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TT Liu, BE280A, UCSD Fall 2013
Bioengineering 280A
Principles of Biomedical Imaging
Fall Quarter 2013
X-Rays Lecture 2
TT Liu, BE280A, UCSD Fall 2013
d
z
Assume z=d/
10 cm
5 cm
d
z
Assume z=d/
10 cm
5 cm
d
z
Assume z=d/
10 cm
10 cm
TT Liu, BE280A, UCSD Fall 2013
Signals and Images
Discrete-time/space signal /image: continuous
valued function with a discrete time/space index,
denoted as s[n] for 1D , s[m,n] for 2D , etc.
Continuous-time/space signal /image: continuous
valued function with a continuous time/space index,
denoted as s(t) or s(x) for 1D, s(x,y) for 2D, etc.
n
t
m
n
y
x
x (^) TT Liu, BE280A, UCSD Fall 2013
Kronecker Delta Function
δ[ n ] =
1 for n = 0
0 otherwise
n
δ[n]
n
δ[n-2]
TT Liu, BE280A, UCSD Fall 2013
Kronecker Delta Function
δ[ 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 2013
Discrete Signal Expansion
g [ n ] = g [ k ] δ[ n − k ]
k =−∞
∞
∑
g [ m , n ] =
l =−∞
∞
∑ g [ k , l ]^ δ[ m^ −^ k , n^ −^ l ]
k =−∞
∞
∑
n
δ[n]
n
1.5δ[n-2]
n
n
g[n]
n
TT Liu, BE280A, UCSD Fall 2013
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 2013
Image Decomposition
g [ m , n ] = a δ[ m , n ] + b δ[ m , n − 1 ] + c δ[ m − 1 , n ] + d δ[ m − 1 , n − 1 ]
= g [ k , l ]
l = 0
1
∑
k = 0
1
∑^ δ[ 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 2013
Representation of 1D Function
€
From the sifting property, we can write a 1D function as
g ( x ) = g ( ξ) δ( x − ξ) d ξ. −∞
∞
∫ To gain intuition,^ consider the approximation
g ( x ) ≈ g ( n Δ x )
1
Δ x
Π
x − n Δ x
Δ x
,
.
/ n =−∞
∞
∑ Δ x.
g(x)
TT Liu, BE280A, UCSD Fall 2013
Representation of 2D Function
€
Similarly, we can write a 2D function as
g ( x , y ) = g ( ξ,η) δ( x − ξ, y − η) d ξ d η. −∞
∞
∫ −∞
∞
∫
To gain intuition, consider the approximation
g ( x , y ) ≈ g ( n Δ x , m Δ y )
1
Δ x
Π
x − n Δ x
Δ x
,
.
/
0 n =−∞
∞
∑
1
Δ y
Π
y − m Δ y
Δ y
,
.
/
0 Δ x Δ y m =−∞
∞
∑.
TT Liu, BE280A, UCSD Fall 2013
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 2013
Bushberg et al 2001
TT Liu, BE280A, UCSD Fall 2013
Full Width Half Maximum
(FWHM) is a measure of resolution.
Prince and Link 2005 (^) TT Liu, BE280A, UCSD Fall 2013
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 ; ξ) = L [ δ ( x 1 − ξ)] 1D Impulse Response
h ( x 2
, y 2
; ξ,η) = L δ x 1
− ξ, y 1 [ ( −^ η)] 2D Impulse Response
x 1
y 1
x 2
y 2
€
h ( x 2
, y 2
; ξ,η)
€
Impulse at ξ,η
TT Liu, BE280A, UCSD Fall 2013
X-Ray Imaging
s(x)
d
z
m
s
x
m
t ( x ) =
M
δ( x )
TT Liu, BE280A, UCSD Fall 2013
Linearity (Addition)
I
1
(x,y)
R(I)
K
1
(x,y)
I
2
(x,y)
R(I)
K
2
(x,y)
I
1
(x,y)+ I 2
(x,y)
R(I)
K
1
(x,y) +K 2
(x,y)
TT Liu, BE280A, UCSD Fall 2013
Superposition Integral
What is the response to an arbitrary function g ( x 1
,y 1
Write g ( x 1
,y 1
) = g ( ξ,η) δ( x 1
∞
∞
∫ −^ ξ,^ y 1 −^ η) d ξ d η.
The response is given by
I ( x 2
, y 2
) = L g 1
( x 1
,y 1
[ )]
= L g ( ξ,η) δ( x 1
∞
∞
∫ −^ ξ,^ y 1 −^ η) d ξ d η
[ ]
= g ( ξ,η) L δ( x 1
− ξ, y 1
[ −^ η)]
∞
∞
∫ d ξ d η
= g ( ξ,η) h ( x 2
, y 2
; ξ,η)
∞
∞
∫ d ξ d η
TT Liu, BE280A, UCSD Fall 2013
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 ( x 2
, y 2
; ξ,η) = h x 2
− ξ, y 2
( −^ η)
TT Liu, BE280A, UCSD Fall 2013
X-Ray Imaging
s(x)
d
z
m
s
x
m
t ( x ) =
M
δ( x )
Is this a linear system?
Is it a space invariant system?
TT Liu, BE280A, UCSD Fall 2013
d
z
Assume z=d/
10 cm
5 cm
d
z
Assume z=d/
10 cm
5 cm
d
z
Assume z=d/
10 cm
10 cm
TT Liu, BE280A, UCSD Fall 2013
Convolution
g [ m ] = g [0] δ[ m ] + g [ 1 ] δ[ m − 1 ] + g [2] δ[ m − 2 ]
h [ m ', k ] = L [ δ[ m − k ]] = h [ m $ − k ]
y [ m '] = L [ g [ m ]]
= L (^) [ g [0] δ[ m ] + g [ 1 ] δ[ m − 1 ] + g [2] δ[ m − 2 ]]
= L (^) [ g [0] δ[ m ]] + L (^) [ g [ 1 ] δ[ m − 1 ]] + L (^) [ g [2] δ[ m − 2 ]]
= g [0] L [ δ [ m ]] + g [ 1 ] L [ δ [ m − 1 ]] + g [2] L [ δ [ 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 2013
1D Convolution
I ( x ) = g ( ξ) h ( x ; ξ) d ξ
∞
∫
= g ( ξ) h ( x − ξ)
∞
∫
d ξ
= g ( x ) ∗ h ( x )
Useful fact:
€
g ( x ) ∗ δ( x − Δ) = g ( ξ) δ( x − Δ − ξ)
∞
∫ d ξ
= g ( x − Δ)
TT Liu, BE280A, UCSD Fall 2013
2D Convolution
I ( x 2
, y 2
) = g ( ξ,η) h ( x 2
, y 2
; ξ,η)
∞
∫
∞
∫
d ξ d η
= g ( ξ,η) h ( x 2
− ξ, y 2
− η)
∞
∫
∞
∫
d ξ d η
= 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 2013
Rectangle Function
Π( x ) =
0 x > 1 / 2
1 x ≤ 1 / 2
x
x
y
Also called rect(x)
Π( x , y ) = Π( x )Π( y )
TT Liu, BE280A, UCSD Fall 2013
X-Ray Imaging
s(x)
d
z
x 0
x 0
Mx 0
m
s
x − Mx 0
m
t ( x ) =
M
δ( x − x 0
€
M ( z ) =
d
z
; m ( z ) = −
d − z
z
TT Liu, BE280A, UCSD Fall 2013
X-Ray Imaging
s
x − Mx 0
m
( = s
x
m
M
δ
x − Mx 0
M
= s ( x / m ) * t
x
M
€
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 2013
X-Ray Imaging
d
z
m
s
x
m
' ∗^ t^
x
M
t ( x ) = rect ( x /10)
s ( x ) = rect ( x /10)
€
M ( z ) =
d
z
; m ( z ) = −
d − z
z
TT Liu, BE280A, UCSD Fall 2013
X-Ray Imaging
m
s
x
m
' ∗^ t^
x
M
' =^ rect ( x^ /^10 )^ ∗^ rect ( x^ /^20 )
m = 1 ; M = 2
TT Liu, BE280A, UCSD Fall 2013
Summary
- The response to a linear system can be
characterized by a spatially varying impulse
response and the application of the superposition
integral.
- A shift invariant linear system can be
characterized by its impulse response and the
application of a convolution integral.
