RECONSTRUCTION FROM SUPPORT FUNCTIONS

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RECONSTRUCTION FROM SUPPORT FUNCTIONS
Richard Gardner
Joint work with Markus Kiderlen, U. Aarhus
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The X-ray Problems (P. C. Hammer)

• Proc. Symp. Pure Math. Vol VII Convexity
• (Providence, RI), AMS, 1963, pp. 498-9

Suppose there is a convex hole in an
otherwise homogeneous solid and that X-ray
pictures taken are so sharp that the darkness
at each point determines the length of a chord
along an X-ray line. (No diffusion, please.) How
many pictures must be taken to permit exact
reconstruction of the body if a. The
X-rays issue from a finite point source?
b. The X-rays are assumed parallel?
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Parallel and Point X-rays
There are sets of 4 directions so that the
parallel X-rays of a planar convex body in those
directions determine it uniquely. (R.J.G. and
P. McMullen, 1980)
A planar convex body is determined by its
X-rays taken from any set of 4 points with no
three collinear. (A. Volcic, 1986)
In these situations a viable algorithm
exists for reconstruction, even from noisy
measurements.
R.J.G. and M. Kiderlen, A solution to Hammers
X-ray reconstruction problem, Adv. Math. 214
(2007), 323-343.
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Geometric Tomography
The area of mathematics dealing with the
retrieval of information about a geometric object
from data about its sections, or projections, or
both. The term geometric object is
deliberately vague a convex polytope or body
would certainly qualify, as would a star-shaped
body, or even, when appropriate, a compact or
measurable set.
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Enter the Brunn-Minkowski theory
width function
brightness function
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Present Scope of Geometric Tomography
Computerized tomography
Discrete tomography
Robot vision
Convex geometry
Point X-rays
Parallel X-rays
Imaging
Sections through a fixed point dual
Brunn-Minkowski theory
Projections classical Brunn-Minkowski theory
?
Integral geometry
Pattern recognition
Minkowski geometry
Stereology and local stereology
Local theory of Banach spaces
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The Support Function
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The Program of Alan WillskyElectrical
Engineering, MIT
Algorithms for reconstructing convex bodies
from noisy support function measurements
J. L. Prince and A. Willsky, Estimating convex
sets from noisy support line measurements, IEEE
Trans. Pattern Anal. and Machine Intell. 12
(1990), 377-389.
A. Lele, S. Kulkarni, and A. Willsky, Convex
polygon estimation from support line measurements
and applications to target reconstruction from
laser radar data, J. Optical Soc. Amer. A. 9
(1992), 1693-1714.
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Consistency
Consistent measurements h1,,h6 and the maximal
polygon
Inconsistent measurements h1, h2, h3
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The Prince-Willsky Algorithm
Input Natural numbers n 2 and k n1 vectors
whose positive hull is Rn noisy support function
measurements i 1,,k of an unknown convex
body K in Rn, where the Xis are independent
N(0,s2) random variables. Task Construct a
convex polytope Pk in Rn that approximates
K. Action Solve the following constrained linear
least squares problem
subject to h1,hk are consistent. Let
h1,,hk be a solution and let Pk P(h1,,hk) be
the corresponding maximal convex polytope.
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Good News and Bad News
Good news When n 2, the consistency
constraint can be expressed by an inequality Ch
0, where h h1,hkT, C is the matrix
and ui (cos ?i, sin ?i), 1 i k.
Bad news When n 3, this is no longer
possible. At best the global consistency
constraint can be replaced by a set of local
consistency conditions, but it is not
computationally feasible to check all such
conditions.
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The New Algorithm
Input Natural numbers n 2 and k n1 vectors
whose positive hull is Rn noisy support function
measurements i 1,,k of an unknown convex
body K in Rn, where the Xis are independent
N(0,s2) random variables. Task Construct a
convex polytope Pk in Rn that approximates
K. Action Solve the following constrained linear
least squares problem
subject to xjTui xiTui for 1
i ? j k. Let x1,,xk be a solution and let Pk
convx1,,xk.
R.J.G. and Markus Kiderlen, A new algorithm
for 3D reconstruction from support function
measurements, submitted.
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New Algorithm Output
Possible output of the new algorithm when s 0
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Typical 2D Reconstructions
Reconstructions of a 7-gon from 17 equally spaced
noisy support function measurements with s 0.1
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Errors
Three ways to measure the error between the input
K and the reconstruction Pk
1. Mean square error
2. L2 distance
3. Hausdorff distance
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Performance - MSE
100 reconstructions of an 11-gon from
equally spaced measurement directions with s 0.1
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Performance Hausdorff and L2
Average Hausdorff error (s 0.1 fixed)
Average L2 error (s 0.1 fixed)
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Performance - Time
Prince-Willsky algorithm k real variables New
algorithm nk real variables (and k(k-1) linear
constraints).
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The New Algorithm (LP version)
Input Natural numbers n 2 and k n1 vectors
whose positive hull is Rn noisy support function
measurements i 1,,k of an unknown convex
body K in Rn, where the Xis are independent
N(0,s2) random variables. Task Construct a
convex polytope Pk in Rn that approximates
K. Action Solve the following linear
program subject to
xjTui xiTui yi for 1 i ? j
k. Let x1,,xk be a solution and let Pk
convx1,,xk.
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Performance (LP) Hausdorff and L2
Average Hausdorff error (s 0.1 fixed)
Average L2 error (s 0.1 fixed)
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Performance (LP) Time and Noise
Average reconstruction time against number of
measurements (s 0.1 fixed)
Average L2 error against noise level s (with
fixed number of measurements 25)
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3D Reconstructions (LS) - Octahedron
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3D Reconstructions (LS) - Ellipsoid
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3D Reconstructions (LP) Prism
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3D Reconstructions (LP) - Octahedron
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Performance L2 error and Time
100 reconstructions of an ellipsoid measurements
from evenly spread directions, fixed noise
level s 0.05
Average L2 error
Average reconstruction time
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Convergence
By using results from the theory of empirical
processes, it can be shown that under mild
assumptions, the rate of convergence of MSE(K,
Pk) and d2(K, Pk), where Pk is an output of the
least-squares algorithms (either Prince-Willsky
or the new algorithm), is
as k? 8, and that of dH(K, Pk) is
provided n 4.
S. van de Geer, Applications of Empirical
Process Theory, Cambridge Univ. Press, Cambridge,
2000.
R.J.G., Markus Kiderlen, and Peyman Milanfar,
Convergence of algorithms for reconstructing
convex bodies and directional measures, Ann.
Statist. 34 (2006), 1331-1374.