Graphics Reference
In-Depth Information
Local PCA (clustered PCA).
As described earlier, local PCA selects basis
vectors for local clusters rather than for the entire data as is done in normal PCA.
As a result, approximation of signals can be performed with a few basis vectors.
This method is well suited to a situation where each portion of the signal distri-
bution has some characteristics, e.g., specular reflection. The method of Muller
et al. described in Section 9.3.5 for compressing and rendering BTF data is an
example of this technique. It is also used for precomputed radiance transfer, as
will be described in Chapter 10. Figure 9.19 illustrates local PCA on a surface
signal.
9.3.7 TensorTextures
The appearance of a textured surface can vary in a complex manner according to
the interaction of surface geometry, material properties, illumination, and imag-
ing. All of these are captured (ideally) in the BTF. However, there is no known
closed form mathematical model of the BTF that, for arbitrary surfaces, captures
the physical interactions between these different factors. Instead, BTF data is
represented in terms of a representative set of images (captured or synthetic); the
statistical approach of PCA largely amounts to a way of reducing the storage re-
quirements. A major limitation of PCA and related methods is that they attempt
to model the overall surface appearance variation without trying to explicitly dis-
tinguish the visual elements by their physical cause.
In the paper “TensorTextures: Multilinear Image-Based Rendering,” M. Alex O.
Vasilescu and Demetri Terzopoulos introduced an advanced statistical model that
accurately approximates arbitrary BTFs in a multifactor manner [Vasilescu and
Terzopoulos 04]. The method, which the authors call
TensorTextures
, captures
how a texture changes with the basic variables of surface position, lighting direc-
tion, and viewing direction in separate ways. Furthermore, it can synthesize the
appearance of a textured surface for new viewpoints, illumination, and to a certain
extent, geometry not observed in the sampled data.
The definition of a tensor depends on the context. Although all the definitions
are essentially equivalent, the common definition (particularly in engineering) is
a multidimensional array. That is, a tensor is a higher-order generalization of a
vector or a matrix. The
order
of the tensor is, loosely speaking, the overall number
of dimensions: a vector is a first-order tensor; a matrix is a second-order tensor.
In this topic, general tensors are denoted by capital script letters.
As usual, the TensorTextures approach starts by capturing a set of images from
different lighting/viewing directions. Each image is “vectorized” by arranging all
the pixels into a single vector. Given a collection of sample images of a textured
surface captured from
I
V
different viewpoints, each under
I
L
different illumination
