Singular Values of the Attenuated Photoacoustic Imaging Operator

Transcription

1 Singular Values of the Attenuated Photoacoustic Imaging Operator Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 1 Computational Science Center, University Vienna, Austria & 2 Radon Institute of Computational and Applied Mathematics (RICAM), Linz, Austria Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 1 / of48com

2 Outline 1 Photoacoustics Applications 2 Modeling aspects Attenuation Examples of attenuation models 3 Inversion of the integrated photoacoustic operator Strongly attenuating media Weakly attenuating media 4 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 2 / of48com

3 Photoacoustics Photoacoustic imaging Lightning and Thunder (L.H. Wang) Specimen is uniformly illuminated by a short/pulsed electromagnetic pulse (visible or near infrared light - Photoacoustics, microwaves - Thermoacoustics) eter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 3 / of48com

4 Photoacoustics Photoacoustic imaging Lightning and Thunder (L.H. Wang) Specimen is uniformly illuminated by a short/pulsed electromagnetic pulse (visible or near infrared light - Photoacoustics, microwaves - Thermoacoustics) Two-step conversion process: Absorbed EM energy is converted into heat Material reacts with expansion Expansion produces pressure waves eter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 3 / of48com

5 Photoacoustics Photoacoustic imaging Lightning and Thunder (L.H. Wang) Specimen is uniformly illuminated by a short/pulsed electromagnetic pulse (visible or near infrared light - Photoacoustics, microwaves - Thermoacoustics) Two-step conversion process: Absorbed EM energy is converted into heat Material reacts with expansion Expansion produces pressure waves Imaging: Pressure waves are detected at the boundary of the object (over time) and are used for reconstruction of conversion parameter (EM energy into expansion/ waves) eter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 3 / of48com

6 Photoacoustics The method in between DOT (diffuse optical tomography), OCT (optical coherence tomography), SPIM (single plane imaging). MAD (mandibular advancement device), 2P (2 photon),... Fig. 1 Citation Steven Y. Leigh, Ye Chen, Jonathan T.C. Liu, "Modulated-alignment dual-axis (MAD) confocal microscopy for deep optical sectioning in tissues," Biomed. Opt. Express 5, (2014); Image 2014 Optical Society of America and may be used for noncommercial purposes only. Report a copyright concern regarding this image. Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 4 / of48com

7 Photoacoustics Schematic representation Acoustic Part Photoacoustic PAI Equations for wave propagation Thermodynamics Light Propagation QPAT Equations for photon density Figure: Three kind of problems Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 5 / of48com

8 Photoacoustics Governing equation of photoacoustics: Standard Wave equation for the pressure 1 c p dj (x, t) p(x, t) = t2 dt (t)h(x) Parameters and functions: H absorbed electromagnetic energy c 0 speed of sound - in most mathematical studies normalized to constant 1 j(t) can be considered to approximate a δ-impulse (Lightning) Alternative if j = δ: Initial value problem for homogeneous wave equation p(x, 0) = H(x), p t (x, 0) = 0 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 6 / of48com

9 Photoacoustics Inverse problem of photoacoustics measurement data: p(x, t) for x M, t > 0 measurement region Reconstruction of H(x) for x Ω Exact Reconstruction Formulae for H depending on measurement geometry Sphere, Cylinder, Plane [Xu, Wang, 2002], [Finch, Patch, Rakesh, 2004] Circle [Finch, Haltmeier, Rakesh, 2005] Universal Backprojection [Wang et al, 2005, Natterer 2012] Palamodov Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 7 / of48com

10 Photoacoustics Applications Setup Figure: Photoacoustic setup: Robert Nuster and Günther Paltauf (University Graz) Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 8 / of48com

11 Photoacoustics Applications Applications: Microscopy Figure: Chicken embryo (5 days old)(1.2 mm) and mouse heart. Data by Berooz Zebian and Wolfgang Drexler, Robert Nuster et al. Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 2 17, Radon 2017Institute 9 / of48com

12 Photoacoustics Applications Medical applications Figure: Human studies of combined OCT (optical coherence tomography) and PAI: data by Boris Hermann and Wolfgang Drexler Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 10 / of48com

13 Modeling aspects Modeling aspects Standard Photoacoustics does not model variable sound speed, attenuation and variable illumination and does not recover physical parameters 1 Quantitative Photoacoustics (separation of parameters in H) [Arridge, Bal, Ken, Scotland, Uhlmann,...] 2 Sound speed variations: [Agranovsky, Hristova, Kuchment, Stefanov, Uhlmann,...] 3 Attenuation and dispersion [Anastasio, Patch, Riviere, Burgholzer, Kowar, S., Ammari, Wahab,...] 4 Variable Illumination [Wang, Bal,...] 5 Finite bandwidth detectors [Haltmeier et al] Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 11 / of48com

14 Modeling aspects Attenuation Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 12 / of48com

15 Modeling aspects Attenuation Attenuation: Model equation A κ p(x, t) p(x, t) = dδ dt (t)h(x) p(x, t) = 0 for t < 0 L: pseudo-differential operator w.r.t. t, with symbol κ 2 (ω) Iκ(ω) attenuation law Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 13 / of48com

16 Modeling aspects Attenuation Solution of forward problem Inverse Fourier-transform with respect to t: ˇf (ω) = 1 2π Solution of attenuated equation: ˇp(ω; x) = R 3 iω 4π 2π f (t)e iωt dt e iκ(ω) x y H(y)dy x y Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 14 / of48com

17 Modeling aspects Attenuation Solution of forward problem Inverse Fourier-transform with respect to t: ˇf (ω) = 1 2π Solution of attenuated equation: ˇp(ω; x) = R 3 iω 4π 2π f (t)e iωt dt e iκ(ω) x y H(y)dy x y Object of interest: Time-integrated photoacoustic operator in frequency domain: 1 R3 ˇP κ H(ω; x) = 4π e iκ(ω) x y H(y)dy 2π x y Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 14 / of48com

18 Modeling aspects Attenuation Formal definition of attenuation operator A κ : S (R R 3 ) S (R R 3 ) is defined by the action on the tensor products φ ψ S(R R 3 ), given by (φ ψ)(t, x) = φ(t)ψ(x): A κ u, φ ψ S,S = u, (F 1 κ 2 Fφ) ψ S,S, where F : S(R) S(R), Fφ(ω) = 1 2π φ(t)e iωt dt denotes the Fourier transform Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 15 / of48com

19 Modeling aspects Attenuation Assumptions on κ κ C (R; H) with H = {z C : Iz > 0} is called attenuation coefficient if 1 all derivatives of κ are polynomially bounded: l N0 κ1 >0,N N κ (l) (ω) κ 1 (1 + ω ) N 2 there exists a continuous continuation κ : H H of κ such that κ : H H is holomorphic. Moreover, κ1 >0,N N κ(z) κ 1 (1 + z )Ñ z H 3 κ( ω) = κ(ω) for all ω R Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 16 / of48com

20 Modeling aspects Attenuation Consequences of these assumptions 1 A κ is well-defined: κ 2 u S (R) u S (R) 2 p S (R R 3 ) is causal: that is supp(p) [0, ) R 3 3 A κ maps real-valued distributions onto itself Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 17 / of48com

21 Modeling aspects Attenuation Finite speed of propagation p S (R R 3 ) propagates with finite speed c > 0 if supp p {(t, x) R R 3 : x ct + R} h supp h B R (0) Lemma Let κ be an attenuation coefficient with the holomorphic extension κ : H H. Then, p propagates with finite speed if and only if κ(iω) lim ω iω > 0 In this case, it propagates with the speed c = lim ω iω κ(iω) Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 18 / of48com

22 Modeling aspects Examples of attenuation models Examples of attenuation models κ C (R; H) is called a strong attenuation coefficient if κ0,β>0,ω 0 0 Iκ(ω) κ 0 ω β, ω R, ω ω 0 weak attenuation coefficient if it is of the form c>0,κ 0 κ C (R) L 2 (R) κ(ω) = ω c + iκ + κ (ω), ω R Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 19 / of48com

23 Modeling aspects Examples of attenuation models Thermo-viscous model Name: Thermo-viscous model ω Coefficient: κ : R C, κ(ω) =, 1 iτω Upper bound: Speed: Type: Range of κ: κ(z) z for all z H iω c = lim ω κ(iω) = lim 1 + τω = ω Strong attenuation coefficient τ > 0, κ(z) = κ(z) i i κ i i Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 20 / of48com

24 Modeling aspects Examples of attenuation models Kowar-S-Bonnefond model Attenuation: Speed: Type: Range of κ: ( ) α κ : R C, κ(ω) = ω 1 +, 1+( iτω) γ κ(z) = κ(z) iω c = lim ω κ(iω) = lim ω 1 + Strong attenuation coefficient 1 α 1+(τω) γ = 1 i i κ i i Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 21 / of48com

25 Modeling aspects Examples of attenuation models Power law Szabo Model: Power law Coefficient: κ : R C, κ(ω) = ω + iα( iω) γ, κ(z) = κ(z) Parameters: γ (0, 1), α > 0 Upper bound: κ(z) z + α z γ α(1 γ) + (1 + αγ) z Propagation speed: iω c = lim ω κ(iω) = lim 1 ω 1 + αω γ 1 = 1 Attenuation type: Strong attenuation coefficient Range of κ: i i κ i i Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 22 / of48com

26 Modeling aspects Examples of attenuation models Modified Szabo s model Coefficient: κ : R C, κ(ω) = ω 1 + α( iω) γ 1 Parameters: γ (0, 1), α > 0 Upper bound: κ(z) 1 2 α(1 γ) + (1 + α 2 (1 + γ)) z iω Speed: c = lim ω κ(iω) = lim 1 = 1 ω 1 + αω γ 1 Type: Strong attenuation coefficient Range of κ: i i κ i i Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 23 / of48com

27 Modeling aspects Examples of attenuation models Nachman-Smith-Waag Coefficient: κ : R C, κ(ω) = ω 1 i τω c 0 1 iτω Parameters: c 0 > 0, τ > 0, τ (0, τ) z 1 i τz Extension: κ : H C, c 0 1 iτz, κ(z) 1 c 0 z for all z H iω 1 + τω τ Speed: c = lim ω κ(iω) = lim c 0 ω 1 + τω = c 0 τ Type: Weak attenuation coefficient Range of κ: i i κ i i Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 24 / of48com

28 Inversion of the integrated photoacoustic operator Photoacoustic equation in F-domain: recalled The solution p of the attenuated wave equation in Fourier domain: ˇp(ω, x) = G κ (ω, x y)h(y)dy R 3 = iω 4π 2π R3 e iκ(ω) x y x y Measurement data of photoacoustics H(y)dy, ω R, x R 3 ˇm(ω, ξ) = ˇp(ω, ξ) ω R, ξ Ω Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 25 / of48com

29 Inversion of the integrated photoacoustic operator Alternative equation: integrated photoacoustic operator Inverse problem of attenuated photoacoustics: Find H such that 1 1 ˇm(ω, ξ) = ˇp(ω, ξ), iω iω ω R, ξ Ω Meaning that the data are integrated in time-domain before inversion! Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 26 / of48com

30 Inversion of the integrated photoacoustic operator Integrated photoacoustic operator Let Ω R 3 be a bounded Lipschitz domain and Ω ε = {x Ω dist(x, Ω) > ε} κ denotes either a strong or a weak attenuation coefficient Definition ˇP κ : L 2 (Ω ε ) L 2 (R Ω), H 1 4π e 2π Ωε iκ(ω) ξ y H(y)dy ξ y is called integrated photoacoustic operator of the attenuation coefficient κ in frequency domain Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 27 / of48com

31 Inversion of the integrated photoacoustic operator Properties of integrated photoacoustic operator Let ˇP κ : L 2 (Ω ε ) L 2 (R Ω) be the integrated photoacoustic operator of a weak or a strong attenuation coefficient κ: Theorem Then, ˇP κ ˇP κ : L 2 (Ω ε ) L 2 (Ω ε ) is a self-adjoint integral operator with kernel F L 2 (Ω ε Ω ε ) given by that is F κ (x, y) = 1 32π 3 Ω e iκ(ω) ξ y iκ(ω) ξ x ds(ξ)dω, ξ y ξ x ˇP κ ˇP κ h(x) = F κ (x, y)h(y)dy Ω ε In particular, ˇP κ ˇP κ is a Hilbert Schmidt operator and thus compact Proof: lengthy Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 28 / of48com

32 Inversion of the integrated photoacoustic operator Strongly attenuating media Singular values of IPO for strongly attenuating media Kernel estimates 1 Lemma For strongly attenuation κ the kernel F κ of ˇP κ ˇP κ satisfies 1 B,b>0 sup sup j j! x,y Ω ε s j F κ (x, y + sv) Bbj j ( N β 1)j j N 0 v S 2 s=0 1 N N denotes the exponent for l = 0 (polynomial condition) and β (0, N] is the exponent in the condition for the strong attenuation coefficient κ Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 29 / of48com

33 Inversion of the integrated photoacoustic operator Strongly attenuating media Singular values of IPO for strongly attenuating media Let Ω be a bounded Lipschitz domain in R 3 and ε > 0 Corollary Let ˇP κ : L 2 (Ω ε ) L 2 (R Ω) be the integrated photoacoustic operator of a strong attenuation coefficient κ. Then, there exist constants C, c > 0 such that the eigenvalues (λ n ( ˇP κ ˇP κ )) n N (in decreasing order) satisfy λ n ( ˇP κ ˇP κ ) Cn m ) n exp ( cn β Nm n N Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 30 / of48com

34 Inversion of the integrated photoacoustic operator Weakly attenuating media Singular values of IPO for weakly attenuating media Split ˇP (0) κ h(ω, ξ) = ˇP (1) κ h(ω, ξ) = ˇP κ = ˇP (0) κ + ˇP (1) κ, 1 4π 2π 1 4π 2π Ωε e i ω c ξ y ξ y e κ ξ y h(y)dy Ωε e i ω c ξ y ξ y e κ ξ y (e iκ (ω) ξ y 1)h(y)dy ( ˇP κ (0), ˇP κ (1) ) photoacoustic operator with constant attenuation and the perturbation Lemma Let κ be a weak attenuation coefficient. Then, there exist constants C 1, C 2 > 0 such that we have C 1 n 2 3 λn ( ˇP κ ˇP κ ) C 2 n 2 3 n N Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 31 / of48com

35 Inversion of the integrated photoacoustic operator Weakly attenuating media Summary Medium type Examples Singular values decay No attenuation κ(ω) = ω c 0 n 2/3 (Palamodov) Weak attenuation Nachman-Smith-Waag κ(ω) = z 1 i τω c 0 1 iτω n 2/3 Strong attenuation Thermo-viscous model κ(ω) = ω 1 iτω e cnc Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 32 / of48com

36 Inversion of the integrated photoacoustic operator Weakly attenuating media References P. Elbau, O. Scherzer, and C. Shi. Singular values of the attenuated photoacoustic imaging operator. Preprint on ArXiv arxiv: , University of Vienna, Austria, R. Kowar and O. Scherzer. Attenuation models in photoacoustics. In H. Ammari, editor, Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographies. Springer Verlag, Berlin Heidelberg, R. Kowar, O. Scherzer, and X. Bonnefond. Causality analysis of frequency-dependent wave attenuation. Math. Methods Appl. Sci., 34: , A. I. Nachman, J. F. Smith, III, and R. C. Waag. An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Amer., 88(3): , V. P. Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Probl. Imaging, 4(4): , T.L. Szabo. Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Amer., 96: , Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 33 / of48com

37 Inversion of the integrated photoacoustic operator Weakly attenuating media Sound Speed Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 34 / of48com

38 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Photoacoustic with variable sound speed Reconstruct H in from measurements of 1 c 2 (x) p (x, t) p(x, t) = 0 in R n (0, ) p(x, 0) = H(x) in R n p (x, 0) = 0 in R n m = p Γ (0, ) Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 35 / of48com

39 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Literature Exact reconstruction formulas Convergence results require non-trapping conditions M. Agranovsky and P. Kuchment. Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed. Inverse Probl., 23(5): , P. Stefanov and G. Uhlmann. Thermoacoustic tomography with variable sound speed. Inverse Probl., 25(7):075011, 16, 2009 Y. Hristova, P. Kuchment, and L. Nguyen. Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media. Inverse Probl., 24(5):055006, Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 36 / of48com

40 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Our approach: Operator equation formulation Let L : H 1 0(Ω) L 2 (Γ (0, )) H p Γ (0, ) We consider the solution of L[H] = m with Landweber iteration H k+1 = H k λl [L[H k ] m] Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 37 / of48com

41 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Adjoint problem Let h C (Γ (0, )). Then the L 2 -adjoint is given z := z(h) = L 2 h, where z solves 1 c 2 z z = 0 in R n \ Γ (0, T ) z(t ) = z (T ) = 0 in R n [z] = 0 on Γ (0, T ) [ ν z] = h on Γ (0, T ) Moreover, [ z L h = 1 ] (0) c 2 Z. Belhachmi, T. Glatz, O.Scherzer A Direct Method for Photoacoustic Tomography with Inhomogeneous Sound Speed. Inverse Problems 4, , 2016 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 38 / of48com

42 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Classical regularization theory Let H be the minimum-norm-solution H { } H1 = min H H1 : L[H] = m Then 1 If m = p(h ). Then H k H. 2 If m δ m δ, then H δ k (δ) H, where k (δ) is the largest index such that p(hk δ ) (δ) mδ > δ C. W. Groetsch. The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston, 1984 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 39 / of48com

43 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Time reversal Let h L 2 (Γ (0, )) and Γ = Ω. Moreover, let ˆL[h] be defined z c 2 z = 0 in Ω (0, T ) z(t ) = 0, z (T ) = 0 in R n z = h on Γ (0, T ) Standard time reversal: Ĥ = ˆL[m] M. Fink. Time reversal of ultrasonic fields. i. basic principles. IEEE Trans. Ultrason., Ferroeletr., Freq. Control, 39(5): , H. Grün, G. Paltauf, M. Haltmeier, and P. Burgholzer. Photoacoustic tomography using a fiber based Fabry-Perot interferometer as an integrating line detector and image reconstruction by model-based time reversal method. Volume 6631 of Proceedings of SPIE, page SPIE, 2007 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 40 / of48com

44 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Time reversal: Stefanov & Uhlmann Let h L 2 (Γ (0, )) and Γ = Ω. Moreover, let L[h] = z(0) be defined z c 2 z = 0 in Ω (0, T ) z(t ) = φ, z (T ) = 0 in R n z = h on Γ (0, T ) where φ = 0 in Ω and φ Γ = h(t ) on Γ Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 41 / of48com

45 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Time reversal of Stefanov & Uhlmann First iteration (time reversal) H = L[m] Moreover, LL = I K (K compact) and H = K m Lm m=0 Probably not minimum norm solution. P. Stefanov and G. Uhlmann. Thermoacoustic tomography with variable sound speed. Inverse Probl., 25(7):075011, 16, 2009 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 42 / of48com

46 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Non-trapping speed, T = 4T 0 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 43 / of48com

47 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Non-trapping speed, T = 1.2T 0 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 44 / of48com

48 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Landweber: Non-trapping speed, T = 1.2T 0 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 45 / of48com

49 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Trapping speed, T = 2T 0. Time reversal; 5th Neumann sum; Landweber iterate 20 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 46 / of48com

50 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Generalizations κ(x)y (x, t) (ρ(x) 1 y(x, t) ) = 0 in R n (0, ) y(x, 0) = H(x) in R n y (x, 0) = 0 in R n. κ compressibility ρ density c(x) = 1 κρ Z. Belhachmi, T. Glatz, O.Scherzer Photoacoustic Tomography With Spatially Varying Compressibility and Density. JIIPP, 2016 Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 47 / of48com

51 If time allows: Time reversal with variable sound speed (joint with Z. Belhachmi (Metz), T. Glatz (UVIE) Thank you for your attention Peter Elbau 1, Otmar Scherzer 1,2, Cong Shi 1 Singular ( 1 Computational Values of the Science Attenuated Center, Photoacoustic University Vienna, Imaging Austria February Operator & 17, 2 Radon 2017 Institute 48 / of48com