Transcript LABRI – University of Bordeaux

Spatial Frequency Analysis Optical Transfer Function
(OTF)
Ofer Hadar
Communication Systems Engineering Dept., BGU
URL: http://www.cse.bgu.ac.il/~hadar
Copyright @2004, O. Hadar
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System classification
Continuous time and discrete time systems
Linear systems: These apply superposition in delay,
addition and multiplication
Time-invariant systems F1: F1[ x(t  t0 )]  y (t  t0 )
Linear, Time-Invariant (LTI) system combines the previous
Non-linear systems, examples
 diodes, transistors, modulators, mixers
Note that a system might appear to be nonlinear in a domain
but linear in some other domain. Thus nonlinear systems can
often be linearized
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Signal transmission and
filtering
Linear systems

y(t )  F[ x (t )]
apply superposition, hence
 delay:
 addition:
y(t   )  F[ x (t   )]
( a  b) y(t )  F[ax (t )  bx (t )]
 multiplication:
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ay(t )  F[ax (t )]
Linear systems are characterized by
impulse response or transfer function
System F[ ] response to
y(t )  F[ (t )]  h(t )
impulse:
When the impulse response
h(t) is known, the response y(t )  h(t )  x(t )
y(t) to some other excitation
x(t) is obtained via
convolution:
at   bt  

 at  b  t dt


Here H(f) is the impulse
H  f    ht  exp  j 2ft dt
response in frequency

domain, that is the transfer
function:
Y( f )  H( f )X ( f )
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System response in frequency
and time domain
Note that system response is easy to calculate in frequency
domain. In time domain:
y(t )  h(t )  x(t )
but in frequency domain convolution replaced by multiplication
Y ( f )  H( f ) X ( f )
Note that the transfer function H(f) has both magnitude and
phase components. For real h(t) transfer function magnitude
has even and phase has odd-symmetry, (the conjugate
symmetry):
H ( )  H * ( )
The table follows* from the definition:
H ( )  H *( )

H ( )   h(t )exp[ j t ]dt
 ( )   ( )
Transfer function and impulse response are easily determined
for electrical circuits by Laplace transforms or Fourier techniques
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*prove left to tutorials
Linear Systems Approach to Imaging
x
Entrance
Window
Isoplanatic
x’
Any Optical System
g ( x' , y ' )   h( x, x' ; y, y ' ) f ( x, y )dxdy
Exit
Window
g ( x' , y ' )   h(mx  x' , my  y ' ) f ( x, y )dxdy
G ( f x / m, f y / m)  H ( f x , f y ) F ( f x , f y )
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Convolution
PSF ( x, y )  O( x, y)  I ( x, y )
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Terminology
h is called the point
spread function (PSF)
H is called the optical
transfer function (OTF)


Magnitude is called
Modulation Transfer
Function (MTF)
Phase is Phase Transfer
Function (PTF)
fx and fy are spatial
frequencies
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g ( x' , y ' )   h(mx  x' , my  y ' ) 
Uobject
f ( x, y )dxdy
Uimage
G ( f x / m, f y / m)  H ( f x , f y ) F ( f x , f y )
Uobject
Uimage
The Fourier Domain
Gaussian
Fourier Modulus
(also Gaussian)
These Fourier modulus of a Gaussian produces another Gaussian. A large object
comprised of low spatial frequencies produces a compact Fourier modulus and a
smaller object with higher spatial frequencies produces a larger Fourier modulus.
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system
f(x,y)=(x)
g(x,y) = LSF
Fourier T.
Fourier T.
S(f)
F
system
1
f
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f
Point spread function (PSF)
Describes the unsharpness that results when a point
in the object is not reproduced as a true point in the
image. The unsharpness is a blurring effect (i.e., a
spreading out of the point image to form a
measurable circle). PSF is the impulse response of
the imaging system.
object
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image
system
f(x,y)=x,y)
g(x,y) = PSF
If the system is space-invariant:
PSF0,0(x,y) = PSFx0,y0(x-x0, y-y0)
If the system is also linear:
g( x, y)   f ( x, y)PSF(  x,   y)dd
G(u,v)=F(u,v) F (PSF)= F(u,v) OTF(u,v)
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Spatial Frequency
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Illustration of Spatial Frequency
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Angular Frequency
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Spatial Frequency Analysis
When we look at a visual scene, we
specify it in terms of spatial locations of
light, dark, contour, and color.
Visual scenes, however, can be broken
down into much smaller components.
According to spatial frequency analysis, or
Fourier analysis, a visual scene can be
broken down into a series of sine waves.
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Sine Waves
A sine wave is a continuous waveform that
undulates in a smooth regular fashion.
In vision, a sine wave refers to a pattern in
which luminance undulates in a smooth
regular fashion.
The sine wave contains a number of
important components.
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Sine Waves (cont’d)
1) Spatial frequency refers to the size of
the image of an object.
Spatial frequency tells us the number of
times a cycle of a sine wave (one light
stripe and one dark stripe) repeat in 1
degree of visual space.
Thus, it is measured in cycles per
degree (c/deg).
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Sine Waves (cont’d)
The spatial frequency is inversely
related to the wavelength of sine wave.
Thus, a low spatial frequency sine wave
has a large wavelength and thus, has
large stripes .
Conversely, a high spatial frequency
sine wave has a small wavelength and
thus, thin stripes.
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Sine Waves (cont’d)
2) Contrast: Refers to the difference in
light intensity between an object and its
surroundings.
In terms of sine waves, it refers to the
difference in light intensity between a
light stripe and a dark stripe.
3) Phase: Refers to the difference in
timing between two or more waves.
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Sine Waves (cont’d)
In other words, it refers to the relative
position of the crests and troughs of
two or more waves.
4) Orientation: Refers to the angle at
which an object is viewed.
Thus, any visual scene can be broken
down into a series of sine waves of
various spatial frequencies, contrasts,
orientations, and phases. This applies
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square
waves.
Spatial Frequency Spectra
It’s easiest to think about Fourier’s
analysis in terms of waves. However,
these sine waves can also be specified
as equations or as spectra.
The location of the spike depends on
the frequency, and the height depends
on the contrast.
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Spatial Frequency Spectra
high
high
contrast
contrast
low
low
low
high
1/
Spatial Frequency
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low
high
1/
Spatial Frequency
MTF
Modulation Transfer
Function
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The Modulation Transfer Function
Any visual scene can be broken down into
component sine waves.
Why is this important?
We can predict how good an image an optical
system can form of any visual scene by
seeing how well it “perceives”…
...Sine waves.
This is the purpose of the modulation transfer
function (MTF).
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Modulation Transfer Function
(cont’d)
An optical system (e.g., lens) is presented
with a series of sine wave gratings of
different spatial frequencies of a specific
contrast level.
The performance of the system is
evaluated by comparing the contrast of the
sine wave gratings to the contrast of the
image that is produced.
The MTF can then be plotted.
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Modulation Transfer Function
(cont’d)
Low SFs are
reproduced well. High
SFs are not. Note the
drop-off.
1.0
Relative
Contrast
0.1
low
high
Spatial Frequency
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Modulation Transfer Function
(cont’d)
1.0
Here is a lens of slightly
poorer quality. The
drop-off is at a lower
SF.
Relative
Contrast
0.1
low
high
Spatial Frequency
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Some Resolution Charts (1)
Sinusoidal Chart
Edge
Bar Charts
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Point and
Lines
Contrast Resolution:
Describes ability to distinguish small value differences.
b
2
b
1
How much is the contrast?
b 2  b1
x100
b 2  b1
b 2  b1
x100
b1
b 2  b1
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Some Resolution Charts (2)
Air Force
ISO
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Bar Charts
1
0.9
0.8
Radial Target and Image
0.7
0.6
0.5
Colorbar for all
Image
Object
20
40
0.4
0.3
0.2
0.1
60
20
80
40
20
100
60
40
120
80
60
140
100
80
160
120
100
180
20
40
60
80
Point-Spread
Function of
System
100
140
120
0
120
140
160
180
160
140
180
20
40
60
80
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100
160
120
140
160
180
180
20
40
60
80
100
120
140
160
180
low
medium
object:
100%
contrast
contrast
image
1
0
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spatial frequency
high
• The modulation transfer function (MTF) indicates the ability of an
optical system to reproduce (transfer) various levels of detail (spatial
frequencies) from the object to the image.
• Its units are the ratio of image contrast over the object contrast as a
function of spatial frequency.
• It is the optical contribution to the contrast sensitivity function (CSF).
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MTF: Cutoff Frequency
cut-off frequency
1 mm
2 mm
4 mm
6 mm
8 mm
modulation transfer
1
0.5
f cutoff
a

57.3  
Rule of thumb: cutoff
frequency increases by
~30 c/d for each mm
increase in pupil size
0
0
50 100 150 200 250
spatial frequency (c/deg)
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300
Effect of Defocus on the MTF
450 nm
650 nm
Charman and Jennings, 1976
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Sharpness
A property of a lens, film, sensor, or system defined by
boundaries between zones
Bar pattern: Sharpness
defined by 10-90%
risetime.
Patterns of increasing spatial frequency (Log scale)
Sine pattern: Sharpness
defined by contrast at a
given spatial frequency.
The top half of each pattern is sharp; the bottom is less sharp.
How is sharpness measured?
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Sharpness and spatial..frequency
response
Sine and bar patterns are shown with and
without rolloff for a high quality 35mm
lens
(on a 0.5 mm virtual target)
Amplitude response of bar pattern
Rise distance (10-90%) difficult to
calculate for compound systems.
The relative contrast of a sine pattern
(pure tone) is called
Spatial Frequency Response (SFR) or
Modulation Transfer Function (MTF);
Multiplicative for compound systems.
Perceived image sharpness strongly correlates
with MTF50, the spatial frequency where contrast is half
its low frequency value. MTF50 is a close approximation
to bandwidth W in Shannon capacity calculations.
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MTF50
Spatial frequency LP/mm
PTF
Phase Transfer
Function
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low
medium
object
phase shift
image
180
0
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spatial frequency
high
Imatest Sharpness calculation
derived from ISO-12233 standard
Results are derived from a slantededge image in a test chart.
Algorithm: Find average edge
location. Put each line into one of four
“bins” based on avg. edge. Find
mean 4x oversampled edge, then
take Fourier transform of the spatial
derivative.
Upper (black) curve is the average edge.
Lower (black) curve is the Spatial
Frequency Response (SFR or MTF).
These results are strongly affected by
sharpening, The dashed red curves are the
edge and MTF response with standardized
sharpening that corrects for oversharpening.
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Liming resolution of optical
system
Contrast
Ratio
(% )
Bad
MTF
Increased Image
Content
Good
MTF
limiting
resolution
large
features
Spatial Frequency
small
features
All hi-res, hi-contrast images are not created
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Contrast Sensitivity Function
The visual system has its own MTF, it’s known as the
contrast sensitivity function (CSF).
Psychophysical techniques are used to measure contrast
threshold at several different spatial frequencies.
 i.e., how much contrast you need to see stripes of a
certain spatial frequency.
The inverse of each threshold is then calculated
(contrast sensitivity) and plotted to form the CSF.
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CSF (cont’d)
Thus, if you have a low contrast threshold,
you have a high contrast sensitivity. It takes
little contrast for you to see the stripes.
If you have a high contrast threshold, you
have a low contrast sensitivity. It takes a lot
of contrast for you to see the stripes.
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CSF (cont’d)
high


Contrast
Sensitivity



low
0.1
1.0
10
Spatial Frequency
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100
Spatial Response
• Fourier transform over multidimensional space
- Continuous space FT (CSFT)
- Discrete space FT (DSFT)
- Sampled space FT (SSFT)
• Frequency domain characterization of video signals
- Spatial frequency
- Temporal frequency
- Temporal frequency caused by motion
• Frequency response of the HVS
- Spatial frequency response
- Temporal frequency response and flicker
- Spatio-temporal response
- Smooth
eye
movement
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CSF (cont’d)
The CSF possesses a characteristic
inverted U shape
The CSF is in essence, a window of
visibility.
 That is, we can detect all
combinations of contrasts and
spatial frequencies under the
curve, but not those above the
curve.
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CSF (cont’d)
The CSF is similar to the MTF in that
there is high frequency attenuation.
 i.e., we don’t see small objects
particularly well.
However, unlike the MTF, the CSF shows
low frequency attenuation.
 We don’t see large objects
particularly well.
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Perceived Image Content
MTF 2
Contrast
Ratio
(% )
MTF 1
MTFA
CTF
Spatial Frequency
Perceived 1080
Reso luti on
(1.0’)
MTF Area (MTFA) Method
first proposed in 1965 by
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Charman and Olin
MTF can greatly effect
perceived resolution, as
well as perceived content