Shape From Moments - 1 -
Shape from Moments An Estimation
Perspective
Michael Elad*, Peyman Milanfar**, and Gene Golub*
SIAM 2002 MeetingMS104 - Linear Algebra in Image Processing
July 12th, 2002
* The CS Department – SCCM Program Stanford University** Peyman Milanfar is with the University of California Santa Cruz
(UCSC).
Shape From Moments - 2 -
Chapter A
Background
Shape From Moments - 3 -
na
A.1 Davis Theorem
P
8z
2z
3z
4z
7z
6z
5z
1z
x
y{The complex plane Z}
,zfzzzz
zzzz
2i
dxdy)z(''fP
N
1nn
n1n
n1n
n1n
n1n
Theorem (Davis
1977) : For any closed
2D polygon, and for any
analytic function f(z) the
following holds
Shape From Moments - 4 -
A.2 Complex Moments
If we use the analytic function , we get from
Davis Theorem that
kzzf
2k
N
1n
knn
P
2k
P
zadxdyz)1k(kdxdy)z(''f
k
2kP
2kk )1k(kdxdyz)1k(k
k
where we define
Shape From Moments - 5 -
A.3 Shape From Moments
M
0k
N
1n
kn
n1n
n1n
n1n
n1nk z
zzzz
zzzz
2i
• Can we compute the vertices from these equations ?
• How many moments are required for exact recovery ?
Shape From Moments
M 0kk N 1nnz
Shape From Moments - 6 -
A.4 Previous Results
• Golub et. al. (1999): – Pencil method replacing the Prony’s - better numerical stability.
– Sensitivity analysis.
• Milanfar et. al. (1995): – (2N-1) moments are theoretically sufficient for computing the N
vertices.
– Prony’s method is proposed.
• Prony’s and the Pencil approaches:– Rely strongly on the linear algebra formulation of the problem.
– Both are sensitive to perturbations in the moments.
– Both will be presented briefly.
Shape From Moments - 7 -
Function of the vertices (and their order)
A.5 To Recap
Known values
N
2
1
MN
M2
M1
2N
22
21
1N
12
11
0N
02
01
M
2
1
0
a
a
a
zzz
zzz
zzz
zzz
Estimate !!! N1nnz
Shape From Moments - 8 -
A.6 Our Focus
• Noisy measurements: What if the moments are
contaminated by additive noise ? How can re-
pose our problem as an estimation task and solve
it using traditional stochastic estimation tools ?
• More measurements: What if there are M>2N-1
moments ? How can we exploit them to robustify
the computation of the vertices ?
Shape From Moments - 9 -
A.7 Related Problems
• It appears that there are several very different
applications where the same formulation is
obtained– Identifying an auto-regressive system from its output,
– Decomposing of a linear mixture of complex cissoids,
– Estimating the Direction Of Arrival (DOA) in array processing,
– and more ... N 1kka • Major difference – are not functions of the
unknowns but rather free parameters.
• Nevertheless, existing algorithms can be of use.
Shape From Moments - 10 -
Chapter B
Prony and Pencil Based Methods
Shape From Moments - 11 -
B.1 Prony’s Relation
N
1n
nNn
NN
1nn zpzzz)z(P
M
0k
N
1n
knnk za
1
1N
N
1M1NMNM
1N32
N21
1N10
M
2N
1N
N
p
p
p
Shape From Moments - 12 -
B.2 Prony’s Methods
M
2N
1N
N
1
1N
N
1M1NMNM
1N32
N21
1N10
p
p
p
d. IQML, Structuted-TLS, Modified Prony, and more.
c. Hankel Constrained SVD, followed by root-finding,
b. Total-Least-Squares, followed by root-finding,
a. Regular Least-Squares, followed by root-finding,
Shape From Moments - 13 -
B.3 Pencil Relation
1M1NMNM
1N32
N21
1N10
0T
M2NM1NM
2N43
1N32
N21
1T
Define
0vTzT n0n1 After some (non-trivial) manipulation we obtain
For some non zero vectors Vn.
Shape From Moments - 14 -
B.4 Non-Square Pencil
T1 - zn T0 vn=0
Shape From Moments - 15 -
B.5 Pencil Methods
0vTzT n0n1
a. Take square portions, solve for the
eigenvalues, and cluster the results,H0Tb. Square by left multiplication with (closely
related to LS-Prony),
c. Hua-Sarkar approach: different squaring methods
which is more robust and related to ESPRIT.
Shape From Moments - 16 -
Chapter C
ML and MAP Approaches
Shape From Moments - 17 -
C.1 What are we Missing ?
• We have seen a set of simple methods that
give reasonable yet inaccurate results.
N 1kka
• In our specific problem we do not exploit the
fact that are vertices-dependent.
• In all the existing methods there is no
mechanism for introducing prior-
knowledge about the unknowns.
Shape From Moments - 18 -
C.2 Recall …
We have the following system of equations
Measured Function of the unknowns
zaza
a
a
zzz
zzz
zzz
zzz
t
N
2
1
MN
M2
M1
2N
22
21
1N
12
11
0N
02
01
M
2
1
0
V
Shape From Moments - 19 -
C.3 Our Suggestion
• If we assume that the moments are contaminated
by zero-mean white Gaussian noise, Direct-
Maximum-Likelihood (DML) solution is given by
22
zzaztMinimize V
• Direct minimization is hard to workout, BUT
• We can use one of the above methods to obtain an
initial solution, and then iterate to minimize the
above function until getting to a local minima.
Shape From Moments - 20 -
C.4 Things to Consider
• Even (complex) coordinate descent with effective
line-search can be useful and successful (in order
to avoid derivatives).
• Per each candidate solution we HAVE TO solve the
ordering problem !!!! Treatment of this problem is
discussed in Durocher (2001).
• If the initial guess is relatively good, the ordering
problem becomes easier, and the chances of the
algorithm to yield improvement are increased.
Shape From Moments - 21 -
C.5 Relation to VarPro
• VarPro (Golub & Pereyra 1973)
– Proposed for minimizing
– The basic idea: Represents the a as and
use derivatives of the Pseudo-Inverse matrix.
22azt V
tza V
• Later work (1978) by Kaufman and Pereyra
covered the case where a=a(z) (linear
constraints).• We propose to exploit this or similar method, and
choose a good initial solution for our iterative
procedure.
Shape From Moments - 22 -
C.6 Regularization
• Since we are minimizing (numerically) the DML
function, we can add a regularization – a penalty term
for directing the solution towards desired properties.
zgzaztMinimize 22
z V
• The minimization process is just as easy.
• This concept is actually an application of the
Maximum A-posteriori-Probability (MAP)
estimator.
Shape From Moments - 23 -
C.7 MAP Possibilities
Kind of Prior Expression for g(*)
Promoting 90° angles
Promoting smoothness (1)
Promoting smoothness (2)
Promoting regularity
Promoting small area
N
1n
2
n1n
n1n
n1n
n1n 1zzzz
zzzz
21
N
1n
2n1n zz
N
1n
2n1n1n z2zz
N
1nn1n
2N
1nn1n zzImzz
N
1nn1n zzIm
Shape From Moments - 24 -
Chapter D
Results
Shape From Moments - 25 -
D.1 Experiment #1
• Compose the following
star-shaped polygon
(N=10 vertices),
• Compute its exact
moments (M=100),
• add noise (=1e-4),
• Estimate the vertices
using various methods.
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Shape From Moments - 26 -
D.1 Experiment #1
LS-Prony method Squared Pencil method Hua-Sarkar method
0.0201 0.0196 0.0174
-1 -0.5 0 0.5 1
-0.8-0.6-0.4-0.2
0.20.40.60.8
-1
0
1
-1 -0.5 0 0.5 1
-0.8-0.6-0.4-0.2
0.20.40.60.8
-1
0
1
-1 -0.5 0 0.5 1
-0.8-0.6-0.4-0.2
0.20.40.60.8
-1
0
1
Mean Squared Error averaged over 100 trials
Shape From Moments - 27 -
D.2 Experiment #2
• For the star-shape polygon
with noise variance =1e-
4, initialize using Hua-
Sarkar algorithm.
• Then, show the DML
function per each vertex,
assuming all other vertices
fixed.
+ Hua-Sarkar result
New local minimum
-1 -0.5 0 0.5 1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0.1 0.2 0.3 0.4 0.5 0.6
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Shape From Moments - 28 -
D.3 Experiment #3
• For the E-shape polygon
with noise variance =1e-3,
initialize using LS-Prony
algorithm.
• Then, show the DML
function per each vertex,
assuming all other vertices
fixed.
+ LS-Prony result
New local minimum
-0.4 -0.2 0 0.2 0.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-0.3 -0.2 -0.1 0 0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Shape From Moments - 29 -
D.4 Experiment #4
• For the E-shape polygon with
noise variance =1e-3, initialize
using LS-Prony algorithm.
• Then, show the MAP function per
each vertex, assuming all other
vertices fixed.
• Regularization – promote 90°
angles.
+ LS-Prony result
New local minimum
-0.4 -0.2 0 0.2 0.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Shape From Moments - 30 -
0 5 10 15
10-2
10-1
D.5 Experiment #5
Error as a function of the iteration number*
DMLMAP
* Using a derivative free coordinate-descent procedure
Shape From Moments - 31 -
D.6 To Conclude
• The shape-from-moments problem is formulated, showing a close resemblance to other problems in array processing, signal processing, and antenna theory.
• The existing literature offers many algorithms for estimating the “vertices” – some of them are relatively simple but also quite sensitive.
• In this work we propose methods to use these simple algorithms as initialization, followed by a refining stage based on the Direct Maximum Likelihood and the MAP estimator.
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