Statistical and Computational Inverse Problems with Applications Part 5B: Electrical impedance tomography

Transcription

1 Statistical and Computational Inverse Problems with Applications Part 5B: Electrical impedance tomography Aku Seppänen Inverse Problems Group Department of Applied Physics University of Eastern Finland Kuopio, Finland Jyväskylä Summer School August 11-13, 2014

2 Electrical impedance tomography (EIT) In EIT electric currents I are applied to electrodes on the surface of the object and the resulting potentials V are measured using the same electrodes. The conductivity distribution σ = σ(x) is reconstructed based on the potential measurements. Diffusive tomography modality

3 Forward model for EIT (σ u) = 0, x Ω u + z l σ u = U l, x e l n σ u e l n ds = I l, l = 1, 2,..., L σ u n = 0, x Ω\ L l=1 e l

4 Forward model & inverse problem in EIT Finite element (FE) approximation of the complete electrode model V = U(σ) Additive noise model V obs = U(σ) + n

5 Examples of forward solutions See examples of EIT forward solutions in Appendix 1. (Don t print it out; huge number of pages & figs.) What do the last examples tell us about the ill-posedness of EIT? Any suggestions for the remedy?

6 MAP estimates In the case of Gaussian likelihood model and Gibbs type prior, the posterior density is of the form π(σ V ) π(v σ)π(σ) ( exp 1 2 (V U(σ))T Γ 1 n (V U(σ)) 1 ) 2 G(σ) And the MAP estimate can be written in the form σ MAP = arg min σ { L n (V U(σ)) 2 + G(σ)} (1) where L T n L n = Γ 1 n. Iterative solution (e.g. Gauss-Newton)

7 MAP estimates with Gaussian models In the case of Gaussian likelihood model and Gaussian prior, the posterior density is of the form ( π(σ V ) exp 1 2 (V U(σ))T Γ 1 n (V U(σ)) 1 ) 2 (σ η σ) T Γ 1 σ (σ η σ ) And the MAP estimate can be written in the form σ MAP = arg min σ { L n (V U(σ)) 2 + L σ (σ η σ ) 2 } (2) where L T n L n = Γ 1 n, L T σ L σ = Γ 1 σ. Iterative solution (e.g. Gauss-Newton)

8 Iteration step 1 Left: estimated conductivity distribution. Right: Measured vs. computed potentials.

9 Iteration step 2 Left: estimated conductivity distribution. Right: Measured vs. computed potentials.

10 Iteration step 3 Left: estimated conductivity distribution. Right: Measured vs. computed potentials.

11 Iteration step 4 Left: estimated conductivity distribution. Right: Measured vs. computed potentials.

12 Iteration step 5 Left: estimated conductivity distribution. Right: Measured vs. computed potentials.

13 Forward model MAP estimates Computational aspects Gaussian prior models MAP estimate Figure: Left: Photo of the true target; Right: estimated conductivity distribution. TV prior

14 Computational aspects Solution of the optimization problem in the MAP estimate typically Gauss-Newton-type iteration Line-search Non-negativity constraint: e.g. for Gaussian priors, P(σ < 0) 0. However, in reality the conductivity is non-negative. In MAP estimates, the non-negativity constraint can be handled by constrained optimization σ MAP = arg min σ 0 { L n(v U(σ)) 2 + L σ (σ η σ ) 2 } (3) Projected line-search (not a good choice...) Interior point method

15 Interior point method for the non-negativity constraint Idea: set a barrier function b(σ) which gives high penalty, when any element of the conductivity vector σ k 0. The MAP estimate with the interior point method σ MAP = arg min{ L n (V U(σ)) 2 + L σ (σ η σ ) 2 + b(σ)} σ Example: logarithmic barrier function b(σ) = µ N ln(σ k ) (4) k where µ is a weighting parameter. Usually µ is adaptively decreased during the iteration.

16 White noise prior A white noise prior is of the form σ N (η σ, γ 2 σi) (5) where η σ and γ 2 σ are the expectation and variance of σ. The prior density ( π(σ) exp 1 exp 2γ 2 σ ) (σ η σ ) T (σ η σ ) ( 1 ) 2γσ 2 (σ η σ ) 2 (6) (7)

17 White noise prior: How to select η σ and γ 2 σ? Gaussian random variable σ σ N (η σ, γ 2 σi) Define σ min = η σ 3γ σ and σ max = η σ + 3γ σ. Then, P(σ min < σ < σ max ) Practical way of selecting η σ and γσ: 2 The expectation of the conductivity η σ can often be assessed based on the knowledge of the physical properties of the target (prior information!) Further, you may also have an idea of "upper limit" of conductivity σ max (loosely speaking!) Then, a reasonable choice for the variance is γσ 2 = ( σ max η σ ) 2 3 Problems: White noise prior is usually not a good model in EIT the conductivity is usually spatially correlated.

18 Uninformative smoothness prior Standard (uninformative) smoothness prior (continuous σ) ( ) π(σ) exp α σ 2 dr Ω Finite dimensional approximation for σ; prior density can be written in the form π(σ) exp ( 12 ) α L σσ 2 = exp ( 12 α σt L TσL ) σ σ 2 Matrix L T σl σ is not invertible Γ σ does not exist. Problems: How to select α? How to control the degree of spatial smoothness?

19 Extensions of the uninformative smoothness prior (Uninformative) anisotropic smoothness prior is defined accordingly (continuos form) ( ) π(σ) exp α A(r) σ 2 dr Ω (8) where A(r) is tensor field. (Uninformative) structural priors can be constructed by selection of A(r) based on structural information (example: anatomical information provided by another imaging modality).

20 An informative smoothness prior Gaussian random variable σ N (η σ, Γ σ ) ( π(σ) exp 1 ) 2 (σ η σ) T Γ 1 σ (σ η σ ) Write the covariance matrix Γ σ as { Γ σ (i, j) = a exp x i x j 2 2 2b 2 } (9) (10) where x i R 2,3 is the spatial coordinate corresponding to a discrete conductivity value σ i (Lieberman, Willcox, Ghattas 2010). Other similar models exist.

21 An informative smoothness prior: How to select a and b? The variance of the conductivity at point x i is var(γ i ) = Γ σ (i, i) = a (11) Selection of the variance: See the white noise prior above. Define the correlation length l as the distance where the cross-covariance Γ σ (i, j) drops to 1% of var(γ i ). Then b = l 2 ln(100). (12)

22 An informative anisotropic smoothness prior Again, Gaussian random variable σ N (η σ, Γ σ ) ( π(σ) exp 1 ) 2 (σ η σ) T Γ 1 σ (σ η σ ) Write the covariance matrix Γ σ as Γ σ (i, j) = a exp 3 k=1 x(k) i 2bk 2 x (k) j 2 2 (13) (14) where x i R 3, x i = (x (1) i, x (2) i, x (3) i ) is the spatial coordinate corresponding to a discrete conductivity value σ i, and coefficients b k define the correlation lengths l k at the directions of the coordinate axes. Other directions by coordinate transformations.

23 Examples of informative smoothness priors For examples of informative smoothness priors, see Appendix 2. Samples corresponding to smoothness priors with different correlation lenghts.

24 A sample based Gaussian prior Assume that you have a set of samples of the conductivity distribution (based on e.g. other experiments or a flow simulation); denote the samples by σ (j), j = 1,..., K. Approximate σ as a Gaussian random variable σ N (η σ, Γ σ ) ( π(σ) exp 1 ) 2 (σ η σ) T Γ 1 σ (σ η σ ) (15) where η σ is chosen to be the sample mean 1 K K j=1 σ(j), and the sample covariance is used as the prior covariance matrix: Γ σ = 1 K 1 K (σ (i) η σ )(σ (i) η σ ) T (16) i=1

25 Total variation prior A couple of different versions of TV prior exists. The following one has certain advantages. Total variation prior (continuous form, 2D case) ( ) ( ) σ π(σ) exp α σ 2 dr = exp α Ω Ω x + σ y dr Finite dimensional approximation π(σ) exp ( α Promotes sparsity of σ. M l=1 ) (L x σ) 2 l + (L yσ) 2 l

26 Total variation prior Hence where Gibbs type prior A(σ) = π(σ) exp ( αa(σ)) M l=1 The posterior density is of the form (L x σ) 2 l + (L yσ) 2 l π(σ V ) π(v σ)π(σ) ( exp 1 ) 2 (V U(σ))T Γ 1 n (V U(σ)) A(σ)

27 Total variation prior MAP estimate σ MAP = arg min{ 1 σ 2 L n(v U(σ)) 2 + A(σ)} Solution: e.g. Gauss-Newton Note: A(σ) is not differentiable. Hence, approximation: A(σ) = M l=1 where β is a small constant. (L x σ) 2 l + (L yσ) 2 l + β

28 An example

29 Sensing skin application

30 Sensing skin application