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A portion of lecture notes from a Bioengineering 280A course at UCSD, Fall 2008. The notes cover topics related to X-Rays, signal expansions, linearity, superposition, and convolution. The professor, TT Liu, discusses the Kronecker delta function, discrete signal expansion, and image decomposition. The document also includes information on Dirac delta functions, rectangle functions, and the representation of 1D and 2D functions.
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TT Liu, BE280A, UCSD Fall 2008
TT Liu, BE280A, UCSD Fall 2008
Topics
TT Liu, BE280A, UCSD Fall 2008
Kronecker Delta Function
"[ n ] =
1 for n = 0
0 otherwise
n
δ[n]
n
δ[n-2]
TT Liu, BE280A, UCSD Fall 2008
Kronecker Delta Function
"[ m , n ] =
1 for m = 0 , n = 0
0 otherwise
δ[m,n] δ[m-2,n]
δ[m,n-2] δ[m-2,n-2]
TT Liu, BE280A, UCSD Fall 2008
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]
n
n
g[n]
n
TT Liu, BE280A, UCSD Fall 2008
2D Signal
a b
c d
0 0
0 d
=
a 0
0 0
0 b
0 0
0 0
c 0
TT Liu, BE280A, UCSD Fall 2008
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
a b
0 0
0 1
=
c
d
a b
1 0
0 0
0 1
0 0
0 0
1 0
TT Liu, BE280A, UCSD Fall 2008
Dirac 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 Function
TT Liu, BE280A, UCSD Fall 2008
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 2008
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 =%&
&
1
1
) y
y % m ) y
) y
,
.
/
0 ) x ) y
m =%&
&
1 .
TT Liu, BE280A, UCSD Fall 2008
Intuition: the impulse response is the response of
a system to an input of infinitesimal width and
unit area.
Impulse Response
Since any input can be thought of as the
weighted sum of impulses, a linear system is
characterized by its impulse response(s).
Blurred Image
Original
Image
TT Liu, BE280A, UCSD Fall 2008
Bushberg et al 2001
TT Liu, BE280A, UCSD Fall 2008
Full Width Half Maximum
(FWHM) is a measure of resolution.
Prince and Link 2005
TT Liu, BE280A, UCSD Fall 2008
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 ( x 2
; ") = L # x 1
[ ] 1D Impulse Response
h ( x 2
, y 2
; ",%) = L # x 1
$ ", y 1
$ % [ ( )] 2D Impulse Response
x 1
y 1
x 2
y 2
!
h ( x
2
, y
2
; ",#)
!
Impulse at ",#
TT Liu, BE280A, UCSD Fall 2008
Pinhole Magnification Example
η
η
a
b
b
a
In this example, an impulse at ",# ( ) will yield an impulse
at m ", m # ( ) where m = $ b / a.
Thus, h x 2
, y 2
( ) = L % x 1
$ ", y 1
[ ( )] = %( x 2
$ m ", y 2
$ m #).
y 1
y 2
TT Liu, BE280A, UCSD Fall 2008
Linearity (Addition)
1
(x,y)
1
(x,y)
2
(x,y)
2
(x,y)
1
(x,y)+ I 2
(x,y)
1
(x,y) +K 2
(x,y)
TT Liu, BE280A, UCSD Fall 2008
What is the response to an arbitrary function g ( x 1
,y 1
Write g ( x 1
,y 1
) = g ( ",#) $( x 1
%
%
' ", y 1
' #) d " d #.
The response is given by
I ( x 2
, y 2
) = L g 1
( x 1
,y 1
= L g ( ",#) $( x 1
%
%
' ", y 1
' #) d " d #
= g ( ",#) L $( x 1
' ", y 1
%
%
d " d #
= g ( ",#) h ( x 2
, y 2
%
%
d " d #
TT Liu, BE280A, UCSD Fall 2008
I ( x 2
, y 2
) = g ( ",#) h ( x 2
, y 2
$
$
d " d #
= C g ( ",#) &( x 2
' m ", y 2
' m #)
$
$
d " d #
I(x 2,
y 2
g(x 1,
y 1
TT Liu, BE280A, UCSD Fall 2008
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 ( x 2
, y 2
; ",#) = h x 2
$ ", y 2
( )
TT Liu, BE280A, UCSD Fall 2008
η
η
a
b
b
a
!
h x 2
, y 2
2
% m ", y 2
% m #).
Is this system space invariant?
TT Liu, BE280A, UCSD Fall 2008
Pinhole Magnification Example
____, the pinhole system ____ space invariant.
TT Liu, BE280A, UCSD Fall 2008
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 [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 2008
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 2008
2D Convolution
I ( x 2
, y 2
) = g ( ",#) h ( x 2
, y 2
$
%
$
%
d " d #
= g ( ",#) h ( x 2
& ", y 2
$
%
$
%
d " d #
= g ( x 2
, y 2
) ** h ( x 2
, y 2
For a space invariant linear system, the superposition
integral becomes a convolution integral.
where ** denotes 2D convolution. This will sometimes be
abbreviated as *, e.g. I (x 2
, y 2
)= g(x 2
, y 2
)*h(x 2
, y 2
TT Liu, BE280A, UCSD Fall 2008
Pinhole Magnification Example
!
I ( x 2
, y 2
) = s ( ",#) h ( x 2
, y 2
; ",#)
$
%
$
%
d " d #
= s ( ",#) &( x 2
' m ", y 2
' m #)
$
%
$
% d " d #
after substituting ( " = m " and (
=
1
m
2
s ( "( / m , #( / m ) &( x 2
' "( , y 2
' #( )
$
%
$
% d " ( d #(
=
1
m
2
s ( x 2 / m , y 2 / m ) )) & x 2 , y 2
=
1
m
2
s ( x 2
/ m , y 2
/ m )
TT Liu, BE280A, UCSD Fall 2008
X-Ray Imaging
s(x)
d
z
m
s
x
m
t ( x ) = "( x )
TT Liu, BE280A, UCSD Fall 2008
X-Ray Imaging
s(x)
d
z
x 0
x 0
Mx 0
m
s
x " Mx 0
m
t ( x ) = "( x # x 0
!
M ( z ) =
d
z
; m ( z ) = "
d " z
z
TT Liu, BE280A, UCSD Fall 2008
X-Ray Imaging
s
x " Mx 0
m
= s
x
m
x
" x 0
= s ( x / m ) * t
x
!
I ( x , y ) = t
x
M
,
y
M
"
$
%
&
' ((
1
m
2
s
x
m
,
y
m
"
$
%
&
'
For off-center pinhole object, the shifted source image can be written as
For the general 2D case, we convolve the magnified object with the impulse response
Note: we have ignored obliquity factors etc.
TT Liu, BE280A, UCSD Fall 2008
X-Ray Imaging
d
z
m
s
x
m
( t
x
t ( x ) = rect ( x /10)
s ( x ) = rect ( x /10)
TT Liu, BE280A, UCSD Fall 2008
X-Ray Imaging
m
s
x
m
( t
x
= rect ( x / 10 ) ( rect ( x / 20 )
m = 1 ; M = 2
TT Liu, BE280A, UCSD Fall 2008
Summary
characterized by a spatially varying impulse
response and the application of the superposition
integral.
characterized by its impulse response and the
application of a convolution integral.
