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Computational Vision
CSCI 363, Fall 2012
Lecture 14
Spatial Frequency in Striate Cortex
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Another viewpoint: V1 cells are
spatial frequency filters
•DeValois (and others)
proposed that V1 cells
are tuned to spatial
frequency.
•V1 simple cells can be
modeled as Gabor
filters.
sin(10x)
G(x)
•A Gabor filter is the
product of a sinewave
sin(10x)G(x)
and a gaussian.
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Fourier Transform of a Gabor
Filter
The Fourier Transform of a Gabor filter is a localized set of
spatial frequencies.
Gabor filters are band-pass filters. They are tuned to spatial
frequency.
If Striate Cortex cells are like Gabor filters, then they are also
acting as band-pass filters.
Fourier
Transform
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2D
Gabor filter
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Simple Cell Receptive Field vs.
Gabor Function
Solid line: Simple Cell Receptive
Field.
Dashed line: Best fitting Gabor
function.
From: DeValois and DeValois, "Spatial Vision", 1988.
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Contrast Sensitivity of V1 Cells
CSF for individual V1 cells
Distribution of tuning bandwidth
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Spatial Profile vs. CSF
The spatial profile of
the simple cell
receptive field is
predicted by taking
the inverse Fourier
transform of the
contrast sensitivity
function for that cell.
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Spatial Frequency Columns
As with orientation and ocular dominance, spatial frequency shows
columnar organization in the cortex.
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Simple vs. Complex Cells
The response of simple cells to drifting gratings shows a big
oscillation over time.
The complex cell response does not oscillate much.
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Classifying simple and complex
cells
•Simple cells have a larger AC (Alternating current) response.
•Complex cells have a larger DC (Direct current) response.
•The ratio of AC/DC allows classification.
AC/DC >1 => simple cells. AC/DC < 1 => complex cells
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Temporal Frequency
•Temporal frequency is the frequency of
oscillation of light intensity over time.
•For a drifting sinewave grating, the
luminance at a single point in space
oscillates over time.
•http://web.mit.edu/~jgolomb/www/dri
fting.gif (Prof. Julie Golomb, MIT)
•Temporal frequency is measured in cycles/sec = Hertz (Hz)
•Temp. Freq. (cycles/sec) = Speed (deg/sec) x Spatial Freq (cycles/deg)
•Faster drift => Higher temporal frequency
•Higher spatial frequency => Higher temporal frequency
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Speed vs. Temporal Frequency
•To test whether V1 cells are tuned to speed or temporal frequency,
Tolhurst and Movshon (and others) examined speed tuning for
sinewave gratings with different spatial frequencies.
•Example: Test spatial frequencies of 1.0, 2.0 and 3.0 cycles/deg.
•If neurons are tuned for speed, the tuning curves will peak at the
same speed , independent of spatial frequency. (For example, if the
neuron prefers 5 deg/sec, all the graphs will have a peak there).
•If neurons are tuned for temporal frequency (e.g. 10 cycles/sec),
then the peaks for speed tuning will depend on the spatial
frequency (The peaks will be at 10, 5 and 3.33 deg/sec).
•Cells in striate cortex appear to be tuned to temporal frequency.
(Cells at higher levels may be tuned to speed).
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V1 cells and Temporal
Frequency
Response of
neuron tuned for
speed.
MT cells match
this pattern.
Response of neuron
tuned for temporal
frequency.
V1 cells match this
pattern
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Simoncelli and Heeger, Nature Neuroscience, 2001
Linear Systems
•Linear functions:
F(x1 + x2) = F(x1) + F(x2)
F(ax) = aF(x)
•Linear systems are nice to work with because you can predict (or
compute) the responses of the system relatively easily.
•For example, if you double the input, the output doubles.
•Fourier Transforms are linear operations. (The Fourier transform
of the sum of two images is the sum of the Fourier transforms of
each image).
•Gabor filters are linear filters.
•Neurons are not linear.
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Threshold and Saturation
Threshold non-linearity: Neurons do not respond until the input
reaches a minimum level (threshold).
Response
Saturation non-linearity: Neurons have a maximum firing rate.
The response saturates after they reach this maximum.
Threshold
Saturation
Linear response region
Input strength
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Phase and Half-wave
Rectification
Phase non-linearity: Complex cells are insensitive to the phase
(position) of a grating within the receptive field.
Complex cells do not sum inputs within the receptive field.
Response
Half-wave Rectification: Cortical cells have a low spontaneous firing
rate. There cannot be as large a negative response as a positive
response. The bottom half of the waveform is clipped off.
This can be alleviated with
pairs of matched cells that are
180 deg out of phase with one
another. The difference in
responses acts like a linear
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time response.
Lateral Inhibition
•There is evidence that a spatial frequency channel is inhibited by
other channels tuned to nearby frequencies. (Also true for
orientation tuning).
•This is accomplished by lateral inhibitory connections within the
cortex, known as lateral inhibition.
•This can cause interesting effects, such as repulsion of perceived
orientation when 2 lines of similar orientation are shown close
together.
•If you adapt 1 spatial frequency, there is an increased sensitivity
at other nearby frequencies.
•Inhibitory interactions can help to make tuning curves narrower.
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