Statistical Multiresolution Estimation for Variati

Authors: Klaus Frick Philipp Marnitz Axel Munk

Publish Date: 2012/07/18

Volume: 46, Issue: 3, Pages: 370-387

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Abstract

In this paper we present a spatiallyadaptive method for image reconstruction that is based on the concept of statistical multiresolution estimation as introduced in Frick et al Electron J Stat 6231–268 2012 It constitutes a variational regularization technique that uses an ℓ ∞type distance measure as datafidelity combined with a convex cost functional The resulting convex optimization problem is approached by a combination of an inexact alternating direction method of multipliers and Dykstra’s projection algorithm We describe a novel method for balancing datafit and regularity that is fully automatic and allows for a sound statistical interpretation The performance of our estimation approach is studied for various problems in imaging Among others this includes deconvolution problems that arise in Poisson nanoscale fluorescence microscopyThe quadratic fidelity in 2 has an essential drawback The information in the residual is incorporated globally that is each pixel value Ku ij −Y ij contributes equally to the estimator hatulambda independent of its spatial position In practical situations this is clearly undesirable since images usually contain features of different scales and modality ie constant and smooth portions as well as oscillating patterns both of different spatial extent A solution hatulambda of 2 is hence likely to exhibit under and oversmoothed regions at the same timeSpecial instances of 6 have been studied recently For the case when mathcalS contains the entire domain G only it has been shown in 8 that 6 is equivalent to 2 if K satisfies certain conditions As mentioned above this approach is likely to oversmooth smallscaled image features such as texture and/or underregularize smooth parts of the image An improved model was proposed in 2 where mathcalS is chosen to consist of a datadependent partition of G that is obtained in a preprocessing step for the numerical simulations in 2 MumfordShah segmentation is considered Under similar conditions on K as in 8 it was shown in 2 that 6 is equivalent to 5 where λ ij is constant on each Sin mathcalS This approach was further developed in 1 where a subset S⊂G is fixed and afterwards mathcalS is defined to be the collection of all translates of S in fact the authors study the convolution of the squared residuals with a discrete kernel The authors propose a proximal point method for the solution of 6 This approach of local constraints wrt a window or kernel of fixed size was also studied in 15 for irregular sampling and regularization functionals other than the total variation were considered In particular it is observed that the difference between results obtained by using the total variation penalty 3 and the Dirichletenergy integrated squared norm of the derivative is not so big when using local constraints This is in accordance to findings in 19 for onedimensional signals In 11 the model of 1 was studied in the continuous function space setting Moreover the authors in 11 provided a fast algorithm for the solution of the constrained optimization problem based on the hierarchical decomposition scheme 34 combined with the unconstrained problem 5In this paper we propose a novel automatic selection rule for the weights c S based on a statistically sound method that is applicable for any prespecified deterministic system of subsets mathcalS We are particularly interested in the case when mathcalS constitutes a highly redundant collection of subsets of G consisting of overlapping subsets of different scales This is a substantial extension to the approaches in 1 11 15 that only consider one fixed predefined scale Our approach will amount to select a single parameter α∈01 with the interpretation that the true signal u 0 satisfies the constraint in 6 with probability α From the definition of 6 it is then readily seen that mathbbP Jhatu leq Ju0 geqalpha for any solution hatu of 6 In other words our method controls the probability that the reconstruction hatu is at least as smooth in the sense of J as the true image u 0 To this aim it will be necessary to gain stochastic control on the nulldistribution Tε where ε=ε ij is a lattice of independent mathcalN0sigma2distributed random variablesWe review sufficient conditions that guarantee existence of SMREs solutions of 6 that is To this end we rewrite 6 into an equality constrained problem and study the corresponding augmented Lagrangian function Sect 21 Moreover we address the important question on how to choose the scale weights c S automatically in Sect 22 Finally we discuss different choices for the system mathcalS that have proved feasible in practice in Sect 23It is well known that the saddlepoints of L and L λ coincide cf 16 Chap III Theorem 21 and that existence of a saddle point of L λ follows from existence of solutions of 9 together with constraint qualifications of the MRstatistic T One typical example for the latter is given in Proposition 21 The result is rather standard and can be deduced eg from 12 Chap III Proposition 31 and Theorem 42 cf also 16 Chap IIIIf J is chosen to be the total variation seminorm 3 then a sufficient condition for the existence of solutions of 9 will be that there exists ij∈S for some SinmathcalS such that K 1 ij ≠0 where 1∈L2Ω is the constant 1function This is immediate from Poincaré’s inequality for functions in BVΩ cf 37 Theorem 5111

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