Forward and inverse wave problems:quantum billiards and brain imaging p.1

Transcription

1 Forward and inverse wave problems: Quantum billiards and brain imaging Alex Barnett Courant Institute Forward and inverse wave problems:quantum billiards and brain imaging p.1

2 The one-minute talk : two areas Simple linear 2 -order PDEs Forward and inverse wave problems:quantum billiards and brain imaging p.2

3 The one-minute talk : two areas Simple linear 2 -order PDEs I. Scaling method for Helmholtz eigenproblem waves: elliptic PDE, time-independent short-wavelength limit numerically hard quantum chaos explains fast new method Forward and inverse wave problems:quantum billiards and brain imaging p.2

4 The one-minute talk : two areas Simple linear 2 -order PDEs I. Scaling method for Helmholtz eigenproblem waves: elliptic PDE, time-independent short-wavelength limit numerically hard quantum chaos explains fast new method II. Brain imaging with diffuse optical tomography diffusion: parabolic PDE, time-dependent ill-posed inverse problem, messy 3D geometry clinical and functional neuroimaging Forward and inverse wave problems:quantum billiards and brain imaging p.2

5 I. Scaling method with Cohen (Ben-Gurion), Heller (Harvard) Domain in L boundary inside domain Dirichlet Want spectrum, eigenfunctions Forward and inverse wave problems:quantum billiards and brain imaging p.3

6 I. Scaling method with Cohen (Ben-Gurion), Heller (Harvard) Domain in L boundary inside domain Dirichlet Want spectrum, eigenfunctions Motivation? Cavities: acoustics, electromag, optics, quantum dots (electron systems), quantum chaos... Forward and inverse wave problems:quantum billiards and brain imaging p.3

7 I. Scaling method with Cohen (Ben-Gurion), Heller (Harvard) Domain in L boundary inside domain Dirichlet Want spectrum, eigenfunctions Motivation? Cavities: acoustics, electromag, optics, quantum dots (electron systems), quantum chaos... Often care about E.g. spectral statistics as (asympotics). Forward and inverse wave problems:quantum billiards and brain imaging p.3

8 Physical examples dielectric laser resonators Tureci Forward and inverse wave problems:quantum billiards and brain imaging p.4

9 Physical examples dielectric laser resonators Tureci liquid surfaces Kudrolli Forward and inverse wave problems:quantum billiards and brain imaging p.4

10 3 approaches 1. Finite Element (FEM) type basis funcs obey BCs, not PDE basis size -sized patches in ). V Forward and inverse wave problems:quantum billiards and brain imaging p.5

11 3 approaches 1. Finite Element (FEM) type basis funcs obey BCs, not PDE basis size -sized patches in ). V 2. Boundary Integral (BIM) type Greens func known use as basis basis obeys PDE, not BCs patches on surface). Forward and inverse wave problems:quantum billiards and brain imaging p.5

12 3 approaches 1. Finite Element (FEM) type basis funcs obey BCs, not PDE basis size -sized patches in ). V 2. Boundary Integral (BIM) type Greens func known use as basis basis obeys PDE, not BCs patches on surface). 3. Measure resonances of real system microwaves cavities, etc... Q-factor, painful! Forward and inverse wave problems:quantum billiards and brain imaging p.5

13 3 approaches 2. Boundary Integral (BIM) type Greens func known use as basis basis obeys PDE, not BCs patches on surface). Forward and inverse wave problems:quantum billiards and brain imaging p.5

14 Scaling sketch quadratic functional is nearly diagonal in the basis : spatially rescaled to same wavenumber (discovered, not explained, Vergini & Saraceno 1994). Forward and inverse wave problems:quantum billiards and brain imaging p.6

15 Scaling sketch quadratic functional is nearly diagonal in the basis : spatially rescaled to same wavenumber (discovered, not explained, Vergini & Saraceno 1994) Diagonalize basis rep of all in range. Forward and inverse wave problems:quantum billiards and brain imaging p.6

16 Scaling sketch quadratic functional is nearly diagonal in the basis : spatially rescaled to same wavenumber (discovered, not explained, Vergini & Saraceno 1994) Diagonalize basis rep of all in range times faster than ubiquitous BIM! (BIM has to search for each ). Forward and inverse wave problems:quantum billiards and brain imaging p.6

17 Scaling sketch quadratic functional is nearly diagonal in the basis : spatially rescaled to same wavenumber (discovered, not explained, Vergini & Saraceno 1994) Diagonalize basis rep of all in range times faster than ubiquitous BIM! (BIM has to search for each Special relies on boundary overlap of ). s... Forward and inverse wave problems:quantum billiards and brain imaging p.6

18 Quasi-orthogonality sketch. Short time correspondence of dynamics M ν WAVE PARTICLE C( ω) ω Power spectrum (ω) of (weighted) classical bounces µ force t heating rate under periodic deformation Forward and inverse wave problems:quantum billiards and brain imaging p.7

19 Quasi-orthogonality sketch. Short time correspondence of dynamics M ν WAVE PARTICLE C( ω) ω Power spectrum (ω) of (weighted) classical bounces µ force t heating rate under periodic deformation Special deformations : no heating as. Forward and inverse wave problems:quantum billiards and brain imaging p.7

20 Results ( ) plane-wave basis, speed: 100 such found per minute Forward and inverse wave problems:quantum billiards and brain imaging p.8

21 New basis for nonconvex new singular basis, Forward and inverse wave problems:quantum billiards and brain imaging p.9

22 Directions Better basis sets for variety of shapes Forward and inverse wave problems:quantum billiards and brain imaging p.10

23 Directions Better basis sets for variety of shapes Understand basis completeness. Forward and inverse wave problems:quantum billiards and brain imaging p.10

24 Directions Better basis sets for variety of shapes Understand basis completeness. Error analysis, creeping solutions Forward and inverse wave problems:quantum billiards and brain imaging p.10

25 Directions Better basis sets for variety of shapes Understand basis completeness. Error analysis, creeping solutions Application to spectral statistics Forward and inverse wave problems:quantum billiards and brain imaging p.10

26 Forward and inverse wave problems:quantum billiards and brain imaging p.11

27 II. Diffuse Optical Tomography with Boas et al. (NMR Center, MGH / Harvard) Source Detector µ a tissue κ Image inside diffusive media? scattering length absorption Learn about, m mm depth few cm Forward and inverse wave problems:quantum billiards and brain imaging p.12

28 It s all about blood Near infrared: small Hemoglobin dominates Hb - deoxy HbO - oxy at many s maps of Hb, HbO Forward and inverse wave problems:quantum billiards and brain imaging p.13

29 It s all about blood Near infrared: small Hemoglobin dominates Hb - deoxy HbO - oxy at many s maps of Hb, HbO Clinical: stroke, trauma, babies, breast tumors... Forward and inverse wave problems:quantum billiards and brain imaging p.13

30 It s all about blood Near infrared: small Hemoglobin dominates Hb - deoxy HbO - oxy at many s maps of Hb, HbO Clinical: stroke, trauma, babies, breast tumors... Neuronal activation Hb, HbO changes Last decade: imaging the brain in action! Forward and inverse wave problems:quantum billiards and brain imaging p.13

31 DOT equipment Forward and inverse wave problems:quantum billiards and brain imaging p.14

32 DOT equipment signals: Many S,D: use all possible pairs s light pulse photon count vs time Forward and inverse wave problems:quantum billiards and brain imaging p.14

33 DOT equipment signals: Many S,D: use all possible pairs s light pulse photon count vs time fmri: 2-4 mm, 1-2 s,, fixed, Hb only Forward and inverse wave problems:quantum billiards and brain imaging p.14

34 DOT equipment signals: Many S,D: use all possible pairs s light pulse fmri: 2-4 mm, 1-2 s, DOT: 1-2 cm, ms, photon count vs time, fixed, Hb only, portable, Hb & HbO. Forward and inverse wave problems:quantum billiards and brain imaging p.14

35 Forward model parameter vector expected signal vector Forward and inverse wave problems:quantum billiards and brain imaging p.15

36 Forward model parameter vector expected signal vector Incoherent waves transport equation diffusion: fluence, Robin BCs. Forward and inverse wave problems:quantum billiards and brain imaging p.15

37 Forward model parameter vector expected signal vector Incoherent waves transport equation diffusion: fluence, Robin BCs. Finite-Difference Time-Domain in 3D ( accuracy, for now... mm lattice) Forward and inverse wave problems:quantum billiards and brain imaging p.15

38 Inverse problem Ill-posed : many have measured measured Forward and inverse wave problems:quantum billiards and brain imaging p.16

39 Inverse problem measured Ill-posed : many have Statistical: incomplete info learn PDF on measured Forward and inverse wave problems:quantum billiards and brain imaging p.16

40 Inverse problem measured Ill-posed : many have Statistical: incomplete info learn PDF on y noise f(x) measured Bayesian inference measured y prior p( x) posterior p( x y) x posterior likelihood prior Embraces Ill-posedness Forward and inverse wave problems:quantum billiards and brain imaging p.16

41 Inverse problem measured Ill-posed : many have Statistical: incomplete info learn PDF on y noise f(x) measured Bayesian inference measured y prior p( x) posterior p( x y) x posterior likelihood prior Embraces Ill-posedness Use realistic noise model: Poisson photon stats forward model error Forward and inverse wave problems:quantum billiards and brain imaging p.16

42 Baseline meas. with MRI help Use geometry from MRI : for skull, scalp, brain. Q: How well can measure absolute brain? Forward and inverse wave problems:quantum billiards and brain imaging p.17

43 Baseline meas. with MRI help Use geometry from MRI : for skull, scalp, brain. Q: How well can measure absolute brain? µ s µ s µ s a) d) g) µ a 1.1 b) e) h) µ a 2.5 c) f) i) SCALP SKULL BRAIN µ a posterior PDF errorbars Gaussian approx to PDF Markov chain Monte Carlo Show detected photons gives 5% in, 20% in even if 20% forward model error Forward and inverse wave problems:quantum billiards and brain imaging p.17

44 Baseline meas. with MRI help Use geometry from MRI : for skull, scalp, brain. Q: How well can measure absolute brain? µ s µ s µ s a) d) g) µ a 1.1 b) e) h) µ a 2.5 c) f) i) SCALP SKULL BRAIN µ a posterior PDF errorbars Gaussian approx to PDF Markov chain Monte Carlo Show detected photons gives 5% in, 20% in even if 20% forward model error is expensive want fewest evaluations Forward and inverse wave problems:quantum billiards and brain imaging p.17

45 Directions Real-world data Forward and inverse wave problems:quantum billiards and brain imaging p.18

46 Directions Real-world data Forward modelling: accuracy vs speed Adjoint Differentiation for Jacobean cerebrospinal fluid clear diffusion bad Forward and inverse wave problems:quantum billiards and brain imaging p.18

47 Directions Real-world data Forward modelling: accuracy vs speed Adjoint Differentiation for Jacobean cerebrospinal fluid clear diffusion bad Imaging the cortex: unknowns Forward and inverse wave problems:quantum billiards and brain imaging p.18

48 Directions Real-world data Forward modelling: accuracy vs speed Adjoint Differentiation for Jacobean cerebrospinal fluid clear diffusion bad Imaging the cortex: Best S,D placement? unknowns Forward and inverse wave problems:quantum billiards and brain imaging p.18

49 Directions Real-world data Forward modelling: accuracy vs speed Adjoint Differentiation for Jacobean cerebrospinal fluid clear diffusion bad Imaging the cortex: Best S,D placement? unknowns AI / Optimization: explore high-dim PDFs Forward and inverse wave problems:quantum billiards and brain imaging p.18