Shape from surface tensors
Uniqueness, stabiliy and reconstruction results
Markus Kiderlen, Aarhus University
CSGB Follow-up Meeting, May 9, 2016
CENTRE FOR STOCHASTIC GEOMETRY
AND ADVANCED BIOIMAGING
M. Kiderlen
Shape from tensors
AU
AARHUS
UNIVERSITY
1 / 10
Summary statistics in stereology
The numbers are the essence of all things.
Pythagoras of Samos
(570 - 500 BC)
M. Kiderlen
Shape from tensors
2 / 10
Summary statistics in stereology
Summary statistics in stereology
’Numbers’
summary statistics in classical stereology:
real-valued: volume, surface area, mean width, . . . of K ⊂ Rn .
M. Kiderlen
Shape from tensors
3 / 10
Summary statistics in stereology
Summary statistics in stereology
’Numbers’
summary statistics in classical stereology:
real-valued: volume, surface area, mean width, . . . of K ⊂ Rn .
’Numbers’
summary statistics in modern stereology: tensors
vector-valued: Volume tensor of rank r = 1
R
(
K
M. Kiderlen
K
x1 dx , . . . ,
R
K
xn dx ) ∈ Rn
center of gravity (location).
Shape from tensors
3 / 10
Summary statistics in stereology
Summary statistics in stereology
’Numbers’
summary statistics in classical stereology:
real-valued: volume, surface area, mean width, . . . of K ⊂ Rn .
’Numbers’
summary statistics in modern stereology: tensors
vector-valued: Volume tensor of rank r = 1
R
(
K
x1 dx , . . . ,
R
K
xn dx ) ∈ Rn
center of gravity (location).
K
array-valued: Volume tensor of rank r > 1
R
(
K
x1r1 · · · xnrn dx )r1 ,...,rn ∈ Rn
r
with r1 + . . . + rm = r (shape + location).
M. Kiderlen
Shape from tensors
3 / 10
Summary statistics in stereology
Summary statistics in stereology
’Numbers’
summary statistics in classical stereology:
real-valued: volume, surface area, mean width, . . . of K ⊂ Rn .
’Numbers’
summary statistics in modern stereology: tensors
bdK
x
ν(x )
M. Kiderlen
Shape from tensors
4 / 10
Summary statistics in stereology
Summary statistics in stereology
’Numbers’
summary statistics in classical stereology:
real-valued: volume, surface area, mean width, . . . of K ⊂ Rn .
’Numbers’
summary statistics in modern stereology: tensors
bdK
Surface tensor of rank r
R
Tr (K ) = (
x
ν(x )
bdK
ν1r1 (x ) · · · νnrn (x ) dx )r1 ,...,rn ∈ Rn
r
with r1 + . . . + rm = r .
Surface tensors (mainly r = 2) are used to describe shape properties of K .
M. Kiderlen
Shape from tensors
4 / 10
Stability and uniqueness
Determination and stability [Astrid Kousholt, 2016]
Let K be a convex particle
(i.e. K ⊂ Rn is convex and compact and has interior points).
Shape from all tensors
Any convex particle K is determined up to translation by all its surface
tensors T0 (K ), T1 (K ), . . ..
M. Kiderlen
Shape from tensors
5 / 10
Stability and uniqueness
Determination and stability [Astrid Kousholt, 2016]
Let K be a convex particle
(i.e. K ⊂ Rn is convex and compact and has interior points).
Shape from all tensors
Any convex particle K is determined up to translation by all its surface
tensors T0 (K ), T1 (K ), . . ..
Stability for finitely many tensors
Let r ∈ N, > 0 and let ρB n ⊆ K , L ⊆ (1/ρ)B n be convex particles. If
Tj (K ) = Tj (L) for j = 0, . . . , r ,
then there is an x ∈ Rn such that the Hausdorff-distance satisfies
1
δ(K , L + x ) ≤ c(n, ρ, )r − 4n + .
M. Kiderlen
Shape from tensors
5 / 10
Stability and uniqueness
Uniqueness results
Which convex particles are uniquely determined up to translation by a
finite number of surface tensors?
M. Kiderlen
Shape from tensors
6 / 10
Stability and uniqueness
Uniqueness results
Which convex particles are uniquely determined up to translation by a
finite number of surface tensors?
Tensors and polytopes
For any convex particle K ⊆ Rn and r ∈ N there is a polytope P such that
Tj (K ) = Tj (P)
for j = 0, . . . , r .
M. Kiderlen
Shape from tensors
6 / 10
Stability and uniqueness
Uniqueness results
Which convex particles are uniquely determined up to translation by a
finite number of surface tensors?
Tensors and polytopes
For any convex particle K ⊆ Rn and r ∈ N there is a polytope P such that
Tj (K ) = Tj (P)
for j = 0, . . . , r .
Corollary
If K ⊆ Rn is uniquely determined up to translation by finitely many
surface tensors, then K is a polytope.
M. Kiderlen
Shape from tensors
6 / 10
Stability and uniqueness
Theorem
A convex particle P ⊆ Rn that is a polytope with at most m facets is
uniquely determined up to translation among all convex bodies in Rn by its
surface tensors T0 (P), . . . , Tm (P).
M. Kiderlen
Shape from tensors
7 / 10
Stability and uniqueness
Theorem
A convex particle P ⊆ Rn that is a polytope with at most m facets is
uniquely determined up to translation among all convex bodies in Rn by its
surface tensors T0 (P), . . . , Tm (P).
For n = 2, the maximal tensor rank m is optimal.
M. Kiderlen
Shape from tensors
7 / 10
Reconstruction from finitely many tensors
A reconstruction algorithm
Can we reconstruct an approximation of K given finitely many tensors?
Given: Tensors T0 (K ), . . . Tr (K ) of an unknown convex particle K .
Wanted: An approximation K̂r with the same tensors up to rank r .
M. Kiderlen
Shape from tensors
8 / 10
Reconstruction from finitely many tensors
A reconstruction algorithm
Can we reconstruct an approximation of K given finitely many tensors?
Given: Tensors T0 (K ), . . . Tr (K ) of an unknown convex particle K .
Wanted: An approximation K̂r with the same tensors up to rank r .
Algorithm: Least squares approach:
Choose K̂r as a minimizer of
f (L) =
r
X
kTs (L) − Ts (K )k2
s=0
on the set of all convex particles in Rn .
M. Kiderlen
Shape from tensors
8 / 10
Reconstructions in R3
Example: Ellipsoid
K̂2
M. Kiderlen
K̂4
Shape from tensors
K̂6
9 / 10
Reconstructions in R3
Example I: Pyramid
K̂2
M. Kiderlen
K̂3
Shape from tensors
K̂4
10 / 10