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Figure 9.20 TensorTexture image organization. The ensemble of acquired images is organized in a
third-order tensor with texels, illumination, and view modes. Although the contents of the
texel mode are vectorized images, for clarity they are displayed as 2D images in this and
subsequent figures. (From [Vasilescu and Terzopoulos 04] c
2004 ACM, Inc. Included
here by permission.)
directions, the TensorTextures approach organizes the vectorized images, each of
which have I x texels (pixels) into a data tensor
. This data tensor is a third-order
tensor, and has dimensions I x I L I V . Figure 9.20 illustrates a data tensor for a set of
captured images.
The obvious next step would be to apply principal component analysis to
the data tensor. This would be done using the singular value decomposition,
but according to the authors there is no true generalization of the SVD to gen-
eral tensors. There are, however, a number of different tensor decompositions.
The authors employ a generalization of the SVD to sections of tensors, known
as the M-mode SVD .The M -mode SVD was first employed by Vasilescu and
Terzopoulos in the context of computer vision. 3 The technique, known as Tensor-
Faces [Vasilescu and Terzopoulos 02] was applied to the problem of face recog-
nition. The authors adapted this work to BTF data in the “TensorTextures” paper.
Describing the M -mode SVD requires more definitions.
A mode-m vector is a subset of a tensor in dimension m ; column vectors are
mode-1 vectors of a matrix (second-order tensor), row vectors are mode-2 vectors.
The general mode-m product is a generalization of a matrix product to tensors.
Although there is a formal definition, it is perhaps easier to understand in terms
of matricized (flattened) tensors. Matricizing a tensor involves arranging all its
mode- m vectors into the columns of a matrix. Figure 9.21 illustrates the three
ways of matricizing an third-order tensor.
D
In general, the mode- m matricized
A
A
tensor
is denoted A [ m ]
.The mode-m product , of a matrix B and a tensor
is
3 Tensor methods were not new in computer graphics; James Arvo used tensors effectively to sim-
ulate non-Lambertian reflectance [Arvo 95].
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