EE263 Summer 2010-11
Laurent Lessard
EE263 homework 5
1.
Curve-smoothing.
We are given a function
F
: [0
,
1]
→
R
(whose graph gives a curve in
R
2
). Our goal
is to find another function
G
: [0
,
1]
→
R
, which is a
smoothed
version of
F
. We’ll judge the smoothed
version
G
of
F
in two ways:
•
Mean-square deviation from
F
,
defined as
D
=
integraldisplay
1
0
(
F
(
t
)
−
G
(
t
))
2
dt.
•
Mean-square curvature,
defined as
C
=
integraldisplay
1
0
G
′′
(
t
)
2
dt.
We want
both
D
and
C
to be small, so we have a problem with two objectives. In general there will be
a trade-off between the two objectives. At one extreme, we can choose
G
=
F
, which makes
D
= 0; at
the other extreme, we can choose
G
to be an affine function (
i.e.
, to have
G
′′
(
t
) = 0 for all
t
∈
[0
,
1]),
in which case
C
= 0. The problem is to identify the optimal trade-off curve between
C
and
D
, and
explain how to find smoothed functions
G
on the optimal trade-off curve. To reduce the problem to a
finite-dimensional one, we will represent the functions
F
and
G
(approximately) by vectors
f, g
∈
R
n
,
where
f
i
=
F
(
i/n
)
,
g
i
=
G
(
i/n
)
.
You can assume that
n
is chosen large enough to represent the functions well. Using this representation
we will use the following objectives, which approximate the ones defined for the functions above:
•
Mean-square deviation,
defined as
d
=
1
n
n
summationdisplay
i
=1
(
f
i
−
g
i
)
2
.
•
Mean-square curvature,
defined as
c
=
1
n
−
2
n
−
1
summationdisplay
i
=2
parenleftbigg
g
i
+1
−
2
g
i
+
g
i
−
1
1
/n
2
parenrightbigg
2
.
In our definition of
c
, note that
g
i
+1
−
2
g
i
+
g
i
−
1
1
/n
2
gives a simple approximation of
G
′′
(
i/n
).
You will only work with this approximate version of the
problem,
i.e.
, the vectors
f
and
g
and the objectives
c
and
d
.
(a) Explain how to find
g
that minimizes
d
+
μc
, where
μ
≥
0 is a parameter that gives the relative
weighting of sum-square curvature compared to sum-square deviation. Does your method always
work? If there are some assumptions you need to make (say, on rank of some matrix, independence
of some vectors, etc.), state them clearly. Explain how to obtain the two extreme cases:
μ
= 0,
which corresponds to minimizing
d
without regard for
c
, and also the solution obtained as
μ
→ ∞
(
i.e.
, as we put more and more weight on minimizing curvature).
(b) Get the file
curve
smoothing.m
from the course web site. This file defines a specific vector
f
that
you will use. Find and plot the optimal trade-off curve between
d
and
c
. Be sure to identify any
critical points (such as, for example, any intersection of the curve with an axis). Plot the optimal
g
for the two extreme cases
μ
= 0 and
μ
→ ∞
, and for three values of
μ
in between (chosen to
show the trade-off nicely). On your plots of
g
, be sure to include also a plot of
f
, say with dotted
line type, for reference. Submit your Matlab code.