Statistics of Lines in Natural Images

(Supplementary material for article)

Natural images are segmented into oriented lines using `steerable filters' (based on the second derivative of a Gaussian and its Hilbert transform)

The following example is taken from M. Sigman et al, PNAS 98, 1935 (2001).

We examined a digital library of natural images, which includes trees, buildings, flowers, leaves, and grass.

• Several precedures were employed for obtaining transverse/longitudinal Fourier power spectra; the results were similar from the different methods.

(i) Vectors defined in half-planes; averaged over 4 half-planes

1. Use standard filters to convert images to an orientation field  S(x,y) (in a half-plane),
2. Average over 4 different half-plane representations ([0,pi], [pi/2, 3 pi,2], [pi, 2 pi], [3 pi/2, pi/2])
3. Rotate images to remove vertical/horizontal bias
4. Fourier transform, and extract the power spectrum, along the
• Longitudinal component  |k . S(k) k . S(k)*|
• Transverse component  |k x S(k) k x S(k)*|

L/T

• Angular averaged results from procedure (i):

(ii) Vectors and absolute values

For a given k , we decompose the orientation field S(x,y) into components parallel and perpendicular to k , and then Fourier transform their absolute value} which are invariant under S(x,y)=>-S(x,y).
The disadvantage is that mirror image orientations are not distinguished, and averaging over orientations does not decouple the two components.

• Angular averaged results from procedure (ii):

(iii) Tensors (as reported in article)

Convert the orientation field  S(x,y) into a tensor    S = S • S , and use standard projection operators to project out the transverse and longitudinal components (see article).

L/T

• Angular averaged results from procedure (iii):