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Section 6.5 Shape from Texture 188

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FIGURE 6.19: Humans obtain information about the shape of surfaces in space from the
appearance of the texture on the surface. The figure on the left shows one common use
for this effect; away from the contour regions, our only source of information about the
surface depicted is the distortion of the texture on the surface. On the right, the texture
gives a clear sense of the orientation of the ground plane, how the plants stand out from
the path, and how far away the building at the back is. Geoff Brightling c Dorling
Kindersley, used with permission.

backprojected texture on it.
Assume that we are viewing a single textured plane in an orthographic camera.
Because the camera is orthographic, there is no way to measure the depth to the
plane. However, we can think about the orientation of the plane. Let us work in
terms of the camera coordinate system. First, we need to know the angle between
the normal of the textured plane and the viewing direction—sometimes called the
slant—and second, the angle the projected normal makes in the camera coordinate
system—sometimes called the tilt (Figure 6.20).
In an image of a plane, there is a tilt direction—the direction in the plane
parallel to the projected normal.
An isotropic texture is one where the probability of encountering a texture
element does not depend on the orientation of that element. This means that a
probability model for an isotropic texture need not depend on the orientation of
the coordinate system on the textured plane.
If we assume that the texture is isotropic, both slant and tilt can be read from
the image. We could synthesize an orthographic view of a textured plane by first
rotating the coordinate system by the tilt and then contracting along one coordinate
direction by the cosine of the slant—call this process a viewing transformation.
The easiest way to see this is to assume that the texture consists of a set of circles,
scattered about the plane. In an orthographic view, these circles will project to
ellipses, whose minor axes will give the tilt and whose aspect ratios will give the
slant (see the exercises and Figure 6.20).
An orthographic view of an isotropic texture is not isotropic (unless the plane
is parallel to the image plane). This is because the contraction in the slant direction
interferes with the isotropy of the texture. Elements that point along the contracted