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*

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

- Longitudinal component

L/T

- Angular averaged results from procedure
**(i):**

**(ii)** *Vectors and absolute values*

For a given *k *,
we decompose the orientation field

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):**

Selected Images from Modern Art