Semi-blind deconvolution in 4Pi-microscopy Robert St uck Institute - PowerPoint PPT Presentation

Semi-blind deconvolution in 4Pi-microscopy Semi-blind deconvolution in 4Pi-microscopy Robert St¨ uck Institute for numerical and applied mathematics University of G¨ ottingen 24.07.2009

Semi-blind deconvolution in 4Pi-microscopy outline 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 outline 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 object ◮ 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 ) cos γ ( γ ˜ x , φ ( x )) ≈ I env ( ˜ z + φ ( x )) , γ = 2 , 4 for one and two photon excitation respectively φ = π φ = π φ = 0 4 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 )) = p ( y − x , φ ( x )) f ( y ) d y . Ω g δ : noisy data φ : phase shift p : psf f : object ◮ define the imaging operator � � � f F := p ( y − x , φ ( x )) f ( y ) d y φ Ω � f � δ ◮ the task now is: find such that φ � � † � � f � δ � f � f � † � � � g − g δ � � � < δ. � � − � is small, with F = g , � � φ φ φ �

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) outline 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 ). � � g − g δ � Find an approximation to x † for right hand side g δ , with � < 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 ′ [ x n ] h n = g δ − F ( x n ) , �� � 2 + α n � h + x n − x 0 � 2 � � F ′ [ x n ]( h ) + F ( x n ) − g δ � h n = argmin h ∈ X then update the iterate solution and the regularization parameter α n +1 = α n x n +1 = x n + h n and e.g. 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 : L 2 (Ω) × H 2 (Ω) → L 2 (Ω) � � � f F := p ( y − x , φ ( x )) f ( y ) d y φ Ω has the Fr´ echet derivative " # „ h f f « Z  p ( y − x , φ ( x )) h f ( y ) + ∂ p ff F ′ ( x ) = ∂φ ( y − x , φ ( x )) f ( y ) h φ ( x ) dy φ h φ Ω ◮ to enhance the reconstruction a constraint has to be added C : convex set defined by the positivity of the object f ≥ 0. „‚ 2 « ‚ F ′ [ x n ]( x ) − F ′ [ x n ]( x n ) + F ( x n ) − g δ ‚ + α n � x n +1 − x 0 � 2 x n +1 = argmin ‚ ‚ ‚ x ∈C

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) convergence a recursive error estimate ◮ Definitions: T := F ′ [ x † T n := F ′ [ x n ] , C ] (1) ◮ � 2 + α n � x − x 0 � 2 � � T n ( x n ) − F ( x n ) + g δ �� J α n ( x ) = � T n ( x ) − � � 2 + α n � x − x 0 � 2 � T n ( x − x † C ) − r n ( x † � � = C − x n ) − ǫ (2) � with F ( x † g δ = C ) + ǫ with � ǫ � ≤ δ (3) F ( x † F ( x n ) + T n ( x † C − x n ) + r n ( x † C ) = C − x n ) r n ( x † C − x n ) : Taylor remainder .

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) convergence a recursive error estimate � � � � � x n +1 − x † � x ref − x † � � � � ≤ � x n +1 − x ref , n � + � x ref , n − x ref � + � (4) C � C � � 2 + α n � x − x 0 � 2 (5) � T n ( x − x † C ) − r n ( x † � � x n +1 := argmin C − x n ) − ǫ � x ∈C � � 2 � T n ( x − x † + α n � x − x 0 � 2 � � x ref , n := argmin C ) (6) � x ∈C � � 2 + α n � x − x 0 � 2 � � T ( x − x † � x ref := argmin C ) (7) � x ∈C

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) convergence Lemma Lemma Let X , Y be Hilbert spaces and T 1 , T 2 : X → Y bounded, linear operators and let C ⊂ X be a closed and convex set. Furthermore let ˜ x ∈ C and x 1 , x 2 ∈ C be the minimizers of the Tikhonov functionals corresponding to T 1 and T 2 with respect to the constraint to the set C : x ) � 2 + α � x − x 0 � 2 , J i ( x ) := � T i ( x − ˜ x i := argmin J i ( x ) , i ∈ { 1 , 2 } , x ∈C with α > 0 . Moreover let the source condition x = P C ( T ∗ ˜ 2 ω + x 0 ) , � ω � ≤ ρ hold for some ω ∈ Y and ρ > 0 . Then the distance of the minimizers is bounded by: √ � x 1 − x 2 � ≤ 2 ρ � T 1 − T 2 � .

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 = P C ( T ∗ ω + x 0 ) , � 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 x n is well defined, the recursive error estimate √ + √ α n ρ L δ � e n � 2 + � e n +1 � ≤ 2 L ρ � e n � + (8) 2 √ α n √ α n � � � x n − x † � � holds. ( � e n � := � ) 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 α n 1 ≤ ≤ r , n →∞ α n = 0 , lim α n > 0 α n +1 with α 0 ≤ 1 and r > 1 . Then one obtains ◮ convergence for exact data: Let δ = 0 � � 1 � x n − x † � � � = O ( α n ) , 2 n → ∞ C ◮ If the stopping index N is chosen such that α N < ηδ ≤ α n , 0 ≤ n < N, with η sufficiently large. Then one obtains convergence with respect to the noise level: � � 1 � x N − x † � � 2 ) , � = O ( δ δ → 0 C

Semi-blind deconvolution in 4Pi-microscopy the iteratively regularized Gauß Newton method (IRGNM) poisson noise incorporation of the Poisson distribution i of the data g δ ∈ R n are drawn from ◮ The Cartesian components g δ independent Poisson distributed random variables G i with mean g i , ( g i ) g δ i.e. P g ( G = g δ ) = � n i i ! e − g i i =1 g δ ◮ Log-likelihood data misfit functional: l ( g ) := − ln P g ( G = g δ ) = � n i =1 g i − g δ i ln g i + 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 ) T H ( g δ )( g − g δ )+ O (( g − g δ ) . 3 ), � �� � 0 with ( ∇ l ( g )) i = 1 − g δ g i and H ( g ) i , j = g δ i i δ i , j i g 2 ◮ This leads to the following weighted l 2 norm in the data space Y : Y = � n � g − g δ � 2 2( g δ ) i ( g − g δ ) 2 1 i =1 i

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