Joseph Fourier (1768-1830)

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Joseph Fourier (1768-1830)

• Mathematical physicist in Napoleons army
• Studied mathematical functions
• Any function (curve) can be decomposed into sine
  waves.
• Any curve can be created by summing sine waves.
• Images are 2-D functions

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Fundamental building blocks

• Any image can be made from fundamental basic
  patterns
• The fundamental patternsthe building blocks of
  any image are
• sine wave gratings.

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Sine wave grating parameters

• Stripe size
• Orientation (horizontal, vertical, etc.)
• Contrast
• Maximum (black on white) 1.0
• Minimum (gray on gray) 0
• Phase shift

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Spatial frequency

• Stripe size
• Repeating pattern of black/white stripes
• One black/white pair one cycle
• Specify the number of cycles (black/white pairs)
  per degree of visual angle.
• Spatial frequency
• Low spatial frequency few big stripes
• High spatial frequency many small stripes

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Orientation
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Contrast
1.00
0.40
0.20
0.00
Michelson formula
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Build an image from 4 gratings




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Fourier analysis of images

• 50,000 sine-wave gratings
• Each with appropriate
• Spatial frequency
• Contrast
• Orientation
• Phase shift

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Test optical performance

• All images are made up of sine wave gratings.
• Optical engineers test how well an optical system
  works by
• testing how it images sine-wave gratings.
• Concept of contrast transfer

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Contrast transfer

• Optical systems transfer light from the object to
  the image.
• Quality is always lost in the imaging process.
• If the object is a sine-wave grating, the
  degraded image will be
• a sine-wave grating with the same spatial
  frequency and orientation
• But, some contrast will be lost.
• Better optical systems transfer contrast better
  (lose less contrast)

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How well do the optics transfer contrast?
1.0
1.0
1.0
1.0
Test pattern
Image
1.0
0.8
0.6
0.2
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Contrast transfer function of the eye
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Spatial frequencies in a image
Low spatial frequencies
High spatial frequencies
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Requirements for good vision

• Good optics good neural processing
• MTF characterizes optics only
• When we test vision we are testing the whole
  visual system (optics neural processing).
• Contrast sensitivity testing
• Like the MTF, uses sine-wave gratings
• Differs in that vision (not only optics) is
  tested.

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Contrast sensitivity
sensitivity
threshold
Spatial frequency
0.01
0.38
0.97
1
Sensitivity
threshold
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Characteristics of normal CS

• Peak at 4 c/d
• Low frequency drop-off
• Drops to zero at about 40-60 c/d
• Cut-off frequency

Non-seeing
CS
seeing
Spatial frequency
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Snellen size equivalent
E
E
CS
E
E
E
E
E
Spatial frequency
VA
1 cycle 2 arc minutes
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Clinical contrast sensitivity tests
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Improving visibility

• Expand the curve
• Correct refractive errors
• Change non-visible objects to bring them inside
  the curve
• Magnify
• Increase contrast

Non-seeing
CS
seeing
Spatial frequency
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Improving visibility by increasing contrast
Photo courtesy of Dr. Ralph Latimer
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Visual system Fourier analyses images
Starry Night by van Gogh
Image without low and high spatial frequencies
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Is VA obsolete? It depends.

• It tests only one point on the contrast
  sensitivity functionthe cut-off frequency.
• Uncorrected refractive errors primarily affect
  high spatial frequencies
• So VA is good for clinical refraction.
• Some diseases may affect low spatial frequencies
  more than high.
• Contrast sensitivity better than VA for this

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A better VA chartETDRS chart

• Early Treatment of Diabetic Retinopathy Study
• logMAR size progressionequal difficulty steps
  between all lines
• Sloan letter setall letters equally difficult to
  read
• Same number of letters (5) in every line
• Same relative spacing between letters
• Every letter counts as 1/5 of a line
• Score VA to the letter (not just the line)

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Scoring logMAR VA

• 20/20 logMAR 0
• Lower score better
• Higher score worse
• Each line 0.1 change
• Each letter 0.02 change
• Score VA to the letter

Negative values
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Examples of logMAR scoring

• Convert 20/402
• 20/40 logMAR 0.3. Each additional letter
  subtracts 0.02 from score.
• Therefore, 20/402 logMAR 0.26
• Convert 20/40-2
• 20/40 logMAR 0.3. Each letter missed adds 0.02
  to score.
• Therefore, 20/40-2 logMAR 0.34

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Snellen lt-gt logMAR conversion
X
x Snellen denominator