Optimal Sampling of Natural Images

By Peyman Khorsand

This post is taken from the following paper,

“Optimal sampling of Natural Images: A design Principle for the Visual Systems?” by W. Bialek, D.L. Ruderman and A. Zee

If the natural scene images can be characterized by $\phi(\mathbf {x})$ and the cells are indexed by integers $n$ and positioned respectively at $\mathbf{x}_n$ . The output of each cell, $Y_n$ , is a linear filter (receptive field) with an additional noisy component

$Y_n=\int d^2 \mathbf{x}F( \mathbf{x}-\mathbf{x}_n)\phi(\mathbf{x})+\eta_n$

The goal is to find a kernel $F$ that maximze the information content of output, $Y_n$ , about the input, $\phi$ . The information content of $Y_n$ assuming that $\phi(\mathbf{x})$ is Gaussian is

$I={1\over 2\ln 2}Tr\ln\left[ \delta_{nm} + {1 \over (2 \pi)^2 \sigma^2} \int d^2\mathbf{k} e^{i \mathbf{k}(\mathbf{x}_n -\mathbf{x}_m)} |\tilde{F}(\mathbf{k})|^2 S(\mathbf{k})\right]$

here $S(\mathbf{k})$ is the power spectrum of the signal. We can approximate $I$ in large noise regime as

$I \approx {N \over 2 (2\pi)^2\sigma^2 \ln 2} \int d^2\mathbf{k} |\tilde{F}(\mathbf{k})|^2 S(\mathbf{k})$

We need to put extra constraints to get solutions that are physically realistic

1) We should fix the filters gain,

$\int d^2 \mathbf{k} |\tilde{F} (\mathbf{k}) |^2 =1$

2) There should be a cost for “long-range interactions in spatial space” or “sharp fluctuations in momentum space”,

$C = \alpha \int d^2 \mathbf{k} \mathbf{k}^2 |\tilde{F}(\mathbf{k}) |^2$

Using variational methods they found that filter should satisfy the schrodinger like equation in $\mathbf{k}$ -space

$-{\alpha\over 2}{\nabla_k}^2\tilde{F}(k)-{1\over 2 \sigma^2\ln 2} S(k)\tilde{F}(k)=\Lambda \tilde{F}(k)$

the ${h}^2/\alpha$ playes the role of the mass $M$ and $- S(k)/ ( 2 \sigma^2 \ln 2 )$ of potential $V$ in the schrodinger equation.

In the case that they are interested the power spectrum of natural images has a power law $S(k)\sim 1/k^2$ and as a result an accidental symmetry. This accidental symmetry allows them to recombine various angular momentum eigenstates and create orientation selective eigenstates (filters).

What interests me the most is that for any linear filter this formalism holds and, the power spectrum of the input signal appears as the potential of the schrodinger equation.

This entry was posted on October 10, 2008 at 12:54 am and is filed under Information Theory. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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Optimal Sampling of Natural Images

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