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Optimal Multivariate Gaussian Fitting with Applications to PSF Modeling Optimal Multivariate Gaussian Fitting with Applications to PSF Modeling in Two-Photon Microscopy Imaging Emilie Chouzenoux, Tim Tsz-Kit Lau, Claire Lefort, Jean-Christophe Pesquet To cite this version: Emilie Chouzenoux, Tim Tsz-Kit Lau, Claire Lefort, Jean-Christophe Pesquet. Optimal Multivariate Gaussian Fitting with Applications to PSF Modeling in Two-Photon Microscopy Imaging. Journal of Mathematical Imaging and Vision, Springer Verlag, 2019, 61 (7), pp.1037-1050. 10.1007/s10851-019- 00884-1. hal-01985663 HAL Id: hal-01985663 https://hal.archives-ouvertes.fr/hal-01985663 Submitted on 18 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. XXX manuscript No. (will be inserted by the editor) Optimal Multivariate Gaussian Fitting with Applications to PSF Modeling in Two-Photon Microscopy Imaging Emilie Chouzenoux1,2 Tim Tsz-Kit Lau3 Claire Lefort4 Jean-Christophe Pesquet1 · · · Received: date / Accepted: date Abstract Fitting Gaussian functions to empirical data is a 1 Introduction crucial task in a variety of scientific applications, especially in image processing. However, most of the existing approa- Fitting Gaussian shapes from noisy observed data points is ches for performing such fitting are restricted to two di- an essential task in various science and engineering applica- mensions and they cannot be easily extended to higher di- tions. In the one-dimensional (1D) case, it lies for instance at mensions. Moreover, they are usually based on alternating the core of spectroscopy signal analysis techniques in physi- minimization schemes which benefit from few theoretical cal science [21,31]. In the two-dimensional (2D) case, where guarantees in the underlying nonconvex setting. In this pa- Gaussian profile parameters are estimated from images, some per, we provide a novel variational formulation of the multi- worth mentioning applications include Gaussian beam char- variate Gaussian fitting problem, which is applicable to any acterization, particle tracking, and sensor calibration [28,37, dimension and accounts for possible non-zero background 15]. In the domain of image recovery, a particularly impor- and noise in the input data. The block multiconvexity of our tant application of Gaussian shape fitting is the modeling of objective function leads us to propose a proximal alternat- Point Spread Functions (PSF) from raw data of optical sys- ing method to minimize it in order to estimate the Gaus- tems (e.g., microscopes, telescopes). The success of image sian shape parameters. The resulting FIGARO algorithm is restoration strategies strongly depends on the accuracy of shown to converge to a critical point under mild assump- the PSF estimation [13]. This estimation is often performed tions. The algorithm shows a good robustness when tested through a preliminary step of image acquisition of normal- on synthetic datasets. To demonstrate the versatility of FI- ized and calibrated objects, associated with a model fitting GARO, we also illustrate its excellent performance in the strategy. The PSF model is chosen as a trade-off between fitting of the Point Spread Functions of experimental raw accuracy and simplicity. Gaussian models often lead to both data from a two-photon fluorescence microscope. tractable and good quality approximations [35,32,1,42,41]. Let L1(RQ) denote the space of real-valued summable RQ Keywords Gaussian fitting Kullback-Leibler divergence functions defined on . In this paper, we address the prob- · · Alternating minimization Proximal methods PSF lem of fitting a Gaussian model to an observed function · · y L1(RQ). We assume that the observed function y can identification Two-photon Fluorescence microscopy ∈ · be modeled as Emilie Chouzenoux [email protected] ( u RQ) y(u)= a + bp(u)+ v(u), (1.1) 1 Center for Visual Computing,· CentraleSupelec,´ INRIA Saclay, ∀ ∈ Universite´ Paris-Saclay, 91190 Gif-sur-Yvette, France where a R is a background term, b (0,+∞) is a scal- 2 Laboratoire d’Informatique Gaspard Monge, UMR CNRS 8049, ∈ ∈ Universite´ Paris-Est Marne-la-Vallee,´ 77454 Marne-la-Vallee´ Cedex 2, ing parameter, p L1(RQ) represents a noiseless version of ∈ France the observed field, and v is a function accounting for acqui- 3 Department of Statistics, Northwestern University, Evanston, IL p 60208, United States of America sition errors. The main assumption is that is close, in a 4 XLIM Research Institute, UMR CNRS 7252, Universite´ de Limo- sense to be made precise, to the probability density function ges, 87032 Limoges, France u g(u, µ,C), of a Q-dimensional normal distribution with 7→ mean µ RQ and precision (i.e., inverse covariance) matrix ∈ 2 Emilie Chouzenoux1,2 et al. C S ++ 1. This distribution is expressed as a two-photon fluorescence microscope, our new computa- ∈ Q tional strategy shows an unprecedented accuracy and relia- RQ µ RQ S ++ bility. ( u )( )( C Q ) ∀ ∈ ∀ ∈ ∀ ∈ In Section 2, the data fitting problem is formulated in C 1 a variational manner. A proximal alternating optimization g(u, µ,C)= | | exp (u µ)⊤C(u µ) , (2π)Q −2 − − method called FIGARO is then proposed in Section 3 for s (1.2) finding a minimizer of the proposed nonconvex cost func- tion. The implementation of the algorithm steps is discussed. where C denotes the determinant of matrix C. The fitting The convergence of the sequence of iterates resulting from | | problem thus consists of finding an estimate (a,b, p, µ,C) FIGARO is established in Section 4. Section 5 illustrates of (a,b, p, µ,C) in accordance with model (1.1) the high robustness of our approach to a model mismatch, Because of its prominent importance in applications,b b b b thereb when compared to a standard nonlinear least squares fitting has been a significant amount of works on this subject [12, strategy on 3D synthetic data. In Section 6, the scope of our 25,24,23,34,42]. To the best of our knowledge, all existing approach is demonstrated through the analysis of the Point works consider that p = g( , µ,C) and they are focused on Spread Function of a 3D two-photon fluorescence micro- · fitting parameters (a,b, µ,C) from y. Two main classes of scope. Finally, Section 7 concludes the paper. methods can be distinguished. The first set of approaches [25,24,34] is basedb onb b theb search for the best fitting pa- rameters minimizing a least-squares cost between the obser- 2 Proposed Variational Formulation vations and the sought model. The minimization process is based on the famous Levenberg-Marquardt alternating min- The key ingredient of our method relies on measuring the imization strategy. However, it is worth mentioning that few closeness of p to the Gaussian probability density functions established convergence guarantees are available for this me- by using the Kullback-Leibler (KL) divergence [5]. Let us thod, which may be detrimental to its reliable use in prac- first recall the definition of KL divergence. Let P denote tice. The second class of methods uses the so-called Caru- the set of probability density functions supported on RQ: ana’s formulation [12]. The idea here is to assume that the background term a is zero and to search for (b, µ,C) which P = q L1(RQ) ( u RQ) q(u) 0 minimize the difference of logarithms between the data and ∈ | ∀ ∈ ≥ n the model [23,1]. The advantage of such a strategyb b b is that q(u)du = 1 . (2.1) Ω it gives rise to a convex formulation, for which efficient and Z reliable optimization techniques can be applied. It is how- o Suppose that (p,q) P2 and q takes (strictly) positive val- ever worth emphasizing that all the aforementioned works ∈ are focused on the resolution of the fitting problem in low ues, the KL divergence from q to p reads dimensions, that is when Q = 1 [12,25,23,34] or Q = 2 [24, p(u) 1,42]. Moreover, except in [34] where a polynomial back- KL (p q)= p(u)log du, (2.2) k RQ q(u) ground is accounted for, the background term a is consid- Z ered as zero. These assumptions however usually do not cor- with the convention 0log0 0. respond to constraints inherent to an experimental setup or = environment. In order to avoid singularity issues, we will assume that The aim of this paper is to propose a new multivariate the Gaussian variances in each direction are bounded above Gaussian fitting strategy which avoids the aforementioned by some maximal values. The spectrum of the precision ma- C limitations. Our method relies on the minimization of a hy- trix is thus bounded from below, in the sense that there exists some ε > 0 such that C = D + εIQ where D belongs brid cost function combining a least-squares data fidelity + Q Q Q to S and IQ R denotes the identity matrix of R . term, a Kullback-Leibler divergence regularizer for improved Q ∈ × robustness, and range constraints on the parameters. This We then propose to define (a,b, p, µ,D) as a minimizer of original variational formulation results in a nonconvex mini- a hybrid cost function, gathering information regarding the mization problem for which we propose a theoretically sound observation model (1.1) andb theb Gaussianb b b shape prior (1.2). and efficient proximal alternating iterative resolution scheme. The minimization problem reads When applied to the analysis of 3D raw data acquired with 1 2 1 Throughout the paper, S ++ will denote the set of symmetric pos- minimize y(u) a bp(u) du Q a A ,b B 2 RQ − − + itive definite matrices of RQ Q, S the set of symmetric positive µ RQ∈p P∈D S + Z × Q , , Q RQ Q S ∈ ∈ ∈ semidefinite matrices of × and Q the set of symmetric matrices µ RQ Q + λKL p g( , µ,D + εIQ) .
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