Natural Images

Giovanni Marelli

Göttingen 27.01.10

Probability distribution and correlation Analysis

Infomation in Natural Images

How much information do they carry? Are they organized in some specific structures? Do they belong to a specific class of universality?

Infomation in Natural Images

How much information do they carry? Are they organized in some specific structures? Do they belong to a specific class of universality?

Infomation in Natural Images

How much information do they carry? Are they organized in some specific structures? Do they belong to a specific class of universality?

Infomation in Natural Images

How much information do they carry? Are they organized in some specific structures? Do they belong to a specific class of universality?

Probability distribution and correlation Analysis

Quantities

A picture is composed by n × m matrix of pixels

Probability distribution and correlation Analysis

Quantities

Every pixel contains an array of R, G, B, (eventually α as transparency) which have values within

0, 255 which define the intensity I of the picture.

Probability distribution and correlation Analysis

Quantities

The hystogram shows the distribution of the intensities in the picture

Probability distribution and correlation Analysis

Quantities

We turn the image into a gray scale image

Probability distribution and correlation Analysis

Quantities

We turn the image into a gray scale image for different coarse grained pixel: 1:4

Probability distribution and correlation Analysis

Quantities

We turn the image into a gray scale image for different coarse grained pixel: 1:16

Probability distribution and correlation Analysis

Quantities

We turn the image into a gray scale image for different coarse grained pixel: 1:32

Probability distribution and correlation Analysis

Quantities

We define the contrast as ϕ() := log(I(x)/I_{0}) where I_{0} : N(ϕ < 0) = N(ϕ > 0)

Probability distribution and correlation Analysis

Quantities

If we calculate the probability distribution

we see that for different coarse grained pixel the plot shows long tails

Probability distribution and correlation Analysis

Quantities

If we calculate ist gradient |∇ϕ|

The distribution is quite precisely exponential

Probability distribution and correlation Analysis

Quantities

If we calculate ist gradient |∇ϕ|

The distribution is quite precisely exponential

Probability distribution and correlation Analysis

Quantities

We calculate the median of the distribution and assign 0 (black) for every intensity below the median and 1 (white) for every intesities above. The pixel are equally populated

Probability distribution and correlation Analysis

Quantities

We calculate the median of the distribution and assign 0 (black) for every intensity below the median and 1 (white) for every intesities above. The pixel are equally populated Coarse grained: 1:4

Probability distribution and correlation Analysis

Quantities

We calculate the median of the distribution and assign 0 (black) for every intensity below the median and 1 (white) for every intesities above. The pixel are equally populated Coarse grained: 1:16

Probability distribution and correlation Analysis

Quantities

The power spectrum in the loglog plot shows a clear power law

Probability distribution and correlation Analysis

Quantities

The full distribution is invariant to block scaling if we coarse-grain and quantize (top) or

quantize and coarse-grain (bottom)

Probability distribution and correlation Analysis

Analysis

We consider a receptor cell with a contrast noise of variance σ^{2} and a Nyquist frequency of k_{c}.

The power spectrum is connected with the autocorrelation (Th. Wiener-Khintchin) < ϕ(x)ϕ(x′) >:= ∫ \frac{d2k}{(2π)2} S_{ϕ}(k)e^{ık(x−x′)}

Among all the distributions, the gaussian has the maximum entropy. In this assumption one can connect the autocorrelation of the picture with the information conveyed I ≤

\frac{πNπ/2}{k2 c} ∫ \frac{d2k}{(2π)2} log_{2} (1 + \frac{(1−k/kc)2}{π2σ2} k_{c}^{2}S_{ϕ}(k))

The signal to noise ration of a single cell is R_{SNR} = \frac{1}{σ2} ∫ \frac{d2k}{(2π)2} (1 − k/k_{c})^{1}S_{ϕ}(k) I ≤ \frac{1}{2} N \frac{π}{2} ∫_{0}^{1} dxx log_{2} (1 + \frac{η(η+1)(η+2)}{π} R_{SNR} \frac{(10x)2}{x2−η} )

A single receptor can produce a spike when the signal overcome a certain threshold The upper bound we have shown tells that the information contained in a natural image is less than a bit per receptor.

A channel that transmit the information to the brain can have capacity lower than 1/2 bit/receptor to convey all the information

D.L. Ruderman, dissertation, University of California at Berkley, 1993 Probability distribution and correlation

Analysis

Analysis

We consider a receptor cell with a contrast noise of variance σ^{2} and a Nyquist frequency of k_{c}.

The power spectrum is connected with the autocorrelation (Th. Wiener-Khintchin) < ϕ(x)ϕ(x′) >:= ∫ \frac{d2k}{(2π)2} S_{ϕ}(k)e^{ık(x−x′)}

Among all the distributions, the gaussian has the maximum entropy. In this assumption one can connect the autocorrelation of the picture with the information conveyed I ≤

\frac{πNπ/2}{k2 c} ∫ \frac{d2k}{(2π)2} log_{2} (1 + \frac{(1−k/kc)2}{π2σ2} k_{c}^{2}S_{ϕ}(k))

The signal to noise ration of a single cell is R_{SNR} = \frac{1}{σ2} ∫ \frac{d2k}{(2π)2} (1 − k/k_{c})^{1}S_{ϕ}(k) I ≤ \frac{1}{2} N \frac{π}{2} ∫_{0}^{1} dxx log_{2} (1 + \frac{η(η+1)(η+2)}{π} R_{SNR} \frac{(10x)2}{x2−η} )

A single receptor can produce a spike when the signal overcome a certain threshold The upper bound we have shown tells that the information contained in a natural image is less than a bit per receptor.

A channel that transmit the information to the brain can have capacity lower than 1/2 bit/receptor to convey all the information

D.L. Ruderman, dissertation, University of California at Berkley, 1993 Probability distribution and correlation

Analysis

Analysis

We consider a receptor cell with a contrast noise of variance σ^{2} and a Nyquist frequency of k_{c}.

The power spectrum is connected with the autocorrelation (Th. Wiener-Khintchin) < ϕ(x)ϕ(x′) >:= ∫ \frac{d2k}{(2π)2} S_{ϕ}(k)e^{ık(x−x′)}

Among all the distributions, the gaussian has the maximum entropy. In this assumption one can connect the autocorrelation of the picture with the information conveyed I ≤

\frac{πNπ/2}{k2 c} ∫ \frac{d2k}{(2π)2} log_{2} (1 + \frac{(1−k/kc)2}{π2σ2} k_{c}^{2}S_{ϕ}(k))

The signal to noise ration of a single cell is R_{SNR} = \frac{1}{σ2} ∫ \frac{d2k}{(2π)2} (1 − k/k_{c})^{1}S_{ϕ}(k) I ≤ \frac{1}{2} N \frac{π}{2} ∫_{0}^{1} dxx log_{2} (1 + \frac{η(η+1)(η+2)}{π} R_{SNR} \frac{(10x)2}{x2−η} )

A single receptor can produce a spike when the signal overcome a certain threshold The upper bound we have shown tells that the information contained in a natural image is less than a bit per receptor.

A channel that transmit the information to the brain can have capacity lower than 1/2 bit/receptor to convey all the information

D.L. Ruderman, dissertation, University of California at Berkley, 1993 Probability distribution and correlation

Analysis

Analysis

We consider a receptor cell with a contrast noise of variance σ^{2} and a Nyquist frequency of k_{c}.

The power spectrum is connected with the autocorrelation (Th. Wiener-Khintchin) < ϕ(x)ϕ(x′) >:= ∫ \frac{d2k}{(2π)2} S_{ϕ}(k)e^{ık(x−x′)}

Among all the distributions, the gaussian has the maximum entropy. In this assumption one can connect the autocorrelation of the picture with the information conveyed I ≤

\frac{πNπ/2}{k2 c} ∫ \frac{d2k}{(2π)2} log_{2} (1 + \frac{(1−k/kc)2}{π2σ2} k_{c}^{2}S_{ϕ}(k))

The signal to noise ration of a single cell is R_{SNR} = \frac{1}{σ2} ∫ \frac{d2k}{(2π)2} (1 − k/k_{c})^{1}S_{ϕ}(k) I ≤ \frac{1}{2} N \frac{π}{2} ∫_{0}^{1} dxx log_{2} (1 + \frac{η(η+1)(η+2)}{π} R_{SNR} \frac{(10x)2}{x2−η} )

A single receptor can produce a spike when the signal overcome a certain threshold The upper bound we have shown tells that the information contained in a natural image is less than a bit per receptor.

A channel that transmit the information to the brain can have capacity lower than 1/2 bit/receptor to convey all the information

D.L. Ruderman, dissertation, University of California at Berkley, 1993 Probability distribution and correlation

Analysis

Analysis

We consider a receptor cell with a contrast noise of variance σ^{2} and a Nyquist frequency of k_{c}.

The power spectrum is connected with the autocorrelation (Th. Wiener-Khintchin) < ϕ(x)ϕ(x′) >:= ∫ \frac{d2k}{(2π)2} S_{ϕ}(k)e^{ık(x−x′)}

Among all the distributions, the gaussian has the maximum entropy. In this assumption one can connect the autocorrelation of the picture with the information conveyed I ≤

\frac{πNπ/2}{k2 c} ∫ \frac{d2k}{(2π)2} log_{2} (1 + \frac{(1−k/kc)2}{π2σ2} k_{c}^{2}S_{ϕ}(k))

The signal to noise ration of a single cell is R_{SNR} = \frac{1}{σ2} ∫ \frac{d2k}{(2π)2} (1 − k/k_{c})^{1}S_{ϕ}(k) I ≤ \frac{1}{2} N \frac{π}{2} ∫_{0}^{1} dxx log_{2} (1 + \frac{η(η+1)(η+2)}{π} R_{SNR} \frac{(10x)2}{x2−η} )

A single receptor can produce a spike when the signal overcome a certain threshold The upper bound we have shown tells that the information contained in a natural image is less than a bit per receptor.

A channel that transmit the information to the brain can have capacity lower than 1/2 bit/receptor to convey all the information

D.L. Ruderman, dissertation, University of California at Berkley, 1993 Probability distribution and correlation

Analysis

Ising

Do the black/white pixels remember to the ↑, ↓ of the Ising model?

Probability distribution and correlation Analysis

Ising

Do the black/white pixels remember to the ↑, ↓ of the Ising model?

Probability distribution and correlation Analysis

Ising

Do the black/white pixels remember to the ↑, ↓ of the Ising model?

Probability distribution and correlation Analysis

Ising

Do the black/white pixels remember to the ↑, ↓ of the Ising model?