January, 1987
LIDS-P-1638
ESTINATION ALCURITHIS FOR RECOOSTRUCRING A XONVEX SET
GIVEN NOISY NEASUREIFIS OF ITS SUPPORT LINES
Jerry L. Prince
and
Alan S. Willsky
ASIRAB Cr:
In many applications, measurements of the support lines of a two
dimensional set are available. From such measurements, the convex hull
of the set may be reconstructed. If, however, the measurements are
noisy, then the set of measurements, taken together, may not correspond
to any set in the plane -- they are inconsistent. This paper describes
the consistency conditions for support line measurements when the angles
of the lines are precisely known, but the lateral displacements are
degraded by noise. We propose three simple algorithms for obtaining
consistent support line estimates from inconsistent measurements and
show examples of the performance of these algorithms.
This research was conducted at the Massachusetts Institute of Technology
Laboratory for Information and Decision Systems, and was supported by
the National Science Foundation grant ECS-8312921 and the U.S. Army
Research Office grant DAAG29-84-K-0005. In addition, the work of the
first author was partially supported by a U.S. Army Research Office
Fellowship.
Laboratory for Information and Decision Systems, Room 35-233, M.I.T.,
77 Massachusetts Avenue, Cambridge, MA, 02139, U.S.A.
1.0 INIRODUCTION
There are several applications that involve the reconstruction of a
2-dimensional set using information about the support lines of the set.
The simplest example of such an application is tactile sensing in
robotics [Schneiter, 1986].
Suppose the typical parallel plate robot
jaw shown in Figure 1 were to close down on a "thick 2-D object" from a
variety of different angles.
Each time the jaw closes down on the
object, two support lines are identified, one for each plate.
Given all
the support lines over the angular interval [0,2Xr) one can then
reconstruct a convex set --
the convex hull of the object --
in which
the object must be contained.
This paper considers the problem in which there are a finite number
of support line measurements for which the angles are known precisely
but the lateral displacements are noisy.
Considering again the example
above, one can imagine obtaining 4 support line measurements, containing
the object in a rectangle (as shown in Figure 2), and then acquiring a
fifth (noisy) measurement for which the measured support line completely
misses the rectangle.
This set constitutes an inconsistent set of
support line measurements (see Section 3); it is this situation that
this paper addresses.
How does one estimate the set (or convex hull of
the set) when faced with inconsistent support line measurements of this
type?
Another area of application, which actually motivated this study,
is computed tomography (see [Herman, 1980], for example).
In computed
tomography, the measurements are (possibly noisy) line integrals in the
plane.
For example, consider the parallel-ray projection geometry shown
- 2 -
Support
Lines
Motion of Jaws
Rotation of
Robot Arm
Figure 1. A parallel plate robot jaw that rotates in the
plane measures support lines of the set S.
- 2a-
L4
L3
L2
L1
Figure 2.
If L 1 - L 4 are known to be true support lines, then
L5 , a noisy measurement, cannot possibly be a support line of
any set support by L 1 - L.4
- 2b -
in Figure 3. For a given angle 0, the line integrals
g(t,O)
=
{
f(x,y) dl
L(t,o)
are obtained for many discrete values of t forming the projection of
f(x,y) at angle 0, as depicted in the figure.
Given a projection of
f(x,y), one may estimate two support lines (corresponding to the dotted
lines in Figure 3) of the set S in which f(x,y) is non-zero.
In a
fashion similar to that described above, a set that contains S may be
built up using the support line estimates from many angular projections.
However, when the observed projections are noisy, the lateral positions
of the support line estimates will be inaccurate; and, as in the
robotics example, the observed support lines may be inconsistent.
This
paper provides several methods for obtaining a consistent set of support
lines from such measurements and, hence, an estimate of the set S.
In
Section 6 we discuss further aspects of the computed tomography problem.
There are many other applications where the algorithms described
herein may be of use.
One can view this problem in terms of the generic
concept of set reconstruction in computational geometry [Preparata and
Shamos, 1985][Greschak, 1985].
Then, in addition to the areas of
tactile sensing and computed tomography, one finds that set reconstruction is also of great interest in robot vision [Horn, 1986],
chemical component analysis [Humel, 1986], and silhouette imaging [Van
Hove and Verly, 1985], for example.
The contribution of this paper to the topic of set reconstruction
is twofold:
1) we expose the fundamental constraint imposed on finite
sets of support lines, and 2) we address the problem of set reconstruction given noisy support line measurements that may violate the
fundamental constraint.
While the particular problems and algorithms
-3-
Support
Lines
\"ttt~~~~~~~~t
(x,y)
L(t,e)
Figure 3.
The projection g(t,O) is formed by integrating
f(x,y) along lines L(t,e).
The dotted lines are support lines
for the set S in which f(x,y) is non-zero.
- 3a -
developed are of potential value, we feel that a more important
contribution of this paper is the initiation of the systematic treatment
of geometric reconstruction in the presence of noise and a priori
information concerning the set to be reconstructed.
The paper is organized as follows.
In Section 2, we present the
basic geometry and some fundamental definitions and properties.
In
Section 3, the definitions and properties related to discrete support
functions are developed.
Section 4 presents three possible algorithms
that take advantage of the constraints developed in Section 3, and
Section 5 presents some simple experiments and simulations, showing the
performance of the two algorithms.
Finally, in Section 6, we discuss
these results and present some possible directions for further research.
-4-
2.0
BACKGROUND
In this section we present the fundamental concepts of object
geometry, followed by the definitions and properties of continuous
support functions and support lines.
2.1
Fundamental Object Geometry.
We refer to the two-dimensional sets under discussion as objects in
the plane.
Although it is not needed, we think of the objects as closed
and bounded convex sets in the plane with no internal holes.
This
allows the convex region formed by the intersection of a finite number
of support half spaces (see Section 3.1) to be a reasonably good
approximation to the true set.
In what follows, points in the plane are denoted by two-dimensional
column vectors.
denoted as uTv
The dot (inner) product of two points u and v will be
(where uT denotes the transpose of vector u).
Since the
unit vector pointing in the direction 0 is often required, it will be
given the symbol w.
Hence,
w = [cosO sinO] T
throughout this paper.
A
line is a set of points that correspond to a shifted one-dimensional
linear subspace.
In the plane, a line may be parametrized by two
variables, t and 0, so that
2 T T.
L(t,O) =w = [cosO
{ xe sin]
| xe = t }
where, as specified above,
2.2
The Support Function and Support Lines
Consider the closed and bounded set S as shown in Figure 4.
In the
figure, two lines are drawn at angle 0 so as to just touch opposite
sides of S. Technically both lines support the set S, but we shall
identify only the solid line, not the dashed line, as the support line
- 5 -
\ S
L
\L
Figure 4.
2
The support line at angle 0 of the set S is L,1
L2.
- 5a -
not
1
at angle 0 of set S.
We make this identification rigorous in the
following two definitions (refer to Figure 5).
Definition:
Support function, support value.
The support function h(O) for the set
=
h(O)
(where
&
= [cos0 sin0]T).
sup
xTS
S C IR2
is given by
c
x
The value of h(S) for some 0 is called
the support value or support distance at angle 0.
Definition:
Support line.
The support line at angle 0 for the set S is given by
LS(0)
=
{ x eC R2
xT
= h(S)
}
Note that given the general definition for the line L(t,e) as
L(t,e)
=
{ x
2 t 1 xT
=
},
an equivalent definition is
Ls(O)
=
L(h(O),O )
It is important to point out that these two definitions succeed in
choosing the solid line in Figure 4 as the support line of set S at
angle 0.
The dashed line may be identified as the support line for the
angle Stir.
In this paper, the measurement of support lines corresponds to
obtaining a measurement of t, a single real number, for a known angle,
0.
Hence, the "noise" of a support line measurement contributes only to
a lateral displacement of the line, not an angular rotation.
- 6 -
h(e)
/
Lsin eJ
'<X
)
Figue5.Theeoer
of suppotlies(
Figure 5. The geometry ofn
support line
Ls(O) is positioned as far in the w direction so as to just
"graze" the set S.
-
6a -
Several simple properties concerning sets and their support lines
and support functions are now stated without proof [Santalo, 1976][Kelly
and Weiss, 1979][Spivak,
1979].
Property 1.
The support function h(O) is periodic with period 2n.
Property 2.
For any point
x E S
it must be true that
xT
< h(O).
This property merely says that S is guaranteed to be on a
particular side of its own support line (see Figure 5).
Property 3.
If S is closed and bounded then the intersection of any
support line of S with S itself cannot be empty.
Unless otherwise
noted, we assume throughout this paper that S is closed and
bounded.
Property 4.
The complete family of support lines (i.e., L (0) with
0 known over any 2ir range) completely specifies the convex hull of
S, denoted hul(S).
If S is known to be convex then
S = hul(S),
and, hence, the family of support lines Ls(0) (or equivalently, the
support function h(O)) completely specifies S.
Property 5.
If h(O) is twice differentiable then S is convex and
the boundary of S is continuous and smooth (i.e., has a
continuously turning normal vector), and is given by the parametric
(vector) equation
[cosO
e(0)
-sinO][h (0)]
=
[sinO
cosO
I
'(0)
where h'(0) is the first derivative of h(O) with respect to 0.
-7-
Property 6.
Suppose the function q(O) is periodic with period 2r
and is twice differentiable.
some set if and only if
Then q(O) is a support function of
q''(0) + q(O) > O.
This proposition is
stated and proved in Santalo [Santalo, 1976], for example.
reasoning goes roughly as follows (refer to Figure 6a).
The
First, we
notice that a support function h(O) defines a family of lines which
must be the envelope of a convex set.
Using the expression in
Property 5 it possible to show that the radius of curvature of the
envelope (boundary) is given by
h''(0) + h(O).
The stated
property then follows because the radius of curvature of the
boundary of a convex set must always be non-negative (see [Spivak,
1979], for example).
In Figure 6b we show a curve which has a
radius of curvature which is first positive, then negative; such a
curve cannot possibly be generated from any support function h(O).
The following Section essentially restates Properties 4 and 5 for the
discrete support function.
As with the continuous case, in the discrete
case we are concerned with 1) the representation of the object geometry
given a discrete support function, and 2) the nature of the constraints
for a discrete function to be a discrete support function.
-8-
N'
h"(e) + h(8)
(a)
(b)
Figure 6.
(a) If h''(0) exists then the boundary of
S (= hul(S)) has a continuously turning normal.
The radius of
curvature at the (unique) boundary points e(O) = Ls(O)
given by h''(0) + h(O), which is alway non-negative.
this "S" shaped curve, the radius of curvature
Mi
n S is
(b) For
changes sign.
Such a curve cannot be the envelope e(O) generated by a
support function h(O).
- 8a -
3.0
SUPPORT LINE CONSTRAINTS AND GEONEIRY
This section contains the bulk of the analytic and geometric
background necessary to understand and appreciate the algorithms
developed in Section 4. We begin by discussing finite sets of support
lines, defining the support vector, and then stating and outlining the
proof of the Support Theorem.
Next, we discuss properties related to
the geometry of the objects in the plane and to the geometry of the
support vector constraints.
3.1
Discrete Support Function
We shall need some further notation.
Let
&i
= [cosOi
the unit vector pointing in the 0i direction and let
the line implied by the value h. at the angle O..
1
1
sinoi]T
be
Li = L(h
i Oi)
be
When referring to the
support angle or support line we have numerous occasion to index the
angle, unit normal, or support line with expressions like i+l, i-i,
etc..
Unless otherwise stated, the index is assumed to be modulo M;
for example, the index M+1 is to be interpreted as simply 1.
3.1.1
Finite sets of support lines.
taken at discrete positions in space.
In most applications data is
For example, in computed
tomography the angular data is collected from a finite set of angles and
a finite set of positions at each angle.
Suppose one were able to
estimate the support values of some set S from observations of its
projections or silhouettes at a finite number of angles.
If the
measurements were noisy, it may be true that the set of estimated
support values does not correspond to any object at all, hence the set
is inconsistent.
-9-
To see how a set of support values (with their corresponding
support angles) could be inconsistent consider Figure 7.
Here two
support lines corresponding to the normals o1 and 03 are assumed to be
known precisely.
We examine the possible location of the third support
line, L2 , corresponding to the normal w2 .
It is obvious that
positioning L2 to the right of the point P 2 would yield an inconsistent
set --
if L 1 and L3 are indeed support lines then L,
2
so positioned,
could not possibly intersect any set S with support lines L 1 and L,3
thus violating property 3 of Section 2.
The consistency of the three
lines in Figure 7 is guaranteed if L 2 is positioned to the left of P 2.
Then it is possible to find some set S which has L 1, L,2
and L3 as
support lines.
In the next section we restrict the support angles 0i to be evenly
spaced over 2w and specify a test to determine the consistency of the
corresponding set of support values.
3.1.2
Consistent support values: the Support Theorem.
Suppose we
are interested in specifying the support values for arbitrary sets in R2
at angles given by
0i = 2ir(i-1)/M, i=1,...,M, where M > 5.
these angles as support angles.
h = [h1
some vector
...
We refer to
The question we wish to answer is given
hM]T , under what conditions may the elements of
h be considered to be the support values at the specified support angles
for some set S?
We will need the following definition in order to more
compactly state the main theorem.
Definition:
support vector.
A vector
h = [h1 ...
hM]T
L.
IR 2
= h. }
= { u
u.
is a support vector if the lines
-
where
10 -
i. = [cosOi
sinOi]T
and
L3
L1
L
Figure 7.
2
Assuming L 1 and L3 are the true support lines then
L 2 can also be a support line (of some set) if it is
positioned to the left of P2 -
- 10a
-
O. = 2r(i-1)/M, for i=1,...,M, are support lines for some set
2
S C IR
Now we are prepared to state the basic theorem of this paper.
Theorem 1:
The Support Theorem.
h CIM
A vector
hT C
<
(M > 5) is a support vector if and only if
1
[O ... O]
(3.1)
where C is a MxM matrix given by
C
1
-k
0
-k
1
-k
0
0
-k
and
0
-k
1
-k
:
0
...
-k
0
0
-k
1
...
0
k = 1/2cos(2v/M).
At this point, it is important to point out the parallel
development of the discrete support vector defined above with the
continuous support function described in Section 2.
One immediately
sees the similarity in the continuous support function constraint,
h''(0) + h(0) 2 O, and the discrete support vector constraint,
In fact, it can be shown that in the limit as M
T
h C < 0
goes to
h''{0) + h(0) > 0
-
'
hTC < O.
the expression
(see Appendix B).
One is led to
view the vector -h C as a type of discrete radius of curvature vector
(which is always positive), in analogy to the continuous case where the
1
A vector inequality such as
x.i < y
T
T
x
< y
where
n
x,y C En
th
for i=,...,n, where x i and yi are the i
vectors x and y respectively.
-
11
-
implies that
elements of the
radius of curvature was found to be
h''(0) + h(O).
This notion is made
more concrete in Section 3.2 where we show that -h C has a direct
geometric interpretation in the discrete case which resembles the
continuous radius of curvature.
The constraint hTC < 0 is more
fundamental than h''(O) + h(O) 2 0, however, since the latter requires
that the second derivative of h exist, which in turn implies that the
set is convex and has continuously turning normals on its boundary.
The
discrete constraint, instead, applies for any bounded set in the plane.
Sketch of Proof:
(See Appendix A for complete proof).
It is relatively straightforward to show the necessity of condition
(3.1).
By hypothesis, h is a support vector of some set S. Now
consider the set D. defined by the two support lines Li_ 1 and Li+1 as
shown in Figure 8. Note that by hypothesis
Oi+1 -0i
1
M > 5, which implies that
< r. This in turn implies that the two lines Li_ 1 and Li+
have a finite intersection point Pi, and that
positive combination of wi-1 and wi+l'
L.
may be written as a
These two facts are necessary in
order to conclude that the support value at angle 0i for the set Di is
T
Pi W.
T
Then,
since
S C D
we must have that
h.
Pi
..
With a bit
of algebraic manipulation (see Appendix A), this inequality may be shown
to be equivalent to the condition given by the ith column of (3.1).
This argument applies to each column independently, which shows the
necessity of (3.1).
The proof of the sufficiency of (3.1) is more complicated.
Here we
assume we have a vector h which satisfies the conditions (3.1); we must
show that h is a support vector.
which h is a support vector.
To do this we construct a set S for
Consider the following two sets
- 12 -
- 1+1
I
//ii~l
defined by lines Li 1 and Li+l and their normal vectors,
and
i+1
The support line with normal vector wi must lie to
the left of the dotted line for consistency.
--
-
-
--
-----·
ii1
-----
·--
-12a -
~1----"--1----~--`--11----i-
constructed from the vector h (see Figure 9):
{
{u
sB-=
e
2IRI*
2
l1
'" M]
<-
[hl h2 "'
hM] }
(3.2)
and
=
S
hul(v1,
v 2. '
(3.3)
M)
where the vi's are points defined as the intersection of two lines
n Li+
vi
Li
L
2
{ uEIR
where
1
uT
1
= hi1 }
(3.5)
and hul(.) denotes the convex hull (of a set of points in this case).
Now suppose that
S B = SV.
support vector of the set
following.
Then it must be true that h is the
S =S B = S .
One method of proof is the
(See Appendix A for an alternative proof of this fact.)
First note from the definition of SB in (3.2), that we must have
sup
x Wi
xES
On the other hand, v.i
<
SV = S B = S
h.
1
1
and
T
v.U.
= h.. Consequently, hi
is the support value at this angle.
What remains to be shown, then is that condition (3.1) implies that
SB = S .
The lengthy details of this part of the proof may be found in
Appendix A.
o
The immediate use of the support theorem is as a test of
consistency.
Given a test vector h we may determine whether h specifies
a consistent set of support lines by evaluating hTC and seeing whether
the elements of the resultant row vector are all negative.
From an
estimation viewpoint, we see that if we are trying to estimate a support
vector h from a set of noisy measurements, then we must make sure that
- 13 -
L3
SV
2
L4
L2
L5
Figure 9. The set SB (shaded region) is the intersection of
the half spaces determined by lines
vectors.
The set S
L1 - L5
and their normal
(whose boundary is given by the thickened
line segments) is the convex hull of the vertex points
U1 - v5'
- 13a -
h
our estimate h satisfies
^T
h C < O.
In the following section we examine
these constraints in more detail.
3.2
Geometry of the Support Cone and Implied Objects
· = { h e EM I hTC < [O ...
The set
0] } is a convex polytope cone
We
which defines the consistent set of M-dimensional support vectors.
shall call I the support cone.
if h is in ·
property of cones:
It is a cone because it obeys the usual
then ah (a > O) is also in I.
It is a
polytope because it is the intersection of a finite number of closed
half spaces in IR.
The algorithms of Section 4 are inherently
constrained optimization algorithms because of this constraint cone;
they are, however, of the simplest type because the constraints are
linear.
In what follows, we study the structure of the support cone and
what this structure reveals about the objects supported by vectors in
the cone.
3.2.1
The Basic Object and the Vertex Points.
Although a support
vector does not define a unique set S in the plane, it is useful for us
to think of a particular object (set) implied by h.
In this paper we
use the largest convex set consistent with the support vector h as the
implied object -- we call this the basic object.
The basic object is a
convex polytope with vertices that may be found directly from the
support vector h.
We now give more precise definitions for these ideas.
The Basic Object.
Let
h = [h1 h2 ...
hM]T
be a support vector.
We define the basic object as the set
SB
=
{ u eC 2 I
uT[wl
M
M...
[h ]...
An example for M = 5 is shown in Figure 10.
- 14 -
hM
.
(3.6)
It should be clear that
L3
V2
'l
Z,
a
LL
Li
L5
Figure 10.
,4
The basic object, SB' is the largest convex set
consistent with support vector h.
support vector h, SB = S'.
necessarily distinct.
For lines Li
defined by a
Note that vertex points are not
This occurs when 3 or more of the Li
have a common intersection point, in this case L 1, L2 , and L.5
Note that this corner of SB is "sharper" than the others.
- 14a -
any set which has h as its support vector is contained in SB.
is the largest set whose support vector is h.
Hence, SB
If the true object, S, is
known to be convex, then as M is increased the set SB becomes a better
If S is not known to be convex, then SB may never
approximation to S.
be a good approximation to S, but will closely approximate the convex
hull of S, denoted hul(S) as M increases (see Figure 11).
In the proof of the Support Theorem we used the set
Vertex points.
S
V
defined as
SV
=
(3.7)
'
"'
v2
hul(v,
where hul(.) denotes the convex hull (of a set of points) and the points
V 1 ... v M are given by
vi
=
Li
n
(3.8)
i=l,...,M
Li+ 1
We proved that when h is a support vector it must be true that
SB
Therefore, the points
vI ,.... vM
object (see Figure 10).
=
SV
must be extreme points of the basic
Ordinarily, one would call such points
vertices, but since the points vi are not necessarily distinct (as
demonstrated in Figure 10), we shall call them vertex potnts.
The
transformation of the support vector h to the set of vertex points is
the most convenient way to describe the basic object, since any point in
the basic object is simply a convex combination of the vertex points.
Let us examine this transformation in some detail.
vi = L i n Li+ 1
Since, by definition,
we see that vi
must
simultaneously satisfy the two linear equations
T
Vi w.
11
h.
1
and
vi
1
T
i+
1+1
=
h
i+l
which is solved as
~~~~T ~-1
Vi
=
hi
hi+l][oi
- 15 -
i+]
*
(3.9)
SB
Figure 11.
he IR5
S
he IR ' °
As M increases, SB becomes a better approximation
to hul(S) (whose boundary is indicated by dotted lines) but
will never be a good approximation to S.
- 15a -
The matrix inverse is found as follows
-1
Doi
oi+l -
cosOi
cosei+l
=sinOi
s
1
= 0i+l-0.
0
=
1
1
sine
Eisine.n
sinOi+l
0
where
= 2-X/M.
-sine. 1
h
Radius of Curvature.
meaning of the vector
-cos0i+
coso.
Hence,
h.-°i+l
[h
i
i+1
sini+
-coso
i
ii-sin0.
1
(3.10)
cose.
1
In this section we interpret the geometric
b = h C.
We show that the negative of each of
the entries of the vector b is proportional to a discrete notion of
"radius of curvature" for the boundary of the basic object corresponding
to the support vector h.
The radius of curvature vector
p = -b
is
used in Section 4.2 as a "smoothness" criterion for the estimated
support vector.
We now give the motivation for such usage.
For two dimensional plane curves, the curvature K is
defined as the
rate of change with respect to arc length along a curve of the angle p
that the unit normal for the curve makes with the x-axis
dO
ds
The radius of curvature A is then the reciprocal of the curvature (see
[Thomas, 1972], for example)
1
The analogy to the basic (polygonal) objects of this paper is
straightforward.
Referring to Figure 12a we see that the unit normal
angle changes by 0 0 over an arclength of f.,
- 16 -
th
the length of the i
face.
&
hi-
..--
fi
eo
(a)
Pi
p
Twi-hi
(b)
Figure 12.
(a) The ith discrete radius of curvature, ri, has
an intuitive meaning as the reciprocal of the change in angle
over arclength.
(b) Pi is proportional to ri through the
geometry.
- 16a-
One may then write, by analogy with the continuous case, the i
"radius
of curvature" as
r.
i
As
AO
=
fi
00
=
-
But fi may be written in terms of familiar quantities.
reminds us of the geometric meaning of the quantity
used in Section 3.1 to derive the Support Theorem.
always non-negative since the vector
p = -h C
p = [P1 P2 ...
where h is a support vector.
Figure 12b
Pi = PiT i - hi'
We note that Pi is
pM]T
PM]T
satisfies
By simple trigonometry we have
that
2P i
tan0O
fi
so that the ith radius of curvature, ri, becomes
2 Pi
r
i
Otan8O
=
Hence, we have that
T
1
2 hC
rtan0
2
...
rM]
(3.11)
establishing the proportionality of p to the discrete radii of
curvature.
To relate the vector p to "smoothness" of the boundary of the basic
object, note that when an element of p, say Pi, is zero, then the ith
face does not exist.
Thus the boundary normal changes by 20o at the
vertex instead of the usual 00 ---
the boundary is sharper.
corresponds precisely to the case in which several of the vi
as illustrated in Figure 10.
is larger.
This
coincide,
A smoother object would be one in which Pi
We see that the smoothest objects are the ones for which the
faces all have equal length, i.e., p = a[l 1 ...
positive constant.
- 17 -
T
1]
where a is
some
3.2.2
Eigen-decomposition of C. A very important property of the
matrix C is that it is circulant.
Because of this property, C is
diagonalizable by the Fourier transform matrix and its eigenvalues are
given by the discrete Fourier transform of the first row of C [Bellman,
1970].
Therefore, the eigenvalues of C are,
M
-j2-r(k-1)(n-1)/M
=
Xk
Cln e
k=l,...,M
(3.12)
n=l
where c.. denotes the element of C in the ith row and jth column.
xj
The
eigenvectors (the columns of the Fourier matrix) are
ek
where
=
WM = exp(
M2).
=
2
WM
WM
WM(k-l)[1
...
WM(M-1)]T
It is straightforward to show that
1 -
.
cos(27r(k-l)/M)k=l,...,M
cos(27r/M)
(3.13)
We immediately see that two eigenvalues (and only two) are identically
zero, X2 = AM
=
0; hence, C is singular.
A basis for the null-space (or
left null-space since C is symmetric) is given by the two eigenvectors
e2
=
eM =
[l
WM
W2
...
WM(M)]T
WM(M-1).[1
WM
WM2
...
WM(M-1)]T
WM
Since the support vectors h are real we are most concerned with the
null-space represented with a real basis.
With some manipulation it may
be seen that an equivalent real basis for the null-space of C is given
by
n1
=
[1
cosO 0
cos200
...
cos(M-1)00]T
n2
=
[O
sinO 0
sin2O0
...
sin(M-)0]T
(3.14a)
and
.
(3.14b)
We return to properties related to the eigenvalues and eigenvectors in
later sections.
singular.
For now we examine the consequences of C being
3.3.3
The proper support cone f
p
In terms of the geometry of the
.
support cone, one consequence of C being singular is that the support
C, cannot be a proper cone --
cone,
there is a linear subspace (of
dimension 2) contained entirely in I.
This implies that the support
cone is composed of the Cartesian product of a proper cone and a linear
subspace.
This linear subspace is obviously given by X = span(nl,n 2 )
where n 1 and n2 are given in Equation 3.14.
Defining the matrix
N = [n1 : n2 ]
(3.15)
we may then identify the proper cone as
'P
=
{hR
0 : 0 0 : 0 0] }
-N : N] < [O ...
hRT[C
(3.16)
.
This then assures that support vectors may always be decomposed into two
orthogonal components as [Dantzig, 1963]
h
where
h EC p
p
p
and
h
n
=
hp
+
h
n
,
(3.17)
C N.
We see in the following section that the nullspace component of a
support vector h has a very appealing geometric interpretation.
3.3.4
Properties of The Nullspace of C.
In this section we show
that the component of h which lies in the nullspace of C corresponds to
a shift
of the implied object in the plane.
A support vector h which
has no component in N corresponds to an object which is centered on the
origin in a sense we will define.
Shift Property.
The shift property of support vectors states that
adding a vector from the nullspace of I to a support vector h, produces
a new support vector whose basic object is a shifted version of h.
see this we first use (3.14a,b) and (3.15) to rewrite (3.6) as
SB
=
{ue
2
uTNT
< hT }
-
19 -
To
Now define the set S as a shifted version of SB
S
=
SB + v
2.
where v is an arbitrary vector in R
then the difference vector
T.T
w - v
Suppose w is an element of S;
must be in SB, hence
T
T
(w-v)TN
hT
<
Then w clearly satisfies
hT
wTNT
+ vTNT
=
(h + Nv)
But since Nv is in X, the nullspace of the support cone, then
h + Nv
is also a support vector, and, in fact, must be the support vector for
the set S.
hn = Nv
The reverse logic holds as well --
adding a null vector
to a support vector h shifts the basic object by v.
From the above arguments we see that the
The "Centered" Object.
nullspace component of a support vector relates to the position of the
It turns out that a useful definition of the
basic object in the plane.
position of a basic object is the average position of its vertex points,
v.
We will show that the support vector is related to v by
M
=(v
=
v
v1Vi=
M
M2 hTN
.
(3.18)
i=l
To see this, we use the expression for the vertex points obtained in
(3.10)
1
V
=
~[sinoi+
-cose i+1
siio [hi hi+ 1 ]
sn0 hi-sinO.
cose.
Expanding (3.18) for vx using (3.19) gives
14
Ux
=M
i=1
1
sinO
sin
[hi
sinO
-[Tsine
2
Msin8O
0
sin
sinOe
hi+ 1]
-sine 10
sinO
sinO
hT sine
sinOeM
- 20 -
2
1
(3.19)
h(e 1 - e2)
Msin
MsinG0
(3.20)
using the fact that the indices are modulo M.
The sum of the y
components expands in a similar way
vy-
M
yL0sinG
M
s 1
[
i+
[cose
cos6
Y
_ _._.
1
[h 3
T
sin
i
T [
hi+l]
h (e3 -
4
e 1 -e
2
i
1
]
(3.21)
e 1 -e2
sin
and
e3 -e4
as follows
sin(G 1+00)-sin(Ol-00)
sinM
e
sinOe
1
)
Now we simplify the expressions for
sin
s
cos M-2
cos81
0MsinS0
r
Msi
=
cos.j
sn
coseo O
cos 221
2sinSO s
01 cOM
1
.cosO
M
sin
[cos
.
=
2sin80 n1
(3.22)
and
coso1
cosO
cos(s +0o)-C°S(Ol-O0)
rCOS-21
os1M-
COS[ M
[cosOj
j
Lcos(OM+GO)-cos(OM- 0O)
sinO
sin 2
2
sin0:
sinO.
-
2sinG
0 n2
.
(3.23)
.
Combining (3.20)-(3.23) and using the definition of N from (3.15) yields
(3.18).
- 21 -
We shall see in Section 4 that (3.18) can be used as a constraint
on estimated support vectors if the position of the true object is known
a priori.
Note, in particular, that when h has no nullspace component,
i.e., h is in ·
,
then
T
hTN = 0
and, therefore,
object is centered on the origin.
- 22 -
v = 0 ---
the basic
4.0
ALGORITHMS
We present three signal processing algorithms based on the ideas
developed in Section 3.
The basic idea is as follows.
Suppose we
obtain noisy measurements of M support values at the angles
i.= 2ir(i-1)/M for i=l,...,M.
It is likely in this case that the
support measurement vector is not a feasible support vector.
Therefore,
a first objective of the following algorithms is to obtain a feasible
support vector estimate from the measurements.
The second objective of
these two algorithms is to use a priori information to guide the
estimate toward "preferable" or "optimal" estimates.
4.1
The Closest Algorithm
The reason this algorithm is called the "Closest algorithm" is
simple:
we obtain as our support estimate that (unique) support vector
which is closest to the measurement.
Not only is this an intuitively
simple thing to do, it is also the "optimal" constrained maximum
likelihood estimate given the data.
In this section we develop these
ideas and describe the algorithm.
Suppose the observed support values are given by
yi
=
hi
+
ni,
i = 1,...,M
(4.1)
where h. are the ideal support values which we are trying to estimate
and ni are samples of independent white Gaussian noise with zero mean
2
and variance a .
concerning
In the absence of any prior probabalistic knowledge
h = [h1 ...
of h given the data
hM]T
y = [Y1
we desire the maximum likelihood estimate
... yM] T
the supportcone defined in Section 3.
- 23 -
and subject to
h E V
where ·
The log likelihood function is
is
given by
L(h)
=
(y-h) - -ln(2r
(y-h)
2
(see [Van Trees, 1968], for example).
2)
(4.2)
The maximum likelihood estimate
of h given y is given by
hML
=
(-
argmax
h
1
2
T
(y-h) (y-h)
(4.3)
hTC < O
which may be found by solving the quadratic programming (QP) problem
minimize
-_ hTh
subject to
hTC
hy
-
(
0
(4.4a)
.
(4.4b)
We see that hML is the support vector in C which is closest (in the
Euclidean metric) to the observation y.
the variance of n, and if y is in ' then
The estimate is independent of
ML = y.
Many efficient general purpose QP algorithms and codes exist (see,
for example [Land and Powell, 1973]) that solve (4.4).
The algorithm we
use is described in Goldfarb and Idnani (1983); it is an active set
"dual" method, leading to speed and efficiency.
taken from M.J.D. Powell (1983).
The FORTRAN code is
The experiments and results of the
Closest algorithm are described in Section 5.
One should note that the algorithm needs only a minor modification
if the noise vector n has an arbitrary covariance, K = E[nnT], rather
than the covariance
a2I
as was assumed above.
In this case, the
optimal constrained maximum likelihood estimate is the solution of the
quadratic program
minimize
subject to
1
hTK-h - hTK-1y
hTC < 0
(4.5a)
(4.5b)
which may be solved using the same code referenced above.
One case
where one would model the noise with this more general covariance might
- 24 -
be in tactile sensing where the robot manipulator is "preloaded" at
certain angles due to gravity, making the effects of backlash angle
dependent.
Another case is in computed tomography, where detection
accuracy of the support lines may depend upon the eccentricity and
orientation of the object.
4.2
The Mini-Max Algorithm
In this section we describe a second algorithm used to estimate the
support vector from noisy observations.
In this case, however, we
assume the noise is known to be bounded.
The optimality criterion we
choose is to maximize the minimum "radius of curvature" of the basic
object implied by the support vector.
This leads to a linear
programming (LP) problem which we solve using a primal simplex method.
Suppose, as before, we have measurements given by
Yi =
hi
+
ni
i=l,....M
(4.6)
where n, are independent, identically distributed noise samples, known
to be uniform over the interval [--,-].
For example, this type of error
might be expected when backlash from the gears of a manipulator arm is
the dominant noise source.
Given the vector of measurements,
T
y = [Yi... .y] , we see that, ignoring the support cone, the true support
vector h, has equal probability of being anywhere in the hypercube
(referred to later as the measurement box)
{x E
M
I
[-r -r
.
T < x -y
<-
and zero probability of being outside this hypercube.
]
(4.7)
Of course, since
we are looking for only feasible h's we insist that the estimate we
produce is also in the support cone, hence, the estimate we seek will
have the following property
hM-M
e
.
n
(4.8)
· n
f
From the development, we can see that
cannot be empty, but that,
in general, (4.8) by itself does not guarantee a unique solution.
As a start towards specifying a unique solution we introduce a new
criterion related to our expectations concerning the shape of the source
object.
Suppose we expect the object to have a "smooth" boundary; i.e.,
we expect to have no sharp corners on the boundary.
One may then choose
to drive the estimate toward a support vector whose basic object has a
boundary that maximizes the minimum radius of curvature, subject to the
condition (4.8), of course.
Thus, we seek the h in
maximizes the minimum radius of curvature.
!
n
v
that
Accordingly, the Mini-Max
support estimate is defined as
hMM
=
T
min ( -h C )
argmax
h
(4.9)
using the definition of radius of curvature given in Section 3 (Equation
3.11).
We now show that the solution to (4.9) may be found by solving the
dual linear program
maximize
uTb
subject to
u A
T
(4.10a)
<
c
where u, b, A, and c are defined below.
must have that
(4.10b)
Since all feasible solutions
h C < 0, (4.9) may be rewritten as
hM M
=
max ( h
argmin
T
T
h,
hTc 2
...''
T
hTc
(4.11)
hEf
where
c 1 ,.... cM
are the columns of C.
The objective function that is
minimized may be identified as
m(h)
=
T
max ( h c
T
1
hTc,
c
2
- 26 -
... ,
T
h c
hM(4.12)
(4.12)
so that m(h) must be a number such that
m(h)
hTc
i=l,...,M
We seek the smallest such number m over all h in
(4.13)
t
n
g.
Consider the
two augmented vectors
u
1
=
b
and
=
(4.14)
Clearly the problem is to maximize uTb; this is the linear objective
function of the dual linear program of Equation (4.10a).
Now consider
the constraints of Equation (4.9) using the augmented vector u.
that it is possible to write the necessary constraints as
We see
uTA < c
(Equation 4.10b) where
C
C
A
,[-1 -1
c
=
I
l
l
=
...
-1]'[O 0 ...
[ 0 0 ...
-I
:(4.15)
l
Ol'[O 0 ...
0 : 0 0 ... 0
:
0]o'0 0 ...
hb
O]
-h
]
(4.16)
where
h
a
=
y
-
[r Y ... r ]T
hb
=
y
+
Iv 7 ...
and
v]T
(4.17a)
(4.17b)
The first partition of A and c defines the objective function
constraints given in (4.13); the second partition specifies the support
constraint; the third and fourth partitions restrict h to lie within the
observation box I.
If any feasible dual solution to (4.9) were known, then the dual
simplex method could be implemented directly to find the optimum
solution (see [Luenberger, 1984], for example).
Unfortunately, no
feasible dual is known a prtort; therefore, we chose to implement the
- 27 -
Mini-Max algorithm by solving the primal linear program (the dual to
Equation 4.9)
T
minimize
subject to
c x
Ax = b
x > 0
and
(4.18)
which we solve using the usual Phase 1/Phase 2 (with Bland's
anticycling) approach of linear programming [Luenberger, 1984].
to the primal problem of (4.18) is found, the
Once the optimum x
Let K be the set of column indices of
dual optimum is found as follows.
A, and Z be the subset of K denoting the elements of x
zero.
which are not
That is
K
=
{ 1, 2
Z
=
{ j
}
... ,
K
xj
and
(4.19)
(4.20)
0
If the cardinality of Z is M+1 (the number of rows in A) then, since x
must correspond to a basic feasible solution, we have that the optimum
dual solution is
u
=
[c
...
c
1
2
c
][a
ZM+
az
1
ZM+1
a
1
2
(4.21)
where the z.'s are the elements of Z, the c's are the elements of the
cost vector c, and the a's are the columns of A.
The optimum support
vector is given by the first M elements of u, according to the
definition of u in Equation 4.14.
If the cardinality of Z is less than M+1 then Z must be augmented
so that it has a total of M+1 elements.
arbitrary elements of K so that
The augmenting indices are
B = [a
independent columns.
...
a
1
2
]
a
has linearly
ZM+1
Then (4.14) is used, as before, to find h.
However, since the columns of A used to augment B are not unique, there
may be several different solutions.
Thus, in general, the Mini-Max
procedure does not yield a unique solution.
We will see in Section 5
that one way in which this non-uniqueness presents itself is as an
- 28 -
arbitrary shift (within a finite range and in a certain direction) of
the basic objects corresponding to the estimates.
In other words, the
estimate that the algorithm produces may be changed by adding a
component from the nullspace of V ---
and the new support vector remains
feasible and has exactly the same cost.
Sparse Data.
The Mini-Max algorithm is applicable with minor
modifications when there are fewer than M support line observations.
One might think of this as obtaining a feasible support vector estimate
which is simultaneously interpolated and optimized for the smoothness
criterion.
For example, given 10 support measurements at angles
corresponding to a 10-dimensional support vector, this method could
obtain an estimate of a 20, 30, 40, etc. -dimensional support vector
that maximizes the minimum radius of curvature subject to the constraint
that the elements corresponding to the 10 measurements must lie within
the uniform noise bound around the measurements.
Consider for simplicity the case in which we wish to reconstruct M
support lines given only N observations, where M/N is integer, and the
observations are evenly distributed in angle.
For example, if
M/N = 3
then the augmented observation vector y would look like
]
Y = [Y1 * * Y4 * * ... YM-2 * *
(4.22)
where the asterisks represent no observations, hence no explicit
information about the support line at the corresponding angle.
The
Mini-Max estimate is still given by Equation 4.9, however, the function
of the observation box must be redefined.
Clearly, only those elements
of hM M that correspond to true observations must be confined by the
observation box; the other elements of the estimate are only restricted
by the support cone constraint.
- 29 -
The problem is still solved using the linear program of (4.10), but
the matrices A and c are slightly different.
c are simple, however.
The modifications to A and
In the last two partitions of A and c (equations
4.15 and 4.16), remove the columns (and thus the constraints)
corresponding to the asterisks (the unobserved angles) of y in Equation
4.22.
The new matrices A and c are shown in Appendix C.
The solution
is found precisely as before using the primal algorithm first, then
solving for the dual optimum, hMM.
4.3
The Close-Min Algorithm
In section 5 we shall see that the Closest algorithm and the
Mini-Max algorithm produce two rather extreme estimates.
The Closest
algorithm produces estimates that strongly resemble the data, and the
Mini-Max algorithm produces estimates that strongly resemble our prior
expectation (circles).
In a way similar to maximum a posteriori (MAP)
estimation [Van Trees, 1968], the Close-Min algorithm is designed to
combine the two criteria, thus producing estimates that will reside
somewhere between these two extremes.
The concept is simple:
we define
a new cost function which is a convex combination of the Closest and
Mini-Max objective functions.
Relating this to MAP estimation, we see
that the Closest objective function plays the role of the measurement
density and the Mini-Max objective funtion plays the role of the
a priori density.
We do not have an optimally defined trade-off between
the two objective functions as is usually the case in MAP estimation;
instead we shall use the convexity parameter, a, which may be varied
between 0 and 1.
- 30 -
The function we maximize in the Closest algorithm is
fC(h)
-~-
=
(y-h)T(y-h)
(4.23)
and in the Mini-Max algorithm we maximize
=
fM(h)
min(-hTC)
(4.24)
We therefore define the Close-Min estimate as
hCM
where
0 < a < 1.
=
afc(h) + (1-a)fM(h)
argmax
h
(4.25)
Since fc(h) is quadratic and fM(h) is linear, we may
solve (4.25) using the QP algorithm.
The statement of the QP is not as
simple, however, because of the necessity to augment h as in the
Mini-Max problem.
Using u and b as in (4.14) we find that (4.25) may be
restated as the QP
minimize
a(
subject to
u A
uT
<
[ ]
u- u
)
+
(-a)
uTb
(4.26a)
c
(4.26b)
here I is the MxM identity matrix and y are the observations.
The
quantities A and c may be as given in (4.15) and (4.16) although
provided
4.4
a X 0
the observation box constraints may be removed.
Shift Corrected Algorithms
As we suggested at the end of Section 3, each of these algorithms
may have a priori positional information included in the estimation
process.
That is, suppose one knows that the true object is centered on
v, in the sense that the average position of its vertex points is v.
Then the estimate should also be centered on v.
But from Equation
(3.18) we see that this may be assured provided that we enforce the
following linear constraint
hTN
=
v
.
Since this is just another linear constraint, (4.27) is easily
- 31 -
(4.27)
incorporated into the three algorithms with minor modifications to the
matrix A and vector c (see Appendix C).
With the addition of the shift correction, we now have 3 distinct
types of linear constraints:
(1)
the support cone constraint which is always imposed,
(2)
the measurement box constraint which is imposed for the
Mini-Max and Close-Min algorithms, and
(3)
the shift constraint which is imposed when the a priori true
object position is known or if a good estimate of the object's
position is available.
Combining these constraints with the three basic objective functions
allows us to produce a variety of estimation algorithms that favor some
object characteristics over others, as we shall see in the following
section.
- 32 -
5.0
EXPERIENTAL RESULTS
The results contained in this section are examples that illustrate
the behavior of the algorithms.
In the first section, Section 5.1, we
discuss the generation of the data and the methods for producing the
figures used to display the results.
results for the three algorithms:
Section 5.2 contains comparative
Closest, Mini-Max, and Close-Min.
Section 5.3 shows the behavior of the sparse data and shift-corrected
modified algorithms.
5.1
The Data and Plots
The advantage gained by using the proposed algorithms can be
demonstrated nicely using a small (low dimensioned) problem.
All the
examples shown herein, have a 10 dimensional observation vector; and in
all cases except one, the restored support vector has 10 elements.
The
exception is in Section 5.3.2, where the "sparse" mini-max algorithm is
used to reconstruct a support vector of dimension 30.
To show graphically the basic object to which a feasible support
vector corresponds, it is sufficient to simply determine the vertex
points {v1 v2 ...
VM) and connect them sequentially.
(This is not a
trivial observation; a proof that the vertex points are leftward turning
allows one to conclude this.)
However, for an infeasible measurement,
such as each of the observation vectors we shall use, one does not have
the condition that
S B = SV.
Ordinarily, one might display S ,
since,
by default, this must be the set if the measurements are assumed to be
accurate.
Instead, however, we shall display the infeasible
measurements in precisely the same way as the feasible support
- 33 -
functions:
connect the vertex points sequentially.
a vertex plot.
We call such a plot
It can be shown that for an infeasible vector, the
resulting curve will not be simple, i.e., the curve will cross itself.
Therefore, for the small examples we show here, it is easy to spot the
infeasible vectors simply by observing the crossing of the vertex plots.
One should also note that it is easy to visually construct both sets SB
and S
from the vertex plot (refer to Figure 9, for example).
To show the behavior of the three algorithms, we use
noise-corrupted measurements of the support vectors of two objects.
The
first of these objects is a circle centered at the origin, and in this
case the noise-free support vector is
h
[r r... r]T
=
c
where r represents the radius of the circle.
The second object is an
ellipse, centered on the origin, with principle axes corresponding to
the x and y axes.
he
=
ei
=
el
In this case the noise-free support vector is
e
...
eM]
where
cos2
([i-1]00
)
+ b2sin2([i-l]00)
and a and b are the ellipse radii corresponding to the x and y axis,
respectively.
The two true figures we use have the same area.
radius 1/2, thus area vr/4;
The circle has
the ellipse has a=3/4 and b=1/3.
The
experiments shown here involve also two different noise levels.
One set
of noise values, denoted by n1 , has elements drawn independently from a
probability density which is uniform over the interval [-r
- = 0.2.
a], where
The second, n2 , has elements drawn independently from a
probability density which is uniform over the interval [-r
- 34 -
a],
a
= 0.4.
We therefore use four support observation vectors:
YC
=
hc + nl
YC
=
h
=
h
=
he + n 2
+ n2
C2
Y
e1
Ye
2
e
+ n
e
The vertex plots for Ycl,
Yc 2
Yel
and Ye
are shown using solid lines
in Figures 13(a), 14(a), 15(a), and 16(a), respectively.
Appearing in
those same figures, using dotted lines, are the corresponding true
support figures h
and he .
The scales are different on each figure to
accomodate the different sized noise vector contributions.
5.2
Comparison of the Algorithms
Figures 13(b), 14(b), 15(b), and 16(b) show the basic figures
corresponding to the estimates produced by the Closest algorithm (dashed
lines), Mini-Max algorithm (solid lines), and Close-Min algorithm for
a = 0.5
figures.
(dotted line) for each of the observations indicated in the (a)
The same scale is used on the (a) and (b) figures
corresponding to the same figure number so that comparisons can be made.
We now proceed to discuss each of the experiments in order.
Unfortunately, it is too confusing to superpose all four vertex plots
onto the same figure; instead we must make careful observations using
the fact that the scales are the same on each figure (a) and (b).
In
order to indicate points on each of the figures, rememember that the
support value corresponds to the right hand vertical face;
increase in a counterclockwise fashion.
- 35 -
1 St
the indices
---- 1
True Object
Observation y=0.2
- I
.I
-1
L
0
'
'
~-- ~Closest ~1
(b)
'
1
'
I
-- 1
t-
-1
--I
Observation y=0.4
I
Figu
e I.
1,
Observation
-
-1
1
--
T=0.4
i
0
Mini-Max
-. Close-Min
- Closest
:i
-1
/.
0
1
(b)
Figure 14. (a) The true object (circle) and the measured
support vector (y=0.4). (b) The estimates produced by
the three basic algorithms.
-
35b -
SB
-- 1
---I
True Object
Observation 7=0.2
.
-1
.
I
I
1
0
(a)
0
-- i1
Mini-Max
.... Close-Min
- - Closest
-1
1
0
(b)
Figure 15. (a) The true object (ellipse) and the measured
support vector (y=0.2). (b) The estimates produced by the
three basic algorithms.
-
35c -
-1Observation =0.4
-1
0
1
(a)
1
--
Mini-Max
-.... Close-Min
-- Closest
-1
-1
0
1
(b)
Figure 16. (a) The true object (ellipse) and the
measured support vector (y=0.4). (b) The estimates
produced by the three basic algorithms.
-
35d -
In Figure 13, the observation vector is almost feasible; we can see
only one spot (involving support lines 8, 9, and 10) where the vertex
plot crosses itself.
The closest support vector is one that just
corrects the point of inconsistency., as can be seen in 13(b).
At first
glance, the Mini-Max estimate appears to be quite good, however, if one
examines the size and position of the basic object one finds that the
figure is too big and shifted upward.
The fact that it is too large
simply reflects the fact that large basic objects tend to have larger
radii of curvature --- that agrees with our chosen criterion.
The fact
that it is shifted up will be shown in Section 5.3 to be at least partly
due to a non-uniqueness in this solution.
We see that, as expected, the
Close-Min algorithm produces an estimate that "blends" the two extreme
solutions of the former algorithms.
Note in particular that in the
Closest estimate, support line 2 corresponds to a degenerate extreme
point of the basic object; the Close-Min algorithm smooths out this
sharp corner.
Figure 14 shows a more severe degradation of the true support
vector; there are 3 points where the vertex plot of the support
measurement cross.
As before, the Closest algorithm produces an
estimate that seems to simply remove the points of inconsistency.
In
this case, one sees that the Closest estimate has only 6 vertices,
therefore 4 vertex points are degenerate.
In fact, the upper vertex of
the figure corresponds to 3 vertex points --in the boundary.
this is a very sharp turn
The Mini-Max algorithm produces the desired
smoothness, but as before, creates a figure that is too large and is
shifted (down in this case).
"blend" once again.
The Close-Min algorithm shows the correct
It is important to point out that the set S B
- 36 -
(crosshatched region in Figure 14(a)) constructed from the raw
measurements, is a bad estimate of the true set, thanks to the extremely
large noise value.
It is too small and is shifted down.
The fact that
it is small comes from the fact that the construction of SB essentially
ignores the support lines that are farthest out.
Each of the algorithms
proposed here use all of the measurements to "pull" the inner support
lines out, if necessary.
There is not much new information about the estimation algorithms
to be derived from Figure 15 except that the Mini-Max estimate tends to
be too circular.
This was also to be expected since a circle has the
smoothest overall boundary.
Figure 16 again shows that the Closest
algorithm has the effect of shifting neighboring support lines to fix
the points of inconsistency.
Also, the Mini-Max algorithms produces an
estimate which is more like an ellipse, but is still too large.
It
appears that the Close-Min algorithm is, as usual, the best estimate of
the true support vector.
Finally, to clarify some of the behavior of the Mini-Max estimates,
it is useful to examine the estimates together with the observation box
bounds ha and hb (see Equation 4.17).
Figure 17 shows the vertex plots
for the Mini-Max estimate (solid line), ha (dotted line), and hb (dashed
line), for the example shown in Figure 13 ---
that is, the noise-free
figure is a circle and the noise range is [-.2 .2].
It should be clear
from the figure that the estimated basic object can be shifted
vertically down and still remain in the observation box.
The
9
th
support line is the first to become infeasible if the figure is shifted
down too far.
Shifting the estimated basic object keeps the estimate
feasible, of course, since this merely corresponds to the addition of a
- 37 -
Mini-Max
-t1
hb
-1
.---
0
1
Figure 17. The Measurement Box. Each element of the Mini-Max
estimate is shown to lie within the corresponding elements
of ha and hb as prescribed by the measurement box constraint.
-
37a -
vector in the nullspace of '.
But, more importantly, such a change in
the estimate does not change the optimum cost of the linear program that
produced the Mini-Max estimate since no radius of curvature element is
changed.
Thus, this solution is not unique ---
any of the shifted
versions discussed above are optimal solutions --happened to find this one first.
the LP algorithm just
In the shift-corrected algorithms
discussed below, this component of non-uniqueness is eliminated by
imposing a known object position.
As we shall see, this simple
correction has dramatic effects on the Mini-Max estimates.
5.3
Modified Algoritbms
In Section 4, we discussed the shift-correction and sparse-data
modifications that may be made to the basic algorithms.
We now show
results of applying these modifications to the example of Figure 13.
5.3.1
Shift Correction.
Figure 18 shows three vertex plots
corresponding to the true support vector (solid line), the Mini-Max
estimate from Figure 13 (dotted line), and the Mini-Max shift-corrected
(for
v = O) estimate (dashed line).
We see immediately that the shift
correction does not simply shift the original MNni-Nax solution down.
To understand this we recall Figure 17.
We saw that due to
non-uniqueness of the solution that we could shift the solution down and
up over a finite range.
But, evidently, none of these shifted positions
causes the sum of the vertex points to be exactly zero.
To allow this
to occur, the (shift-corrected) Mini-Max algorithm was forced to shrink
the estimate as well.
We see from this result that prior information
about the position of the object may have a very strong influence on the
performance of the algorithms.
- 38 -
True Object
.... Mini-Max
--1
--
-I
-1
Shift-Corrected Mini-Max
,
.
I
I
1
0
Figure 18. Shift-Corrected Mini-Max estimate.
-
38a -
5.3.2
Sparse Data.
Now we implement the sparse-data Mini-Max
algorithm for the example of Figure 13.
In Figure 19 we show the
original Mini-Max estimate for M=10 (dashed line), and the sparse-data
Mini-Max estimate for M=30 (dotted line) as well as the measurement box
bounds, ha and hb.
With more degrees of freedom in the estimate one
would expect to see more circular objects in the sparse-data case, and,
indeed, that is so.
Since the 10 measured support lines are still
constrained, the estimates cannot get too much larger than the original,
but within the bounds of ha and hb , the new and old lines are arranged
in a more circular fashion.
We also see that the non-uniqueness related
to the shift of the estimates is still prominent, as the two trials
happened to produce estimates at opposite ends of the feasible shift
range.
- 39 -
1
--1
-
ha (inside) and hb (outside)
---
Mni-Max (M=10
....
Mini-Max (M=30)
-1
Figure 19.
0
1
Sparse-data Mini-Max estimates.
-
39a -
6.0
DISCUSSION
We have seen that knowledge of the basic geometrical constraint,
h C < 0, can be used to advantage when recontructing sets from noisy
observations of their support lines.
We have described and compared
three algorithms which utilize this support constraint as well as other
The Closest algorithm gives the constrained
criteria and constraints.
maximum likelihood estimate assuming the noise is Gaussian.
It requires
the minimum amount of prior knowledge about the set to be reconstructed,
and is implemented in a straightforward manner using quadratic
programming techniques.
The Mini-Max algorithm assumes bounded
independent uniform noise;
it must be guided towards a unique solution
by prior knowledge about the object; and it uses linear programming
techniques.
Finally, the Close-Min algorithm blends the former two
objective functions (similar to MAP estimation) to produce estimates
guided by prior information but not as strongly as in Mini-Max.
The examples studied in Section 5 show that:
1)
As an estimate of the basic object, SB,
in not very good.
It
always produces an estimate that is too small, because it, in effect,
ignores the support observations that are farthest out, thus placing all
the emphasis on innermost observations.
2)
For the Closest algorithm, small amounts of noise result in
estimates that closely resemble the observations; the algorithm merely
corrects the points of inconsistency.
3)
The Mini-Max algorithm is highly biased toward the prior
expectation ---
a large circle, in this case.
Also, an observation
which is just slightly outside of the support cone may cause the
- 40 -
estimate to differ greatly from the observation.
4)
The Closest algorithm always yields a unique solution; the
Mini-Max algorithm does not, in general.
5)
Prior shift information can substantially improve the
estimates.
6)
The Close-Min algorithm permits one to blend the prior
expectation of boundary smoothness with the criteria that the estimate
be close to the observations, thus implementing a kind of MAP criterium.
The primary contribution of this paper is in the formulation of the
problem as a constrained optimization problem which includes the
fundamental support vector constraint, a priori information, and
uncertainty in the measurements.
Many extensions are possible, both in
the inclusion of additional constraints imposed by prior knowledge or in
the development of more elaborate objective functions.
Among the
possible extensions is the inclusion into the estimation process of a
known object area or bounds on the areas.
The area of a basic object is
a quadratic function of h, however, so that, as a constraint, the
inclusion of this type of information makes the algorithms more
complicated.
A simpler extension involving the constraints may arise if
one only has partial information regarding the position of the object in
the plane.
If the position was known to be bounded, then instead of
having two lineae equality constraints as in the shift-corrected
algorithms, one would have four linear inequality constraints.
Other
extensions may include improvements on the objective function.
For
example, if the position was known only approximately, but could not be
considered to be bounded, then a quadratic penalty could easily be
- 41 -
incorporated into the objective function to try to keep the estimate
near the expected position.
As promised in the introduction, we wish to discuss more how we
believe that this research can be of benefit in the discipline of
computed tomography (CT).
We now know how, given estimates of a set of
support lines, we may reconstruct an estimate of the convex hull of an
object.
Figure 3 showed how one might seek to estimate such support
lines in the CT setting.
But given an estimate of the support of a
function f(x,y), does this aid in reconstructing f(x,y) from its
projections?
There is some literature that indicates how known convex support
constraints on f(x,y) can be used to aid in reconstructing f(x,y).
Among these methods are the method of "projection onto convex sets"
(POCS) [Sezan and Stark, 1984][Trussell and Civanlar, 1984][Youla and
Webb, 1982], dual methods of POCS
[Leahy and Goutis, 1985], and
constrained simulated annealing methods [Prince, 1986].
The difference
here is that we begin with an estimate of the support of f(x,y).
Thus,
a two-step hierarchical reconstruction method is envisioned:
(1) estimate the region of support, and
(2) estimate f(x,y) using the above estimate with a
constrained reconstruction algorithm.
There is much precedence in using estimates of certain parameters to aid
in reconstruction.
For example, the "dc" bias is quite commonly
estimated from the data and used to provide the "dc" bias of the final
estimate (see [Herman, 1980]).
here.
But there are many unanswered questions
For example, how does one estimate the support values from the
- 42 -
projections?, and should the estimated support be imposed as a
constraint or merely a penalty on the subsequent reconstruction process?
It is interesting to consider the above reconstruction process if
one were to know a priort that the function f(x,y) has the value 1 on S
and vanishes elsewhere, and that S is convex.
the reconstruction process;
Then step (1) completes
i.e., the problem reduces to a straight set
reconstruction problem, as discussed in the introduction.
Thus, this
work may be considered to be extending the model-based approaches to
reconstruction begun by Rossi and Willsky [Rossi and Willsky, 1984] and
Bresler and Macovski [Bresler and Macovski, 1984a,b,c], by considering a
slightly more general object (convex and binary with unknown boundary)
and using fundamental constraints related to the consistency of the
Radon transform [Louis, 1980] to form optimal estimates of the modeled
object.
- 43 -
APPENDIX A
The Support Theorem (Theorem 1) is proven in this appendix.
We
begin with two lemmas which are needed in the proof.
A.1:
Lemnm
R2
SB = { u
Let the set
I uTU < h }
Q= [w1
for w. = [cos i
sinoi]T
0
02
where
M]
."'
= (i-1)2r/M,
M > 5
and
Then SB is
bounded and convex.
Proof:
We show that SB is convex first.
For any
u,v E SB
0 < a < 1
and
we have
auT
+
] 2
=
auT++(1-a)v]
auQ)h
=h
Therefore, all the points on the line segment joining u and v are in SB.
Hence, SB is convex.
exist vectors
a > O.
u e SB
This implies
Now suppose S B is unbounded.
and
v E R
(u + av) T
such that
< h
Then there must
u + av
is in SB for all
vT
and, hence, that
< [0...0].
(Points v are called ray points of SB; we must prove that these cannot
exist.)
Writing v in polar coordinates as
v = [rcosp
rsinc]
T
we see
that the inner product of v with the columns of £ is given by the row
vector
p = [rcos(01 -p)
rcos(0 2 -p)
...
rcos(OM-q)]
Since the angles 0i are evenly distributed samples over 2w and
M > 5
we see that it is impossible to sample the cosine function so that all
- 44 -
sample values are negative; therefore, p cannot have all non-positive
elements.
o
Hence, SB cannot be unbounded.
Lemma A.2:
Let S be a set in R2 and L be the line
L = { u C R2I
uTW < t )
T
xw < t
(i)
(ii) L n S
for
where w is a unit vector.
x E S,
Suppose
and
is not empty.
Then L is a support line of S.
Proof:
Since w is a unit vector in JR it may be written as
T
w = [cosO sinO] .
Then, to show that L is a support line of S we must
show that the support value at angle 0 of the set S, given by
=
h(O)
sup
xTw
is equal to t. From (i) we see that
that there exists a point
y C S
h(O) < t. But from (ii) we have
such that
T
y ¢ = t. Therefore,
h(O) = t, which proves the lemma.
a
For convenience we repeat the Support Theorem and then commence the
complete proof.
Theorem 1:
The Support Theorem.
h C
ER
A vector
h
C
(M 2 5) is a support vector if and only if
< [O ... O]
(A.1)
- 45 -
where C is the MxM matrix given by
1
-k
-k
0
0
...
0
0
0
...
-k
and
-k
1 -k
-k
1
: -k
0
-k
0
1
k = 1/2cos(27r/M).
Proof of Theorem 1
(Support Theorem):
First we prove that (A.1) is necessary.
vector h is a support vector.
2
L. = { u E
We are given that the
Therefore, by definition,
I uT i = hi }, i=l,...,M, are support lines for some
(non-empty) set S C R . Then it must be true that S is contained within
the polytope
D.
D
(see Figure 4 in the main text)
2
xT[&i 1
=
xER
i1
[hi-1
h
1]
i+]
(A.2)
A2
for any i, where the index arithmetic is taken to be modulo M so that
the indices stay in the range
M 2 5, which implies that
Li_
1
1 < i < M. Note that by hypothesis
0i+1l-Oi
1
< r. This in turn implies that
and Li+ 1 have a finite point of intersection Pi, and that wi may be
written as a positive combination of ci-1 and Wi+1 ' These two facts are
necessary in order to conclude that the support value at angle 0i for
the set D. is Pi wT. Then, since
1
i'
S C D.1 we must have that
hiI < Pi1
We now show that the condition
th
i
constraint of
1.
hi <P.Ti
T
hTC < O. We have that
T
Pi
T[i-1
Oi+ll
c
=
[hi-l
(A.3)
hi+l]
- 46 -
is equivalent to the
Pi =L
L1
Li+
therefore,
Solving for Pi
T
gives
T
-1
Pi
[hi-1
hi+l1
[h i-1
hi+]
[wi-1i :
i+
80 = 2r/M.
where
i+l1
cosO i
1
cosoi+ 1
sinOi
1
sine
1
sin2
sinOi+l
-sinei 1
0
The inner product Pi
*i
T
is
=
[hi[h
hi+
hi1sinel
hi-1i
sin2O0
sinco 1
-sine
i0
1
i+1
-c9sO
i+
cosOe
I
|cosi
sine
sini
hi+1l
[hi-1
i
found as
i+1
1_
Pi
-cosOi+l
coseO
sin(h0
0
sinO0
sin200
(hi-1 + hi+l
)
Using the double angle formula sin200 = 2cosO0sinO 0 and the inequality
hi < Pi '
-h.
(Equation A.3) we see that
h
hi-1
i+1
+ 2cose0
2cos
hi1
th
which is precisely the condition specified by the i
(A.1).
column of C in
Since this applies to each i independently, this proves the
necessity of (A.1).
Next we show that
hTC < 0
is a sufficient condition for h to be
S C R .
the support vector of some set
We do this by construction.
Consider the two sets
SB
=
{u
Sv
=
hul(v1 , v2 .....
IR2| uT[
1
22 ... UoM]
<
[hi h 2 ...'h2M
(A.4)
and
M)
(A.5)
- 47 -
where hul(.) indicates the convex hull of the set of points v.,
i=l,...,M, given by
n
v
=
Li
L
=
{ u C
Li+1
(A.6)
where
2
u T.i=
h. } for i=l,...,M.
(A.7)
Now consider what would happen if the two sets SB and S
v
Then letting
were equal.
S = S B = S-, we see from the definition of SB that each
point x in S has the property
from the definitions of S ,
i. < h.
x
for i=l,...,M.
Furthermore,
v i, and Li, each line Li intersects S in at
S n Li
least one point vi, yielding the fact that
is
not empty.
Hence, from Lemma A.2 we may conclude that each line Li, i=l,...M, is a
support line of S.
vector for
Then it follows immediately that h is a support
S = S B = S-,
which proves the sufficiency of (A.1).
To complete the theorem, then,
S B = S.
that
S
we must show that (A.1) implies
This is done in two stages.
First we show that
SB C S ; then
C SB -
Since SB is a bounded (convex) polytope (by Lemma A.1), it may be
written as
SB
=
hul(e1 , e 2
..-.
ep)
(A.8)
where e. are the extreme points of S B (which are guaranteed to be finite
1
in number since S B is formed as the intersection of a finite number of
closed halfspaces).
Consider one particular extreme point of S ,B
ej.
It must satisfy with equality at least two inequalities in Equation A.4,
the defining expression for SB.
indexed by k.
Let one of those inequalities be
Then we have
eTk
U
hk
- 48 -
'
(A.9)
i.e., e. lies on the line Lk.
Two of the vi's also lie on Lk:
Vk-1 and
v k·
Now suppose e. could be written as the convex combination of Vk_ 1
and vk.
Then any extreme point of SB could be written as the convex
combination of 2 points in S .
then we must have that
And since both SB and S are convex,
SB C SB proving this stage of the theorem.
We now show that e. can, indeed, be written as the convex
combination of vk_ 1 and vk.
Vk- 1 = Vk
Vk_
and
1
Here, there are two possibilities:
X Vk.
Each of these cases require some
development.
In the case where
vk_1 = vk, we show that
ej = Vkl = vk
e. is clearly a convex combination of vk_ 1 and vk.
J
Vk are on the line perpendicular to 'k'
e.
where
=
&k
[
=
vk
+
First, since e. and
J
we may write e. as
Pklk
is the perpendicular to
O]k
so that
Wk.
Taking inner
products of both sides of the above expression with uk-1 and using the
fact that e. is in S B we may write
Ok- lTe.
k-1
=
+
Ok-1
and, similarly, for Wk+1
T
k+le
=
hk+1
+
6k
hk-1
TI
k+1
k
k+1
Hence,
TI
PWk-1
0
k
-<
and
P
k+lwk
<
0
After simplifying the above expressions using the definitions of Wk 1
-"-"
- 49 " I ""~"---~---~11----------~I---~-
0
--
Wk+l'
and wk '
we are led to the contradictory equations
P(-sin 0 O)
<
0
3( sin0O)
<
0
hence, p must be zero, and therefore
In the case where
Vk_1 • v k
ej = vk = Vk_ 1 ' as required.
we first need an auxiliary result
relating the unit vectors &k-l' Wk'
and Wk+l'
From the geometry it is
easy to verify that
k
0 0 = 2~/M.
where
=
2cosO
(k-1
Next, since ej, Vk_, 1
k+l)
+
(A.10)
and Vk all lie on the same
line, Lk, and vk_ 1 and vk are distinct points, we may express ej as a
linear combination of vk_ 1 and vk using the single parameter a
ej
=
auk 1 + (1-a)vk
(A.11)
.
Taking the inner product of both sides of (A.11) with
T
ej
T
k-1
=
auk-l1
k-1_
we have
T
k-1 + (1-a)uvk
a hk-1
k-1
+ (1-a)vkl k-1
<
hk-1
(A.12)
The last inequality results from the fact that ej is, by definition, in
SB .
Now we eliminate uk-1 from (A.12) using (A.10) yielding
ahkl + (l-a)vkT( 2 cos
o&k8
- wk+l )
hk-1
which may be further reduced to
(1-a)(2 cosOohk - hk_ 1 - hk+l)
Since from (A.1) the quantity
•
0
2cosO h k - hk_ 1 - hk+1
non-positive we immediately recognize that
(A.13)
.
must be
a < 1.
Taking the inner product of both sides of (A.11) with Wk+l and
using a similar sequence of steps leading to (A.13) one may show that
a( 2 cosOohk - hk_1 - hK+ 1)
from which we conclude that
a > 0.
•
0
Hence, we have that
- 50 -
(A.14)
0 ( a < 1
and
e.
is, in fact, a convex combination of vk_-1 and vk.
proof that
S B C SV.
Bv
Now we commence the proof that
vi E SB for each i=l,...,M.
that
This completes the
S
Therefore, we intend to show that
for all i=l,...,M.
•
wM]
...
In what follows, we show
Since SB is convex this is sufficient
to prove that S v is contained in SB .
ViT[w1
C SB.
...
[h
I
(A.15)
h 1]
Expanding (A.15) using
[cosoi.
=
(.
sinOi
and
T
Vi
-1
=
[hi
hi+l] [i
'i+l
]
and simplifying yields
1
sinOO [qil
]
qiM
qi2
h2
...
h
(A.16)
where
q..i
=
)
hisin( i+1 -
-
hi+lsin(0i - o)
Our task is to show that (A.16) is true given
(A.17)
hTC < 0.
Equation A.16 is true if each term is separately true.
Hence, we
must show
sin sinO
0for
hsin(Oi+
1
- 0j)
oj)
(each vi) and
i=l,...,M
-i
hi+lsin(O
j=l,...,M
-
)
(
h.3
(A.1S)
(each term in A.12).
Because
of the rotational symmetry of the problem we may, without loss of
generality, choose j=1 and prove that (A.18) is true for i=l,...,M.
0i = (i-1)2X/M = (i-1)0 0 ,
Since
we may simplify (A.18) to
sine0 ( h.sin(iOo) - hi+lsin((i-1) 0o) )
h
.
Denoting the left-hand side of (A.19) by Ei we have for i=l that
E
1sin
(
hlinO
-
)
=
h-
- 51 -
(A.19)
The general expression E. for
which satisfies (A.19) trivially.
j=2,...,M may be related to Ej_ 1 using the relation h C < 0 as follows.
From (A.19) we have that
E
=
- hi+lsin((i-1)0 )
sine(io)
Using the double angle formula
sinia
2sin(i-1)acosa - sin(i-1)a
=
this becomes
sin1
sine0
E.
1
[ hi[2sin((i-1)0 0 )cosO0 - sin((i-2)00] L0]
hi+lsin((i-1)00)
sine
0
-
[
(2hicoso
- hi+ 1 )in((i-l)00)
(A.20)
- hi sin((i-2)00
Now we notice that the it h constraint in
hi - h
11
i+l
2cos0i
<
hTC
h
0
may be written as
i-1
Using this inequality in (A.20) yields
sin
hi-1 sin((i-1)00)
0
- hi sin((i-2) 0 ]
which may be reduced to
Ei-1
Ei
This is the result that we sought.
EM
<
EM_
1
<
...
<
E2
<
Now we may conclude that
E1
= h1
which concludes the proof of sufficiency and, hence, the theorem.
- 52 -
(A.21)
APPENDIX B
We show here that in the limit as
the discrete support vector constraint
h''(0) + h(6) > 0,
M
the expression for the
-,
hTC < 0
goes to
the constraint for the continuous support function.
Referring to Theorem 1, we have for three consecutive support values the
relation
-hi_1 + 2cos(M
)hi - hi+
<
0
(B.1)
which simply rewrites the constraint implied by the ith column of C in
the expression
hTC < O.
Now assume that the i t h support value h. has
1
associated with it the angle 0, so that
L i = L(hi,O )
is the
corresponding support line. Then Li_1 = L(hi_1 0
M ) and
L_-=L(h
6-i_ 2=
iLi
L(h i'
- 2)
are the support lines for support values hi.
and hi+l, respectively.
and Li_
1
1
Then any support function which has Li_ 1, Li,
as support lines must satisfy
h(e-0O ) - 2cos( 0o)h(6) + h(0+
0 o)
2>
(B.2)
where we have negated both sides of (B.1) and substituted 60 =
To show that (B.2) approaches h''(0) + h(6) 2 0 as 00 - O,
2Mr
first we
substitute the the 2n d order Taylor series approximation to cos( 0O)
cos( 0o)
1 - 00 2/2
(B.3)
into (B.2) to obtain
h(e-0o) - 2h(e) + 0o2 h(e) + h(0+
0 o)
>
0
(B.4)
.
Dividing both sides of (B.4) by 0O2 (which cannot change the sign of the
inequality since 602 is always positive) and rearranging we obtain
h(0+0 0 )-h(6)
h(O)-h(O-0o)
O00
+
h(6) >
0
.
(B.5)
60
Taking the limit of (B.5) as 00 goes to zero gives the desired result.
- 53 -
APPENDIX C
We show here the modifications to the matrices A and c in Equation
(4.10b) necessary to implement the sparse data and shift-corrected
modified algorithms.
Sparse Data (see end of Section 4.2).
We show here the methods
used to obtain A and c for M/N = 3; the reasoning for any integer
scaling other than 3 is similar.
h C ER
from
N = M/3
Recall that we seek an estimate
observations.
Since h is in IR we begin by
setting up the problem as if there were M observations.
Denoting A
using four submatrices (refer to Equation 4.15)
A = [A1 : A 2 : A3
: A4 ]
(C.1)
we see that the submatrices A1 and A2 will remain unchanged, since these
constraints do not involve the observations.
The matrices A3 and A 4
must change since they do involve the observations.
We easily see that
since certain elements of h (i.e, 2,3 and 5,6 etc.) are not observed
there should be no constraints imposed.
Hence, the modification to A3
and A4 is to decimate (remove) certain columns of each matrix.
N = M/3
Since
we will remove all columns except columns
1, 4, 7,..., 3(N-1)+l.
This applies to both matrices A3 and A4.
Denoting the decimated matrices as A' and A'
we have the new sparse
data A matrix given by
A
s
=
[AI
A2 : A' : A'
[A1
2
3
A4]
(C.2)
(C.2)
Since A' and A' each have N columns corresponding to the constraints on
the measured data, it turns out that the definition of the vector c
(Equations 4.16 and 4.17) remains unchanged.
- 54 -
Shift Correction (see Section 4.4).
The shift corrected algorithms
require the imposition of the linear constraint
T
h N =
M
M- v.
Since
this is an equality constraint and the constraints of each algorithm
discussed have been inequalities we must split the expression into two
inequalities
hTN
<
hTN
>
M
(C.3)
and
(C.4)
.
Then, after multiplying (C.4) by -1, we have the shift constraint in the
required form.
The new shift-corrected A matrix requires a fifth
partition
Asc
=
[A
A1
I A2
I A 3 I A4 ]
(C.5)
where A1 through A 4 are as before, and A0 is given by
A0
=
[N : -N]
(C.6)
In this case, the vector c must also be augmented as
Csc
=
[CO : C1
: C2 : C3
C4 ]
(C.7)
where C 1 through C4 are the former 4 partitions of c (see Equation 4.16)
and C O is given by
CO
=
[v
- 55 -
-v]
(C.8)
REFERENCES
[Bellman,
1970]
Bellman, R.,Introduction to Matrix Analysis, New
York: McGraw-Hill, 1970.
[Bresler and Macovski, 1984a]
Bresler Y. and Macovski A., "3-D Reconstruction From
Projections Based on Dynamic Object Models," Proc.
IEEE Int. Conf. on Acoustics, Speech & Signal
Processing, San Diego, California, March 1984.
[Bresler and Macovski, 1984b]
Bresler Y. and Macovski A., "Estimation of 3-D Shape
of Blood Vessels from X-Ray Images," Proc IEEE Comp.
Soc. Int. Symp. on Medical Images and Icons,
Arlington, Virginia, July 1984.
[Bresler and Macovski, 1984c]
Bresler Y. and Macovski A., "A Hierarchical Bayesian
Approach to Reconstruction From Projections of a
Multiple Object 3-D Scene," Proc. 7th International
Conference on Pattern Recognition, Montreal, Canada,
August 1984.
[Dantzig, 1963]
Dantzig G.B., Linear Programming and Extensions,
Princeton, N.J.: Princeton University Press, 1963.
[Goldfarb and Idnani, 1983]
Goldfarb D., Idnani A., "A numerically stable dual
method for solving strictly convex quadratic
programs", Mathematical Programming, v.27, pp.1-33,
1983.
[Greschak, 1985]
Greschak J.P., "Reconstructing Convex Sets", Ph.D.
Dissertation in Electrical Engineering, Mass.
Institute of Technology, February 1985.
[Herman, 1980]
Herman G.T., Image Reconstruction From Projections,
New York: Academic Press, 1980.
[Horn, 1986]
Horn, B.K.P., Robot Vision, Cambridge, MA: MIT Press,
1986
[Humel, 1986]
Humel, J.M., "Resolving bilinear data arrays", S.M.
Thesis, M.I.T., Cambridge, MA, 1986.
[Kelly and Weiss, 1979]
Kelly P.J., Weiss M., Geometry and Convexity -- a
Study in Mathematical Methods, New York: John Wiley
and Sons, 1979.
[Land and Powell, 1973]
Land, A.H., and Powell, S., Fortran Codes for
Mathematical Programming, London: Wiley-Interscience,
1973.
-
56 -
[Leahy and Goutis, 1985]
Leahy R.M. and Goutis C.E., "An optimal technique for
constraint based image restoration and
reconstruction," Preprint, subm. to IEEE Trans. ASSP,
July 1, 1985.
[Louis, 1980]
Louis A.K., "Picture Reconstruction from Projections
in Restricted Range," Math. Meth. in the Appl. Sci.,
v.2, pp.209-220, 1980.
[Luenberger, 1984]
Luenberger D.G., Linear and Nonlinear Programming,
2nd edition, Reading Massachusetts: Addison-Wesley
Publishing Company, 1984.
[Powell, 1983]
Powell M.J.D., "ZQPCVX: A Fortran subroutine for
convex quadritic programming", Report
DAMTP/1983/NA17, University of Cambridge, 1983.
[Preparata and Shamos, 1985]
Preparata, F.P. and Shamos, M.I., Computattonal
Geometry, New York: Springer-Verlag, 1985.
[Prince, 1986]
Prince J.L., "Geometric model-based Bayesian
estimation from projections", Proposal for Ph.D.
research in Electrical Engineering, Mass. Institute
of Technology, April 1986.
[Rossi and Willsky, 1984]
Rossi D.J. and Willsky A.S., "Reconstruction from
projections based on detection and estimation of
objects--Parts I and II: Performance analysis and
robustness analysis," IEEE Trans. ASSP, v.ASSP-32,
n.4, pp.886-906, August 1984.
[Santalo, 1976]
Santalo L.A., Integral Geometry and Geometric
Probability, Encyclopedia of mathematics and its
applications I., Reading MA: Addison-Wesley.
[Schneiter, 1986]
Schneiter J., "Automated Tactile Sensing for Object
Recognition and Localization,", Ph.D. Dissertation in
Mechanical Engineering, Massachusetts Institute of
Technology, June 1986.
[Sezan and Stark, 1984]
Sezan M.I. and Stark H., "Tomographic image
reconstruction from incomplete view data by convex
projections and direct Fourier inversion," IEEE
Trans. Med. Imag., v.MI-3, n.2, pp.91-98, June 1984.
[Spivak, 1979]
Spivak M., A Comprehensive Introduction to
Differential Geometry, Volume II, Berkeley: Publish
or Perish, Inc., 1979.
[Thomas, 1972]
Thomas G.B., Calculus and Analytic Geometry, Reading,
Massachusetts: Addison-Wesley Publishing Company,
Inc., 1972.
-
57 -
[Trussell and Civanlar, 1984]
Trussell H.J., Civanlar M.R., "The feasible solution
in signal restoration", IEEE Trans. Acoust., Speech,
and Sig. Proc., v.ASSP-32, n.2, pp.201-212, April
1984.
[Van Hove and Verly, 1985]
Van Hove P.L. and Verly J.G., "A silhouette-slice
theorem for opaque 3-D objects," ICASSP, March 26-29,
1985, pp. 9 3 3 - 9 3 6 .
[Van Trees, 1968]
Van Trees H.L., Detection, Estimation, and Modulation
Theory, Part I, New York: John Wiley and Sons, 1968.
[Youla and Webb, 1982]
Youla D.C., Webb H., "Image restoration by the method
of convex projection: Part 1 -theory", IEEE Trans.
Med. Imaging, v.MI-1, pp.81-94, October 1982.