Graphics Reference
In-Depth Information
directions). If the new lighting direction does not lie strictly between that of
A
and
B
, then the synthesized image could be interpolated from
A
and
B
and a third
image
C
, and possibly even more images. In general, deciding which images to
interpolate is a nontrivial problem in itself. The interpolation problem becomes
more complicated still when a different viewing direction is considered along with
a different lighting direction.
The BTF synthesis methods described above do not directly interpolate pixels;
rather, they perform a kind of extrapolation from a reference object. In the basic
synthesis method of Liu et al., this reference object is a reference image taken
from the BTF image data set. The template image is synthesized from recovered
geometry, which is obtained from a kind of interpolation. In the texton method,
the reference object is the set of textons, which were likewise formed by a kind of
interpolation (filtering by all the filters in the filter bank can be viewed as a kind
of interpolation.)
The key to image-based representations is how the acquired image data is
interpolated. Recent research has looked to algebraic methods to improve inter-
polation. The basic idea of algebraic analysis is to look for a kind of basis set for
the image data. Armed with such a basis, an image from an arbitrary viewing (and
lighting) direction can be synthesized from a combination of the basis elements.
The texton method of Section 9.2.3 employs one kind of basis: the basis elements
are the textons, or more properly, the appearance vectors.
Matrix analysis is a more algebraic way of looking at the basis for a set of
images. Consider a basic set of BTF images, i.e., a set of images of an object or
surface sample captured from various lighting/viewing directions. Each captured
image can be represented by a single vector by “flattening” the image pixels from
a 2D array to a 1D array (a vector). There is one such vector for each image in
the data set, and the collection of all these vectors can be placed into the columns
of a matrix. This matrix is large: the matrix for a typical BTF data set has several
hundred columns, and as many as several million rows. Each row represents the
variation in the appearance of a point as the lighting/viewing directions change;
each column represents the appearance of points over the surface of the object for
a fixed lighting/viewing direction. This matrix is known as the
response matrix
R
; the vector columns are
response vectors
. Suppose there are
n
captured images,
each of which contains measurements for
m
points. The response matrix
R
is an
m
n
matrix in this case.
The appearance of the object for an arbitrary lighting/viewing direction can
be represented as an
m
-dimensional column vector just like the other columns of
the response matrix. Such a vector is a
synthesized
response vector. For definite-
ness, let
V
denote the set of response vectors for all lighting/viewing directions.
×
