.
Bayesian inference framework for Inverse problems Ali Mohammad-Djafari Laboratoire des Signaux et Syst`emes (L2S) UMR8506 CNRS-CentraleSup´elec-UNIV PARIS SUD SUPELEC, 91192 Gif-sur-Yvette, France http://lss.centralesupelec.fr Email:
[email protected] http://djafari.free.fr http://publicationslist.org/djafari
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
1/59
Contents 1. Inverse problems examples I I I I
Instrumentation Imaging systems to see outside of a body Imaging systems to see inside of a body Other imaging systems (Acoustics, Radar, SAR,...)
2. Analytical/Algebraic methods 3. Deterministic regularization methods and their limitations 4. Bayesian approach 5. Two main steps: Priors and Computational aspects 6. Case studies: Instrumentation, X ray Computed Tomography, Microwave imaging, Acoustic source localisation, Ultrasound imaging, Satellite image restoration, etc.
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
2/59
Inverse Problems examples I
Example 1: Instrumentation: Measuring the temperature with a thermometer Deconvolution I I
I
Example 2: Seeing outside of a body: Making an image using a camera, a microscope or a telescope: Image restoration I I
I
f (x, y ) real scene g (x, y ) observed image
Example 3: Seeing inside of a body: Computed Tomography usng X rays, US, Microwave, etc.: Image reconstruction I I
I
f (t) input of the instrument g (t) output of the instrument
f (x, y ) a section of a real 3D body f (x, y , z) gφ (r ) a line of observed radiographe gφ (r , z)
Example 4: Seeing differently: MRI, Radar, SAR, Infrared, etc.: Fourier Synthesis I I
f (x, y ) a section of body or a scene g (u, v ) partial data in the Fourier domain
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
3/59
Measuring variation of temperature with a therometer I
f (t) variation of temperature over time
I
g (t) variation of length of the liquid in thermometer
I
Forward model: Convolution Z g (t) = f (t 0 ) h(t − t 0 ) dt 0 + (t) h(t): impulse response of the measurement system
I
Inverse problem: Deconvolution Given the forward model H (impulse response h(t))) and a set of data g (ti ), i = 1, · · · , M find f (t)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
4/59
Measuring variation of temperature with a therometer Forward model: Convolution Z g (t) = f (t 0 ) h(t − t 0 ) dt 0 + (t)
f (t)−→
Thermometer h(t) −→
g (t)
Inversion: Deconvolution f (t)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
g (t)
ICEEE 2015, Sharif Univ. Tehran, Iran,
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Seeing outside of a body: Making an image with a camera, a microscope or a telescope I
f (x, y ) real scene
I
g (x, y ) observed image
I
Forward model: Convolution ZZ g (x, y ) = f (x 0 , y 0 ) h(x − x 0 , y − y 0 ) dx 0 dy 0 + (x, y ) h(x, y ): Point Spread Function (PSF) of the imaging system
I
Inverse problem: Image restoration Given the forward model H (PSF h(x, y ))) and a set of data g (xi , yi ), i = 1, · · · , M find f (x, y )
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
6/59
Making an image with an unfocused camera Forward model: 2D Convolution ZZ g (x, y ) = f (x 0 , y 0 ) h(x − x 0 , y − y 0 ) dx 0 dy 0 + (x, y ) (x, y ) f (x, y ) - h(x, y )
? - + -g (x, y )
Inversion: Image Deconvolution or Restoration ? ⇐=
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
7/59
Seeing inside of a body: Computed Tomography I
f (x, y ) a section of a real 3D body f (x, y , z)
I
gφ (r ) a line of observed radiography gφ (r , z)
I
Forward model: Line integrals or Radon Transform Z gφ (r ) = f (x, y ) dl + φ (r ) L
ZZ r ,φ = f (x, y ) δ(r − x cos φ − y sin φ) dx dy + φ (r ) I
Inverse problem: Image reconstruction Given the forward model H (Radon Transform) and a set of data gφi (r ), i = 1, · · · , M find f (x, y )
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
8/59
2D and 3D Computed Tomography 3D
2D
Z gφ (r1 , r2 ) =
Z f (x, y , z) dl
gφ (r ) =
Lr1 ,r2 ,φ
f (x, y ) dl Lr ,φ
Forward probelm: f (x, y ) or f (x, y , z) −→ gφ (r ) or gφ (r1 , r2 ) Inverse problem: gφ (r ) or gφ (r1 , r2 ) −→ f (x, y ) or f (x, y , z) A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
9/59
Computed Tomography: Radon Transform
Forward: Inverse:
A. Mohammad-Djafari,
f (x, y ) f (x, y )
−→ ←−
Bayesian inference framework for inverse problems,
g (r , φ) g (r , φ)
ICEEE 2015, Sharif Univ. Tehran, Iran,
10/59
Microwave or ultrasound imaging Measures: diffracted wave by the object g (ri ) Unknown quantity: f (r) = k02 (n2 (r) − 1) Intermediate quantity : φ(r) ZZ
Gm (ri , r0 )φ(r0 ) f (r0 ) dr0 , ri ∈ S D ZZ φ(r) = φ0 (r) + Go (r, r0 )φ(r0 ) f (r0 ) dr0 , r ∈ D g (ri ) =
D
Born approximation (φ(r0 ) ' φ0 (r0 )) ): ZZ g (ri ) = Gm (ri , r0 )φ0 (r0 ) f (r0 ) dr0 , ri ∈ S D
r
r
r r ! ! L r , aa r , E - E r e φ0r (φ, f )% r % r r r r g r r
Discretization: g = H(f) g = Gm Fφ −→ with F = diag(f) φ= φ0 + Go Fφ H(f) = Gm F(I − Go F)−1 φ0 A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
11/59
Fourier Synthesis in X ray ZZ Tomography f (x, y ) δ(r − x cos φ − y sin φ) dx dy
g (r , φ) = Z G (Ω, φ) =
g (r , φ) exp [−jΩr ] dr ZZ
F (u, y ) = F (v , y ) = G (Ω, φ) y 6 s
f (x, y ) exp [−jvx, yy ] dx dy for
@ I @ @ @ @ (x, y ) @ f @ @ @ φ @ H H @ @ @
u = Ω cos φ
r
α
@ I @ @
Ω
@
F (ωx , @ ωy ) @
@ φ @ @
-
x
g (r , φ)–FT–G (Ω, φ) @ @
A. Mohammad-Djafari,
and v = Ω sin φ v 6
Bayesian inference framework for inverse problems,
-
u
@ @ @
ICEEE 2015, Sharif Univ. Tehran, Iran,
12/59
Fourier Synthesis in X ray tomography ZZ G (u, v ) =
f (x, y ) exp [−j (ux + vy )] dx dy
? =⇒
Forward problem: Given f (x, y ) compute G (u, v ) Inverse problem: Given G (u, v ) on those lines estimate f (x, y ) A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
13/59
Fourier Synthesis in Diffraction tomography
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
14/59
Fourier Synthesis in Diffraction tomography ZZ G (u, v ) =
f (x, y ) exp [−j (ux + vy )] dx dy
? =⇒ Forward problem: Given f (x, y ) compute G (u, v ) Inverse problem : Given G (u, v ) on those semi cercles estimate f (x, y ) A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
15/59
Fourier Synthesis in different imaging systems ZZ G (u, v ) =
X ray Tomography
f (x, y ) exp [−j (ux + vy )] dx dy
Diffraction
Eddy current
SAR & Radar
Forward problem: Given f (x, y ) compute G (u, v ) Inverse problem : Given G (u, v ) on those algebraic lines, cercles or curves, estimate f (x, y ) A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
16/59
Linear inverse problems I
Deconvolution Z f (τ )h(t − τ ) dτ
g (t) = I
Image restoration Z g (x, y ) =
f (x 0 , y 0 )h(x − x 0 , y − y 0 ) dx dy
I
Image reconstruction in X ray CT Z g (r , φ) = f (x, y )δ(r − x cos φ − y sin φ) dx dy
I
Fourier synthesis Z g (u, v ) =
I
f (x, y ) exp [−j(ux + vy )] dx dy
Unified linear relation Z g (s) =
A. Mohammad-Djafari,
f (r) h(s, r) dr
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
17/59
Linear Inverse Problems Z g (si ) = I
h(si , r) f (r) dr + (si ),
i = 1, · · · , M
f (r) is assumed to be well approximated by N X f (r) ' fj φj (r) j=1
with {φj (r)} a basis or any other set of known functions Z N X g (si ) = gi ' fj h(si , r) φj (r) dr, i = 1, · · · , M Z j=1 g = Hf + with Hij = h(si , r) φj (r) dr I
H is huge dimensional I
I
1D: 103 × 103 ,
2D: 106 × 106 ,
3D: 109 × 109
Due to ill-posedness of the inverse problems, Least squares (LS) methods: bf = arg minf {J(f)} with J(f) = kg − Hfk2 do not give satisfactory result. Need for regularization methods: J(f) = kg − Hfk2 + λkfk2
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
18/59
Regularization theory Inverse problems = Ill posed problems −→ Need for prior information Functional space (Tikhonov): g = H(f ) + J(f ) = ||g − H(f )||22 + λ||Df ||22 Finite dimensional space (Philips & Towmey): g = Hf + J(f) = kg − Hfk2 + λkfk2 • Minimum norme LS (MNLS): • Classical regularization: • More general regularization: or
J(f) = ||g − H(f)||2 + λ||f||2 J(f) = ||g − H(f)||2 + λ||Df||2
J(f) = Q(g − H(f)) + λΩ(Df)
J(f) = ∆1 (g, H(f)) + λ∆2 (Df, f 0 ) Limitations: • Errors are considered implicitly white and Gaussian • Limited prior information on the solution • Lack of tools for the determination of the hyperparameters A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
19/59
Inversion: Probabilistic methods Taking account of errors and uncertainties −→ Probability theory I
Maximum Likelihood (ML)
I
Minimum Inaccuracy (MI)
I
Probability Distribution Matching (PDM)
I
Maximum Entropy (ME) and Information Theory (IT)
I
Bayesian Inference (Bayes)
Advantages: I
Explicit account of the errors and noise
I
A large class of priors via explicit or implicit modeling
I
A coherent approach to combine information content of the data and priors
Limitations: I
Practical implementation and cost of calculation
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
20/59
Bayesian estimation approach M: g = Hf + Observation model M + Hypothesis on the noise −→ p(g|f; M) = p (g − Hf) I A priori information p(f|M) p(g|f; M) p(f|M) I Bayes : p(f|g; M) = p(g|M) Link with regularization : I
I
I
Maximum A Posteriori (MAP) : bf = arg max {p(f|g)} = arg max {p(g|f) p(f)} f f = arg min {J(f) = − ln p(g|f) − ln p(f)} f Regularization: bf = arg min {J(f) = Q(g, Hf) + λΩ(f)} f with Q(g, Hf) = − ln p(g|f) and λΩ(f) = − ln p(f)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
21/59
Case of linear models and Gaussian priors g = Hf + I
Prior knowledge on the noise: ∼
I
1 → p(g|f) ∝ exp − 2 kg − Hfk2 2σ
Prior knowledge on f: f∼
I
N (0, σ2 I)
N (0, σf2 (D0 D)−1 )
1 → p(f) ∝ exp − 2 kDfk2 2σf
A posteriori:
1 1 p(f|g) ∝ exp − 2 kg − Hfk2 − 2 kDfk2 2σ 2σf bf = arg max {p(f|g)} = arg min {J(f)} f f
I
MAP :
I
with J(f) = kg − Hfk2 + λkDfk2 , λ = σσ2 f Advantage : characterization of the solution b with bf = PH b 0 g, P b = H0 H + λD0 D −1 p(f|g) = N (bf, P)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
2
ICEEE 2015, Sharif Univ. Tehran, Iran,
22/59
Regularization versus Bayesian
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
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Main advantages of the Bayesian approach I
MAP = Regularization
I
Posterior mean ? Marginal MAP ?
I
More information in the posterior law than only its mode or its mean
I
Tools for estimating hyper parameters
I
Tools for model selection
I
More specific and specialized priors, particularly through the hidden variables and hierarchical models More computational tools:
I
I
I I
I
Expectation-Maximization for computing the maximum likelihood parameters MCMC for posterior exploration Variational Bayes for analytical computation of the posterior marginals ...
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
24/59
Bayesian Estimation: Simple priors I I
Linear model: g = Hf + Gaussian case: p(g|f, θ1 ) = N (Hf, θ1 I) b → p(f|g, θ) = N (bf, P) p(f|θ2 ) = N (0, θ2 I) with
(
b = (H0 H + λI)−1 , P bf = PH b 0g
λ=
θ1 θ2
I
bf = arg min {J(f)} with J(f) = kg − Hfk2 + λkfk2 2 2 f Generalized Gaussian prior & MAP:
I
bf = arg min {J(f)} with J(f) = kg − Hfk2 + λkfkβ 2 f Double Exponential (β = 1): bf = arg min {J(f)} with J(f) = kg − Hfk2 + λkfk1 2 f
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
25/59
Full (Unsupervised) Bayesian approach M:
I I I I I
I
I I
g = Hf +
Forward & errors model: −→ p(g|f, θ 1 ; M) Prior models −→ p(f|θ 2 ; M) Hyperparameters θ = (θ 1 , θ 2 ) −→ p(θ|M) p(g|f ,θ ;M) p(f |θ ;M) p(θ |M) Bayes: −→ p(f, θ|g; M) = p(g|M) b = arg max {p(f, θ|g; M)} Joint MAP: (bf, θ) (f ,θ ) R p(f|g; M) = R p(f, θ|g; M) dθ Marginalization: p(θ|g; M) = p(f, θ|g; M) df ( RR bf = f p(f, θ|g; M) dθ df RR Posterior means: b = θ θ p(f, θ|g; M) df dθ Evidence of the model: ZZ p(g|M) = p(g|f, θ; M)p(f|θ; M)p(θ|M) df dθ
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
ICEEE 2015, Sharif Univ. Tehran, Iran,
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Hierarchical models Simple case (1 layer): g = Hf + θ2 θ1 p(f|g, θ) ∝ p(g|f, θ 1 ) p(f|θ 2 ) ? ? Objective: Infer on f f MAP: b f = arg maxf {p(f|g,Zθ)} H ? Posterior Mean (PM): bf = p(f|g, θ) df g
Unsupervised case (2 layers): β0
α0
? ?
θ2
θ1
p(f, θ|g) ∝ p(g|f, θ 1 ) p(f|θ 2 ) p(θ) Objective: Infer on f, θ
b = arg max JMAP: (bf, θ) ? ? (f ,θZ) {p(f, θ|g)}
f Marginalization: p(θ|g) = H
?
p(f, θ|g) df
VBA: Approximate p(f, θ|g) by q1 (f) q2 (θ)
g
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Two main steps in the Bayesian approach I
Prior modeling I
I
I
I
Separable: Gaussian, Gamma, Sparsity enforcing: Generalized Gaussian, mixture of Gaussians, mixture of Gammas, ... Markovian: Gauss-Markov, GGM, ... Markovian with hidden variables (contours, region labels)
Choice of the estimator and computational aspects I I I I
I
MAP, Posterior mean, Marginal MAP MAP needs optimization algorithms Posterior mean needs integration methods Marginal MAP and Hyperparameter estimation need integration and optimization Approximations: I I I
A. Mohammad-Djafari,
Gaussian approximation (Laplace) Numerical exploration MCMC Variational Bayes (Separable approximation)
Bayesian inference framework for inverse problems,
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Different prior models for signals and images: Separable
Gaussian p(fj ) ∝ exp −α|fj |2
Generalized Gaussian p(fj ) ∝ exp [−α|fj |p ] , 1 ≤ p ≤ 2
Gamma p(fj ) ∝ fjα exp [−βfj ]
Beta p(fj ) ∝ fjα (1 − fj )β
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Bayesian inference framework for inverse problems,
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Sparsity enforcing prior models I
Sparse signals: Direct sparsity
I
Sparse signals: Sparsity in a Transform domain
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Bayesian inference framework for inverse problems,
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Sparsity enforcing prior models I
Simple heavy tailed models: I I I I I
I
Generalized Gaussian, Double Exponential Symmetric Weibull, Symmetric Rayleigh Student-t, Cauchy Generalized hyperbolic Elastic net
Hierarchical mixture models: I I I I I I
Mixture of Gaussians Bernoulli-Gaussian Mixture of Gammas Bernoulli-Gamma Mixture of Dirichlet Bernoulli-Multinomial
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Bayesian inference framework for inverse problems,
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Which images I am looking for?
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Bayesian inference framework for inverse problems,
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Which image I am looking for?
Gauss-Markov
Generalized GM
Piecewize Gaussian
Mixture of GM
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Bayesian inference framework for inverse problems,
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Different prior models for signals and images: Separable I
Simple Gaussian, Gamma, Generalized Gaussian X p(f) ∝ exp φ(f j ) j
I
Simple Markovian models: Gauss-Markov, Generalized Gauss-Markov X X p(f) ∝ exp φ(f j − f i ) j
I
j∈N (i)
Hierarchical models with hidden variables: Bernouilli-Gaussian, Gaussian-Gamma X X p(f|z) ∝ exp p(f j |z j ) and p(z) ∝ exp p(z j ) j
j
with different choices for p(f j |z j ) and p(z j ) A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Hierarchical models and hidden variables I
Student-t model
ν+1 log 1 + f 2 /ν St(f |ν) ∝ exp − 2 I
Infinite Scaled Gaussian Mixture (ISGM) equivalence Z ∞ St(f |ν) ∝= N (f |, 0, 1/z) G(z|α, β) dz, with α = β = ν/2 0
p(f|z) p(z|α, β) p(f, z|α, β) A. Mohammad-Djafari,
i h 1P 2 z f N (f |0, 1/z ) ∝ exp − j j j j j j 2 Q Q (α−1) = j G(z hPj |α, β) ∝ j z j iexp [−βz j ] ∝ exp (α − 1) ln z j − βz j h jP i ∝ exp − 21 j z j f 2j + (α − 1) ln z j − βz j =
Q
j p(f j |z j ) =
Q
Bayesian inference framework for inverse problems,
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Gauss-Markov-Potts prior models for images
f (r)
z(r)
c(r) = 1 − δ(z(r) − z(r0 ))
p(f (r)|z(r) = k, mk , vk ) = N (mk , vk ) X p(f (r)) = P(z(r) = k) N (mk , vk ) Mixture of Gaussians I I
k Q Separable iid hidden variables: p(z) = r p(z(r)) Markovian hidden variables: p(z) Potts-Markov: X p(z(r)|z(r0 ), r0 ∈ V(r)) ∝ exp γ δ(z(r) − z(r0 )) r0 ∈V(r) X X p(z) ∝ exp γ δ(z(r) − z(r0 )) r∈R r0 ∈V(r)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Four different cases To each pixel of the image is associated 2 variables f (r) and z(r) I
f|z Gaussian iid, z iid : Mixture of Gaussians
I
f|z Gauss-Markov, z iid : Mixture of Gauss-Markov
I
f|z Gaussian iid, z Potts-Markov : Mixture of Independent Gaussians (MIG with Hidden Potts)
I
f|z Markov, z Potts-Markov : Mixture of Gauss-Markov (MGM with hidden Potts)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
f (r)
z(r) ICEEE 2015, Sharif Univ. Tehran, Iran,
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Hierarchical models (3 layers) p(f, z, θ|g) ∝ p(g|f, θ 1 ) p(f|z, θ 2 ) p(z|θ 3 ) p(θ) p(θ) = p(θ 1 |α0 ) p(θ 2 |β0 ) p(θ 3 |γ0 ) Objective: Infer on f, z, θ
γ0 ?
β0
θ3
α0
? ? ?
JMAP: b = arg max (bf, b z, θ)
(f ,z,θ ) {p(f, z, θ|g)}
Marginalization: z θ2 θ1 Z @ ? ? p(z, θ|g) = p(f, z, θ|g) df Z R f @ p(θ|g) = p(z, θ|g) dz H
?
g
Z Z or p(f|g) =
p(f, z, θ|g) dz dθ
VBA: Approximate p(f, z, θ|g) by q1 (f) q2 (z) q3 (θ) A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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JMAP, Marginalization, VBA I
JMAP: p(f, θ|g) optimization
I
−→ bf b −→ θ
Marginalization p(f, θ|g) −→
p(θ|g)
b −→ p(f|θ, b g) −→ bf −→ θ
Joint Posterior Marginalize over f I
Variational Bayesian Approximation
p(f, θ|g) −→
A. Mohammad-Djafari,
Variational Bayesian Approximation
Bayesian inference framework for inverse problems,
−→ q1 (f) −→ bf b −→ q2 (θ) −→ θ ICEEE 2015, Sharif Univ. Tehran, Iran,
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VBA: Choice of family of laws q1 and q2 Case 1 : −→ Joint MAP n o ( e M) ef = arg max p(f, θ|g; b1 (f|ef) = δ(f − ef) q f n o e = δ(θ − θ) e−→θ= e arg max p(ef, θ|g; M) b2 (θ|θ) q θ
I
I
I
(
Case 2 : −→ EM e M)i b1 (f) q ∝ p(f|θ, g) Q(θ, θ)= hln p(f, θ|g; q1 (o f |θe ) n −→ e = δ(θ − θ) e θ e e b2 (θ|θ) q = arg maxθ Q(θ, θ) Appropriate choice for inverse problems
e g; M) Accounts for the uncertainties of b1 (f) ∝ p(f|θ, q −→ b e θ for bf and vise versa. b2 (θ) ∝ p(θ|f, g; M) q Exponential families, Conjugate priors
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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JMAP, EM and VBA JMAP Alternate optimization Algorithm: n o e ef = arg max p(f, θ|g) e −→ef −→ bf θ (0) −→ θ−→ f ↑ ↓ n o b ←− θ←− e e = arg max p(ef, θ|g) ←−ef θ θ θ EM: e θ (0) −→ θ−→ ↑ b ←− θ←− e θ
−→q1 (f) −→ bf ↓
e g) q1 (f) = p(f|θ, e = hln p(f, θ|g)i Q(θ, θ) q1o (f ) n e = arg max Q(θ, θ) e θ θ
←− q1 (f)
VBA: h i θ (0) −→ q2 (θ)−→ q1 (f) ∝ exp hln p(f, θ|g)iq2 (θ ) −→q1 (f) −→ bf ↑ ↓ h i b θ ←− q2 (θ)←− q2 (θ) ∝ exp hln p(f, θ|g)iq1 (f ) ←−q1 (f) A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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y 6
Computed Tomography: Discretization Hij
f (x, y )
Q Q
f1 Q QQ fjQ Q Q Q Qg
@ @ -
x
HH
i
fN
@
Z g (r , φ) =
f (x, y ) dl PL f (x, y ) = j fj bj (x, y ) 1 if (x, y ) ∈ pixel j bj (x, y ) = 0 else N X gi = Hij fj + i j=1
g = Hf + A. Mohammad-Djafari,
Case study: Reconstruction from 2 projections R g1 (x) = R f (x, y ) dy , g2 (y ) = f (x, y ) dx Very ill-posed inverse problem f (x, y ) = g1 (x) g2 (y ) Ω(x, y ) RΩ(x, y ) is a Copula: R Ω(x, y ) dx = 1 Ω(x, y ) dy = 1
Bayesian inference framework for inverse problems,
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Simple example
1 2 3 g1 g2 g3 g4
3 4 ? 4 6 ? 7 3 1 0 = 1 0
I
? 4 f1 f3 g3 1 -1 ? 6 f2 f4 g4 -1 1 7 g1 g2 0 0 f1 f4 f1 1 0 0 f2 f2 f5 0 1 1 f3 f6 f3 0 1 0 g1 g2 f4 1 0 1 Hf = g −→ bf = H−1 g if H invertible.
I
H is rank deficient: rank(H) = 3
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Problem has infinite number of solutions.
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How to find all those solutions ?
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Which one is the good one? Needs prior information.
I
To find an unique solution, one needs either more data or prior information.
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
0 0
-1 1 0
1 0 -1 0 0
f7 g4 f8 g5 f9 g6 g3
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Application in CT: Reconstruction from 2 projections
g|f g = Hf + g|f ∼ N (Hf, σ2 I) Gaussian
f|z iid Gaussian or Gauss-Markov
z iid or Potts
c q(r) ∈ {0, 1} 1 − δ(z(r) − z(r0 )) binary
p(f, z, θ|g) ∝ p(g|f, θ 1 ) p(f|z, θ 2 ) p(z|θ 3 ) p(θ)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Results
Original
Backprojection
Filtered BP
Gauss-Markov+pos
GM+Line process
GM+Label process
c
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
LS
z
ICEEE 2015, Sharif Univ. Tehran, Iran,
c
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Application in Acoustic source localization (Ning Chu et al.)
Wind tunnel
Beamforming
DAMAS
Proposed VBA inference
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Microwave Imaging for Breast Cancer detection (L. Gharsalli et al.)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Microwave Imaging for Breast Cancer detection CSI: Contrast Source Inversion, VBA: Variational Bayesian Approach, MGI: Independent Gaussian mixture, MGM: Gauss-Markov mixture
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Images fusion and joint segmentation (with O. F´eron) gi (r) = fi (r) + i (r) 2 p(fi (r)|z(r) Q = k) = N (mi k , σi k ) p(f|z) = i p(f i |z)
g1
−→
bf 1 b z
g2
A. Mohammad-Djafari,
bf 2
Bayesian inference framework for inverse problems,
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Data fusion in medical imaging (with O. F´eron) gi (r) = fi (r) + i (r) 2 p(fi (r)|z(r) Q = k) = N (mi k , σi k ) p(f|z) = i p(f i |z)
g1
−→
bf 1 b z
g2 A. Mohammad-Djafari,
bf 2 Bayesian inference framework for inverse problems,
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Conclusions I
I
I I
I
I I
Inverse problems arise in many science and engineering applications Deterministic Algorithms: Optimization of a two terms criterion, penalty term, regularization term Probabilistic: Bayesian approach Hierarchical prior model with hidden variables are very powerful tools for Bayesian approach to inverse problems. Gauss-Markov-Potts models for images incorporating hidden regions and contours Main Bayesian computation tools: JMAP, MCMC and VBA Application in different imaging system (X ray CT, Microwaves, PET, Ultrasound, Optical Diffusion Tomography (ODT), Acoustic source localization,...)
Current Projects: I
Efficient implementation in 2D and 3D cases
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Bayesian Blind deconvolution θ3
θ2
θ1
g = h ∗ f + = Hf + = Fh +
? ? ?
Simple priors: h f p(f, h|g, θ) ∝ p(g|f, h, θ 1 ) p(f|θ 2 ) p(h|θ 3 ) @ Objective: Infer on f, h R @ ? JMAP: (bf, b z) = arg max(f ,z) {p(f, z|g)} g VBA: Approximate p(f, h|g) by q1 (f) q2 (h)
γ0
γ0
α0
Unsupervised: ? ? ? p(f, h, θ|g) ∝ p(g|f, h, θ 1 ) p(f|θ 2 ) p(h|θ 3 ) p(θ)
θ3 θ2 θ1 Objective: Infer on f, h, θ ? ? ? JMAP: b = arg max h f (bf, b z, θ) {p(f, z, θ|g)} (f ,z,θ ) VBA: @ R @ ? Approximate p(f, h, θ|g) by q1 (f) q2 (h) q3 (θ) g
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Bayesian Blind deconvolution with hierarchial models g = h ∗ f + = Hf + = Fh + Simple priors: p(f, h|g, θ) ∝ p(g|f, h, θ 1 ) p(f|θ 2 ) p(h|θ 3 )
γ h0
γf 0
? ?
Unsupervised: p(f, h, θ|g) ∝ p(g|f, h, θ 1 ) p(f|θ 2 ) p(h|θ 3 ) p(θ)
θ3 θ2 α Sparsity enforcing prior for f: 0 p(f, zf , h, θ|g) ∝ ? ? ? p(g|f, h, θ 1 ) p(f|zf ) p(zf |θ 2 ) p(h|θ 3 ) p(θ) z
z
θ
h f 1 Sparsity enforcing prior for h: ? ? ? p(f, h, zh , θ|g) ∝ p(g|f, h, θ1 ) p(f|θ2 ) p(h|z) p(z|θ3 ) p(θ) h f Hierarchical models for both f and h: @ p(f, zf , h, zh , θ|g) ∝ R @ ? p(g|f, h, θ 1 ) p(f|zf ) p(zf |θ 2 ) p(h|zh ) p(zh |θ 3 ) p(θ) g JMAP: n o b b b = arg max b b (bf, b zf , h, zh , θ) p(f, zf , h, zh , θ|g) b b (f ,zf ,h,zh ,θ ) VBA: Approximate p(f, zf , h, zh , θ|g) by q1 (f) q2 (zf ) q3 (h) q4 (zh ) q5 (θ) A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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Thanks to: Present PhD students: I L. Gharsali (Microwave imaging for Cancer detection) I M. Dumitru (Multivariate time series analysis for biological signals) I S. AlAli (Diffraction imaging for geophysical applications) Freshly Graduated PhD students: I C. Cai (2013: Multispectral X ray Tomography) I N. Chu (2013: Acoustic sources localization) I Th. Boulay (2013: Non Cooperative Radar Target Recognition) I R. Prenon (2013: Proteomic and Masse Spectrometry) I Sh. Zhu (2012: SAR Imaging) I D. Fall (2012: Emission Positon Tomography, Non Parametric Bayesian) I D. Pougaza (2011: Copula and Tomography) I H. Ayasso (2010: Optical Tomography, Variational Bayes) Older Graduated PhD students: I S. F´ ekih-Salem (2009: 3D X ray Tomography) I N. Bali (2007: Hyperspectral imaging) I O. F´ eron (2006: Microwave imaging) A. Mohammad-Djafari, I
Bayesian inference framework for inverse problems,
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Thanks to: Older Graduated PhD students: I H. Snoussi (2003: Sources separation) I Ch. Soussen (2000: Geometrical Tomography) I G. Mont´ emont (2000: Detectors, Filtering) I H. Carfantan (1998: Microwave imaging) I S. Gautier (1996: Gamma ray imaging for NDT) I M. Nikolova (1994: Piecewise Gaussian models and GNC) I D. Pr´ emel (1992: Eddy current imaging) Post-Docs: I J. Lapuyade (2011: Dimentionality Reduction and multivariate analysis) I S. Su (2006: Color image separation) I A. Mohammadpour (2004-2005: HyperSpectral image segmentation) Colleagues: I B. Duchˆ ene & A. Joisel (L2S)& G. Perruson (Inverse scattering and Microwave Imaging) I N. Gac (L2S) (GPU Implementation) I Th. Rodet A. Mohammad-Djafari, Bayesian inference framework for inverse problems, ICEEE 2015, Sharif Univ. Tehran, Iran, (L2S) (Computed Tomography)
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Thanks to: National Collaborators I A. Vabre & S. Legoupil (CEA-LIST), (3D X ray Tomography) I E. Barat (CEA-LIST) (Positon Emission Tomography, Non Parametric Bayesian) I C. Comtat (SHFJ, CEA) (PET, Spatio-Temporal Brain activity) I J. Picheral (SSE, Sup´ elec) (Acoustic sources localization) I D. Blacodon (ONERA) (Acoustic sources separation) I J. Lagoutte (Thales Air Systems) (Non Cooperative Radar Target Recognition) I P. Grangeat (LETI, CEA, Grenoble) (Proteomic and Masse Spectrometry) I F. L´ evi (CNRS-INSERM, Hopital Paul Brousse) (Biological rythms and Chronotherapy of Cancer) International Collaborators I K. Sauer (Notre Dame University, IN, USA) (Computed Tomography, Inverse problems) I F. Marvasti (Sharif University), (Sparse signal processing) I M. Aminghafari (Amir Kabir University) (Independent Components Analysis)
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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References 1 I
A. Mohammad-Djafari, “Bayesian approach with prior models which enforce sparsity in signal and image processing,” EURASIP Journal on Advances in Signal Processing, vol. Special issue on Sparse Signal Processing, (2012).
I
A. Mohammad-Djafari (Ed.) Probl` emes inverses en imagerie et en vision (Vol. 1 et 2), Hermes-Lavoisier, Trait´ e Signal et Image, IC2, 2009,
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A. Mohammad-Djafari (Ed.) Inverse Problems in Vision and 3D Tomography, ISTE, Wiley and sons, ISBN: 9781848211728, December 2009, Hardback, 480 pp.
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A. Mohammad-Djafari, Gauss-Markov-Potts Priors for Images in Computer Tomography Resulting to Joint Optimal Reconstruction and segmentation, International Journal of Tomography & Statistics 11: W09. 76-92, 2008.
I
A Mohammad-Djafari, Super-Resolution : A short review, a new method based on hidden Markov modeling of HR image and future challenges, The Computer Journal doi:10,1093/comjnl/bxn005:, 2008.
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H. Ayasso and Ali Mohammad-Djafari Joint NDT Image Restoration and Segmentation using Gauss-Markov-Potts Prior Models and Variational Bayesian Computation, IEEE Trans. on Image Processing, TIP-04815-2009.R2, 2010.
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H. Ayasso, B. Duchene and A. Mohammad-Djafari, Bayesian Inversion for Optical Diffraction Tomography Journal of Modern Optics, 2008.
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N. Bali and A. Mohammad-Djafari, “Bayesian Approach With Hidden Markov Modeling and Mean Field Approximation for Hyperspectral Data Analysis,” IEEE Trans. on Image Processing 17: 2. 217-225 Feb. (2008).
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H. Snoussi and J. Idier., “Bayesian blind separation of generalized hyperbolic processes in noisy and underdeterminate mixtures,” IEEE Trans. on Signal Processing, 2006.
A. Mohammad-Djafari,
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References 2 I
O. F´ eron, B. Duch` ene and A. Mohammad-Djafari, Microwave imaging of inhomogeneous objects made of a finite number of dielectric and conductive materials from experimental data, Inverse Problems, 21(6):95-115, Dec 2005.
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M. Ichir and A. Mohammad-Djafari, Hidden Markov models for blind source separation, IEEE Trans. on Signal Processing, 15(7):1887-1899, Jul 2006.
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F. Humblot and A. Mohammad-Djafari, Super-Resolution using Hidden Markov Model and Bayesian Detection Estimation Framework, EURASIP Journal on Applied Signal Processing, Special number on Super-Resolution Imaging: Analysis, Algorithms, and Applications:ID 36971, 16 pages, 2006.
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O. F´ eron and A. Mohammad-Djafari, Image fusion and joint segmentation using an MCMC algorithm, Journal of Electronic Imaging, 14(2):paper no. 023014, Apr 2005.
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H. Snoussi and A. Mohammad-Djafari, Fast joint separation and segmentation of mixed images, Journal of Electronic Imaging, 13(2):349-361, April 2004.
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A. Mohammad-Djafari, J.F. Giovannelli, G. Demoment and J. Idier, Regularization, maximum entropy and probabilistic methods in mass spectrometry data processing problems, Int. Journal of Mass Spectrometry, 215(1-3):175-193, April 2002.
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H. Snoussi and A. Mohammad-Djafari, “Estimation of Structured Gaussian Mixtures: The Inverse EM Algorithm,” IEEE Trans. on Signal Processing 55: 7. 3185-3191 July (2007).
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N. Bali and A. Mohammad-Djafari, “A variational Bayesian Algorithm for BSS Problem with Hidden Gauss-Markov Models for the Sources,” in: Independent Component Analysis and Signal Separation (ICA 2007) Edited by:M.E. Davies, Ch.J. James, S.A. Abdallah, M.D. Plumbley. 137-144 Springer (LNCS 4666) (2007).
A. Mohammad-Djafari,
Bayesian inference framework for inverse problems,
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References 3 I
N. Bali and A. Mohammad-Djafari, “Hierarchical Markovian Models for Joint Classification, Segmentation and Data Reduction of Hyperspectral Images” ESANN 2006, September 4-8, Belgium. (2006)
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M. Ichir and A. Mohammad-Djafari, “Hidden Markov models for wavelet-based blind source separation,” IEEE Trans. on Image Processing 15: 7. 1887-1899 July (2005)
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S. Moussaoui, C. Carteret, D. Brie and A Mohammad-Djafari, “Bayesian analysis of spectral mixture data using Markov Chain Monte Carlo methods sampling,” Chemometrics and Intelligent Laboratory Systems 81: 2. 137-148 (2005).
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H. Snoussi and A. Mohammad-Djafari, “Fast joint separation and segmentation of mixed images” Journal of Electronic Imaging 13: 2. 349-361 April (2004)
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H. Snoussi and A. Mohammad-Djafari, “Bayesian unsupervised learning for source separation with mixture of Gaussians prior,” Journal of VLSI Signal Processing Systems 37: 2/3. 263-279 June/July (2004)
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F. Su and A. Mohammad-Djafari, “An Hierarchical Markov Random Field Model for Bayesian Blind Image Separation,” 27-30 May 2008, Sanya, Hainan, China: International Congress on Image and Signal Processing (CISP 2008).
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N. Chu, J. Picheral and A. Mohammad-Djafari, “A robust super-resolution approach with sparsity constraint for near-field wideband acoustic imaging,” IEEE International Symposium on Signal Processing and Information Technology pp 286–289, Bilbao, Spain, Dec14-17,2011
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