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Section 6.6 Notes 193

because not much is known about how to proceed.
There is a long tradition of using marked point processes as texture models
(explicitly in, for example (Ahuja and Schachter 1983a, Ahuja and Schachter 1983b,
Blake and Marinos 1990, Schachter 1980, Schachter and Ahuja 1979) and implicitly
in pretty much all existing literature). A Poisson model has the property that
the expected number of elements in a domain is proportional to the area of the
domain. The constant of proportionality is known as the model’s intensity.A
texture is isotropic if the choice of element rotation is uniform and random, and is
homogeneous if the density from which texture elements are drawn is independent
of position on the surface.
There are surprisingly few shape from texture methods. Global methods
attempt to recover an entire surface model, using assumptions about the distri-
bution of texture elements. Appropriate assumptions are isotropy (Witkin 1981)
(the disadvantage of this method is that there are relatively few natural isotropic
textures) or homogeneity (Aloimonos 1986, Blake and Marinos 1990).
Texture deformation can be exploited in global methods, with some assump-
tions about the element—see the methods in (Lee and Kuo 1998, Sakai and Finkel
1994, Stone and Isard 1995)). Alternatively, one observes that the per-element
imaging transformations are going to affect the spatial frequency components on
the surface; this means that if the texture has constrained spatial frequency prop-
erties, one may observe the orientation from the texture gradient (Bajcsy and
Lieberman 1976, Krumm and Shafer 1990, Krumm and Shafer 1992, Sakai and
Finkel 1994, Super and Bovik 1995).
Local methods recover some differential geometric parameters at a point on
a surface (typically, normal and curvatures). This class of methods, which is due
to Garding (1992), has been successfully demonstrated for a variety of surfaces by
Malik and Rosenholtz (1997) and Rosenholtz and Malik (1997); a reformulation in
terms of wavelets is due to Clerc and Mallat (1999). The methods have a crucial
flaw; it is necessary either to know that texture element coordinate frames form
a frame field that is locally parallel around the point in question, or to know the
differential rotation of the frame field (see Garding (1995) for this point, which is
emphasized by the choice of textures displayed in Rosenholtz and Malik (1997); the
assumption is known as texture stationarity). For example, if one were to use
these methods to recover the curvature of a doughnut dipped in chocolate sprinkles,
it would be necessary to ensure that the sprinkles were all parallel on the surface
(or that the field of angles from sprinkle to sprinkle was known).
One might construct a generative model, where object texture is modelled
with a parametric random model, then choose a geometry and parameters that
minimizes the difference between either a predicted image and the observed im-
age (Choe and Kashyap 1991) or a predicted image density and the observed image
density (Lee and Kuo 1998).
More recent local methods emphasize repetition. Forsyth (2001) infers shape
from slant estimates only, establishing an analogy with shape from shading. Forsyth
(2002) shows that element repetition is sufficient to get normal estimates up to an
ambiguity, with a cleaner version in (Lobay and Forsyth 2006); Loh and Hartley
(2005) give a method to reconstruct a surface in this case; and Lobay and Forsyth
(2004) demonstrate that repetition of textons gives cues to illumination.