Graphics Reference
In-Depth Information
Matrix
∑
Matrix
U
Matrix
V
T
Matrix
V
σ
1
v
1
v
2
σ
2
O
M
=
σ
k
v
k
O
σ
N
v
N
u
1
u
2
u
k
u
N
v
1
v
2
v
k
v
N
Figure 9.9
A graphical representation of the singular value decomposition. A matrix
M
is decomposed
into the product of three matrices:
M
=
U
Σ
V
T
;
U
has left-singular vectors in its columns,
V
T
has right-singular vectors in its rows, and
has the singular values on its diagonal. The
matrix
V
T
is so written because the rows of this matrix are the relevant singular vectors,
while usually a matrix is regarded as collection of column vectors.
Σ
V
T
with the last few singular values
set to zero results in a matrix that is fairly close to
M
, but has lower rank. That
is, the rows and columns have a smaller linear span. Setting the smallest singular
values to zero therefore removes the least important basis elements. In effect, this
is a kind of lossy data compression algorithm for the span of the matrix. The
number of singular values set to zero depends on the particular application. For
example, when the matrix
M
corresponds to pixels in a grayscale image, there is
often a particular SVD threshold where the grayscale image of the approximate
matrix ceases to be recognizable.
PCA is often used for analyzing physical responses to a discrete input signal,
such as the response to sound or the response to an external force acting on an
object in the finite-element method. Rendering an object can be regarded as com-
puting its response to light at a specific position on the surface of the object, due
to a specific lighting/viewing direction. A PCA approximation isolates the most
significant parts of the response.
singular vector. Consequently, the product
U
Σ
9.3.3 Eigen-Textures
An early paper that uses PCA to approximate responses that represent the ap-
pearance of an object, “Face Recognition Using Eigenfaces” by Matthew Turk,
and Alex Pentland was published in the field of image recognition in 1991 [Turk
and Pentland 91]. This paper describes the application of PCA to the image data
of various human faces with various facial expressions, in order to create faces
(called “eigenfaces”) that best represent the characteristics of all the faces in the
data set. These eigenfaces are then used as a way to identify a person from a new
face image. The paper was considered a landmark result in the field of image
recognition, and is one of the most cited papers in the field. The paper “Eigen-
Texture Method: Appearance Compression and Synthesis Based on a 3D Model”
