____, the pinhole system ____ space invariant.
TT Liu, BE280A, UCSD Fall 2006
Convolution
!
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= 0
2
$
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TT Liu, BE280A, UCSD Fall 2006
1D 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 $%)
TT Liu, BE280A, UCSD Fall 2006
2D Convolution
!
I(x2,y
2) = g(",#)h(x
2,y
2;",#)
-$
$
%-$
$
% d"d#
= g(",#)h(x2&",y
2&#)
-$
$
%-$
$
% d"d#
= g(x2,y
2) **h(x
2,y
2)
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).
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TT Liu, BE280A, UCSD Fall 2006
Rectangle Function
!
"(x) =0 x >1/2
1 x #1/2
$ % &
-1/2 1/2
1
-1/2 1/2
1/2
x
-1/2
x
yAlso called rect(x)
!
"(x,y) = "(x)"(y)
TT Liu, BE280A, UCSD Fall 2006
2D Convolution Example
-1/2 12x
y
h(x)=rect(x,y)
y
x
g(x)= δ(x+1/2,y) + δ(x,y)
xI(x,y)=g(x)**h(x,y)
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TT Liu, BE280A, UCSD Fall 2006
2D Convolution Example
TT Liu, BE280A, UCSD Fall 2006
Image of a point object
!
Id (x,y) = limm"0
ks(x /m,y /m)
= #(x,y)
s(x,y)
s(x,y)
!
Id (x,y) = s(x,y)
m=1
!
Id (x,y) =1
m2s(x /m,y /m)In general,
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TT Liu, BE280A, UCSD Fall 2006
Pinhole 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
=1
m2
s( ( " /m, ( # /m)&(x2 ' ( " ,y2 ' ( # )-$
$
%-$
$
% d ( " d ( #
=1
m2s(x2 /m,y2 /m)))& x2,y2( )
=1
m2s(x2 /m,y2 /m)
TT Liu, BE280A, UCSD Fall 2006
Magnification of Object
!
M(z) =d
z
=Source to Image Distance (SID)
Source to Object Distance (SOD)
Bushberg et al 2001
zd
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TT Liu, BE280A, UCSD Fall 2006
Image of arbitrary object
!
limm"0
Id (x,y) = t(x,y)
s(x,y)
s(x,y)
!
Id (x,y) = ???
m=1
!
Id (x,y) =1
m2s(x /m,y /m) **t(x /M,y /M)
t(x,y)
t(x,y)
Note: we are ignoring obliquity, etc.
TT Liu, BE280A, UCSD Fall 2006
X-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
=1
m2
s( - " /m, - # /m)tx2 & - "
M,y2 & - #
M
'
( )
*
+ ,
-$
$
%-$
$
% d - " d - #
=1
m2s(x2 /m,y2 /m)..t x2 /M,y2 /M( )
Note: we have ignored obliquity factors etc.
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TT Liu, BE280A, UCSD Fall 2006 Macovski 1983
TT Liu, BE280A, UCSD Fall 2006
Summary1. The response to a linear system can be
characterized 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 integral.
3. Dirac delta functions are generalized functions.
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TT Liu, BE280A, UCSD Fall 2006
Computed Tomography
Suetens 2002
TT Liu, BE280A, UCSD Fall 2006
Computed Tomography
Suetens 2002
ParallelBeam
Fan Beam
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TT Liu, BE280A, UCSD Fall 2006
Computed Tomography
Suetens 2002
TT Liu, BE280A, UCSD Fall 2006
Computed Tomography
Suetens 2002
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TT Liu, BE280A, UCSD Fall 2006
Computed Tomography
Suetens 2002
TT Liu, BE280A, UCSD Fall 2006
CT Line Integral
!
Id = S0 E( )E0
Emax" exp # µ s; $ E ( )0
d
" ds( )dE
Monoenergetic Approximation
Id = I0 exp # µ s;E ( )0
d
" ds( )
gd = #logId
I0
%
& '
(
) *
= µ s;E ( )0
d
" ds
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TT Liu, BE280A, UCSD Fall 2006
CT Number
!
CT_number = µ "µ
water
µwater
#1000
Measured in Hounsfield Units (HU)
Air: -1000 HUSoft Tissue: -100 to 60 HUCortical Bones: 250 to 1000 HUMetal and Contrast Agents: > 2000 HU
TT Liu, BE280A, UCSD Fall 2006
CT Display
Suetens 2002
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TT Liu, BE280A, UCSD Fall 2006
Direct Inverse Approach
µ4µ3
µ2µ1 p1
p2
p3 p4
p1= µ1+ µ2p2= µ3+ µ4p3= µ1+ µ3p4= µ2+ µ4
4 equations, 4 unknowns. Are these the correct equations to use? !
p1
p2
p3
p4
"
#
$ $ $ $
%
&
' ' ' '
=
1 1 0 0
0 0 1 1
1 0 1 0
0 1 0 1
"
#
$ $ $ $
%
&
' ' ' '
µ1
µ2
µ3
µ4
"
#
$ $ $ $
%
&
' ' ' '
No, equations are not linearly independent.p4= p1+ p2- p3Matrix is not full rank.
TT Liu, BE280A, UCSD Fall 2006
Direct Inverse Approach
µ4µ3
µ2µ1 p1
p2
p3 p4
p1= µ1+ µ2p2= µ3+ µ4p3= µ1+ µ3p4= µ2+ µ4
4 equations, 4 unknowns. Are these the correct equations to use? !
p1
p2
p3
p4
"
#
$ $ $ $
%
&
' ' ' '
=
1 1 0 0
0 0 1 1
1 0 1 0
0 1 0 1
"
#
$ $ $ $
%
&
' ' ' '
µ1
µ2
µ3
µ4
"
#
$ $ $ $
%
&
' ' ' '
No, equations are not linearly independent.p4= p1+ p2- p3Matrix is not full rank.
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TT Liu, BE280A, UCSD Fall 2006
Direct Inverse Approach
µ4µ3
µ2µ1 p1
p2
p3 p4
p1= µ1+ µ2p2= µ3+ µ4p3= µ1+ µ3p5= µ1+ µ4
4 equations, 4 unknowns. These are linearly independent now.In general for a NxN image, N2 unknowns, N2 equations.This requires the inversion of a N2xN2 matrixFor a high-resolution 512x512 image, N2=262144 equations.Requires inversion of a 262144x262144 matrix! Inversion process sensitive to measurement errors.
!
p1
p2
p3
p5
"
#
$ $ $ $
%
&
' ' ' '
=
1 1 0 0
0 0 1 1
1 0 1 0
1 0 0 1
"
#
$ $ $ $
%
&
' ' ' '
µ1
µ2
µ3
µ4
"
#
$ $ $ $
%
&
' ' ' '
p5
TT Liu, BE280A, UCSD Fall 2006
Iterative Inverse ApproachAlgebraic Reconstruction Technique (ART)
43
21 3
7
4 6 5
2.52.5
2.52.5 5
5
3.53.5
1.51.5 3
7
5 5
43
21 3
7
5 5
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TT Liu, BE280A, UCSD Fall 2006
Backprojection
Suetens 2002
000030000 0
3
0
30303
000111000
100121001
110131011
111141111
TT Liu, BE280A, UCSD Fall 2006
In-Class Exercise
Suetens 2002
µ4µ3
µ2µ1 5.7
11.3
8.2 8.8 10.1
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TT Liu, BE280A, UCSD Fall 2006
Projections
Suetens 2002
!
r
s
"
# $ %
& ' =
cos( sin(
)sin( cos(
"
# $
%
& ' x
y
"
# $ %
& '
x
y
"
# $ %
& ' =
cos( )sin(
sin( cos(
"
# $
%
& ' r
s
"
# $ %
& '
TT Liu, BE280A, UCSD Fall 2006
Projections
Suetens 2002
!
I" (r) = I0 exp # µ(x,y)dsLr ,"$%
& ' (
) *
= I0 exp # µ(rcos" # ssin",rsin" + scos")dsLr ,"$%
& ' (
) *
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TT Liu, BE280A, UCSD Fall 2006
Projections
Suetens 2002
!
I" (r) = I0 exp # µ(rcos" # ssin",rsin" + scos")dsLr ,"$%
& ' (
) *
!
p" (r) = #lnI" (r)
I0
= µ(rcos" # ssin",rsin" + scos")dsLr ,"$
Sinogram
TT Liu, BE280A, UCSD Fall 2006
Sinogram
Suetens 2002
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TT Liu, BE280A, UCSD Fall 2006
Backprojection
Suetens 2002
!
b(x,y) = B p r,"( ){ }
= p(x cos" + y sin",")d"0
#
$
x
y
!
b(x0,y) = p r," = 0( )#"
= p(x0)#"
x0
r
TT Liu, BE280A, UCSD Fall 2006
Backprojection
Suetens 2002
!
b(x,y) = B p r,"( ){ }
= p(x cos" + y sin",")d"0
#
$
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25
TT Liu, BE280A, UCSD Fall 2006
Backprojection
Suetens 2002!
b(x,y) = B p r,"( ){ } = p(x cos" + y sin",")d"0
#
$