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Section 6.5 Shape from Texture 189

Viewing Plane
direction normal

Image
plane
Projected
normal

Textured
plane

Tilt

FIGURE 6.20: The orientation of a plane with respect to the camera plane can be given
by the slant, which is the angle between the normal of the textured plane and the viewing
direction, and the tilt, which is the angle the projected normal makes with the camera
coordinate system. The figure illustrates the tilt, and shows a circle projecting to an
ellipse. The direction of the minor axis of this image ellipse is the tilt, and the slant is
revealed by the aspect ratio of the ellipse. However, the slant is ambiguous because the
foreshortening is given by cos σ,where σ is the slant angle. There will be two possible
values of σ for each foreshortening, so two different slants yield the same ellipse (one is
slanted forwards, the other backwards).

direction get shorter. Furthermore, elements that have a component along the
contracted direction have that component shrunk. Now corresponding to a viewing
transformation is an inverse viewing transformation (which turns an image
plane texture into the object plane texture, given a slant and tilt). This yields
a strategy for determining the orientation of the plane: find an inverse viewing
transformation that turns the image texture into an isotropic texture, and recover
the slant and tilt from that inverse viewing transformation.
There are various ways to find this viewing transformation. One natural
strategy is to use the energy output of a set of oriented filters. This is the squared
response, summed over the image. For an isotropic texture, we would expect the
energy output to be the same for each orientation at any given scale, because the
probability of encountering a pattern does not depend on its orientation. Thus, a
measure of isotropy is the standard deviation of the energy output as a function of
orientation. We could sum this measure over scales, perhaps weighting the measure
by the total energy in the scale. The smaller the measure, the more isotropic the
texture. We now find the inverse viewing transformation that makes the image
looks most isotropic by this measure, using standard methods from optimization.
The main difficulty with using an assumption of isotropy to recover the orientation
of a plane is that there are very few isotropic textures in the world.