Overview on pyramid wavefront sensor: forward ...

Introduction

Linear models and reconstructors

Non-linear models and reconstructors

Summary

Overview on pyramid wavefront sensor: forward models, reconstruction algorithms, practical issues Iuliia Shatokhina, Victoria Hutterer, Andreas Obereder, Stefan Raffetseder, Ronny Ramlau Industrial Mathematics Institute, JKU, Linz

WaveFront Sensing in the VLT/ELT era II, Padova, October 2-4, 2017

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Introduction



Summary

Outline

1

Introduction

2

Linear models and reconstructors Roof WFS: linearized and simplified model Algorithms in closed-loop simulations: quality, speed and spiders Extension of algorithms to other linear models

3

Non-linear models and reconstructors Roof WFS: nonlinear transmission mask model Algorithms in closed-loop simulations: quality and speed Extenstion of algorithms to other non-linear models

4

Summary

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Outline

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Introduction

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4

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Introduction



Summary

Pyramid and Roof WFS

´ Credit: C. Verinaud Sx (x, y) =

[I1 (x, y) + I2 (x, y)] − [I3 (x, y) + I4 (x, y)] I0

Sy (x, y) =

[I1 (x, y) + I4 (x, y)] − [I2 (x, y) + I3 (x, y)] I0

I0 – average intensity per subaperture.

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Introduction



Summary

Inverse problem & Remarks

Task: to reconstruct the unknown wavefront φ from non- / modulated pyramid WFS data Sx , Sy Sx = Px φ, Sy = Py φ Forward operators Px , Py are nonlinear singular integral operators Details omitted (e.g., Sx only) Any modulation meant; αλ will denote the modulation parameter αλ =

2πr 2πα = , λ D

d ∈ R+

Omit aperture sometimes (for clarity) Finite sampling From simple approximate to complicated models

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Outline

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Introduction

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4

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Introduction



Summary

Roof WFS: linearized and simplified model

{n,l,c}

Roof WFS: linearized and simplified operator R s

{n,l,c}

Sx

{n,l,c}

(R s

φ)(x, y)

=

1 π

{n,l,c}

=

Rs

X Z(y)

φ(x 0 , y)k{n,l,c} (x 0 − x)

φ

x − x0

dx 0

−X (y)

= {n,l,c}

mx

h

{n,l,c}

φ ∗ mx

(x, y) :=

i

(x, y)

k{n,l,c} (x)δ(y) πx

kn (x) = 1 kl (x) = sinc(αλ (x)) kc (x) = J0 (αλ (x)) J0 – the zero-order Bessel function of the first kind. 7 / 32

Introduction



Summary


{n,l,c}

Inversion of R s

in spatial domain

Inversion of Finite Hilbert transform: (modulation 0 only, i.e., R ns ) Finite Hilbert Transform Reconstructor (FHTR) [1] {l,c} Singular Value Type Reconstructor (SVTR) [2] → SVTR for R s ?

Iterative algorithms: adjoint operator Conjugate Gradient for the Normal Equation (CGNE) [1,3,4] Steepest Descent (SD) [3,4] Pyramid Kaczmarz Iteration (PKI) [3,4] [1] I. Shatokhina, “Fast wavefront reconstruction algorithms for extreme adaptive optics,” Ph.D. thesis (Johannes Kepler University Linz, 2014). [2] V. Hutterer, R. Ramlau, Wavefront Reconstruction from Non-modulated Pyramid Wavefront Sensor Data using a Singular Value Type Expansion, Inverse Problems, submitted. [3] V. Hutterer, R. Ramlau, Iu. Shatokhina, Real-time AO with pyramid wavefront sensors: Theoretical analysis of pyramid forward model, in preparation. [4] V. Hutterer, R. Ramlau, Iu. Shatokhina, Real-time AO with pyramid wavefront sensors: Accurate wavefront reconstruction with iterative methods, in preparation.

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Introduction



Summary


FHTR & SVTR

Invert Rsn = Tx φ = [φ ∗ mxn ] sxn = Rsn φ = Tx−1 sx

SVTR: FHTR: 1 φ(x) = − π

Z1 −1

s

1 − x 2 sx (x 0 ) 0 dx 1 − x 02 x − x 0

singular value type system (σk , fk , gk )k ≥0 , fk , gk – weighted Chebyshev polynomials, σk = 1 ∀k

φ (x, y) = −2

∞ X 1 hsx (·, y) , gk iω fk (x). σk k =0

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Introduction



Summary


Iterative algorithms: CGNE & SD & PKI Well-known (in applied mathematics) iterative methods Application of adjoint operators Z 1 Ψ(x 0 , y) · k {n,c,l} (x 0 − x) {n,c,l} ∗ Rs Ψ (x, y) = − p.v . dx 0 , π x0 − x Ωy

Due to discretization largely precomputed −→ fast!

Algorithm: Landweber-Kaczmarz iteration choose Φ0 , set attenuation coefficients β1 , β2 for i = 1, . . . K do Φi,0 = Φi−1 Φi,1 = Φi,0 + β1 R ∗x sx − R x φi,0 ∗ Φi,2 = Φi,1 + β2 R y sy − R y φi,1 Φi = Φi,2 endfor 10 / 32

Introduction



Summary


{n,l,c}

Inversion of R s

in Fourier domain {n,l,c}

Fourier domain representation of R s

φ:

{n,l,c} )(u) = (Fx φ)(u) · g{n,l,c} (u) (Fx Sx {n,l,c} (u) g{n,l,c} (u) = Fx mx

Fourier domain based algorithms: Preprocessed Cumulative Reconstructor with domain Decomposition (P-CuReD) [1,2] Convolution with the Linearized Inverse Filter (CLIF) [2,3] Pyramid Fourier Transform Reconstructor (PFTR) [2,3] [1] Iu. Shatokhina, A. Obereder, R. Rosensteiner, R. Ramlau. Preprocessed cumulative reconstructor with domain decomposition: a fast wavefront reconstruction method for pyramid wavefront sensor, Applied Optics 52(12), 2640-2652 (2013). [2] I. Shatokhina, R. Ramlau. Convolution and Fourier transform based reconstructors for pyramid wavefront sensor, Applied Optics 56(22), 6381-6390 (2017). [3] I. Shatokhina, “Fast wavefront reconstruction algorithms for extreme adaptive optics,” Ph.D. thesis (Johannes Kepler University Linz, 2014).

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Introduction



Summary


P-CuReD & CLIF & PFTR

(Fx Sx )(u) = (Fx φ)(u) · g(u) PFTR: −1

P-CuReD: SH

SH

(Fx S)pyr (u) = (Fx φ)(u) · gpyr (u)

(Fx φ)(u) = (Fx Sx )(u) · g (u) h i −1 −1 φ(x, y) = Fx (Fx Sx )(u) · g (u) (x, y)

(Fx SSH )(u) = (Fx Spyr )(u) · gSH/pyr (u) gSH/pyr (u) :=

gSH (u) gpyr (u) −1

SSH (x) = (Spyr ∗ Fx gSH/pyr )(x) | {z } pSH/pyr

φ(x, y) = CuReD(SSH )

CLIF: h i −1 −1 φ(x, y) = Sx ∗ Fx g (x, y) −1 −1 p(x, y ) = Fx g (x) δ(y) φ(x, y) = [Sx ∗ p] (x, y )

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Introduction



Summary

Algorithms in closed-loop simulations: quality, speed and spiders

Considered AO systems

XAO (EPICS on ELT)

SCAO (METIS on ELT)

aim: direct imaging of exoplanets

D = 37 m telescope

D = 42 m telescope

pyramid WFS with 74x74 subapertures, with circular modulation

pyramid WFS with 200x200 subapertures, with circular modulation DM update 3000 times per second!

DM update 500-1000 times per second!

time for reconstruction: 0.3 ms

time for reconstruction: 1-2 ms

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Introduction



Summary


Comparison of quality: linear algorithms

Algorithm Photon flux Frame rate Matrix-Vector Multiplication (MVM) MMSE (YAO)

Quality in end-to-end simulations (OCTOPUS) METIS mod 0 10000 ph/pix/it 1kHz

METIS mod 4 10000 ph/pix/it 1kHz

≈ 0.62 [1]

0.80 [2] 0.89 [3]

Preprocessed CuReD (P-CuReD) Conv. with Linearized Inverse Filter (CLIF) Pyramid FTR (PFTR) Finite Hilbert Transform Rec. (FHTR) Singular Value Type Reconstructor (SVTR) Steepest Descent (SD) Pyramid Kaczmarz Iteration (PKI)

0.89

XAO mod 0 50 ph/pix/it 3kHz

XAO mod 4 50 ph/pix/it 3kHz 0.96

0.91 0.88 0.88

0.77

0.85 0.88

≥ 0.79 0.81

0.90 0.92

0.96 0.94 0.94

[1] M. Le Louarn et. al., Latest AO simulation results for the E-ELT, poster AO4ELT5. [2] Results provided by ESO. [3] MMSE reconstructor in YAO, results provided by Stefan Hippler.

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Introduction



Summary


Comparison of complexities: linear methods

Algorithm no

Modulation small large

Complexity 2

Remarks

Matrix-Vector Multiplication (MVM) Fourier Transform Reconstructor (FTR)

+ –

+ –

+ +

O(n ) O(n log n)

baseline; geometrical model

Preprocessed CuReD (P-CuReD) Conv. with Linearized Inverse Filter (CLIF) Pyramid FTR (PFTR)

+ + +

+ + +

+ + +

O(n) O(n3/2 ) O(n log n)

Fourier domain based (iteartive)

Finite Hilbert Transform Rec. (FHTR) Singular Value Type Reconstructor (SVTR)

+ +

– –

– –

O(n3/2 ) O(n3/2 )

inversion of finite Hilbert transform

Conjugate Gradient for Normal Eq. (CGNE) Steepest Descent (SD) Pyramid Kaczmarz Iteration (PKI)

+ + +

+ + +

+ + +

O(n3/2 ) O(n3/2 ) O(n3/2 )

iterative algorithms, adjoint operators

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Introduction



Summary


Comparison of computational load: linear methods

Algorithm

Number of operations in XAO setting MVM 4na n 3.4120e+09 P-CuReD (4c − 2)n + 20n 1.3248e+06 √ CLIF 4n n + n 1.9579e+07 √ 4 · 5.8996e + 07 SD(K = 4) K · (12n n + 12n + 4) √ PKI(K = 5) K · (8n n + 2n) 5 · 3.9158e + 07 na = 29618, n = 28800, c = 7

100% 0.0388% 0.5738% 4 · 1.73% = 6.92% 5 · 1.15% = 5.75%

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Introduction



Summary


Reconstruction in presence of spiders

Residual segmented piston −→ low LE There are possibilities to control differential piston four methods some provide acceptable quality extremely fast! Add 6x2na FLOPs

Poke matrix inversion −→ see talk by A. Obereder tomorrow @ 12.40 ”Keep it simple – Poke Matrix Inversion for a (stable) piston segment reconstruction” Split approaches −→ see talk by V. Hutterer tomorrow @ 12.00 ”Direct piston reconstruction approaches to control segmented ELT-mirrors”

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Introduction



Summary


Important questions Spiders – theoretical understanding / explanation sign (not possible from linearized roof sensor models) full pyramid model, take interference terms into account? reconstruction of pistons from intensities I{1,2,3,4} ? identify segments between which piston jumps occur criteria to identify if the sign of piston between neighbouring segments is the same, or the opposite criteria for piston sign – work in progress

What is the best possible reconstruction quality?

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Introduction



Summary

Extension of algorithms to other linear models

Other linear models Model

{n,l,c}

(R l

{n,l,c}

(P s

φ)(x, y)

{n,l,c}

φ)

=

Linear

Pl

φ)(x, y )

φ)(x, y)

(R s

{n,l,c}

(R l

{n,l,c}

Ps

pc (˜ x , y˜ ) :=

(P l

Pyramid WFS

{n,l,c} Rl

{n,l,c}

=

{n,l,c}

Rs

{n,l,c}

(R s

Roof-WFS

Linear simplified

1 T

φ) −

=

π

{n,l,c}

x − x0

{n,l,c}

φ)(x, y) − φ(x, y)(R s

X Z(y )

1 π3

φ(x 0 , y)k{n,l,c} (x 0 − x)

−X (y)

(R s

φ)(x, y ) −

T Z/2

X Z(y)

1

=

{n,l,c}

Y Z(x)

Y Z(x)

−X (y ) −Y (x) −Y (x)

dx

0

1)(x, y )

φ(x 0 , y 0 )p{n,c} (x 0 − x, y 0 − y 00 ) (x − x 0 )(y − y 0 )(y − y 00 )

00

0

dy dy dx

0

cos[αλ x˜ sin(2πt/T )] cos[αλ y˜ cos(2πt/T )]dt

−T /2

1 π3

X Z(y )

Y Z(x)

Y Z(x)

−X (y ) −Y (x) −Y (x)

[φ(x 0 , y 0 )−φ(x, y 00 )]p{n,c} (x 0 − x, y 0 − y 00 ) (x − x 0 )(y − y 0 )(y − y 00 )

00

0

dy dy dx

0

i-FHTR, i-SVTR; i-PFTR, i-CLIF; CGNE, SD, PKI 19 / 32

Introduction



Summary


Other linear models – Fourier domain representation Fourier domain representation of {n,l,c}

Rs

φ: P-CuReD, PFTR, CLIF {n,l,c}

)(u) = (Fx φ)(u) · g{n,l,c} (u) {n,l,c} g{n,l,c} (u) = Fx mx (u)

(Fx Sx

{n,l,c}

Rl

φ: i-PFTR, i-CLIF

{n,l,c} (Fx Sx )(u)

h i (Fx φ)(u) · g{n,l,c} (u) − (Fx φ)(u) ∗ (Fx XΩy ×Ωx )(u) · g{n,l,c} (u)

=

{n,l,c}

Ps

Non-modulated case: i-PFTR, i-CLIF n

(Fxy Sx )

= −

n

(Fxy φ) · (Fxy mx ) h i h i n n (Fxy φ) · (Fxy mxy ) ∗ (Fxy XΩy ×Ωx ) · (Fxy my )

Modulated case: ? {n,l,c}

Pl

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Introduction



Summary



Model

Roof-WFS

Ps

Linear

{n,l,c} Rl

Pl

{n,l,c}

Forward operator

Algorithm

Remarks

FHTR → iFHTR iSVTR

Hilbert transform; singular functions

P-CuReD

Fourier domain based

{n,l,c}

Rs

{n,l,c}

{n,l,c}

→ Rl → P ns → P nl {n,l,c} {n,l,c} Rs → Rl → P ns → P nl {n,l,c}

{n,l,c}

Rs

R ns → R nl → P ns → P nl

Rs

Rs

Pyramid WFS

{n,l,c}

Linear simplified

{n,l,c}

→ Rl

{n,l,c}

→ Ps

{n,l,c}

→ Pl

CLIF → i-CLIF PFTR → i-PFTR CGNE

iterative algorithms,

SD PKI

adjoint operators

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Outline

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Introduction



Summary

Roof WFS: nonlinear transmission mask model

Roof WFS: nonlinear transmission mask model

{n,l,c}

Sx

{n,l,c}

(R t

φ)(x, y)

=

1 XΩy ×Ωx (x, y ) π

=

X (y ) Z

{n,l,c}

Rt

φ

sin φ(x 0 , y) − φ(x, y) k{n,l,c} (x 0 − x) x − x0

dx

0

−X (y )

= −

k{n,l,c} (·)δ(y ) XΩy ×Ωx (x, y ) cos(φ(x, y)) · XΩy ×Ωx (·, y) sin(φ(·, y )) ∗ π· k{n,l,c} (·)δ(y) XΩy ×Ωx (x, y ) sin(φ(x, y)) · XΩy ×Ωx (·, y) cos(φ(·, y )) ∗ π·

Inversion: Nonlinear Landweber method, nonlinear CG, nonlinear SD, ... ∗ {n,l,c} 0 {n,l,c} φk +1 = φk + Rt sx − R t φk , k ∈N

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Introduction



Summary

Algorithms in closed-loop simulations: quality and speed

Comparison of quality Algorithm Photon flux Frame rate Matrix-Vector Multiplication (MVM)

Quality in end-to-end simulations (OCTOPUS) METIS mod 0 10000 ph/pix/it 1kHz

METIS mod 4 10000 ph/pix/it 1kHz

≈ 0.62 [1] (1000ph)

0.80 [2] (1000ph) 0.89 [3]

Preprocessed CuReD (P-CuReD) Conv. with Linearized Inverse Filter (CLIF) Pyramid FTR (PFTR)

0.89

XAO mod 0 50 ph/pix/it 3kHz

XAO mod 4 50 ph/pix/it 3kHz 0.96 0.96

0.91 0.88 0.88

Finite Hilbert Transform Rec. (FHTR) Singular Value Type Reconstructor (SVTR)

0.77

0.85 0.88

Conjugate Gradient for Normal Eq. (CGNE) Steepest Descent (SD) Pyramid Kaczmarz Iteration (PKI)

≥ 0.79 0.81

0.90 0.92

Nonlinear Landweber (NL)

≥ 0.83

0.96 0.94 0.94

[1] M. Le Louarn et. al., Latest AO simulation results for the E-ELT, poster AO4ELT5. [2] Results provided by ESO. [3] MMSE reconstructor in YAO, results provided by Stefan Hippler.

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Introduction



Summary

Algorithms in closed-loop simulations: quality and speed

Comparison of computational load

Algorithm no

Modulation small large

Complexity

Remarks

Matrix-Vector Multiplication (MVM) Fourier Transform Reconstructor (FTR)

+ –

+ –

+ +

O(n2 ) O(n log n)

baseline; geometrical model

Preprocessed CuReD (P-CuReD) (i)-Conv. with Linearized Inverse Filter (CLIF) (i)-Pyramid FTR (PFTR)

+ + +

+ + +

+ + +

O(n) O(n3/2 ) O(n log n)

Fourier domain based (iteartive)

Hilbert Transform Reconstructor (HTR) (i)-Finite Hilbert Transform Rec. (FHTR) (i)-Singular Value Type Reconstructor (SVTR)

+ + +

– – –

– – –

O(n log n) O(n3/2 ) O(n3/2 )

(iterative) inversion of finite Hilbert transform; singular functions

Steepest Descent (SD) Pyramid Kaczmarz Iteration (PKI)

+ +

+ +

+ +

O(n3/2 ) O(n3/2 )

adjoint operators

Nonlinear Landweber (NL)

+

+

+

O(n3/2 )

´ Frechet derivative its adjoint

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Introduction



Summary

Extenstion of algorithms to other non-linear models

Non-linear models

Model Nonlinear transmission mask Nonlinear phase mask without interference Nonlinear phase mask withi interference

Roof-WFS {n,l,c}

Rt

{n,l,c} Rp {n,l,c} Ri

Pyramid WFS {n,l,c}

Pt

{n,l,c}

Pp

{n,l,c}

Pi

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Introduction



Summary


Pyramid WFS: nonlinear transmission mask model {n,l,c}

(P t

φ)(x, y)

= −

{n,l,c}

(R t 1 π3

φ)(x, y )

X (y ) Z

Y (x) Z Y (x) Z

sin[φ(x 0 , y 0 )−φ(x, y 00 )]p{n,c} (x 0 − x, y 0 − y 00 ) (x − x 0 )(y − y 0 )(y − y 00 )

00

0

dy dy dx

0

−X (y ) −Y (x) −Y (x)

Inversion: nonlinear Landweber, nonlinear CG, nonlinear SD, ...

ψdet (x, y) =

1 t ψaper ∗ F −1 {OTFpyr } (x, y) 2π

t OTFpyr (ξ, η) =

1 X 1 X

T mn (ξ, η)

m=0 n=0

T mn (ξ, η) = H2d (−1)m · ξ, (−1)n · η I(x, y) ≈

1 X 1 X

ψn,m (x, y) · ψn,m (x, y)

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Introduction



Summary


Roof & Pyramid WFS: nonlinear phase mask model w/o interference Nonlinear, phase mask, no interference: 1 p ψaper ∗ F −1 {OTFpyr } (x, y) ψdet (x, y) = 2π p

t OTFpyr (ξ, η) = exp(−i · Π(ξ, η)) · OTFpyr (ξ, η)

I(x, y) ≈

1 X 1 X


n=0 m=0

Nonlinear, phase mask, with interference: 1 p ψaper ∗ F −1 {OTFpyr } (x, y) ψdet (x, y) = 2π p

t OTFpyr (ξ, η) = exp(−i · Π(ξ, η)) · OTFpyr (ξ, η)

I(x, y)

=

1 X 1 X


n=0 m=0

+

2

1 X 1 X

1 X

1 X

Re[ψn,m (x, y) · ψn0 ,m0 (x, y)]

n=0 m=0 n0 =0,n0 6=n m0 =0,m 6=m 28 / 32

Introduction



Summary



Model

Roof-WFS {n,l,c}

Rt

Nonlinear transmission mask Nonlinear phase mask without interference Nonlinear phase mask withi interference

Forward operator Rt

{n,l,c}

→ Rp

{n,l,c}

Pt

{n,l,c}

→ Pp

{n,l,c}

{n,l,c} Rp {n,l,c} Ri

Pyramid WFS {n,l,c}

Pt

{n,l,c}

Pp

{n,l,c}

Pi

Algorithm

Remarks

→ Ri

NL, nCG, ...

´ Frechet derivative,

→ Pi

NL, nCG, ...

adjoint

{n,l,c}

{n,l,c}

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Outline

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Introduction

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Introduction



Summary

Summary

Roof −→ pyramid Linearized models −→ non-linear models A wide spectrum of algorithms developed and studied: linear and non-linear Quality and speed better than MVM ! Can handle spiders with a linear method ! Open questions: best reconstruction quality model, ncpa, deeper understanding of spiders Go on-sky...

Urban Bitenc et al., On-sky tests of the CuReD and HWR fast wavefront reconstruction algorithms with CANARY. Monthly Notices of the Royal Astronomical Society 448(2), 1199-1205 (2015).

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Introduction



Summary

Thanks

Thank you for attention!

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