Physics-based Prior modeling in Inverse Problems

MURI Meeting 2013

M Usman Sadiq, Purdue University

Charles A. Bouman, Purdue University

In collaboration with:

Jeff Simmons, AFRL Venkat Venkatakrishnan, Purdue

Marc De Graef, CMU ���

1  

Inverse Problems in Imaging •  Recover information from indirect measurement*

Other  Unknowns  (Nuisance  Parameters)  

Regularity  Condi<ons  (Prior  knowledge)  

Image  and  system  models  are  cri1cal  to  accurate  inversion  

Inversion Method

Data  

Physical System Linear/Nonlinear

Deterministic/Stochastic Unknown  Quan<ty  

Es<mate  

x y x̂

φ

2  

Model Based Iterative Reconstruction •  General framework for solving inverse problems

Prior Model:

p(x) Forward model : g(.)

Physical system

Difference

x̂

y

g(x)

x

Optimization Engine

x̂( )← argmaxx

p x y( ){ }= argminx

− log p y x( )− log p x( ){ }

p y x( ) : Likelihood

p(x) : Prior Model

3  

Popular models for the Prior •  Neighborhood based or Local priors: – Penalize ‘dissimilarity’ between voxels:

•  Markov Random Fields •  Bilateral Filtering

•  Non-local priors: – Exploit image information from non-local voxels:

•  Non-Local means •  K-SVD •  BM3D

Sheep lung image and its learned dictionary with 256 atoms*

*Qiong Xu, ‘Low-dose CT reconstruction via Dictionary Learning’

1688 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 31, NO. 9, SEPTEMBER 2012

Fig. 1. Construction of a global dictionary. (a) Sheep lung image with a dis-play window [ , 800] HU reconstructed from the normal-dose sinogramby the FBP method, which is used to extract the training patches. (b) Learneddictionary consisting of 256 atoms. The attenuation coefficient of water was as-sumed as 0.018 to convert the reconstructed image (a) into HU and other imagesthroughout this paper.

low-dose images were quite different from that of the baselineimage. Also, the physiological motion of the sheep most likelyintroduced structural differences. As such, this group of sino-grams actually offers a challenging opportunity to evaluate therobustness of GDSIR.First, a set of overlapping patches were extracted from the

lung region in the baseline image. The patch size was of 8 8pixels. The patches with very small variance were removedfrom the extracted patch set. The direct current (dc) componentwas removed from each patch. Then, a global dictionary of256 atoms was constructed using the online dictionary learningmethod with a fixed sparsity level [40]. The lungregion and the final dictionary are shown in Fig. 1. Finally, a dcatom was added to the dictionary.3) Low-Dose Results: For comparison, five reconstruction

techniques were applied to the aforementioned low-dose sino-grams. As the benchmark, low dose images were reconstructedusing the FBP method. The corresponding reconstructionsusing the other four reconstruction techniques were describedas follows.First, the GDSIR algorithm with a prelearned global dictio-

nary (Section IV-A2) was employed to reconstruct low-doseimages with the following empirical parameters: ,

, and . The initial image was fromthe FBP method. An ordered-subset strategy was used [8]. Thenumber of subsets was 40. The iterative process was stoppedafter 50 iterations.Then, the ADSIR algorithm was tested with the same low-

dose sinograms. In each iteration, the dictionary was learned inreal-time from the set of patches extracted from an intermediateimage. The parameters for dictionary learning were the same asthose in Section IV-A2. Because there was strong noise in an in-termediate image, the atoms in this dictionary are noisy. There-fore, the error control item for ADSIR was made smaller thanthat for GDSIR in which the dictionary was learned from thenormal-dose image. On the other hand, since the dictionary waslearned from the reconstructed image itself, there was no needto use many atoms to capture the structures. The sparsity levelparameter was made much smaller than that for GDSIR.

Taking all these factors into account, the parameter were em-pirically chosen as , , and . Thenumber of subsets was again 40. The iterative process was alsostopped after 50 iterations.Third, the popular TV regularization algorithm was included

to demonstrate the merits of the proposed methods. For that pur-pose, the TVminimization constraint was used as the regulariza-tion term in (12), and enforced using the soft-threshold filteringbased alternating minimization algorithm [17], [45]. We denotethis method as TVSIR.Fourth, to evaluate the effect of the statistical recon-

struction technique in this dictionary learning based recon-struction framework, we replaced the log-likelihood term

in (13) with an unweighted

-norm data fidelity term l . The global dictio-nary based algorithm was modified with this constant weightingscheme, which we denote as GDNSIR. The adaptive dictionarybased algorithm can be modified in a similar way. The regu-larization parameter was empirically set to given theweight change of the data fidelity term in the objective function.The results from a representative low-dose sinogram are in

Fig. 2. It can be seen that there is strong noise in the FBP recon-struction, and streak artifacts along high attenuation structures,such as around bones. This kind of streak artifacts can be easilyidentified from the difference between the FBP image and theresults with the SIR methods. The dictionary learning basedalgorithms generally performed well with low-dose data. WhileGDSIR did better in preserving structures and suppressing noise,ADSIR kept slightlymore structures thanGDSIR (see the regionindicated by the arrow “A”). ADSIR generated a little less uni-formity thanGDSIR in thewhole image (see the region indicatedby the arrow “B”), and some edgeswere obscurer than thosewithGDSIR. The performance of GDNSIR was not much differentfrom that of GDSIR. However, there were some streak artifactswith GDNSIR as in the FBP reconstruction (see the differencefrom the FBP image), especially around the bone (see the regionindicated by the arrow “C”). The image reconstructed by TVSIRhad much less noise than the FBP result, but it was a littleblocky and had an inferior visibility compared to the dictionarylearning based methods (see the regions indicated by the arrows“D” and “E”). Some bony structures in the TVSIR result wereobscure or invisible (see the region indicated by the arrow “F”).4) Few-View Test: Reducing the number of projection views

is an important strategy to reduce image time and radiationdose, giving the few-view problem. To evaluate the proposeddictionary learning based algorithms for few-view tomography,the number of low-dose views was down-sampled from 1160to 580, 290 and 116, respectively. The GDSIR and ADSIRmethods were then applied. Also, the FBP and TVSIR methodswere performed for comparison. The results are in Fig. 3.It is seen that the FBP reconstruction results became worse

and worse when the number of views was gradually decreasedfrom 1160 to 116. The GDSIR, ADSIR and TVSIR results weremuch better than the FBP reconstruction. In the case of 580views, the GDSIR and ADSIR results were almost as good asthat reconstructed from 1160 views in Fig. 2. However, in thecases of 290 and 116 views, some details were lost. The TVSIR

ρ(xi − x j ) :Penalty on the differencexi

4  

Physics-based Prior •  For some inverse problems, Physics can provide more

information than local or non-local priors. •  Example: Microstructure evolution in materials is described by

Phase-field model.

•  We explore the idea of using a Physics-based prior in such inverse problems.

Microstructure evolution in Cu-Al alloy 5  

Cahn-Hilliard Equation as Prior

•  Cahn-Hilliard equation governs the temporal and spatial evolution in binary fluids. •   We use Cahn-Hilliard equation as the prior for inverse problems. •   As a first step, we apply the Cahn-Hilliard prior to Image de-noising problem.  

6  

Cahn Hilliard Equation •  The Cahn-Hilliard equation for a binary fluid is: where

x(r, t) is the concentration of the fluid between (0-1), with 0representing one phase and 1 representing the other.

f *(x) is the dimensionless free energy of the fluid.a and b are parameters of the equation.

H (x,θ ) = ∂x∂t+ a∇4x − b∇2 ∂f *

∂x= 0

7  

Image de-noising in presence of Cahn-Hilliard prior

§  Image de-noising problem statement:

- x, the unknown image - y, the noisy input image

- , a diagonal matrix with - , Cahn-Hilliard equation

D

x̂ = argminx,θ

|| y− x ||D2

subject to H (x,θ ) = 0

di =12σ 2

H x,θ( ) = 0

8  

De-noising Cost Function

•  For de-noising problem, we form the following cost function:

•  Penalize deviation from , i.e. deviation from the Physical behavior. •  MAP Estimate:

Lλ (x,θ ) =12σ 2 y− x 2

+λ H (x,θ ) 2

H (x,θ ) = 0

x̂ = argminx,θ

Lλ (x,θ )

9  

Alternate Minimization using ICD

•  Set number of iterations •  Initialize . •  Low-pass filter to get and initialize •  For each iteration

–  Update to minimize •  For each pixel

Minimize for between (0-1)

–  Update to minimize •  Find least-square estimate .

Lλ (x,θ )x

θ H (x,θ )

x← yy ylp θ← argminθ H (ylp,θ )

θ← argminθ H (x,θ )

Lλ (x,θ ) xs

Lλ (x,θ ) =12σ 2 y− x 2

+λ H (x,θ ) 2

xs

10  

Experiments Generate phantom images that satisfy the Cahn-Hilliard

equation. Generate noisy images from : - Add i.i.d. Gaussian noise with

Apply ICD to minimize jointly over

H (x,θ ) = 0

x

σ = 0.05 and 0.1

x and θ

xy

Lλ (x,θ ) =12σ 2 y− x 2

+λ H (x,θ ) 2

11  

   De-noising results for

σ = 0.05(5% noise)

12  

De-noising Comparison

13  

   De-noising results for σ = 0.1(10% noise)

14  

De-noising Comparison  

15  

Current and future work •  Reconstruction in the presence of Cahn-Hilliard

Prior: where - A, a matrix implementing the linear forward model

•  Reconstruction with time-interleaving and limited

projections.

minx,θ

|| y− Ax ||D2

subject to H (x,θ ) = 0

16  

Questions ?

17  

 Supplementary  Slides  

18  

Cahn Hilliard Equation •  The Cahn-Hilliard equation for a binary fluid is: where

is the concentration of the alloy between (0-1), with 0 representing one phase and 1 representing the other.

is the free energy of the alloy. Assuming a double-well potential energy functional, we have and

),( tru

f (u)

γ : controls rate of growth of the phase [µm2 / s]

ε : controls width of the transition region [µm]

f (u) = u2 (u−1)2

∂u∂t= γ∇2 −ε 2∇2 (u)+ ∂f

∂u

$

%&

'

()

∂f∂u

=4u3 − 6u2 + 2u

0 1

)(* uf

Cahn Hilliard Equation •  The Cahn-Hilliard equation for a binary fluid is: where

is the concentration of the fluid between (0-1), with 0 representing one phase and 1 representing the other. is the dimensionless free energy of the fluid. Assuming a double-well potential energy functional, we have and

),( tru

)(* uff *(u) = u2 (u−1)2

∂u∂t= −a∇4u+ b∇2 ∂f *

∂u

$

%&

'

()

∂f *∂u

=4u3 − 6u2 + 2u

a [µm4 / s] and b [µm2 / s] are parameters of the equation

0 1

)(* uf

20  

Discrete form of Cahn Hilliard Equation

•  Consider 2D spatial coordinates, and let be the discrete realization of u at (i, j) spatial coordinates and time frame, where is time step and is the

spatial step. •  Finite Difference formulation of CH-equation is: where is the

discrete space Laplace operator and

un, i, j = u(iΔx, jΔx,nΔts )

un+1,i, j −un,i, j

Δts= −aD2 (un,Δx)i, j + bD(4un

3 − 6un2 + 2un,Δx)i, j (1)

D(un,Δx)i, j =un,i+1, j +un,i−1, j +un,i, j+1 +un,i, j−1 − 4un,i, j

(Δx)2

thn Δts[sec] Δx [µm]

D2 (un,Δx)i, j = D(D(un,Δx),Δx)i, j21  

Parameterization - Discrete form of Cahn Hilliard Equation

•  Re write Cahn Hilliard equation (1) as where •  So the Cahn-Hilliard regularization, is:

un+1,i, j −un,i, j = − aD2 (un )i, j + bD(4un

3 − 6un2 + 2un )i, j (2)

a = a(Δx)4 Δts, b =

b(Δx)2 Δts are unitless parameters.

H (un+1,un,θ )

H (un+1,un,θ ) = un+1 −un + aD2 (un )− bD(4un

3 − 6un2 + 2un )

22  

Stability Constraints on discretization

•  Some discretization schemes of the Cahn Hilliard are known to be more stable[1]. –  Implicit Euler Scheme:

–  Linearly Stabilized Splitting Scheme[1]:

- Splits the free energy into concave and convex parts

- Treats the convex part implicitly and the concave parts explicitly .

uijn+1 −uij

n

Δt= γD(−ε 2D(uij

n+1)+ (uijn+1)3 −uij

n+1)

uijn+1 −uij

n

Δt= γD(−ε 2D(uij

n+1)+ 2uijn+1)+D((uij

n )3 −3uijn )

)()()( 21 uEuEuE +=2)(

4)()(

24 uuuE −=

[1]:  D.  Eyre,  An  uncondiDonally  stable  one-­‐step  scheme  for  gradient  systems,  1997.  

Cost per pixel vs. iterations

24  

Regularization per pixel vs. iterations

Regularization per pixel after 50 iterations = 8.4469 × 10−525  

Comparison with Standard de-noising methods

•  RMSE for de-noising with 5% noise: –  BM3D: 0.012011 –  BM4D: 0.006212 –  Cahn-Hilliard prior: 0.02614

26  

BM4D De-noising Results

27  

De-noising Comparison

28  

De-noising Comparison  

29