... have
and
Exercise: verifythatthis solution satisfies boththe differentialequation (6. 22) and the initialvalueequation y y .
Thus, the solutions to system (6. 22) for and for have different forms. ... other hand, depend on the constants as follows:
Im Re Im Re
where Re, Im denote the real and imaginary part and where the two-argument function is defined as follows
for
if
if
if...
... the procedure, due to Hestenes and Stiefel (Methods of conjugate gradients for solving linear systems,J.
Res. Bureau National Standards, section B, Vol 49, pp. 40 9-4 36, 1952), which also incorporates ... most problems in robotics, vision, and arguably every other science or endeavor
take on the form of optimization problems. One is that the desired goal may not be achievable, an...
... In other words, the curve-to-curve transformation from circle to
ellipse is unique, but the point-to-pointtransformation is not. Matrices represent point-to-pointtransformations.
The eigenvalue ... repeated for the last equation, therefore forcing , and so forth.
In summary, the equation x x impliesthat , that is, that the vectors x x
are linearly independent.
For Hermitian matrices (an...
... collection of mathematical tools for both understandingand solving problems
in robotics and computer vision. Several classes at Stanford cover the topics presented in this class, and do so in ... want to understandrobotics or vision, youshould take classes in these subjects, since this course
is not on robotics or vision.
On the other hand, if you do plan to study robotics,...
... the equality ofleft- and right-hand
side.
When this process is finished, is in echelon form. In particular, if the matrix is square and if all columns have a
pivot, then is upper-triangular.
“Stop” ... triangularization step, row 2
multiplied by 6/ 3 is subtracted from row 3 for both and c to yield
, c c
There is one zero row in the left-hand side, and the rank of and that of is ,...
... following
components:
for
for
When written as a function of the vector c, this is
y c
Notice that there is no other choice for y, which is therefore unique: minimum residual forces the choice of ,
and minimum-norm ... the
discrepancies between left- and right-hand sides of the equations.
3.2. THE SINGULAR VALUE DECOMPOSITION 27
where
and . Equivalently,
Proof. This proof is adapted...
... STATE ESTIMATION 89
k | k-1
^
y
H
k
x
k | k-1
^
x
^
k | k
k | k
P
x
^
P
k+1 | k
k+1 | k
propagatepropagate
x
^
P
k-1 | k-1
k-1 | k-1
y
k
y
k-1
y
k+1
k
update
k | k-1
P
k-1 k+1
time
Figure 7.2: ... “Kalman filter-based algorithms for estimating depth from image sequences,” InternationalJournal of
Computer Vision, 3(3):20 9-2 36, September 1989; and T.J. Broida, S. Chandrashekhar, and R...
... collection of mathematical tools for both understandingand solving problems
in robotics and computer vision. Several classes at Stanford cover the topics presented in this class, and do so in ... want to understandrobotics or vision, youshould take classes in these subjects, since this course
is not on robotics or vision.
On the other hand, if you do plan to study robotics,...
... following
components:
for
for
When written as a function of the vector c, this is
y c
Notice that there is no other choice for y, which is therefore unique: minimum residual forces the choice of ,
and minimum-norm ... the
discrepancies between left- and right-hand sides of the equations.
3.4. LEAST-SQUARES SOLUTION OF A HOMOGENEOUS LINEAR SYSTEMS 33
which is the projection of the righ...
... be nonzero for some . We have
a a a 0
so that
a a (2 .6)
as desired. The converse is proven similarly: if
a a
“iff” means “if and only if.”
CS 205
Mathematical Methods for Robotics and Vision
Carlo ... want to understandrobotics or vision, youshould take classes in these subjects, since this course
is not on robotics or vision.
On the other hand, if you do plan to st...