Data‐Compatible T‐Matrix Completion (DCTMC)

A new numerical method for solving nonlinear inverse problems

Howard W. Levinson Department of Mathematics, University of Pennsylvania and University of

Michigan (Sept 2016‐)

Vadim A. MarkelInstitut Fresnel, University of Aix‐Marseille

Dept. of Radiology, University of Pennsylvania (on leave)

Motivation 1: Local Minima

[ ] 2guess[ ]D F V Dχ λ ⎡ ⎤= −Φ + Φ −Φ⎣ ⎦

[ ]F V = Φ

( )V rWave in

Wave out

DataTheoretical model

Object

χ

V

Motivation 2: Large Data Sets[ ] 2

guess[ ]D F V Dχ λ ⎡ ⎤= −Φ + Φ −Φ⎣ ⎦

ΩΣd

Σs

rd

rs

Surface of detectors Surface of sources

L

L

L

L2

2

4

s

d

s d

N L

N L

N N N L

=

=

= =

Algebraic Structure of the Inverse Problem1( )A I V VB−− Γ = Φ

0, , -- different restrictions of the same unperturbed Green's function A B GΓ

Every variable is a matrix!

1( )A I V VB−− Γ = Φ

1

1

[ ] ( )[ ] ( )T V I V VV T I TV T

−

−

= − Γ

= +

ATB = Φ

Let us view T‐matrix as the fundamental unknown and use the one‐to‐one Correspondence between T and V

Main Idea

The Experimental T‐Matrix

1

1

d

s

NA A A

NB B B

A f g

B f g

μ μ μμ

μ μ μμ

σ

σ

=

=

=

=

∑

∑ATB = Φ ,

11

A B

d

s

TNN

μ ν μν μνσ σ

μν

= Φ

≤ ≤≤ ≤

* 1

= , 1 ,

1

= , 1 ,

[ ] , [ ]

A Bd

s

A Bv

A B

f g N

N

T g f N

T R TR T T T

μν μ ν

μν μ ν

μ

ν

μ ν−

Φ Φ ≤ ≤

≤ ≤

Φ ≤ ≤

= = =R R

21 , if > =

Unknown , otherwise

A BA BT μν μ νμ νμν

σ σ εσ σ⎧ Φ⎪⎨⎪⎩

Tàexp =

ûAö û

B÷

Φà ö÷

0 0Tà = UNKNOWN

UNKNOWN

UNKNOWN

UNKNOWN

UNKNOWN

ûAö û

B÷

Φà ö÷ MA

MB

MA

MB 0expT A B+ += Φ

Data-compatible T-matrixRepresentation: SV

kT

Data-compatible T-matrixRepresentation: RS

kT

Diagonal interaction-compati-ble T-matrixRepresentation: RS kT ′

Diagonal interaction-compati-ble T-matrixRepresentation: SV kT ′

Data-compatibleinteraction matrix(off-diagonal)Representation: RS

kVDiagonal interaction matrix(not data-compatible)Representation: RS

kD

Shortcut 1: Fast Rotations

Shortcut 2: Fast T→D

expTΦData

First iterationSubsequentiterations

SVD of A and B

1.Rà1

2.T à13.D

4.T

The Bo

ttlene

ck

5.R 6.Ok→ k+ 1

Exit if conditionis met

Computational Shortcut: Fast Rotations

1

11 1

5: [ ]

6: [ ]

1: [ ]

k k

k k

k k

T T

T T

T T+

−+ +

′ ′=

′=

=

R

O

R

exp exp[ ] [ ] [ ]

[ ] [ ]

T T T T T T

T T T

= + = − +

+ =

O M N

M N

1 11 exp[ [ [ ]]] - [ [ [ ]]]k k k kT T T T T− −+ ′ ′ ′= = +R O R R N R

1 * *[ [ [ ]]] ( )A A B BT P P TP P− =R N R

Operation of “Diagonalization” and Linear Reconstructions

[ ]( )ij ij ij ij

j

D VD w r Vδ=

= ∑D

If ( ) , we can analyse the algorithm in the linear regime:ij ijw r δ=

1 exp ( )k kv v I W v+ = + − * *( ) ( )ij A A ij B B jiW P P P P=

Vector containingthe diagonal part of V

1expFixed point: v W v−

∞ =

2Tikhonov regularization: W W Iλ→ +

Practical tip: Richardson iteration is a very slow way to arrive at the linearized solution.Use direct solver of CG to compute linearized solution and then use this result as an initial guessFor the nonlinear iterations.

( )0

Diffraction tomography: Contrast:

exp ( ) 1( , ) ( )4

ikG εχ

π

′− −′ = =′−

r r rr r rr r

No improvements: 900 iterations

With improvements: 70 iterations

• Start from linearized reconstruction (can be computed fast using our method)• Use weighted summation to the diagonal for “force‐diagonalization”• Use reciprocity of source‐detector pairs to improve symmetry of the experimental T‐matrix•Method starts to break down due to incorrect assignment of non‐interacting voxels (this can be avoid altogether – not a problem of convergence)

( ) 00

0

Diffusion tomography: Contrast: Optical depth: Noise:

exp ( )( , ) ( ) 2 2%k

G kLα αδαα

′− − −′ = = ≈′−

r r rr r rr r

Target far Target near

FR

NL

Target far

FR

NL

Target near

DCTMC works when Newton‐Gauss fails (Convergence of Levenberg‐Marquardt Iterations for the inverse diffraction problem, moderate contrast 0.0175)

CONCLUSIONS• DCTMC works for nonlinear ISP with fairly strong nonlinearity• DCTMC is, unfortunately, a complicated method: it requires

many tweaks, attention to detail, and good programming to work

• As any other method, DCTMC breaks at some point. Not every nonlinear ISP can be solved!

• Please listen to Howard Levinson at 15:15 today (this room)

Preprints: arXiv:1401.3319, arXiv:1505.06777Under consideration in Phys. Rev. E