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Part 1: Waves & Fourier Analysis 1.The cycles of waves can be described by four features, or “parameters”. 2.These are Frequency, Amplitude, Phase, and Orientation. 3.A helpful acronym is F.A.P.O.. 4.Let’s see examples of how each parameter…
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Outline Of Today’s Discussion1. Review of Wave Properties, and Fourier Analysis
2. The Contrast Sensitivity Function
3. Metamers
4. Selective Adaptation And The Size Aftereffect
Part 1
Review of Wave Properties
And
Fourier Analysis
Part 1: Waves & Fourier Analysis
1. The cycles of waves can be described by four features, or “parameters”.
2. These are Frequency, Amplitude, Phase, and Orientation.
3. A helpful acronym is F.A.P.O..
4. Let’s see examples of how each parameter…
These Differ In Frequency
Low Spatial FrequencyFat Bars:
Few Cycles Per DegreeC.P.D.
High Spatial FrequencyThin Bars:
Many Cycles Per DegreeC.P.D.
These Differ In Amplitude (or Contrast)
Low Amplitude(Low Contrast)
High Amplitude(High Contrast)
Zero Phase(“Start With Black”)
180 Degree Phase Shift(“Start With White”)
These Differ In Phase (Relative Position)
These Differ In Orientation
Vertical Orientation
Horizontal Orientation
Part 1: Waves & Fourier Analysis
This is a Square Wave Grating:
The Luminance ChangesAbruptly.
Sine Wave GratingChanges Slowly
Square Wave GratingChanges Abruptly
Part 1: Waves & Fourier Analysis
Sine Wave GratingChanges Slowly
Square Wave GratingChanges Abruptly
Part 1: Waves & Fourier Analysis
This is SinusoidallyModulated in Luminance.
0
100
Sine Wave in Luminance
Space
Left Right
Lum
inan
ce
510
700
Sine Wave in Wave Lengths
Space
Left Right
Wav
e Le
ngth
(nm
)
Green
Yellow
Red
A Red-Green Grating:Sinusoidally Modulated Wavelengths
Now, Back To Luminance Profiles…
Space Domain First,
Then The Frequency Domain.
Part 1: Waves & Fourier Analysis
This is the profile in the “Space Domain”Space is on the X-Axis.
Bottom: profile in the “Frequency Domain”Frequency is on the X-Axis
Joseph (Jean Baptiste) Fourier
According to Fourier, we should be able toconstruct a square wave stimulus
(or any other stimulus), by combining sine waves of the correct
F.A.P.O..
Part 1: Waves & Fourier Analysis
TheseAddedTogetherMake This
TheseAddedTogetherMake This
Eventually,You’ll Make This
A square wave can be built from componentsine waves, if the sine waves all have the same phase.
What happens if you introduce a phase shift(say 180 degrees or a half cycle)?
Part 1: Waves & Fourier Analysis
TheseAddedTogetherMake This
TheseAddedTogetherMake This
Eventually,You’ll Make This
Let’s take a very close look atthe square wave and triangle wave,
side-by-side….
Part 1: Waves & Fourier Analysis
SquareWave
TriangleWave
Note the slightdifferences instarting phase(in red circles)
The phasedifferenceshould be180 deg,but the schematicshows a90 deg difference(quartercycle ratherthan a half cycle).Sorry about That.
By shifting the components 180 degrees,a different image is produced, namely,a triangle wave (not a square wave).
So, Phase Matters!
Part 1: Waves & Fourier Analysis
Part 1: Waves & Fourier Analysis
1. Different spatial frequencies specify how light is distributed at various spatial scales.
2. Low spatial frequencies specify the most global spatial scales (i.e., ground versus sky). “Low pass” images appear blury, and lack fine detail.
3. High spatial frequencies specify the finest spatial scales. “High pass” images appear as outlines, showing the boarders between objects.
4. Intermediate spatial frequencies specify information at scales between the two extremes.
Part 1: Waves & Fourier Analysis
Some More Examples
Potential Pop Quiz Question: In your own wordsExplain what is happening in the diagram below.
It is a FACT that any image can bedecomposed into its “Fourier Components”
But is it true that our visual systemsconduct a Fourier Analysis on the retinal image?
Part 1: Waves & Fourier Analysis
Here’s The RF Of A Visual Neuron
+ +
+ ++
+ ++ ++
- --
-
-
-
-
---
--
- -
+ +
+ ++
+ ++ ++
- --
-
-
-
-
---
--
- -
+ +
+ ++
+++ +
+
--
-
-
-
-
-
---
--
--
+ ++ +
++++ ++
- --
-
-
-
-
---
--
- -
RFs Vary In Size,And Size Corresponds To
Spatial Frequency.
Visual neurons respond best when the size (SF) of the stimulus matches
the size (SF) of the receptive field.
Stimulus “b” is the best match here.
V1 Is Organized By Spatial Frequency
Part 1: Waves & Fourier Analysis
1. In principle, the visual system could respond in two ways to the retinal image.
2. One possibility is that the visual system responds to the Fourier components (i.e., a spatial-frequency analysis).
3. Another possibility is that the visual system responds to the point-by-point distribution of light.
4. Either way is a an acceptable PHYSICAL description of the stimulus. Let’s see an example of a point-by-point stimulus description….
Point-By-Point Luminance Values
Sample Test QuestionWrite the point-by-point
luminance profile for these stimuli.
Sample Test QuestionPotential Pop Quiz Question:
Write the point-by-pointluminance profile for these stimuli.
Sample Test QuestionPotential Pop Quiz Question:
Write the point-by-pointluminance profile for these stimuli.
Sample Test QuestionPotential Pop Quiz Question:
Write the point-by-pointluminance profile for these stimuli.
This Photo of Einstein Contained 65,500 Luminance Values, Point-By-Point.
The Same Photo Can Be Readily Identified With Many Fewer Fourier (sine wave) Components. A Fourier Analysis Would Be Neurally Economical.
Part 2
The Contrast Sensitivity Function
C.S.F.
Part 2: The CSF
The contrast sensitivity functioncan be thought of
as a graph that indicateshow easily different SFs are seen.
The Human Contrast Sensitivity Function
Human CSF: Day, Dusk, and Night
The CSF For Different Species
The Human CSF: Infant (3 to 6 months) versus Adult
Potential Pop Quiz Question:Draw Two CSFs, one for the
“1 month” condition below, and One for the “8 months” condition below.
Label your axes. (No need for exactQuantities, I’m just looking for the pattern.)
The Effect of Agingon the Adult Human CSF
Part 3
Metamers
Part 3: Metamers
Metamers are physically differentstimuli that are
perceptually indistinguishable.
Part 3: Metamers
Metamers reveal a failurein discrimination!
Part 3: Metamers
Because the human CSF differsfrom the cat CSF,
stimuli that are “metameric”for cats are not metameric
for humans (and vice versa).
Part 3: Metamers
As an example, the following photos look different to you,
but would appear indistinguishable to a cat.
Part 3: Metamers
Demo Here
Part 3: MetamersNow, let’s change the
spatial frequency content.
Specifically, let’s increasethe SF of both stimuli until
the difference between themfalls outside our “window of visibility”,
making them metameric.
These Differ In Frequency
Low Spatial FrequencyFat Bars:
Few Cycles Per DegreeC.P.D.
High Spatial FrequencyThin Bars:
Many Cycles Per DegreeC.P.D.
Essentially, we just “moved” the Cat stimulifrom left to right in the frequency domain,
making differences invisible at the highest frequencies.
Part 3: MetamersThe artist Charles (Chuck) Close
takes advantage of the humanCSF in his art.
His art looks one way at one scale (SF),and very different at a another scale (SF).
Demo Here
Part 4
Selective Adaptation
And
The Size Aftereffect
Selective Adaptation:
After adapting to a single SF,contrast sensitivity is reduced
at or near that SF,but NOT elsewhere.
This creates a “notch” in the CSF.
Selective Adaptation and the CSF
The Size Aftereffect:
The size aftereffect is conceptuallysimilar to the tilt aftereffect.
There is an illusion of size (rather than orientation)
after adaptation to a single spatial frequency.
The Size Aftereffect: Pre-Adaptation
The Size Aftereffect:
Now, have the subject adaptto a (low) spatial frequency
at or near “A”.
The Size Aftereffect: Post-Adaptation
The Size Aftereffect: Before & After Adaptation
Notethehigherfrequency
The Size Aftereffect:
Like the tilt aftereffect (an illusion of orientation),the size aftereffect arises from and adaptation-induced bias
in the POPULATION’S response.
So, you can “fatigue” an orientation column,or a spatial frequency column!