Phase contrast tomography at ESRF : Algorithms
Pierre Paleo ESRF
16 Sepember 2015
Outline
1 Iterative methods for limited data
Algorithms Wavelets as a sparsifying transform
2 Common problems and approaches
Rings artefacts Paganin with non constant δ/β Local tomography
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Pierre Paleo
1. Plan
1 Iterative methods for limited data
Algorithms Wavelets as a sparsifying transform
2 Common problems and approaches
Rings artefacts Paganin with non constant δ/β Local tomography
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Pierre Paleo
1. Iterative methods • Iterative methods get the most of the data • Limited data • Noise • Usually boils down to
argmin x
n o kP x − dk22 + g(x)
P : projection operator d : data x : slice g(x) : prior
• Sparsity-promoting prior : g(x) = λ kHxk1
• Our approach : • High-throughput (parallel/distributed) implementation • “Simple”, GPU-friendly algorithms
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Pierre Paleo
1.1 Plan
1 Iterative methods for limited data
Algorithms Wavelets as a sparsifying transform 2 Common problems and approaches
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1.1 Algorithms
Two types of formulations :
Analysis formulation
Synthesis formulation
kP x − dk22 + λ kHxk1
kP H ∗ w − dk22 + λ kwk1
• Algorithms are simpler for the Synthesis formulation • Ex: proxk·k is straightforward, proxkH·k is not 1 1 • Both are equivalent iff H is invertible • Wavelets, Dictionary Learning
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1.1 Proximal methods • Minimization of F = f + g : proximal methods • Forward-Backward : FISTA xk = proxγg (yk−1 − γ∇f (yk−1 )) αk = (k − 1)/(k + a) yk = xk + αk (xk − xk−1 ) This modified version : [CD14] • Primal-Dual : Chambolle-Pock
Minimize F (x) = f˜(x) + g˜(Kx) x ˜k = proxτ f˜ (xk−1 − τ K ∗ (uk−1 )) u ˜k = proxσ˜g∗ (uk−1 + σK(2˜ xk − xk−1 ) xk = xk−1 + ρ(˜ xk − xk−1 ) uk = uk−1 + ρ(˜ uk − uk−1 ) This relaxed version : [Con13] 4/25
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1.1 Proximal methods
• Proximal methods involve the computation of
proxγg (y) = argmin x
1 2 kx − yk2 + γg(x) 2
• For simple functions g, proxγg (·) is an element-wise operation ˜ • If not, split again the problem F (x) = f˜(x) + g˜(Kx) + h(x)
[Con13] • Simple operations + variables kept on GPU Python prototype of Chambolle-Pock for TV regularization (Deconvolution, Tomographic reconstruction) : github.com/pierrepaleo/ChambollePock
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1.1 A new “Conjugate subgradient” method
• Conjugate Gradient (CG) is fast for L2 minimization • for L2-L1 optimization, F (x) = kP x − dk22 + λ kHxk1 is not
differentiable • Subgradient methods : xk+1 = xk − γk pk
pk ∈ ∂F (xk )
• “Slow” • Not clear how to chose γk • Not a descent method (energy can increase in the process)
• Build a set of conjugate directions for a non-smooth function ?
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1.1 A new “Conjugate subgradient” method L2-L1 optimization : F (x) = kAx − bk22 + λ kxk1 | {z } | {z } f (x)
g(x)
Dictionary-Learning : F (x) = kP D∗ w − dk22 + λ kwk1 • Choice of the subdifferential :
( sign (x) if x 6= 0 ∂g(x) = sign (∇f (x)) if x = 0
→ Only one choice of subgradient 7/25
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1.1 A new “Conjugate subgradient” method
• Adaptive preconditioner based on |∇f |
[MP15]
• Robust for ill-posed problems
• Python prototype : github.com/pierrepaleo/csg
(Very) fast convergence Component-wise operations : GPU-friendly Current implementation is memory-consuming • Significant amount of time : CPU-GPU transfers
Sum on millions of elements 8/25
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(calculation of β)
: loss of accuracy
1.2 Plan
1 Iterative methods for limited data
Algorithms Wavelets as a sparsifying transform 2 Common problems and approaches
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1.2 Wavelets as a sparsifying transform
• Total Variation Fast Adapted only for simple (piecewise-constant) slices • Dictionary Learning Adapted to all images Have to learn an appropriate dictionary Slower
→ Trade-off between these two regularizations ?
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1.2 Wavelets as a sparsifying transform
Wavelet are attractive : • Most images are sparse in a wavelet basis • No need to learn a dictionary • Fast transforms • (Bonus) Multi-resolution processing Credits: ID16-a
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1.2 Wavelets : implementation • DWT is not translation invariant : synthesis artifacts • Achieve translation invariance by random shifts • undecimated WT and dual-tree complex WT use more memory
regular DWT
FBP
DWT with random shifts
• Choice of the wavelet : few vanishing moments 11/25
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1.2 Iterative methods : Execution time
Test on an Intel Xeon E5-2667 + NVidia Tesla K20m (ESRF scisoft12)
• Operators are simple for TV and Wavelets : iterations are fast • For big volumes, the bottleneck the Projection and
Backprojection routines 12/25
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1.2 Iterative methods in PyHST • Three sparsity-promoting methods are implemented • One or more optimization algorithms for each
• SIRT is implemented via the “SIRT-filter” approach [PB15] • Tip: using FBP instead of P in iterative methods
dramatically increases the convergence rate • In the Fourier grid, the low frequencies are over-represented.
The ramp is a preconditioner. 13/25
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2.0 Plan
1 Iterative methods for limited data
Algorithms Wavelets as a sparsifying transform
2 Common problems and approaches
Rings artefacts Paganin with non constant δ/β Local tomography
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Pierre Paleo
2.1 Plan
1 Iterative methods for limited data 2 Common problems and approaches
Rings artefacts Paganin with non constant δ/β Local tomography
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2.1 Rings artefacts
• Rings artefacts have many possible causes • defective pixels in the detector • dust on the scintillator • vibration of the monochromator • tilt of the rotation axis • ... • Sinogram pre-processing vs slice post-processing • Sum along angles and filter [Riv] [BH06] • Fourier-Wavelet de-striping [MTMS09] • Sinogram regularization [MROB14]
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2.1 Rings artefacts : iterative correction • Sinogram pre-processing can induce new artefacts in the reconstruction • Rings correction can be included in the objective function [PM15] 1 2 argmin F (x) = kP x + U r − dk2 + λ kHxk1 + λr krk1 2 x,r
Syntactic foam. Credits: Elodie Boller and Paul Tafforeau, ID19
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2.2 Plan
1 Iterative methods for limited data 2 Common problems and approaches
Rings artefacts Paganin with non constant δ/β Local tomography
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2.2 Paganin with non constant δ/β • “Flesh + Bones” samples : the “bones” parts are strongly
absorbing, preventing the visualization of surrounding features • Perform two Paganin phase retrieval with different distances
for bones and flesh
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2.2 Non constant δ/β : Multi-Paganin
Looking for cartilage in a Rabbit knee. Credits: Johannes Stroebel and Paola Coan, ID17
• Circumvents the strong absorption of some parts • Works if the heterogeneous parts can be segmented !
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2.3 Plan
1 Iterative methods for limited data 2 Common problems and approaches
Rings artefacts Paganin with non constant δ/β Local tomography
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2.3 Local tomography : cupping • Sinogram padding strategies [PDC13]
• “Sinogram straightening”
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2.3 Local tomography : cupping
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2.3 Local tomography : cupping
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2.3 Drift of the intensity values in the volume
• For elongated samples, the quantity of material outside ROI is
varying along the vertical axis • This leads to a gradient of contrast in the reconstructed
volume • Possible strategies • Appropriate padding ? • Estimate the material outside ROI ?
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2.3 Iterative methods and artefacts removal • Slice: 2048 × 2048 pixels • 250 (very) noisy projections Osmium sample. Credits: Alexandra Pacureanu, ID16
FBP after cupping removal 22/25
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2.3 Iterative methods and artefacts removal • Slice: 2048 × 2048 pixels • 250 (very) noisy projections Osmium sample. Credits: Alexandra Pacureanu, ID16
β = 20 22/25
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2.3 Iterative methods and artefacts removal • Slice: 2048 × 2048 pixels • 250 (very) noisy projections
β = 60 22/25
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3.0 Conclusion
• Iterative methods are not just esthetic • “Compressed sensing makes every pixel count” • Estimation of regularization parameter ? Quality assessment ? • Local tomography challenges • Cupping • Gradient of contrast
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Thanks you for your attention !
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3.0 References I Mirko Boin and Astrid Haibel. Compensation of ring artefacts in synchrotron tomographic images. Opt. Express, 14(25):12071–12075, Dec 2006. Antonin Chambolle and Charles Dossal. On the convergence of the iterates of “fista”. Preprint hal-01060130, September, 2014. Laurent Condat. A primal–dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms. Journal of Optimization Theory and Applications, 158(2):460–479, 2013. A. Mirone and P. Paleo. A conjugate subgradient algorithm with adaptive preconditioning for lasso minimization. To appear in Computational Mathematics and Mathematical Physics, Russian Academy of Science, 2015. E. X. Miqueles, J. Rinkel, F. O’Dowd, and J. S. V. Berm´ udez. Generalized titarenko’s algorithm for ring artefacts reduction. Journal of Synchrotron Radiation, 21:1333–1346, 2014. Beat M¨ unch, Pavel Trtik, Federica Marone, and Marco Stampanoni. Stripe and ring artifact removal with combined wavelet—fourier filtering. Opt. Express, 17(10):8567–8591, May 2009. D. M. Pelt and K. J. Batenburg. Accurately approximating algebraic tomographic reconstruction by filtered backprojection. In Proceedings of the 2015 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, 2015. Yongsheng Pan and Francesco De Carlo. A new padding scheme for local tomograpy in tomographic microscopy, 2013. 24/25
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3.0 References II
Pierre Paleo and Alessandro Mirone. Ring artifacts correction in compressed sensing tomographic reconstruction. Journal of Synchrotron Radiation, pages 1268–1278, 2015. M. Rivers. tomorecon.
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3.0 Iterative rings correction : details (1) argmin x,r
1 F (x) = kP x + U r − dk22 + λ kHxk1 + λr krk1 2
For the theoretical derivations, the variables are vectorized (i.e stacked into a column-vector). Symbol x d r P H
Name slice sinogram rings variables projector sparsf. transform
Shape N 2, 1 Np × Nb , 1 Nb , 1 Np × Nb , N 2 N 2 × N 2, N 2
N : number of rows (or columns) of the slice Nb : number of detector bins Np : number of projection angles
r1 T repeats the ring .. vector on each . = r 1 · · · r Nb · · · r 1 · · · r Nb projection angle r Nb
IdNb . U × r = .. ×
IdNb
U T × d = IdNb · · · IdNb
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Σ1 .. × = . dNp ×Nb ΣNb
Pierre Paleo
d1 .. .
sums along the columns
3.0 Iterative rings correction : details (2)
argmin
F (x) =
x,r
1 kP x + U r − dk22 + λ kHxk1 + λr krk1 2
• Lines assumed constant along the projection angles → Variable r of Nb components, repeated along the Np angles
• Only a few lines in the sinogram : penalize krk1
• The proximal is separable wrt x and r g(x, r) = u(x)+v(r)
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⇒ proxg (x, r) = (proxu (x) , proxv (r))
Pierre Paleo
3.0 Conjugate Gradient
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3.0 Conjugate Sub-gradient (1)
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3.0 Conjugate Sub-gradient (2)
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