Essay On Expectation-Maximicy

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The expectation-maximization (EM) algorithm uses incomplete data to estimate the parameters of probabilistic models, and it has been widely used in machine learning. In this paper, EM techniques are applied to Time-domain fluorescence lifetime imaging microscopy (FLIM) systems for estimating fluorescence lifetimes without measuring the instrument response functions (IRF). The results of Monte Carlo simulations indicate that the proposed approach can obtain comparable or better accuracy and precision performances than the previously reported method.

Introduction: Time-correlated single-photon counting (TCSPC) has excellent timing performances, and it is routinely used for fluorescence lifetime imaging microscopy (FLIM) system [1, 2]. FLIM
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This method is widely used for parameter estimation with incomplete or missing data [10, 11, 12]. In this paper, a new EM-based Lifetime Estimation (EMLE) algorithm is proposed to simultaneously estimate the IRF and lifetime, and it shows better photon efficiency compared with EKF in [9].
Theory: According to the EM theory [10, 11, 12], we assume that {xi, i = 0, 1, …, N-1} are observation values of a random variable x whose density function shown in Fig.1(a) is , (1) where f(x|ξi) is the density function of ith component with a parameter ξi , λj is the component weight satisfying Σλj= 1, j = 0,1, …, L-1, and ϕ = [ξ0, λ0, ξ1, λ1,, ξL-1, λL-1]. ξi and λj can be estimated using {x0, x0, …, xN-1}. An expectation step (E-step) and a maximization step (M-step) are performed iteratively [10, 11], shown in Fig. 1(b).
In E-step, the posterior probability pij is . (2)
In M-step, ϕ that maximizes the expected log-likelihood in E-step can be calculated from , (3) where ϕi = [ξi, λi] is the parameter for the ith component.

a b
Fig. 1 Overview of EMLE a Mixture of the density function b EMLE processing flow.

In a TCSPC-FLIM experiment, we assume that the fluorescence density function is g(t), the IRF is IRF(t), and the measured fluorescence decay is y(t). y(t) is the sum of an additive Poisson noise, v(t) and the convolution of g(t) and
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