maximum likelihood estimation of regularisation β¦ icip vidal 3oct18.pdfΒ Β· imaging inverse...
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Imaging Inverse Problems
OBJECTIVE: to estimate an unknown image π₯ from an observation π¦.
A +
π§~π(0, π2πΌ)
π₯ β βπ
UNKNOWNIMAGE
π¦ β βπ
OBSERVATIONLINEAR
OPERATOR
NOISE
Some canonical examples:β’ image deconvolution β’ compressive sensing β’ super-resolution β’ tomographic reconstructionβ’ inpainting
CHALLENGE: not enough information in π¦ to accurately estimate π₯.
For example, in many imaging problems π¦ = π΄π₯ + π§ where the operator π΄ is either:
High-dimensional problems β π~106
RANK-DEFICIENT
SMALL EIGENVALUESπ΄
not enough equations to solve for π₯
NOISE AMPLIFICATION unreliable estimates of π₯
REGULARISATION: We can render the problem well-posed by using prior knowledge about theunknown signal π₯.
How do we choose the regularisation parameter π?The regularisation parameter controls how much importance we give to prior knowledge andto the observations, depending on how ill posed the problem is and on the intensity of the noise.
π½
PRIORKNOWLEDGE
OBSERVEDDATA
POSSIBLE APPROACHES FOR CHOOSING π:
NON BAYESIANβ’ Cross-validation β Exhaustive search method
β’ Discrepancy Principle
β’ Stein-based methods βMinimise Steinβs Unbiased Risk Estimator (SURE), a surrogate of the MSE
β’ SUGAR β More efficient algorithms that uses gradient of SURE
Limitation: mostly designed for denoising problems. Difficult to implement
BAYESIANβ’ Hierarchical β Propose prior for π and work with hierarchical model
β’ Marginalisation β Remove π from the model π(π₯|π¦) = +β π π₯, π π¦ ππ
Limitation: only for homogeneous π(π₯) or cases with known πΆ(π)
β’ Empirical Bayes β Choose π by maximising marginal likelihood π(π¦|π)
Difficulty: π(π¦|π) becomes intractable in high-dimensional problems
Our strategy: Empirical Bayes
The challenge is that π(π¦|π) is intractable as it involves solving two integrals in βπ (to marginalise π₯ and to compute
πΆ π ). This makes the computation of π ππΏπΈ extremely difficult.
OUR CONTRIBUTIONWe propose a stochastic optimisation scheme to compute the maximum marginal likelihood estimator of theregularisation parameter. Novelty: the optimisation is driven by proximal Markov chain Monte Carlo (MCMC) samplers.
We want to find π that maximises the marginal likelihood π(π¦|π):
ΰ΅
π(π¦|π) = βπ π π¦, π₯ π ππ₯
πππΏπΈ = ππππππ₯π β Ξ
π(π¦|π)
We should be able to reliably estimate π from π¦ as π¦ is very high-dimensional and π is a
scalar parameter
Proposed stochastic optimisation algorithmIf π(π¦|π) was tractable, we could use a standard projected gradient algorithm:
STOCHASTIC APPROXIMATION PROXIMAL GRADIENT (SAPG) ALGORITHM
ππ‘+1 = ππππ¦Ξ
[ππ‘ + πΏπ‘π
ππlog π(π¦|ππ‘)] where πΏπ‘ verifies
To tackle the intractability, we propose a stochastic variant of this algorithm based on a noisy estimate of π
πππππ(π(π¦|ππ‘). Using Fisherβs identity we have:
π
ππlog π π π = πΈπ|π,π
π
ππlog π π, π π = πΈπ|π,π βπ π₯ β
π
πππΏππ πΆ π
If πΆ π is unknown we can use the identity β π
πππΏππ πΆ π = πΈπ|π π π
The intractable gradient becomes π
ππlog π π π =πΈπ|π,π βπ π₯ + πΈπ|π π π
Now we can approximate πΈπ|π,π βπ π₯ and πΈπ|π π π with Monte Carlo estimates.
We construct a Stochastic Approx. Proximal Gradient (SAPG) algorithm driven by two Markov kernels Mπ and Kπtargeting the posterior π π, π π and the prior π π₯ π respectively.
πΏπ+π~Mππ‘ X π¦, ππ‘ , ππ‘
ππ‘+1 = ππππΞ
ππ‘ + πΏπ‘ π πΌπ+π β π πΏπ+π
πΌπ+π~Kππ‘ U ππ‘ , ππ‘
π π₯ π¦, πππΏπΈ
π π₯ πππΏπΈ
πππΏπΈ
CONVERGEJOINTLY TO
EMPIRICAL BAYES POSTERIOR
STOCHASTIC GRADIENT
How do we generate the samples?
MOREAU-YOSIDA UNADJUSTED LANGEVIN ALGORITHM (MYULA)
π₯π‘+1 = π₯π‘ β πΎ (π»ππ¦ π₯π‘ +π₯π‘ β ππππ₯π
ππ π₯π‘
π) + 2πΎ ππ‘+1
π» πππ(π₯)
We use the MYULA algorithm for the Markov kernels Kπ and Mπ because they can handle:βͺ High-dimensionality !βͺ Convex problems with a non-smooth π(π)
βͺ The MYULA kernels do not target π π₯ π¦, π and π π₯ π exactly.βͺ Sources of asymptotic bias:
βͺ Discretisation of Langevin diffusion: controlled by πΎ and π.βͺ Smoothing of non differentiable π(π₯): controlled by π.
βͺ πΎ,π must be < inverse of Lipchitz constant of the gradient driving each diffusion respectively.βͺ More information about how to select each parameter can be found in [2].
WHERE DOES MYULA COME FROM?
ππ π‘ =1
2π» log π π π‘ |π¦ ππ‘ + ππ π‘
LANGEVIN DIFFUSION
π₯π‘+1 = π₯π‘ + πΎ π» πΏππ π π₯π‘ π¦, π + 2πΎ ππ‘+1
ππ‘+1~π 0, π
UNADJUSTED LANGEVIN ALGORITHM (ULA)
If π(π₯) is not Lipschitz differentiable ULA is unstable.
The MYULA algorithm uses Moreau-Yosida regularization to replace the non-smooth term π(π₯) with its Moreau Envelope ππ(π₯).
Moreau-Yosida regularisation controlled by π
π π₯ β πβ|π₯|
ππ π‘
ππ‘β
π₯π‘+1βπ₯π‘
πΎ
EULER MARUYAMA APPROXIMATION
β exp{ β ππ¦(π₯) β π π(π₯) } BROWNIAN MOTION
ResultsWe illustrate the proposed methodology with an image deblurring problem using a total-variation prior.
We compare our results with those obtained with:βͺ SUGARβͺ Marginal maximum-a-posteriori estimationβͺ Optimal or oracle value πβ
EVOLUTION OF π THROUGHOUT ITERATIONS
BOAT IMAGE
QUANTITATIVE COMPARISON
β Generally outperforms the other approaches
For each image, noise level, and method, we compute the MAP estimator and we display on this table the average results for the 6 test images.
EB performs remarkably well at low SNR values
Longer computing time only justified if marginalisation canβt be used
SUGAR performs poorly when the main problem is not noise
Close-up for SNR=20db
METHOD COMPARISON
Over-smoothing: too much regularisation
Ringing:
not enough regularisation
PRIOR
DATA
DEGRADED EMPIRICAL BAYES MAP ESTIMATOR
SNR=40db
Future workβͺ Detailed analysis of the convergence properties.
βͺ Extending the methodology to problems involvingmultiple unknown regularisation parameters.
βͺ Reducing computing times by accelerating the Markovkernels driving the stochastic approximation algorithm:
βͺ Via parallel computingβͺ By choosing a faster Markov kernel
βͺ Adapt algorithm to support some non-convex problems.
βͺ Use the samples from π π π, π½ π΄π³π¬ to perform
uncertainty quantification.
References[1] Marcelo Pereyra, JosΓ© M Bioucas-Dias, and MΓ‘rio ATFigueiredo, βMaximum-a-posteriori estimation with unknownregularisation parameters,β in 23rd European Signal ProcessingConference (EUSIPCO), 2015. IEEE, pp. 230β234, 2015.
[2] Alain Durmus, Eric Moulines, and Marcelo Pereyra, βEfficientBayesian computation by proximal Markov Chain Monte Carlo:when Langevin meets Moreau,β SIAM Journal on ImagingSciences, vol. 11, no. 1, pp. 473β506, 2018.
[3] Ana Fernandez Vidal and Marcelo Pereyra. "MaximumLikelihood Estimation of Regularisation Parameters." In 2018 25thIEEE International Conference on Image Processing (ICIP), pp.1742-1746. IEEE, 2018.
MAXIMUM LIKELIHOOD ESTIMATION OF REGULARISATION PARAMETERSAna Fernandez Vidal, Dr. Marcelo Pereyra (af69@hw.ac.uk, m.pereyra@hw.ac.uk)
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS United Kingdom
The observation π is related to π by a statistical model as a random variable.
We use a prior density to reduce uncertainty about and deliver accurate estimates.
The Bayesian Framework
π(π₯) penalises undesired properties and the regularisation parameter π controls the intensity of the penalisation.
Example: random vector πrepresenting astronomical images.
π π₯|π =πβπ π π₯
πΆ(π)
HYPERPARAMETER
We model the unknown image π₯ as a random vector with
prior distribution π π₯|π promoting desired properties about π₯.
PRIOR INFORMATION
π π¦ π₯ β πβππ¦(π₯)
The observation π¦ is related to π₯ by a statistical model:
OBSERVED DATA
LIKELIHOOD
Observed and prior information are combined by using Bayes' theorem:
POSTERIOR DISTRIBUTION
π π₯ π¦, π β πβ ππ¦ π₯ + π π π₯
π π₯ π¦, π = π π¦ π₯ π π₯|π /π(π¦|π)
ΰ·π₯ππ΄π = argmaxπ₯ββπ
π π₯ π¦, π = argminπ₯ββπ
ππ¦ π₯ + π π(π₯)
The predominant Bayesian approach in imaging is MAP estimationwhich can be computed very efficiently by convex optimisation:
We will focus on convex problems where:- π(π₯) is lower semicontinuous,
proper and possibly non-smooth- ππ¦(π₯) is Lipschitz differentiable
with Lipschitz constant L
ππ¦(π₯) = π¦βπ΄π₯ 2
2
2π2)π¦~π(π΄π₯, π2πΌ
NORMALIZING CONSTANT
Conclusionsβͺ We presented an empirical Bayesian method to estimate regularisation parameters in convex inverse imaging problems.βͺ We approach an intractable maximum marginal likelihood estimation problem by proposing a stochastic.
optimisation algorithm.βͺ The stochastic approximation is driven by two proximal MCMC kernels which can handle non-smooth regularisers efficiently.βͺ Our algorithm was illustrated with non-blind image deconvolution with TV prior where it:
β achieved close-to-optimal performanceβ outperformed other state of the art approaches in terms of MSEΓ had longer computing times.
βͺ More details can be found in [3].
EXPERIMENT DETAILSβ’Recover π from a blurred noisy observation π , π = π΄π + π and π )~N(0, π2πΌβ’π΄ is a circulant uniform blurring matrix of size 9x9 pixelsβ’π π = ππ π β isotropic total-variation pseudo-normβ’We use 6 test images of size 512x512 pixelsβ’We compute the MAP estimator for each using different values of π obtained with different
methodsβ’We compare methods by taking average value over 6 images of the mean squared error
(MSE) and the computing time (in minutes)
ORACLE ORIGINAL
in minutes
Γ Increased computing timesβ Achieves close-to-optimal performance
SNR=40dB
Close-up for SNR=20dB
DEGRADEDORIGINAL EMP. BAYES MARGINAL MAP SUGAR
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