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Sparsity in signal and image processing: from modeling and representation to reconstruction and processing Ali Mohammad-Djafari Laboratoire des Signaux et Syst`emes, UMR8506 CNRS-SUPELEC-UNIV PARIS SUD 11 SUPELEC, 91192 Gif-sur-Yvette, France http://lss.supelec.free.fr Email:
[email protected] http://djafari.free.fr Seminar at NUDT, Changsha, China, Nov. 28, 2013 A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 1/53
Contents Sparse signals and images First ideas for using sparsity in signal processing Modeling for sparse representation Bayesian Maximum A Posteriori (MAP) approach and link with Deterministic Regularization 5. Priors which enforce sparsity 1. 2. 3. 4.
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Heavy tailed: Double Exponential, Generalized Gaussian, ... Mixture models: Mixture of Gaussians, Student-t, ... Hierarchical models with hidden variables General Gauss-Markov-Potts models
6. Computational tools: Joint Maximum A Posteriori (JMAP), MCMC and Variational Bayesian Approximation (VBA) 7. Applications: X ray Computed Tomography, Microwave and Ultrasound imaging, Sattelite and Hyperspectral image processing, Spectrometry, CMB, ... A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 2/53
1. Sparse signals and images ◮
Sparse signals: Direct sparsity 3
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 3/53
Sparse signals and images ◮
Sparse signals in a Transform domain 1
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 4/53
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Sparse signals in Fourier domain Time domain
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 5/53
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Sparse signals and images ◮
Sparse signals: Sparsity in a Transform domaine
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Sparse signals and images
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Sparse signals and images
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 8/53
2. First ideas: some history ◮
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1948: Shannon: Sampling theorem and reconstruction of a band limited signal 1993-2007: ◮
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Mallat, Zhang, Cand`es, Romberg, Tao and Baraniuk: Non linear sampling, Compression and reconstruction, Fuch: Sparse representation Donoho, Elad, Tibshirani, Tropp, Duarte, Laska: Compressive Sampling, Compressive Sensing
2007-2012: Algorithms for sparse representation and compressive Sampling: Matching Pursuit (MP), Projection Pursuit Regression, Pure Greedy Algorithm, OMP, Basis Poursuit (BP), Dantzig Selector (DS), Least Absolute Shrinkage and Selection Operator (LASSO),... 2003-2012: Bayesian approach to sparse modeling Tipping, Bishop: Sparse Bayesian Learning, Relevance Vector Machine (RVM), ...
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 9/53
3. Modeling and representation ◮
Modeling via a basis (codebook, overcomplete dictionnary, Design Matrix) g(t) =
N X
f j φj (t), t = 1, · · · , T −→ g = Φ′ f
j=1
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When T ≥ N
2 N T X X b g(t) − f j φj (t) −→ f j = arg min fj t=1 j=1 b = arg min kg − Φ′ f k2 = [ΦΦ′ ]−1 Φg f f b = Φg When orthogonal basis: ΦΦ′ = I −→ f fbj =
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g(t) φj (t) =< g(t), φj (t) >
t=1
Application in Compression, Transmission and Decompression
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 10/53
Modeling and representation ◮
When overcomplete basis N > T : Infinite number of solutions for Φ′ f = g. We have to select one: b = arg min f kf k22 ′ f : Φ f =g or writing differently:
minimize kf k22 subject to Φ′ f = g resulting to: ◮ ◮ ◮
b = Φ[Φ′ Φ]−1 g f b = Φg. Again if Φ′ Φ = I −→ f No real interest if we have to keep all the N > T coefficients. Sparsity: minimize kf k0 subject to Φ′ f = g or minimize kf k1 subject to Φ′ f = g
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 11/53
Sparse decomposition ◮
Strict sparsity and exact reconstruction minimize kf k0 subject to Φ′ f = g kf k0 is the number of non-zero elements of f ◮ ◮ ◮ ◮
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Matching Pursuit (MP) [Mallat & Zhang, 1993] Orthogonal Matching Pursuit (OMP) [Lin, Huang et al., 1993] Projection Pursuit Regression Greedy Algorithms
Sparsity enforcing and exact reconstruction minimize kf k1 subject to Φ′ f = g ◮ ◮
Basis Pursuit (BP) Block Coordinate Relaxation
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 12/53
Sparse decomposition ◮
Strict sparsity and approximate reconstruction
minimize kf k0 subject to kg − Φ′ f k2 < c b = arg min kf k0 + µkg − Φ′ f k2 = arg min kg − Φ′ f k2 + λkf k0 f f f ◮
Sparsity enforcing and approximate reconstruction b = arg min kg − Φ′ f k2 + λkf k1 f f
J(f ) = kg − Φ′ f k2 + λkf k1 = kg − Φ′ f k2 + λ
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Main Algorithm: LASSO [Tibshirani 2003] minimize kg − Φ′ f k2 subject to kf k1 < τ
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 13/53
Sparse Decomposition Algorithms ◮
LASSO: J(f ) = kg − Φ′ f k2 + λ
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Other Criteria ◮
Lp J(f ) = kg − Φ′ f k2 + λ1
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1 0 DE(f |a) = exp {−a|f |} = 2 0 n o Q Q P = j p(fj |zj ) = j N (fj |0, zj ) ∝ exp − 12 j fj2 /zj p(f |z) nP 2 o Q 2 2 a = j E(zj | a2 ) ∝ exp p(z| a2 ) j 2 zj o n P 2 2 p(f , z| a2 ) ∝ exp − 21 j fj2 /zj + a2 zj ◮
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With these models we have: ◮
Simple priors p(f , θ|g) ∝ p(g|f , θ1 ) p(f |θ 2 ) p(θ)
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Hierarchical priors p(f , z, θ|g) ∝ p(g|f , θ1 ) p(f |z, θ2 ) p(z|θ3 ) p(θ)
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 29/53
Bayesian Computation and Algorithms ◮
When the expression of p(f , θ|g) or of p(f , z, θ|g) is obtained, we have following options:
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Joint MAP: (needs optimization algorithms) b = arg max {p(f , θ|g)} b , θ) (f (f ,θ )
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MCMC: Needs the expressions of the conditionals p(f |z, θ, g), p(z|f , θ, g), and p(θ|f , z, g)
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Variational Bayesian Approximation (VBA): Approximate p(f , z, θ|g) by a separable one q(f , z, θ|g) = q1 (f ) q2 (z) q3 (θ) and do any computations with these separable ones.
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 30/53
Joint MAP p(f , θ|g) ∝ p(g|f , θ 1 ) p(f |θ 2 ) p(θ) ◮
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Objective:
b = arg max {p(f , θ|g)} b , θ) (f (f ,θ ) Alternate optimization: n o b = arg max b f p(f , θ|g) fn o b , θ|g) b = arg max p(f θ θ b θ (0) −→ θ−→ ↑
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n o b g) b = arg max θ, f p(f | f n o b = arg max p(θ|f b , g) θ θ
Uncertainties are not propagated.
b −→f ↓
b ←− f
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 31/53
MCMC based algorithm p(f , z, θ|g) ∝ p(g|f , z, θ) p(f |z, θ) p(z) p(θ) General scheme (Gibbs Sampling): ◮
Generate samples from the conditionals: b g) −→ z b , θ, b g) −→ θ b, z b ∼ p(f |b b ∼ (θ|f b ∼ p(z|f b, g) f z , θ,
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Waite for convergency
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Compute empirical statistics (means, modes, variances) from the samples E {f } ≈
1 X (n) f N n
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 32/53
Variational Bayesian Approximation ◮
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Approximate p(f , θ|g) by q(f , θ|g) = q1 (f |g) q2 (θ|g) and then continue computations. Criterion KL(q(f , θ|g) : p(f , θ|g)) ZZ ZZ q q1 q2 KL(q : p) = q ln = q1 q2 ln p p Iterative algorithm q1 −→ q2 −→ q1 −→ q2 , · · · n o q1 (f ) ∝ exp hln p(g, f , θ; M)i q2 (θ ) o n q2 (θ) ∝ exp hln p(g, f , θ; M)i q1 (f )
n o (0) qb2 −→ qb2 −→ q1 (f ) ∝ exp hln p(g, f , θ; M)iq2 ↑
n o b ← qb2 (θ)←− q2 (θ) ∝ exp hln p(g, f , θ; M)i θ q1 ◮
b −→b q1 (f ) −→ f ↓
←− qb1
Uncertainties are propagated (Message Passing methods)
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 33/53
Summary of Bayesian approach ◮
Simple priors
↓ α, β
Hyper prior model p(θ|α, β) p(θ 2 |α2 , β2 ) p(θ 1 |α1 , β1 ) ❄
p(f |θ 2 )
⋄ p(g|f , θ 1 ) −→p(f , θ|g, α, β) → MCMC
Prior ◮
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Likelihood
Hierarchical priors
Joint Posterior
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b −→ f b −→ θ
↓ α, β, γ Hyper prior model p(θ|α, β, γ) p(θ 3 |α3 , β3 ) ❄ ⋄ p(z|θ3 ) Hidden variable
p(θ2 |α2 , β2 ) p(θ1 |α1 , β1 ) JMAP b −→ f ❄ ❄ b p(f |z, θ2 ) ⋄ p(g|f , θ1 ) −→ p(f , z, θ|g) −→ MCMC −→ z b −→ θ VBA Prior Likelihood Joint Posterior
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 34/53
Advantages of the Bayesian Approach ◮
More possibilities to model sparsity
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More tools to handle hyperparameters
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More tools to account for uncertainties
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More possibilities to understand and to control many ad hoc deterministic algorithms Hierarchical models give still more modeling possibilities
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Bernouilli-Gaussian: strict sparsity Bernouilli-Gamma: strict sparsity + positivity Bernouilli-Multinomial: strict sparsity + discrete values (finite states) Independent Mixture models: sparsity enforcing Mixture of multivariate models: group sparsity enforcing Gauss-Markov-Potts models: sparsity in transform domains
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 35/53
Examples of Hierarchical models 3
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 36/53
Which class of images I am looking for? 50 100 150 200 250 300 350 400 450 50
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 37/53
Which class of signals I am looking for?
Gauss-Markov
Generalized GM
Piecewize Gaussian
Mixture of GM: Gauss-Markov-Potts
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 38/53
Gauss-Markov-Potts prior models for images
f (r)
z(r)
q(r) = 1 − δ(z(r) − z(r′ ))
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
k Q Separable iid hidden variables: p(z) = r p(z(r)) ◮ Markovian hidden variables: p(z) Potts-Markov: X ′ ′ ′ p(z(r)|z(r ), r ∈ V(r)) ∝ exp γ δ(z(r) − z(r )) ′ r ∈V(r ) X X p(z) ∝ exp γ δ(z(r) − z(r ′ )) r ∈R r ′ ∈V(r)
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 39/53
Four different cases To each pixel of the image is associated 2 variables f (r) and z(r) ◮
f |z Gaussian iid, z iid : Mixture of Gaussians
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f |z Gauss-Markov, z iid : Mixture of Gauss-Markov
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f |z Gaussian iid, z Potts-Markov : Mixture of Independent Gaussians (MIG with Hidden Potts)
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f |z Markov, z Potts-Markov : Mixture of Gauss-Markov (MGM with hidden Potts)
f (r)
z(r)
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 40/53
Application of CT in NDT Reconstruction from only 2 projections
g1 (x) = ◮
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g2 (y) =
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Given the marginals g1 (x) and g2 (y) find the joint distribution f (x, y). Infinite number of solutions : f (x, y) = g1 (x) g2 (y) Ω(x, y) Ω(x, y) is a Copula: Z Z Ω(x, y) dx = 1 and Ω(x, y) dy = 1
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 41/53
Application in CT
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g|f f |z g = Hf + ǫ iid Gaussian or g|f ∼ N (Hf , σǫ2 I) Gaussian Gauss-Markov
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 42/53
Proposed algorithm p(f , z, θ|g) ∝ p(g|f , z, θ) p(f |z, θ) p(θ) General scheme: b ∼ p(f |b b g) −→ z b , θ, b g) −→ θ b ∼ (θ|f b, z b ∼ p(z|f b, g) f z , θ,
Iterative algorithme: ◮
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b g) ∝ p(g|f , θ) p(f |b b Estimate f using p(f |b z , θ, z , θ) Needs optimisation of a quadratic criterion. b , θ, b g) ∝ p(g|f b, z b p(z) b, θ) Estimate z using p(z|f
Needs sampling of a Potts Markov field. ◮
Estimate θ using b, z b , σ 2 I) p(f b |b b, g) ∝ p(g|f p(θ|f z , (mk , vk )) p(θ) ǫ Conjugate priors −→ analytical expressions.
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 43/53
Results
Original
Backprojection
Gauss-Markov+pos
Filtered BP
GM+Line process
LS
GM+Label process
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 44/53
Application in Microwave imaging g(ω) = ZZ
g(u, v) =
Z
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f (x, y) exp {−j(ux + vy)} dx dy + ǫ(u, v) g = Hf + ǫ
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 45/53
Images fusion and joint segmentation (with O. F´eron) gi (r) = fi (r) + ǫi (r) 2 p(fi (r)|z(r) Q = k) = N (mik , σi k ) p(f |z) = i p(f i |z)
g1
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b z
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 46/53
Data fusion in medical imaging (with O. F´eron) gi (r) = fi (r) + ǫi (r) 2 p(fi (r)|z(r) Q = k) = N (mik , σi k ) p(f |z) = i p(f i |z)
g1
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A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 47/53
Joint segmentation of hyper-spectral images (with N. Bali & A. Mohammadpour) gi (r) = fi (r) + ǫi (r) 2 p(fi (r)|z(r) Q = k) = N (mik , σi k ), k = 1, · · · , K p(f |z) = i p(f i |z) mik follow a Markovian model along the index i
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 48/53
Segmentation of a video sequence of images (with P. Brault) gi (r) = fi (r) + ǫi (r) 2 p(fi (r)|zi (r) Q = k) = N (mik , σi k ), k = 1, · · · , K p(f |z) = i p(f i |z i ) zi (r) follow a Markovian model along the index i
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 49/53
Image separation in Sattelite imaging (with H. Snoussi & M. Ichir) N X Aij fj (r) + ǫi (r) gi (r) = j=1
p(fj (r)|zj (r) = k) = N (mj k , σj2 k ) p(A ) = N (A , σ 2 ) ij 0ij 0 ij
f
g
b f
b z
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 50/53
Conclusions Sparsity: a great property to use in signal and image processing ◮ Origine: Sampling theory and reconstruction, modeling and representation Compressed Sensing, Approximation theory ◮ Deterministic Algorithms: Optimization of a two termes criterion, penalty term, regularization term ◮ Probabilistic: Bayesian approach ◮ Sprasity enforcing priors: Simple heavy tailed and Hierarchical with hidden variables. ◮ 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 and microwave imaging) Current Projects and Perspectives : ◮ Efficient implementation in 2D and 3D cases ◮ Evaluation of performances and comparison between MCMC ◮
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 51/53
Some references ◮ ◮ ◮ ◮ ◮ ◮ ◮ ◮ ◮
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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, A. Mohammad-Djafari (Ed.) Inverse Problems in Vision and 3D Tomography, ISTE, Wiley and sons, ISBN: 9781848211728, December 2009, Hardback, 480 pp. H. Ayasso and Ali Mohammad-Djafari Joint NDT Image Restoration and Segmentation using Gauss-Markov-Potts Prior Models and Variational Bayesian Computation, To appear in IEEE Trans. on Image Processing, TIP-04815-2009.R2, 2010. H. Ayasso, B. Duchene and A. Mohammad-Djafari, Bayesian Inversion for Optical Diffraction Tomography Journal of Modern Optics, 2008. 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. 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. 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. M. Ichir and A. Mohammad-Djafari, Hidden markov models for blind source separation, IEEE Trans. on Signal Processing, 15(7):1887-1899, Jul 2006. 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. 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. H. Snoussi and A. Mohammad-Djafari, Fast joint separation and segmentation of mixed images, Journal of Electronic Imaging, 13(2):349-361, April 2004. 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.
A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 52/53
Thanks, Questions and Discussions Thanks to:
My graduated PhD students:
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H. Snoussi, M. Ichir, (Sources separation) F. Humblot (Super-resolution) H. Carfantan, O. F´ eron (Microwave Tomography) S. F´ ekih-Salem (3D X ray Tomography)
My present PhD students:
◮ ◮ ◮ ◮ ◮
H. Ayasso (Optical Tomography, Variational Bayes) D. Pougaza (Tomography and Copula) —————– Sh. Zhu (SAR Imaging) D. Fall (Emission Positon Tomography, Non Parametric Bayesian)
My colleages in GPI (L2S) & collaborators in other instituts:
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B. Duchˆ ene & A. Joisel (Inverse scattering and Microwave Imaging) N. Gac & A. Rabanal (GPU Implementation) Th. Rodet (Tomography) —————– A. Vabre & S. Legoupil (CEA-LIST), (3D X ray Tomography) E. Barat (CEA-LIST) (Positon Emission Tomography, Non Parametric Bayesian) C. Comtat (SHFJ, CEA)(PET, Spatio-Temporal Brain activity)
Questions and Discussions A. Mohammad-Djafari, Tutorial: Sparsity in signal and image processing, November 28, 2013, NUDT, Changsha, China, 53/53
