Download Sparse recovery and compressed sensing in inverse problems

Sparse recovery and compressed sensing in inverse problems
Sparse recovery and compressed sensing in
inverse problems
Gerd Teschke
(joint work with Evelyn Herrholz)
Institute for Computational Mathematics in Science and Technology
Neubrandenburg University of Applied Sciences, Germany
Konrad-Zuse-Institute Berlin (ZIB), Germany
June 7, 2010
Gerd Teschke
(7. Juni 2010)
1/68
Sparse recovery and compressed sensing in inverse problems
1 Motivation
2 Sparse recovery for inverse problems
3 Compressively sensed data and recovery
4 Ill-posed sensing model and sparse recovery
5 Numerical experiment
6 Short summary
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Sparse recovery and compressed sensing in inverse problems
Motivation
Motivation
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Motivation
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Motivation
References
E. Candes, The restricted isometry property and its implications for compressed sensing, Academie des
sciences, 2008.
E. J. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory, 51(12),
4203-4215, 2005.
Y. Eldar, Compressed sensing of analog signals in shift invariant space, IEEE Trans. on Signal Processing,
57(8), 2009.
I. Daubechies, M. Fornasier and I. Loris, Accelerated projected gradient method for linear inverse problems
with sparsity constraints, J. Fourier Anal. Appl., 2008.
G. Teschke and C. Borries, Accelerated projected steepest descent method for nonlinear inverse problems
with sparsity constraints, Inverse Problems, 26, 025007, 2010.
E.Herrholz and G.Teschke, A compressed sensing approach for inverse problems, preprint, 2010.
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Motivation
Solve Ax = y or F (x) = y
(A, F : X → Y , X , Y Hilbert spaces)
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Motivation
Solve Ax = y or F (x) = y
(A, F : X → Y , X , Y Hilbert spaces)
Often noisy data ky δ − y k ≤ δ
Gerd Teschke
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Motivation
Solve Ax = y or F (x) = y
(A, F : X → Y , X , Y Hilbert spaces)
Often noisy data ky δ − y k ≤ δ
The inverse problem is ill-posed
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Motivation
Solve Ax = y or F (x) = y
(A, F : X → Y , X , Y Hilbert spaces)
Often noisy data ky δ − y k ≤ δ
The inverse problem is ill-posed
Often: incomplete data or no data sampled at adequate rate
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Motivation
Solve Ax = y or F (x) = y
(A, F : X → Y , X , Y Hilbert spaces)
Often noisy data ky δ − y k ≤ δ
The inverse problem is ill-posed
Often: incomplete data or no data sampled at adequate rate
Some a-priori knowledge required:
solution has sparse representation / is compressible
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Sparse recovery
for inverse problems
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Tikhonov approach for linear/nonlinear problems:
min kF (x) − y δ k2 + 2αkxkpp
x
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Tikhonov approach for linear/nonlinear problems:
min kF (x) − y δ k2 + 2αkxkpp
x
Linear case: [Daubechies,Defrise,DeMol 2004]
x n+1 = Sα (x n + γA∗ (y − Ax n ))
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Tikhonov approach for linear/nonlinear problems:
min kF (x) − y δ k2 + 2αkxkpp
x
Linear case: [Daubechies,Defrise,DeMol 2004]
x n+1 = Sα (x n + γA∗ (y − Ax n ))
Nonlinear case: [Ramlau, T. 2005, 2007]
x n+1 = Sα (x n + γF 0 (x n+1 )∗ (y − F (x n )))
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Tikhonov approach for linear/nonlinear problems:
min kF (x) − y δ k2 + 2αkxkpp
x
Linear case: [Daubechies,Defrise,DeMol 2004]
x n+1 = Sα (x n + γA∗ (y − Ax n ))
Nonlinear case: [Ramlau, T. 2005, 2007]
x n+1 = Sα (x n + γF 0 (x n+1 )∗ (y − F (x n )))
Many alternatives and improvements (incomplete list ...)
steepest descent, domain decomposition, semi-smooth Newton
methods, projection methods [Bredies, Daubechies, Fornasier, Lorenz, Loris, ...]
adaptive iteration
Gerd Teschke
[Ramlau, T., Zhariy and Dahlke, Fornasier, Raasch, ...]
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Sparsity through projection + acceleration
Problem:
x n+1 = Sα (x n + γA∗ (y − Ax n ))
overshooting (dynamics discussed by Daubechies, Fornasier,
Loris 2008)
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Sparsity through projection + acceleration
Problem:
x n+1 = Sα (x n + γA∗ (y − Ax n ))
overshooting (dynamics discussed by Daubechies, Fornasier,
Loris 2008)
Alternative:
(Daubechies et.al. 2008, Borries, T. 2009 (nonlinear case))
min {kAx − y k2 } where BR := {x ∈ `2 ; kxk1 ≤ R}
x∈BR
x n+1 = PR (x n + γ n A∗ (y − Ax n )).
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Vector-valued setup and joint sparsity formulation:
min
ky δ − Axk2
p,q
d∈CR
where
CRp,q


!q
m
p


X
X
≤R
= x ∈ (`2 (Λ))m : Ψp,q (x) :=
|x`,λ |p


`=1
Gerd Teschke
λ∈Λ
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Vector-valued setup and joint sparsity formulation:
min
ky δ − Axk2
p,q
d∈CR
where
CRp,q


!q
m
p


X
X
≤R
= x ∈ (`2 (Λ))m : Ψp,q (x) :=
|x`,λ |p


`=1
λ∈Λ
Special case p = 2, q = 1:
PR (x) = (Sµ ({x1,λ }λ∈Λ ), . . . , Sµ ({xm,λ }λ∈Λ ))
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Sparse recovery for inverse problems
Vector-valued setup and joint sparsity formulation:
min
ky δ − Axk2
p,q
d∈CR
where
CRp,q


!q
m
p


X
X
≤R
= x ∈ (`2 (Λ))m : Ψp,q (x) :=
|x`,λ |p


`=1
λ∈Λ
Special case p = 2, q = 1:
PR (x) = (Sµ ({x1,λ }λ∈Λ ), . . . , Sµ ({xm,λ }λ∈Λ ))
with
Sµ ({x`,λ }λ∈Λ ) =
Gerd Teschke
{x`,λ }λ∈Λ
max(k{x`,λ }λ∈Λ k`2 (λ) − µ, 0)
k{x`,λ }λ∈Λ k`2 (λ)
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Compressively sensed data
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Compressed Sensing Idea:
Suppose there is a signal
x.
x
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Compressed Sensing Idea:
=
Suppose we measure
just a few linear
samples.
Can we reconstruct x?
Ã
y
x
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Compressed Sensing Idea:
=
Yes! If x is sparse in a
given “uncoherent”
basis or frame,i.e.
Ã
y
B
x =Bd
d
Gerd Teschke
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Compressed Sensing Idea:
=
Yes! If x is sparse in a
given “uncoherent”
basis or frame,i.e.
Ã
y
B
x =Bd
d
à is p × m matrix with p m
Gerd Teschke
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Compressed Sensing Idea:
=
Yes! If x is sparse in a
given “uncoherent”
basis or frame,i.e.
Ã
y
B
x =Bd
d
à is p × m matrix with p m
d k-sparse, then p ≥ 2k data points are required
(stability: p ≥ ck log(m/p))
Gerd Teschke
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Compressed Sensing Idea:
=
Yes! If x is sparse in a
given “uncoherent”
basis or frame,i.e.
Ã
y
B
x =Bd
d
à is p × m matrix with p m
d k-sparse, then p ≥ 2k data points are required
(stability: p ≥ ck log(m/p))
A = ÃB has to fulfill a restricted isometry property
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Definition (restricted isometry property∼ RIP)
For each integer k = 1, 2, . . . , define the isometry constant δk of a
sensing matrix A as the smallest number such that
(1 − δk )kdk2`2 ≤ kAdk2`2 ≤ (1 + δk )kdk2`2
holds for all k-sparse vectors d. A vector is said to be k-sparse if it
has at most k non-vanishing entries.
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Optimization problem:
min kdk`1 subject to y = Ad or ky δ − Adk ≤ δ
d
Gerd Teschke
(∗)
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Compressively sensed data and recovery
Optimization problem:
min kdk`1 subject to y = Ad or ky δ − Adk ≤ δ
d
(∗)
(E. Candes 2008)
√
Let A satisfy RIP with δ2k < 2 − 1 and d k denote the best
k-term approximation, then the solution d ∗ to (∗) obeys
kd ∗ − dk`2 ≤ C0 k −1/2 kd k − dk`1
or in case of noisy data y δ with ky δ − y k`2 ≤ δ
kd ∗ − dk`2 ≤ C0 k −1/2 kd k − dk`1 + C1 δ
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Ill-posed sensing model
and sparse recovery
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Indirect measurement Kx:
=
Ã
y
K
B
d
Consequences
K may be ill-conditioned ; RIP is usually lost.
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Indirect measurement Kx:
=
Ã
y
K
B
d
Consequences
K may be ill-conditioned ; RIP is usually lost.
Question: How to overcome this problem and how to adjust
the finite-dimensional CS setting to inverse problems ?
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
(setup by Y. Eldar, 2009)
Let X be a Hilbert space and Xm ⊂ X the reconstruction space
(
)
m X
X
m
Xm = x ∈ X , x =
d`,λ a`,λ , d ∈ (`2 (Λ))
`=1 λ∈Λ
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
(setup by Y. Eldar, 2009)
Let X be a Hilbert space and Xm ⊂ X the reconstruction space
(
)
m X
X
m
Xm = x ∈ X , x =
d`,λ a`,λ , d ∈ (`2 (Λ))
`=1 λ∈Λ
Sparse structure: support set I ⊂ {1, . . . , m}




X X
Xk = x ∈ X , x =
d`,λ a`,λ , d ∈ (`2 (Λ))m


`∈I,|I|=k λ∈Λ
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Analysis and synthesis operators
Tφ : X → `2 via x 7→ x = {hx, φλ i}λ∈Λ
Tφ∗ : `2 → X via x 7→
X
xλ φλ .
λ∈Λ
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
representation of solution: x = Ta∗ d
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
representation of solution: x = Ta∗ d
data model: y = Tv KTa∗ d
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
representation of solution: x = Ta∗ d
data model: y = Tv KTa∗ d
compressed sampling:




v1,λ
s1,λ
 . 
 .. 
 .  = A  .. 
vm,λ
sp,λ
Gerd Teschke
for all λ ∈ Λ
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
representation of solution: x = Ta∗ d
data model: y = Tv KTa∗ d
compressed sampling:




v1,λ
s1,λ
 . 
 .. 
 .  = A  .. 
vm,λ
sp,λ
for all λ ∈ Λ
compressed data model: y = Ts KTa∗ d
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
compressed data model: y = Ts KTa∗ d
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
compressed data model: y = Ts KTa∗ d
Assume
hv`,λ , Ka`0 ,λ0 i = κ`,λ δ`,`0 δλ,λ0
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
compressed data model: y = Ts KTa∗ d
Assume
hv`,λ , Ka`0 ,λ0 i = κ`,λ δ`,`0 δλ,λ0
then for all λ ∈ Λ:
yλ = ADλ dλ with Dλ = diag (κ1,λ , . . . , κm,λ )
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
compressed data model: y = Ts KTa∗ d = ADd
Assume
hv`,λ , Ka`0 ,λ0 i = κ`,λ δ`,`0 δλ,λ0
then for all λ ∈ Λ:
yλ = ADλ dλ with Dλ = diag (κ1,λ , . . . , κm,λ )
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
compressed data model: y = Ts KTa∗ d = ADd
Assume
hv`,λ , Ka`0 ,λ0 i = κ`,λ δ`,`0 δλ,λ0
then for all λ ∈ Λ:
yλ = ADλ dλ with Dλ = diag (κ1,λ , . . . , κm,λ )
Problem: AD does not satisfy RIP
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
For all λ ∈ Λ consider optimization problem
min kyλδ − ADλ dλ k2`2 + αkdλ k2`2 ,
dλ ∈BR
where BR = {dλ ∈ `2 : kdλ k`1 ≤ R}
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
For all λ ∈ Λ consider optimization problem
min kyλδ − ADλ dλ k2`2 + αkdλ k2`2 ,
dλ ∈BR
where BR = {dλ ∈ `2 : kdλ k`1 ≤ R}
Corresponding model operator L2 := Dλ A∗ ADλ + αI , leading
to stabilized RIP
(κ2min (1 − δk ) + α)kxk2 ≤ kLxk2 ≤ (κ2max (1 + δk ) + α)kxk2
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Theorem (Herrholz, T. 2010)
Define dλ† as the BR -best approximation to the true solution.
Assume R was chosen such that the solution dλ 6∈ BR and that
√
√
(1 + 2)κ2min − κ2max + 2α
√
.
0 ≤ δ2k <
(1 + 2)κ2min + κ2max
Then the minimizer dλ∗ satisfies
√
kdλ∗ −dλ k`2 ≤ C0 k −1/2 kdλk −dλ k`1 +C1 δ+C2 kL(dλ† −dλ )k`2 +C3 αR.
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Treatment of all λ simultaneously by a joint sparsity formulation:
min ky δ − ADdk2(`2 (Λ))p + αkdk2(`2 (Λ))m
d∈CR2,1
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Treatment of all λ simultaneously by a joint sparsity formulation:
min ky δ − ADdk2(`2 (Λ))p + αkdk2(`2 (Λ))m
d∈CR2,1
where
CR2,1 = {d ∈ (`2 (Λ))m : Ψ2,1 (d) ≤ R}
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Theorem (Herrholz, T. 2010)
Let d † be the BR -best approximation to the true solution, R as
before and
√
√
(1 + 2)κ2min − κ2max + 2α
√
.
0 ≤ δ2k <
(1 + 2)κ2min + κ2max
Then the minimizer d ∗ satisfies
kd ∗ − dk(`2 (Λ))m
Gerd Teschke
≤ C0 k −1/2 Ψ2,1 (d k − d) + C1 δ
√
+C2 kL(d † − d)k(`2 (Λ))m + C3 αR.
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Discussion on the balancing on δ2k of α:
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Discussion on the balancing on δ2k of α:
Given K and sensing matrix A:
(1 + δ2k )κ2max − (1 +
√
2
Gerd Teschke
√
2)(1 − δ2k )κ2min
<α
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Discussion on the balancing on δ2k of α:
Given K and sensing matrix A:
(1 + δ2k )κ2max − (1 +
√
2
√
2)(1 − δ2k )κ2min
<α
If δ2k < 1, a stabilization becomes necessary if
√
1 + δ2k κ2max
· 2 >1+ 2 .
1 − δ2k κmin
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Discussion on the balancing on δ2k of α:
Given K and sensing matrix A:
(1 + δ2k )κ2max − (1 +
√
2
√
2)(1 − δ2k )κ2min
<α
If δ2k < 1, a stabilization becomes necessary if
√
1 + δ2k κ2max
· 2 >1+ 2 .
1 − δ2k κmin
If δ2k ≥ 1, lower bound is always positive
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Discussion on the balancing on δ2k of α:
Given K and sensing matrix A:
(1 + δ2k )κ2max − (1 +
√
2
√
2)(1 − δ2k )κ2min
<α
If δ2k < 1, a stabilization becomes necessary if
√
1 + δ2k κ2max
· 2 >1+ 2 .
1 − δ2k κmin
If δ2k ≥ 1, lower bound is always positive
Specializing the case δ2k < 1 to κ2max = κ2min = 1:
√
δ2k > 2 − 1
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Given K , stabilize first and suppose A can be chosen
accordingly:
√
κ2max − (1 + 2)κ2min
√
<α.
2
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Given K , stabilize first and suppose A can be chosen
accordingly:
√
κ2max − (1 + 2)κ2min
√
<α.
2
No stabilization necessary (α = 0):
1 ≤ κ2max /κ2min < 1 +
Gerd Teschke
√
2
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Given K , stabilize first and suppose A can be chosen
accordingly:
√
κ2max − (1 + 2)κ2min
√
<α.
2
No stabilization necessary (α = 0):
1 ≤ κ2max /κ2min < 1 +
δ2k
Gerd Teschke
√
2
√
1 + 2 − κ2max /κ2min
√
<
1 + 2 + κ2max /κ2min
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Given K , stabilize first and suppose A can be chosen
accordingly:
√
κ2max − (1 + 2)κ2min
√
<α.
2
No stabilization necessary (α = 0):
1 ≤ κ2max /κ2min < 1 +
δ2k
√
2
√
1 + 2 − κ2max /κ2min
√
<
1 + 2 + κ2max /κ2min
Stabilization necessary (α > 0):
√
1 + 2 ≤ κ2max /κ2min
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Sparse recovery and compressed sensing in inverse problems
Ill-posed sensing model and sparse recovery
Given K , stabilize first and suppose A can be chosen
accordingly:
√
κ2max − (1 + 2)κ2min
√
<α.
2
No stabilization necessary (α = 0):
1 ≤ κ2max /κ2min < 1 +
δ2k
√
2
√
1 + 2 − κ2max /κ2min
√
<
1 + 2 + κ2max /κ2min
Stabilization necessary (α > 0):
√
1 + 2 ≤ κ2max /κ2min
δ2k <
Gerd Teschke
1+
√
√
2 − κ2max /κ2min + 2α/κ2min
√
1 + 2 + κ2max /κ2min
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Sparse recovery and compressed sensing in inverse problems
Numerical experiment
Numerical experiment
(MATLAB)
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(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Numerical experiment
a`,λ - sinc- wavelets, ` ∈ {−3, −2, . . . , 4, 5}, i.e. m = 9
K convolution operator → Dλ
x is 2-sparse,
d1,21 = 3 , d4,20 = −1 , d4,25 = 2
yλδ = ADλ dλ + δ
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Sparse recovery and compressed sensing in inverse problems
Numerical experiment
K corresponds to diagonal entries κ`,λ :
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Sparse recovery and compressed sensing in inverse problems
Numerical experiment
minimization of
min ky δ − ADdk2(`2 (Λ))p + αkdk2(`2 (Λ))m
d∈CR2,1
yields:
Gerd Teschke
(7. Juni 2010)
65/68
Sparse recovery and compressed sensing in inverse problems
Numerical experiment
Algorithmic aspects:
Gerd Teschke
(7. Juni 2010)
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Sparse recovery and compressed sensing in inverse problems
Short summary
Summary:
Algorithms for sparse recovery:
- projected steepest descent, step length control, vector-valued,
joint sparsity
Suggestion for a setup for incomplete/compressed data:
- semi-infinite dimensional
In the compressed data model:
- balance between the sensing properties of A and stabilization by α
- choice of α restricted (typically α 6→ 0 to ensure CS recovery)
Future work:
- apply infinite CS model (A. Hansen, 2010)
- combine compressed data model with adaptive operator evaluation
Gerd Teschke
(7. Juni 2010)
67/68
Sparse recovery and compressed sensing in inverse problems
Short summary
Workshop on CS, Sparsity and Inverse Problems
September 6-7, 2010 at TU Braunschweig, Germany
http://www.dfg-spp1324.de/nuhagtools/event/make.php?event=cssip10
Gerd Teschke
(7. Juni 2010)
68/68