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Sparse tomography
Samuli Siltanen
Department of Mathematics and StatisticsUniversity of Helsinki, Finland
Minisymposium:Fourier analytic methods in tomographic image reconstruction
Applied Inverse Problems ConferenceDaejeon, Korea, July 1, 2013
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Finland
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http://wiki.helsinki.fi/display/inverse/Home
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This is a joint work with
Keijo Hämäläinen, University of Helsinki
Aki Kallonen, University of Helsinki
Ville Kolehmainen, University of Eastern Finland
Matti Lassas, University of Helsinki
Esa Niemi, University of Helsinki
Kati Niinimäki, University of Eastern Finland
Eero Saksman, University of Helsinki
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Outline
Sparse sampling and tomography
Total variation regularization
Discretization-invariance
Besov space regularization
Parameter choice: the S-curve method
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Outline
Sparse sampling and tomography
Total variation regularization
Discretization-invariance
Besov space regularization
Parameter choice: the S-curve method
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Let us study a simple two-dimensional example oftomographic imaging
4 4 5
1 3 4
1 0 2
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Tomography is based on measuring densities ofmatter using X-ray attenuation data
13 (=4+4+5)4 4 5
1 3 4
1 0 2
X-ray source• -
Detector
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A projection image is produced by parallel X-raysand several detector pixels (here three pixels)
13 (=4+4+5)
8 (=1+3+4)
3 (=1+0+2)
4 4 5
1 3 4
1 0 2
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Detector
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For tomographic imaging it is essential to recordprojection images from different directions
4 4 5
1 3 4
1 0 2
6 7 11
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The length of X-rays traveling inside each pixel isimportant, thus here the square roots
4 4 5
1 3 4
1 0 2
√2
√2
9 √2
8 √2
5 √2
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The direct problem of tomography is to find theprojection images from known tissue
4 4 5
1 3 4
1 0 2
√2
√2
9 √2
8 √2
5 √2
6 7 11
13
8
3
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The inverse problem of tomography is toreconstruct the interior from X-ray data
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? ? ?
? ? ?
√2
√2
9 √2
8 √2
5 √2
6 7 11
13
8
3
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We write the reconstruction problemin matrix form and assume Gaussian noise
f1 f4 f7
f2 f5 f8
f3 f6 f9@
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Our measurement model is m = Af + ε with independently distributedGaussian noise (white noise) with standard deviation σ > 0.
f =
f1f2f3f4f5f6f7f8f9
, m =
m1m2m3m4m5m6
,
m = Af
m1
m2
m3
m4
m5
m6
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Outline
Sparse sampling and tomography
Total variation regularization
Discretization-invariance
Besov space regularization
Parameter choice: the S-curve method
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We reconstruct the unknown pixel values usingtotal variation regularization and non-negativity
We consider the minimizer of the anisotropic TV functional
‖Af −m‖22 + α {‖LHf ‖1 + ‖LVf ‖1}
where LH and LH are horizontal and vertical first-order differencematrices. [Rudin, Osher and Fatemi 1992]
Primal-dual algorithms Chambolle, Chan, Chen, Esser, Golub, Mulet,Nesterov, ZhangThresholding Candès, Chambolle, Chaux, Combettes, Daubechies,Defrise, DeMol, Donoho, Pesquet, Starck, Teschke, Vese, WajsBregman iteration Cai, Burger, Darbon, Dong, Goldfarb, Mao, Osher,Shen, Xu, Yin, ZhangSplitting approaches Chan, Esser, Fornasier, Goldstein, Langer,Osher, Schönlieb, Setzer, WajsNonlocal TV Bertozzi, Bresson, Burger, Chan, Lou, Osher, Zhang
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We use quadratic programming (QP) to computethe non-negative minimizer of the TV functional
The minimizer of the functional
argminf ∈Rn+
{‖Af −m‖22 + α‖LHf ‖1 + α‖LVf ‖1
}can be transformed into the standard form
argminz∈R5n
{12zTQz + cT z
}, z ≥ 0, Ez = b,
where Q is symmetric and E implements equality constraints.
Large-scale primal-dual interior point QP method was developed inKolehmainen, Lassas, Niinimäki & S (2012) andHämäläinen, Kallonen, Kolehmainen, Lassas, Niinimäki & S (2013).
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Reduction to argminz∈R5n
{12z
TQz + cT z}
Denote horizontal and vertical differences by
LHf = u+H − u−H and LVf = u+V − u−V ,
where u±H , u±V ≥ 0. TV minimization is now
argminf ∈Rn+
{f TATAf − 2f TATm + α1T (u+H + u−H + u+V + u−V )
},
where 1 ∈ Rn is vector of all ones. Further, we denote
z =
f
u+Hu−Hu+Vu−V
, Q =
1σ2
ATA 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0
, c =−2ATmα1α1α1α1
.
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We collected X-ray projection data of a walnutfrom 1200 directions
The data was collected by Keijo Hämäläinen andAki Kallonen at University of Helsinki.
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This is the reconstruction using all 1200projections and filtered back-projection
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When only few projection angles are available,TV regularization performs better than FBP
23 angles 15 angles 10 angles
FBP
TV
These images were computed by Kati Niinimäki.
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Outline
Sparse sampling and tomography
Total variation regularization
Discretization-invariance
Besov space regularization
Parameter choice: the S-curve method
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X-ray tomography: Continuum model
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X-ray tomography: Practical measurement model
m ∈ Rk
k = 4
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X-ray tomography: Computational model
m = Afm ∈ Rk
f ∈ Rn
k = 4n = 48
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The numbers k and n are independent
m = Afm ∈ Rk
f ∈ Rn
k = 8n = 48
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The numbers k and n are independent
m = Afm ∈ Rk
f ∈ Rn
k = 8n = 156
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The numbers k and n are independent
m = Afm ∈ Rk
f ∈ Rn
k = 8n = 440
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The numbers k and n are independent
m = Afm ∈ Rk
f ∈ Rn
k = 16n = 440
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The numbers k and n are independent
m = Afm ∈ Rk
f ∈ Rn
k = 24n = 440
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Bayesian inversion using total variation priordoes not have a well-defined continuous limit
Theorem (Lassas and S 2004)Total variation prior is not discretization-invariant.
Sketch of proof: Apply a variant of the central limit theorem tothe independent, identically distributed random consecutivedifferences.
New numerical experiments are reported inKolehmainen, Lassas, Niinimäki and S (2012) andLucka (2012).
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Bayesian inversion using Besov space priorshas a well-defined continuous limit
Theorem (Lassas, Saksman and S 2009)Besov space priors are discretization-invariant.
Sketch of proof: Construction of well-defined Bayesian inversiontheory in infinite-dimensional Besov spaces that allow wavelet bases.Discretizations are achieved by truncating the wavelet expansion.
Numerical experiments are reported inKolehmainen, Lassas, Niinimäki and S (2012).
Deterministic Besov space regularization was first introduced inDaubechies, Defrise and De Mol (2004); this applies toBayesian MAP estimates.
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Outline
Sparse sampling and tomography
Total variation regularization
Discretization-invariance
Besov space regularization
Parameter choice: the S-curve method
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The wavelet transform divides an image into threetypes of details at different scales
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This is how the wavelet decomposition is definedfor two-dimensional periodic images
We can represent periodic functions by wavelet expansion
f (x , y) =2J0−1∑k1=0
2J0−1∑k2=0
cJ0~k φJ0,~k(x , y)+J−1∑j=J0
3∑`=1
2j−1∑k1=0
2j−1∑k2=0
wj~k` ψ`j ,~k(x , y),
where ~k = (k1, k2) and the coefficients cJ0~k and wj~k` are defined by
cJ0~k = 〈f , φJ0~k〉 =∫T2
f (x , y)φj~k(x , y)dxdy ,
wj~k` = 〈f , ψ`j~k〉 =
∫T2
f (x , y)ψ`j~k(x , y)dxdy .
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Besov space norm can be definedin terms of the wavelet coefficients
A function f belongs to Bspq(T2) if and only if the followingexpression is finite:
2J0−1∑k1=0
2J0−1∑k2=0
|cJ0~k |p
1p+ ∞∑
j=J0
2jq(s+1−2p )
3∑`=1
2j−1∑k1=0
2j−1∑k2=0
|wj~k`|p
qp
1q
We focus on the case p = 1 = q and s = 1:
‖f ‖B111(T2) =∑~k
|cJ0~k |+∑j ,`,~k
|wj~k`|.
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We promote sparsity by minimizinga sum of `2 and `1 norms
In the case of Besov space penalty we minimize
f MAP = arg minf ∈Rn+
{1
2σ2‖Af −m‖2`2 + α‖f ‖B111(T2)
},
where ψj are the wavelet basis functions.
The above kind of minimization task has been studied inChambolle, DeVore, Lee & Lucier 1998,Daubechies, Defrise & De Mol 2004,Candès, Romberg & Tao 2006,Klann, Maaß & Ramlau 2006,Grasmair, Haltmeier & Scherzer 2008,Kolehmainen, Lassas, Niinimäki & S 2013.
We use constrained quadratic programming for the minimization.
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History of multiresolution tomography
1994 Olson and DeStefano1994 Sahiner and Yagle1995 Delaney and Bresler1996 Berenstein and Walnut1996 Bhatia, Karl and Willsky1997 Rashid-Farrokhi, Liu,Berenstein and Walnut1997 Zhao, Welland and Wang1998 Louis, Maaß and Rieder1999 Candès and Donoho1999 Madych2000 Smith and Adhami2000 Bonnet, Peyrin, Turjmanand Prost
2002 Das and Sastry2002 Frese, Bouman and Sauer2004 Soleski and Walter2004 Zhong, Ning and Conover2006 Sastry and Das2006 Soleski and Walter2006 Lee and Lucier2006 Rantala, Vänskä, Järvenpää,Kalke, Lassas, Moberg and S2007 Niinimäki, S and Kolehmainen2008 Soussen and Idier2009 Vänskä, Lassas and S2011 Klann, Ramlau and Reichel2011 Terzija and McCann2011 Frikel
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Besov regularized sparse-data reconstructionscompared to filtered back-projection
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Outline
Sparse sampling and tomography
Total variation regularization
Discretization-invariance
Besov space regularization
Parameter choice: the S-curve method
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We took photographs of walnuts cut in half
These photos are used for estimating the expected number of nonzerowavelet coefficients in a two-dimensional tomographic reconstruction.Special thanks go to Esa Niemi for his careful job in sawing the walnuts.
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The S-curve method determines a regularizationparameter value giving the right sparsity level
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Kolehmainen, Lassas, Niinimäki & S 2012
Num
berof
nonzerowavelet
coeffi
cients
Regularization parameter α
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Something needs to be done to deal with theerratic behaviour of the “raw” S-curve
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Reconstruction from 30 projections usinginterpolated S-curve
α = 0.019
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Interpolated S-curve method for TV
These unpublished images were computed by Kati Niinimäki.
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All Matlab codes freelyavailable on a website!
Part I: Linear Inverse Problems1 Introduction2 Naïve reconstructions and inverse crimes3 Ill-Posedness in Inverse Problems4 Truncated singular value decomposition5 Tikhonov regularization6 Total variation regularization7 Besov space regularization using wavelets8 Discretization-invariance9 Practical X-ray tomography with limited data10 Projects
Part II: Nonlinear Inverse Problems11 Nonlinear inversion12 Electrical impedance tomography13 Simulation of noisy EIT data14 Complex geometrical optics solutions15 A regularized D-bar method for direct EIT16 Other direct solution methods for EIT17 Projects
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Inverse Days, Dec 11–13, 2013, Inari, Finlandhttp://inverse-problems.org/id2013/
Organizers:Maarten de HoopMatti LassasMarkku LehtinenLassi RoininenS. S.Gunther Uhlmann
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Thank you for your attention!
Preprints available at www.siltanen-research.net.
Sparse sampling and tomographyTotal variation regularizationDiscretization-invarianceBesov space regularizationParameter choice: the S-curve method
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