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Section 6.6 Notes 191

information (the proof will take us far out of our way; it is in Lobay and Forsyth
(2006)). We must now fit a surface to this information. Doing so is complicated,
because we need to resolve the ambiguity at each surface normal. This can be done
by assuming that the surface is smooth (so that elements that lie near one another
tend to share normal values), and by assuming we have some geometric constraints
on the surface.
Interestingly, modeling a texture as a set of repeated elements reveals illumi-
nation information. If we can find multiple instances of an element on a surface,
then the reason for their different image brightnesses is that they experience dif-
ferent illumination (typically, because they are at different angles to the incoming
light). We can estimate surface irradiance directly from this information, even if
the illumination field is complex (Figure 6.21).

6.6 NOTES
The idea that textures involve repetition of elements is fundamental, and appears
in a wide variety of forms. Under some circumstances, one can try to infer the
elements directly, as in Liu et al. (2004). Image compression can take advantage of
the repetitions created by texture. If we have large, plane figures in view (say, the
faces of buildings), then it can be advantageous to model viewing transformations
to compress the image (because then the same element repeats more often). This
means that, on occasion, image compression papers contain a shape from texture
component (for example, Wang et al. (2008)).

Filters, Pyramids and Efficiency
If we are to represent texture with the output of a large range of filters at many
scales and orientations, then we need to be efficient at filtering. This is a topic that
has attracted much attention; the usual approach is to try and construct a tensor
product basis that represents the available families of filters well. With an appropri-
ate construction, we need to convolve the image with a small number of separable
kernels, and can estimate the responses of many different filters by combining the
results in different ways (hence the requirement that the basis be a tensor product).
Significant papers include Freeman and Adelson (1991), Greenspan et al. (1994),
Hel-Or and Teo (1996), Perona (1992), (1995), Simoncelli and Farid (1995), and
Simoncelli and Freeman (1995b).

Pooled Texture Representations

The literature does not seem to draw the distinction between local and pooled
texture representations explicitly. We think it is important, because quite differ-
ent texture properties are being represented. There has been a fair amount of
discussion of just what should be vector quantized to form these representations.
Typically, one evaluates the goodness of a particular representation by its dis-
criminative performance in a texture classification task; we discuss this topic in
Chapter 16. Significant papers include Varma and Zisserman (2003), Varma and
Zisserman (2005), Varma and Zisserman (2009), Leung and Malik (2001), Leung
and Malik (1999), Leung and Malik (1996), Schmid (2001),