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PHYS 6720  Lecture V
1
Lecture V
Imaging System Analysis
1. Linear imaging systems
2. Fourier transform and discrete sampling
3. Image contrast and MTF
4. Effect of scattering
5. Other imaging quality parameters
PHYS 6720  Lecture V
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V1
Linear imaging systems
Imaging system requirement
Æ
An imaging system maps an input signal I(
r
) (object) to an
output signal I’(
r
’) (image) in real space
Æ
A signal I can often be acquired at a location
r
of a plane
image sensor and (I,
r
) make up a
pixel
(
pic
ture
se
l
ement)
Æ
A “clear” image for replica imaging
requires a onetoone
relation existing
between a pixel in the object I(
r
) to the
conjugate pixel I’(
r
’) in the image
PHYS 6720  Lecture V
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Linear imaging systems
Imaging system requirement
Imaging
System
Object space (
r
)
image space (
r’
)
PHYS 6720  Lecture V
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V1
Linear imaging systems
Imaging system requirement
The object and images spaces,
r
and
r
’, can share the same
dimensions, such as a pinhole camera
Object space (
r
)
image space (
r’
)
PHYS 6720  Lecture V
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V1
Linear imaging systems
Imaging system requirement
The object and image spaces can have different scales
Object space (
r
)
image space (
r’
)
PHYS 6720  Lecture V
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V1
Linear imaging systems
Imaging system requirement
The object and image spaces share the same real space but still use
different symbols to separate them out
Object space (
r
)
image space (
r’
)
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PHYS 6720  Lecture V
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V1
Linear imaging systems
Dimensionality of
r
and
r
’
Æ
1D spaces: signals are function of one spatial coordinate (e.g.,
infinitely large grating of blackwhite stripes, linear array
detectors).
Æ
2D spaces: signals are functions of two spatial coordinates (e.g., all
conventional films or panel detectors).
Æ
3D spaces: signals are functions of three spatial coordinates
(through reconstruction).
PHYS 6720  Lecture V
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Linear imaging systems
Linear imaging systems
Let I(
r
)=input signal for an object, I’(
r
’)=output signal for an image,
then an imaging system can be described mathematically as a
mapping functional H:
I’(
r
’)=H{I(
r
)}.
If the following relation is true
aI’
1
(
r
’)+bI’
2
(
r
’)=H{aI
1
(
r
)+ bI
2
(
r
)}
,
where a and b are scalar constants, then the imaging system is defined
as
linear
V1
Linear imaging systems
Why do we like linear systems?
Æ
If linear, it can be proved that the mapping functional H must be a
convolution operator:
I’(
r
’) = h(
r’
,
r
)
⊗
I(
r
)
≡
where h is also called a system response function.
Æ
For a linear system with a given h function, any output can be
predicted from the input signal.
('
,)()
hI
d
∫
rr r r
A.C. Kak, M. Slaney, “Principles of computerized tomographic imaging”, (IEEE
Press, 1988) (pdf file is available at:
http://www.slaney.org/pct/
)
PHYS 6720  Lecture V
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Linear imaging systems
Spatial invariant systems
If an imaging system with object and image spaces have the same
scale and is spatial invariant, then
h(
r’
,
r
)=h(
r’

r
)
Imaging
System
PHYS 6720  Lecture V
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V1
Linear imaging systems
Magnification
For a 2D linear imaging system, a magnification M is defined
through a scaling relation:
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 Spring '10
 HU
 MTF, Linear imaging systems

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