Optical sensing techniques and signal processing

1. 1
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Chap 6 Frequency analysis of optical
imaging systems

2. 2
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Outline
• 6.1 Generalized treatment of imaging systems
• 6.2 Frequency response for diffraction-limited coherent
image
• 6.3 Frequency response for diffraction-limited incoherent
image

3. 3
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6.1.1 A Generalized Model
• To specify the properties of the lens system, we adopt
the point of view that all image

4. 4
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6.1.2 Effects of diffraction on the image
• Diffraction effect plays a role only during passage of
light from the object to the entrance pupil, or
alternatively and equivalently, form the exit pupil to
the image.
• There are two points of view that regard image
resolution as limited by
– (1) The finite entrance pupil
– (2) The finite exit pupil

5. 5
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• According to the Abbe theory, only a certain portion
of the diffracted components generated by a
complicated object are intercepted by this finite pupil.
The components not intercepted are precisely those
generated by the high-frequency component of the
object amplitude transmittance.

6. 6
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• This notational simplification yields a convolution equation
(called amplitude convolution integral)
ggi UhddUvuhvuU ⊗=−−= ∫ ∫
∞
∞−
ηξηξηξ ~~
)~,
~
()~,
~
(),(

7. 7
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
where the ideal image (i.e., the geometrical-optics prediction of
the image) for a perfect imaging system is expressed as
)
~
,
~
(
1
)~,
~
(
MM
U
M
U og
ηξ
ηξ =
and the impulse response (called amplitude impulse response)
is given by
dxdyvyux
z
jyxP
z
A
vuh
ii
)](
2
exp[),(),( +−= ∫ ∫
∞
∞− λ
π
λ
where the pupil function P is unity inside and outside the
projected aperture.

8. 8
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• Thus, for a diffraction-limited system, we can regard
the image as being a convolution of the image
predicted by geometrical optics with an impulse
response that is the Fraunhofer diffraction pattern of
the exit pupil (i.e., the Fourier transform of the exit
pupil)

9. 9
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6.1.3 Polychromatic illumination: the coherent
and incoherent cases
• When the object illumination is coherent, the various impulse
responses in the image plane vary in unison, and they must be
added on a complex amplitude basis. Therefore, a coherent
imaging system is linear in complex amplitude.
• It follows that an incoherent imaging system is linear in
intensity and the impulse response of such a system (called
intensity impulse response) is the squared magnitude of the
amplitude impulse response.

10. 10
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• An incoherent imaging system is linear in intensity and the
impulse response of such a system (called intensity impulse
response) is the squared magnitude of the amplitude impulse
response.
222
2
~~
)~,
~
()~,
~
(),( gggi UhIhddIvuhvuI ⊗=⊗=−−= ∫ ∫
∞
∞−
ηξηξηξκ
2
h
gI
Thus, for incoherent illumination, the image intensity is
found as a convolution of the intensity impulse response
with the ideal image intensity.

11. 11
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6.2 Frequency response for diffraction-limited
• As emphasized previously, a coherent imaging is
linear in complex amplitude. This implies, of course,
that such a system provides a highly nonlinear
intensity mapping. If frequency analysis is to be
applied in its usual form, it must be applied to the
linear amplitude.

12. 12
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6.2.1 The amplitude transfer function
• In a coherent system, a space-invariant form of the
amplitude mapping is given from the manipulation of
convolution. One can anticipate, then the transfer-
function concepts can be applied to the system,
provided it is the convolution is done on an amplitude
basis.

13. 13
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
From the convolution equation, it is found that
),(),(),( YXgYXYXi ffGffHffG =
where the frequency spectra of the input and output are
respectively expressed as
)](2exp[).()}.({),( dudvvfufjvuUvuUFffG YXggYXg +−== ∫ ∫
∞
∞−
π
dudvvfufjvuUvuUFffG YXiiYXi )](2exp[).()},({),( +−== ∫ ∫
∞
∞−
π
and
The Fourier transform of a object.
The Fourier transform of an image.
and the amplitude transfer function (ATF )
dudvvfufjvuhvuhFffH YXYX )](2exp[).()},({),( +−== ∫ ∫
∞
∞−
π
is the Fourier transform of space-invariant amplitude
impulse response.

14. 14
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• Since the impulse response is a scaled Fourier transform of the
pupil function, we have
).,()(
})](
2
exp[),({),(
YiXii
ii
YX
fzfzPzA
dxdyvyux
z
jyxP
z
A
FffH
λλλ
λ
π
λ
−−=
+−= ∫ ∫
∞
∞−
For notational convenience, we set the constant izAλ
and ignore the negative signs in the arguments of
equal to unity
P
applications of interest to us here have pupil functions that are
symmetrical in x and y). Thus
(almost all
),(),( YiXiYX fzfzPffH λλ=

15. 15
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6.2.2 Examples of amplitude transfer
),(),( YiXiYX fzfzPffH λλ=

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6.3.1 The optical transfer function (OTF)
• Recall the imaging systems that use incoherent illumination
have been seen to obey the intensity convolution integral
ηξηξηξκ ~~
)~,
~
()~,
~
(),(
2
ddIvuhvuI gi ∫ ∫
∞
∞−
−−=
• Such systems should therefore be frequency-analysis as linear
mappings of intensity distributions.

17. 17
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Application of the convolution theorem to the above equation
then yields the frequency-domain relation
),(),(),( YXgYXYXi ffffff GHG =
where the normalized frequency spectra of gI and are respectivelyiI
defined by
∫ ∫
∫ ∫
∞
∞−
∞
∞−
+−
=
dudvvuI
dudvvfufjvuI
ff
g
YXg
YXg
),(
)](2exp[),(
),(
π
G
∫ ∫
∫ ∫
∞
∞−
∞
∞−
+−
=
dudvvuI
dudvvfufjvuI
ff
i
YXi
YXi
),(
)](2exp[),(
),(
π
G
and

18. 18
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
And the normalized transfer function of the system is similarly
defined by
∫ ∫
∫ ∫
∞
∞−
∞
∞−
+−
=
dudvvuh
dudvvfufjvuh
ff
YX
YX 2
2
),(
)](2exp[),(
),(
π
H
By international agreement, the function is known as the optical
transfer function (OTF) of the system and it is also the normalized
autocorrelation of the amplitude transfer function.
Its modulus H is known as the modulation transfer function (MTF).
H
)(
)(
PTFi
e
MTFOTF =
Where PTF is phase transfer function.

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6.3.2 General properties of the OTF
• The most important of these properties are as follows:
)0,0(),(.3
),(),(.2
1)0,0(.1
*
HffH
ffHffH
H
YX
YXYX
≤
=−−
=
• Property 1 follow directly by substitution of )0,0( == YX ff
• The proof of Property 2 is left as an exercise for the reader, it
being no more than a statement that the Fourier transform of a real
function has Hermitian symmetry.
• To proof Property 3 we use Schwarz’s inequality.