Structured illumination microscopy (SIM) enables high resolution in fluorescence images by using spatially varying illumination patterns, under the constraint that such patterns be perfectly known and controlled
[Gustafsson 2005]. In practice, however, controlling illumination is a difficult question. A recent breakthrough has been obtained by Mudry et al. [2012], whose approach (called blind-SIM) achieves high resolution using unknown excitation patterns (e.g. uncontrolled random speckles), provided that their average be roughly homogeneous over the sample. A key element of blind-SIM is to simultaneously estimate the fluorescence image and the illumination patterns by solving a joint least-squares (JLS) optimization problem in both the fluorescence density and the patterns, which are alternately updated. Practical experiments reported in Mudry et al. [2012] allow to conclude that blind-SIM does produce the expected high resolution effect. However, such a positive effect remains to be better understood on an information-theoretic basis. From a statistical perspective, blind-SIM belongs to the family of joint estimation methods, where the quantity of interest (here, the fluorescence image) is obtained jointly with nuisance parameters (here, the unknown excitation patterns). In general, such joint estimators must be considered with suspicion, since their statistical behaviour is often pathological – in particular, they are likely to be inconsistent [Little and Rubin 1983].
This thesis aims at building alternate solutions for image reconstruction with blind structured illumination microscopy, with more robust statistical properties. The resulting estimators should then achieve high resolution with stronger guarantees than the procedure proposed by Mudry et al. [2012]. In particular, Maximum Likelihood (ML) estimation, and more generally M-estimation, lead to consistent results in a general and well-understood statistical framework [Little and Rubin 1983, van der Vaart 2000]. Contrary to the JLS approach, the latter estimation principles rely on a likelihood function (and more generally on a constrast function) which depends on the statistics of the nuisance parameters, but not on their specific values. While ML estimation relies on the maximization of the probability density of the data given the parameters of interest, M-estimation is a more general and tolerant setting where the maximized contrast function may result, for instance, from an approximation of the true statistics of the nuisance parameters.
The super-resolution capacity of blind SIM imaging should also be studied from a theoretical perspective. Fisher information is a mathematical tool of choice to determine the precision limit that can be achieved in blind SIM. As a corollary, the experimental setup can be adjusted to maximize Fisher information within practical constraints, according to an optimal design scheme. A recent and inspiring example of such a scheme can be found in the related context of computational photography [Trouvé 2012]. This strategy allows, for instance, the determination of the number of speckle illuminations required to reach a given precision in the reconstruction of the fluorescence density. It also gives the opportunity to optimize the
statistical parameters of the speckle (e.g., its spatial correlation) used for illumination.
Implementation issues will also be central in this thesis, since the computation of such estimators leads to optimization of marginal criteria, with more complex structures than the former JLS criterion. In this per-
spective, a first algorithmic structure is the well-known Expectation-Maximization (EM) strategy [Dempster et al. 1977, Fuhrmann 2007], that usually yields low cost iterations, but at the expense of slow convergence.
Accelerated variants could be sought within the extended family of majorize-minimize schemes [Hunter and Lange 2004, Chouzenoux et al. 2012]. On the other hand, it is known that the gradient of the marginal
criterion can be obtained as a by-product of the EM equations [Cappé et al. 2005]. Therefore, other first-order algorithms could be devised, with potentially faster convergence rate than the EM algorithm. In particular, the L-BFGS algorithm [Nocedal and Wright 1999] and interior point methods suited to large-scale problems [Armand and Segalat 2003] could represent very efficient alternatives. Finally, Markov Chain Monte Carlo (MCMC) approaches should be studied in this context as an interesting stochastic alternative to maximum likelihood estimation obtained by local optimization. MCMC algorithms allow to approximate the posterior mean for the quantities of interest, which is a marginal estimator with statistical properties comparable to maximum likelihood [Gelman et al. 2003]. The distinctive interest of the MCMC approach is to avoid the analytical integration step needed by the ML type solutions. Instead, both the object and the illumination patterns would be sampled according to their conditional posterior probability, following an updating scheme that is much alike a stochastic version of the JLS blind-SIM algorithm.
Supervisors : Jérôme Idier, CNRS - IRCCyN (40%), Sébastien Bourguignon, Ecole Centrale de Nantes - IRCCyN (30%), Marc Allain, Aix-Marseille Université - Institut Fresnel (30%).
Keywords : estimation theory, information theory, optimization and stochastic algorithms, fluorescence microscopy imaging
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