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Computational Vision
CSCI 363, Fall 2012
Lecture 10
Spatial Frequency
1
Ocular Dominance Columns
Input from the LGN to layer 4C is segregated by eye. Inputs
alternate between the left and right eyes.
2
Ocular Dominance columns on
the cortical surface
Labeling the cortex with a radioactive tracer from one eye
reveals that the ocular dominance columns form stripes on the
cortical surface.
3
Ocular Preference
1 => responds only
to the contralateral
eye.
7 => responds only
to the ipsilateral
eye.
4 => responds
equally to both
eyes.
In V1, many cells respond to both eyes, but most prefer one eye over
4
the other.
Orientation Columns
Preferred orientation varies smoothly across the cortex.
Cells in columns perpendicular to the surface have the same
orientation preference.
5
Blobs and interblobs
When the cortex is stained with cytochrome oxidase, patches on
the surface stain more darkly than the rest of the cortex.
The patches are called "blobs".
Cells within the blobs have non-oriented receptive fields. They
respond well to particular wavelengths of light (color).
Blobs overlap the orientation and ocular dominance columns.
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A Hypercolumn
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Cortical Magnification
About 25% of striate cortex processes
the central 2.50 of the visual field.
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Periodic Functions
Periodic functions repeat with a given frequency:
Amplitude
f(x) = Asin(2pwx)
x (deg)
period
Frequency = number of cycles per unit distance (deg or cm)
Frequency = 1/period = w
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Fourier Series
Any periodic function can be written as the sum of sine and cosine
functions, whose frequencies are an integer multiple of the base
frequency of the function.
For a function with a base frequency of w, we can write:

f (x)  a 0   (a k cos 2 pwkx  bk sin 2 pwkx)
k 0

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Example: A square wave
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f(x) = 4/p(sin(f) + 1/3sin(3f) + 1/5sin(5f) + ...
Graphing frequency
components
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Phase
The phase of a sine wave is the amount of lateral shift.
f(x) = Asin(2pwx + f)
f is the phase.
phase
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Combining sine waves with
different phases
Asinq + Bcosq = Csin(q + f)
A function can be written as the
sum of sines and cosines or as
the sum of sines that have phase
shifts.
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Complex representation
It is convenient mathematically to represent sines and cosines in
the form:
Aeiq = A(cosq + isinq)
where: i 
1
Complex numbers are written B + Ci, where B is the real part
and C is the imaginary part.

Note that A, above, may be a complex number (B + Ci)
The sine or cosine function is the real part of the above complex
function.
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The Fourier Transform
We can represent non-periodic functions with a continuous Fourier
spectrum that represents all possible frequencies.
We find the function representing the frequency components of a
function by taking the Fourier transform of the function:

F( )   f (x)e i2 px dx

Given the Fourier transform of a function, we can find the original
function by taking the inverse Fourier transform:


f (x)   F( )e i2 px d

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2D Fourier Transform
2D images are composed of sine waves that vary in amplitude,
phase, and orientation. We can find these frequency components
with a 2D Fourier transform.
 
F(,  )    f (x, y)e i2 p (x  y ) dxdy
 
The 2D inverse Fourier transform:

 
f (x, y)    F(,  )e i2 p (x  y ) dd
 
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Examples of 2D frequency
spectra
Square wave
Checkerboard
Plaid
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Filtering Einstein
Low frequencies only
Higher frequencies added in
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Visual Psychophysics
Psychophysics is an approach used to study human vision.
The idea is as follows:
Present observers with a well-defined visual stimulus.
Examine their ability to perform tasks with respect to that
stimulus (e.g. detect a sinewave grating).
From the data, we can learn much about the capabilities and
requirements of the human visual system.
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Contrast Sensitivity
Luminance
How much contrast is required to distinguish a sinewave grating
from a uniform field of gray?
Contrast = (Lmax - Lmin)/(Lmax + Lmin)
Threshold = amount of contrast needed to detect the grating.
Sensitivity = 1/Threshold
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Contrast Sensitivity Function
The contrast sensitivity function (CSF) is determined by
measuring the contrast sensitivity for many different frequencies.
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Frequency Adaptation
Adaptation of the visual system
occurs after prolonged
exposure to a given stimulus.
If an observer views a single
spatial frequency grating for a
long time (e.g. 1 minute), he
will become less sensitive to
that frequency (adaptation).
Other frequencies are
unaffected.
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Multiple Spatial Frequency
Channels
The human visual system
appears to have multiple
mechanisms (or channels)
tuned to different spatial
frequencies.
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