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Computational Vision

CSCI 363, Fall 2012Lecture 10

Spatial Frequency

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Ocular Dominance Columns

Input from the LGN to layer 4C is segregated by eye. Inputs alternate between the left and right eyes.

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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.

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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 the other.

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Orientation Columns

Preferred orientation varies smoothly across the cortex.Cells in columns perpendicular to the surface have the same orientation preference.

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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:

period

Amplitude

x (deg)

Frequency = number of cycles per unit distance (deg or cm)Frequency = 1/period =

f(x) = Asin(2x)

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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.

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f (x) = a0 + (ak cos2πωkx + bk sin 2πωkx)k= 0

∞

∑

For a function with a base frequency of , we can write:

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Example: A square wave

f(x) = 4/(sin(f) + 1/3sin(3f) + 1/5sin(5f) + ...

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Graphing frequency components

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Phase

The phase of a sine wave is the amount of lateral shift.

f(x) = Asin(2x + )

is the phase.

phase

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Combining sine waves with different phases

Asin + Bcos = Csin( + )

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 representationIt is convenient mathematically to represent sines and cosines in the form:

Aei = A(cos + isin)

where:

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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i = −1

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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:

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F(α ) = f (x)e− i2παxdx−∞

∞

∫

Given the Fourier transform of a function, we can find the original function by taking the inverse Fourier transform:

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f (x) = F(α )e i2παxdα−∞

∞

∫

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2D Fourier Transform2D images are composed of sine waves that vary in amplitude, phase, and orientation. We can find these frequency components with a 2D Fourier transform.

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F(α ,β ) = f (x,y)e− i2π (αx+βy )dxdy−∞

∞

∫−∞

∞

∫

The 2D inverse Fourier transform:

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f (x,y) = F(α ,β )e i2π (αx+βy )dαdβ−∞

∞

∫−∞

∞

∫

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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 SensitivityHow much contrast is required to distinguish a sinewave grating from a uniform field of gray?

Lum

inan

ce

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 AdaptationAdaptation 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.