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Semi-blind deconvolution in 4Pi-microscopy

Semi-blind deconvolution in 4Pi-microscopy

Robert St¨ uck

Institute for numerical and applied mathematics University of G¨

• ttingen

24.07.2009

Semi-blind deconvolution in 4Pi-microscopy

• utline

introduction 4Pi-microscopy the unknown phase shift the iteratively regularized Gauß Newton method (IRGNM) the method convergence poisson noise measurements and reconstructions

Semi-blind deconvolution in 4Pi-microscopy introduction

• utline

introduction 4Pi-microscopy the unknown phase shift the iteratively regularized Gauß Newton method (IRGNM) the method convergence poisson noise measurements and reconstructions

Semi-blind deconvolution in 4Pi-microscopy introduction 4Pi-microscopy

confocal fluorescence microscopy

the principle

◮ focused light excites

fluorescence markers in the

• bject

◮ the markers emit photons

which are detected

◮ optical scanning microscopy

advantages

◮ higher resolution compared to other optical microscopes ◮ imaging of 3-dimensional objects possible ◮ imaging of living objects possible

Semi-blind deconvolution in 4Pi-microscopy introduction 4Pi-microscopy

4Pi-microscopy

the principle

◮ confocal fluorescence microscopy ◮ interference of two laser beams in the

focus

◮ interference of the objects photons in

the detector

S.W. Hell, E.H.K. Stelzer, J. Opt. Soc.

• Am. A Vol. 9, Nr: 12, 1992

→ less illuminated volume and thus higher resolution 4Pi psf 2 point object 4Pi data

Semi-blind deconvolution in 4Pi-microscopy introduction 4Pi-microscopy

microtubular network in a PtK-2 cell

M.C. Lang, T. Staudt, J. Engelhardt, S.W. Hell, New Journal of Physics 10(2008)

Semi-blind deconvolution in 4Pi-microscopy introduction the unknown phase shift

the influence of a varying refractive index

◮ the 4Pi-psf: p(˜

x, φ(x)) ≈ Ienv(˜ x) cosγ(γ˜ z + φ(x)) , γ = 2, 4 for one and two photon excitation respectively φ = 0 φ = π

4

φ = π

2 ◮ leads to bad image recovery, namely side lobes if linear

deconvolution is applied

Semi-blind deconvolution in 4Pi-microscopy introduction the unknown phase shift

mathematical formulation of the imaging process

◮ The process of taking the image can be described by a

quasi-convolution with a position-dependent psf E(g δ(x)) =

• Ω

p(y − x, φ(x))f (y)dy. g δ : noisy data φ : phase shift p : psf f : object

◮ define the imaging operator

F

• f

φ

• :=
• Ω

p(y − x, φ(x))f (y)dy

◮ the task now is: find

f φ δ such that

• f

φ δ − f φ †

• is small, with F

f φ † = g,

• g − g δ

< δ.

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM)

• utline

introduction 4Pi-microscopy the unknown phase shift the iteratively regularized Gauß Newton method (IRGNM) the method convergence poisson noise measurements and reconstructions

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) the method

the setting

Let X, Y be Hilbert spaces, D(F) ⊂ X open, and F : D(F) → Y continuously Fr´ echet differentiable. Given the (nonlinear) operator equation F(x) = g with solution x† ∈ D(F). Find an approximation to x† for right hand side g δ, with

• g − g δ

< δ.

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) the method

the IRGN method

Find a regularized solution to the linearized equation F ′[xn]hn = g δ − F(xn), hn = argmin

h∈X

• F ′[xn](h) + F(xn) − g δ

2 + αnh + xn − x02 then update the iterate solution and the regularization parameter xn+1 = xn + hn and e.g. αn+1 = αn 2 .

• A. B. Bakushinskii, Comput. Maths Math. Phys. 32 1353-9, 1992

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) the method

IRGNM for 4Pi-microscopy

◮ the imaging operator F : L2(Ω) × H2(Ω) → L2(Ω)

F

• f

φ

• :=
• Ω

p(y − x, φ(x))f (y)dy has the Fr´ echet derivative

F ′ " f φ # „hf hφ « (x) = Z

Ω

 p(y − x, φ(x))hf (y) + ∂p ∂φ(y − x, φ(x))f (y)hφ(x) ff dy

◮ to enhance the reconstruction a constraint has to be added

C: convex set defined by the positivity of the object f ≥ 0.

xn+1 = argmin

x∈C

„‚ ‚ ‚F ′[xn](x) − F ′[xn](xn) + F(xn) − g δ‚ ‚ ‚

2

+ αnxn+1 − x02 «

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) convergence

a recursive error estimate

◮ Definitions:

Tn := F ′[xn], T := F ′[x†

C]

(1)

◮

Jαn (x) =

• Tn(x) −
• Tn(xn) − F(xn) + g δ

2 + αn x − x02 =

• Tn(x − x†

C) − rn(x† C − xn) − ǫ

• 2

+ αn x − x02 (2) with g δ = F(x†

C) + ǫ with ǫ ≤ δ

(3) F(x†

C)

= F(xn) + Tn(x†

C − xn) + rn(x† C − xn)

rn(x†

C − xn)

: Taylor remainder.

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) convergence

a recursive error estimate

• xn+1 − x†

C

• ≤

xn+1 − xref ,n + xref ,n − xref +

• xref − x†

C

• (4)

xn+1 := argmin

x∈C

• Tn(x − x†

C) − rn(x† C − xn) − ǫ

• 2

+ αn x − x02(5) xref ,n := argmin

x∈C

• Tn(x − x†

C)

• 2

+ αn x − x02 (6) xref := argmin

x∈C

• T(x − x†

C)

• 2

+ αn x − x02 (7)

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) convergence

Lemma

Lemma

Let X, Y be Hilbert spaces and T1, T2 : X → Y bounded, linear

• perators and let C ⊂ X be a closed and convex set. Furthermore let

˜ x ∈ C and x1, x2 ∈ C be the minimizers of the Tikhonov functionals corresponding to T1 and T2 with respect to the constraint to the set C: Ji (x) := Ti(x − ˜ x)2 + α x − x02 , xi := argmin

x∈C

Ji(x), i ∈ {1, 2}, with α > 0. Moreover let the source condition ˜ x = PC(T ∗

2 ω + x0),

ω ≤ ρ hold for some ω ∈ Y and ρ > 0. Then the distance of the minimizers is bounded by: x1 − x2 ≤ √ 2ρ T1 − T2 .

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) convergence

a recursive error estimate

Assumptions:

◮ source condition: For some ω ∈ Y and ρ > 0 let

x†

C = PC(T ∗ω + x0),

ω ≤ ρ. (A1)

◮ Lipschitz condition: Let F ′[x] − F ′[y] ≤ L x − y for some L > 0

and for all x, y ∈ D(F). (A2)

Lemma

If the assumptions (A1),(A2) are satisfied and xn is well defined, the recursive error estimate en+1 ≤ L 2√αn en2 + √ 2Lρ en + δ √αn + √αnρ (8) holds. (en :=

• xn − x†

C

• )

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) convergence

convergence result

Theorem (Hohage, St¨ uck)

Let the assumptions (A1),(A2) be satisfied and assume that ρ is sufficiently small and 1 ≤ αn αn+1 ≤ r, lim

n→∞ αn = 0,

αn > 0 with α0 ≤ 1 and r > 1. Then one obtains

◮ convergence for exact data: Let δ = 0

• xn − x†

C

• = O(α

1 2

n ),

n → ∞

◮ If the stopping index N is chosen such that

αN < ηδ ≤ αn, 0 ≤ n < N, with η sufficiently large. Then one

• btains convergence with respect to the noise level:
• xN − x†

C

• = O(δ

1 2 ),

δ → 0

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) poisson noise

incorporation of the Poisson distribution

◮ The Cartesian components g δ i of the data g δ ∈ Rn are drawn from

independent Poisson distributed random variables Gi with mean gi, i.e. Pg(G = g δ) = n

i=1 (gi)gδ

i

g δ

i ! e−gi

◮ Log-likelihood data misfit functional:

l(g) := − ln Pg(G = g δ) = n

i=1 gi − g δ i ln gi + c, where c is

independent of g.

◮ Taylor expansion of l(g) =

l(g δ)+(∇l)(g δ)

• (g −g δ)+ 1

2(g −g δ)TH(g δ)(g −g δ)+O((g −g δ).3),

with (∇l(g))i = 1 − g δ

i

gi and H(g)i,j = g δ

i

g 2

i δi,j

◮ This leads to the following weighted l2 norm in the data space Y :

g − g δ2

Y = n i=1 1 2(g δ)i (g − g δ)2 i

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

• utline

introduction 4Pi-microscopy the unknown phase shift the iteratively regularized Gauß Newton method (IRGNM) the method convergence poisson noise measurements and reconstructions

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

unconstraint reconstruction

psf data 20 40 60 80 phase reconstruction

• 2

2 4

• bject reconstruction
• 200
• 100

100 200 300

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

constraint reconstruction - start phase 0

psf data 20 40 60 80 phase reconstruction

• 1.5
• 0.5

0.5 1.5

• bject reconstruction

100 200 300

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

constraint reconstruction

psf data 20 40 60 80 phase reconstruction

• 2

2 4

• bject reconstruction

100 200 300

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

future work

◮ test algorithm on 3d real data ◮ reliable choice of regularization parameters for H2 and object

penalization

◮ automatic estimation of the phase shift initial guess ◮ combination with BV regularisation

Semi-blind deconvolution in 4Pi-microscopy measurements and reconstructions

future work

◮ test algorithm on 3d real data ◮ reliable choice of regularization parameters for H2 and object

penalization

◮ automatic estimation of the phase shift initial guess ◮ combination with BV regularisation

Thank you for your attention.