Humble attempt to depict line spread function measurement

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Humble attempt to depict line spread function
measurement
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4 mm wide perpendicular scans in x direction
100 horizontal scans collected in y direction
0.8 mm long
What about some Gaussian values to put in these
boxes.
JSA How about these?
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4 mm wide perpendicular scans in x direction
0.8 mm long
100 horizontal scans collected in y direction
JSA Illustrates optical dot gain variation, or
variation in kp, which makes a variation in Yule
-Nielsen n.
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Interrogation

• Is the diagram correct?
• JSA Yes
• At what point in the 4 mm scan does the shadow
  disappear and become a white paper reading?
  
• JSA The shadow approaches the paper white
  asymptotically. It is like asking where the
  "bell curve" goes to zero. This is why we
  describe the width in terms of a standard
  deviation instead of the width of the "bell
  curve" at the base.
• How many measurements in horizontal direction?
• JSA What does "how many" mean? The diagram
  shows 12 "measurement" boxes from 0 through etc.
  We could double the measurements if we used twice
  as many boxes, each half the width. This would
  give us a higher resolution look at the edge.
  Experimentally, we use a camera with "boxes" that
  are image pixels. The higher resolution camera
  gives the better data.
• JSA Or maybe "how many" means how many boxes to
  we average vertically for our horizontal scan.
  For an MTF - line spread measurement, we use as
  many as we can get, limited by the camera
  resolution.

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• For a simple chemist, do you have a picture of
  the line spread function and then a picture of
  the MTF that is calculated from it? Where would
  the magical point of MTF 0.5 usually be located
  on a 4 mm scan?
• JSA I made some slides (below) with data I have
  for a coated paper. Don't recall what paper.
  
• For paper this would normally be 0.2 mm?
• JSA Yes, between 0.15 and 0.35 millimeters
  covers all the paper samples I ever saw.
  
• Can I use the same picture to jump to your
  experiment in which 1200 rows and 1200 columns
  were scanned and then analyzed.
  
• JSA Yes! This stuff really is simple enough
  for us chemists to do!
  
• Is scanned the right term? Or captured?
• JSA I "capture" the image and do the "scan" in
  software. The scan is the process of averaging
  each vertical column of pixels and plotting the
  result vs the horizontal location.
• In the illustration what is the size of each spot
  you are measuring reflectance of?
  
• JSA I don't know exactly. I do know exactly in
  the illustration below. Each pixel "box"
  measures 0.00323 millimeters in the paper image.
  This is called the "pixel size projected onto the
  paper" and is not the same as the physical size
  of the pixel on the detector in the camera.

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• Is there some advantage to MTF mathematically, in
  other words a convenient inflexion point or max
  that provides additional information?
  
• JSA MTF doesn't really give you any more
  practical information than just measuring the
  width of the spread function. MTF is useful
  for
• (1) Theoretical mechanistic analysis. It is
  like a rate constant in chemical kinetics. k 3
  sec-1 means the reaction is about half over in
  1/3 sec. Then, things like Arrhenius theory can
  predict k vs temperature. Similar stuff in
  optical theory for the MTF. Linear Systems
  theory tells how the spread function MTF
  propagates through other imaging systems, such as
  cameras, dots that are spread out, etc.
• (2) The Geek factor. If you slip the term "MTF"
  or "Fourier Transform" into a talk, and point
  quickly to big equations, it gives that Geek
  effect without really being informative. I'm
  happy to hear you are trying to avoid that. It
  sure isn't helpful!
• Below is an example analysis I did on a piece of
  coated paper. Sorry I don't know the
  
• origin of the paper or other things about it.

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A coated sheet with an image of an edge projected
onto it. This is a good way to do the MTF analys
is. Start with a scan of the edge.
Call it the "edge spread function"
1 millimeters
I is the relative intensity of the
reflected light, where I1
is the paper itself.
Note the locations marked s and kp.
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The edge spread function.
LSF slope of I vs x.
Calculate and plot the slope at each point on the
curve. (Take the derivative) The result is calle
d the line spread function.
0.04
LSF
s 0.05 millimeters 2s 0.10 millimeters
2s
0
x in millimeters
A practical measure of the line spread function
is s. This is (approximately)
the half the width of the curve measured half way
down the peak.
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Line Spread Function
0.04
LSF
2s
0
x in millimeters
More rigorously, we treat the line spread
function as a probability density function for la
teral light travel in the paper. The so called
"bell shaped curve". Then s is defined
statistically as the average distance light trave
ls before returning to the surface as reflected
light. The type of "average" is the "RMS" avera
ge, or the square Root of the Mean Squared distan
ce of travel. s is also called the
second statistical moment about the mean.
Anyway, it is a useful indicator of how far ligh
t scatters laterally around the edge.
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Line Spread Function
0.04
LSF
2s
0
0.5
0.5
x in millimeters
6s 0.30 millimeters
Instead of the average distance, you generally
want to know the practical boundary for how far
light CAN scatter. This would be
measured as the base width of the curve. But the
curve approaches zero asymptotically, so what is
a useful way to estimate a base width?
One good way is to use some multiple of s. This
is where "six sigma" comes from. Nice for market
ing people, but it isn't really useful
as a way to characterize practical scattering.
Instead, we can use Fourier Transform theory.
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Apply the Fourier Transform by giving the data to
a computer. Leave the understanding of the math
to the mathematicians.
Line Spread Function
0.04
LSF
2s
0
x in millimeters
0.5
0.5
MTF Fourier Transform of LSF vs x.
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Plot it as frequency, 1/x, instead of distance
, x.
MTF
0.5
0
0
1
2
3
4
frequency, 1/x
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1
MTF
0.5
0
0
1
2
3
4
frequency, 1/millimeters
1/millimeters is often called "cycles" per
millimeter.
All we did is apply some math to the original
data, I vs x. The information
contained in the MTF is identical to the
information contained in the original edge spread
function, I vs x. We do this because it helps
us extract useful parameters. In this case, the
useful parameter we want is the frequency half wa
y down the MTF. This gives us the value
1/mm 3.0. We invert this and call it kp 1/3
0.33 millimeters. Experimentally, it is easie
r to measure a point half way down a curve than
at some place along the asymptote. This is why
kp is a useful measurement.
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The distance kp indicates the practical limit of
light scattering distance.
s
1
kp0.33 mm
I
0.5
0
x in millimeters
kp is a practical estimate of the base width of
the line spread function.
0.04
LSF
s 0.05 millimeters 2s 0.10 millimeters
2s
0
kp0.33 mm
0.5
0.5
x in millimeters
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The distance kp indicates the practical limit of
light scattering distance.
1
I
0.5
0
x in millimeters
Placing the edge (halftone dot) directly on the
paper chops off half of our edge data and the res
t is the Yule-Nielsen effect, or the dot "shadow"
.