Image Processing Reference
In-Depth Information
yielding:
c
=(2
/
4
,
1
/
4
,
1
/
4
,
0)
T
(3.43)
By summing the basis matrices
U
i
weighted with the coefficients
c
i
,
F
can be re-
constructed.
Tensors
represent physical or geometric quantities, e.g., a force vector field in a
solid, that are viewpoint-invariant in that they have invariant representations w.r.t. a
coordinate frame. For practical manipulations they need to be represented in a coor-
dinate system, however. The data corresponding to these representations are usually
stored as arrays having multiple indices, e.g., as ordinary vectors and matrices of lin-
ear algebra, although their representation does not critically depend on the choice of
the coordinate system more than a
change of basis
,the
viewpoint transformation
.In
other words, the numerical representation of a tensor field given in a coordinate frame
should be recoverable from its representation in another known frame. For example,
the directions and the magnitudes of the internal forces in a body will be fixed with
respect to the body, whereas the body itself and thereby the force field can be viewed
in different coordinate systems. Once observed and measured in two different coordi-
nate systems, the two measurements of the force fields will only differ by a viewpoint
transformation and
nothing else
because the forces (which stayed fixed relative the
body) are not influenced by the coordinate system. The principal property of the ten-
sors is that the influence of the coordinate system becomes nonessential. The tensors
look otherwise like ordinary arrays having multiple indices, e.g., matrices.
Geometric or physical quantities are characterized by the degrees of freedom they
have. For example, the mass and the temperature are scalars having zero degrees of
freedom. A velocity or a force is determined by an array of scalars that can be ac-
cessed with one index, a vector. Force and velocity accordingly have one degree of
freedom. The polarization and the inertia are quantities that are described by matri-
ces; therefore they require two indices.
3
They have then two degrees of freedom. The
elements of arrays representing physical quantities are real or complex scalars and
are accessed by using 1, 2, 3, 4, etc. indices. The number of indices or the degree
of freedom is called the
order of a tensor
. Accordingly, the temperature, the veloc-
ity, and the inertia are tensors with the orders of 0, 1, and 2, respectively. First- and
second-order tensors are arrays with 1 and 2 indices, respectively, but an array with
1 or 2 indices is a tensor if the array elements are not influenced by the observing
coordinate system by more than a viewpoint transformation.
Tensors consitute Hilbert spaces too since their realizations are arrays. It is most
interesting to study the first- and second-order tensors as being
tensor valued pixels
in the scope of this topic. That is to say, apart from the space-time indices, which
are the points where the tensor elements are either measured or computed, we will
have at most two indices representing a tensor as a pixel value. For example, in a
color image sequence there are four indices, the last of which represents the “color
vector”, which is encoded with one index. Further in the topic, Sect. 9.6 and Chap.
3
There are quantities that would require more than two indices, i.e., they are to be viewed as
generalized matrices.
