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    Particle Filter: Exploring Particle Filters in Computer Vision
    Particle Filter: Exploring Particle Filters in Computer Vision
    Particle Filter: Exploring Particle Filters in Computer Vision
    Ebook105 pages45 minutes

    Particle Filter: Exploring Particle Filters in Computer Vision

    By Fouad Sabry

    ()

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    About this ebook

    What is Particle Filter


    Particle filters, or sequential Monte Carlo methods, are a set of Monte Carlo algorithms used to find approximate solutions for filtering problems for nonlinear state-space systems, such as signal processing and Bayesian statistical inference. The filtering problem consists of estimating the internal states in dynamical systems when partial observations are made and random perturbations are present in the sensors as well as in the dynamical system. The objective is to compute the posterior distributions of the states of a Markov process, given the noisy and partial observations. The term "particle filters" was first coined in 1996 by Pierre Del Moral about mean-field interacting particle methods used in fluid mechanics since the beginning of the 1960s. The term "Sequential Monte Carlo" was coined by Jun S. Liu and Rong Chen in 1998.


    How you will benefit


    (I) Insights, and validations about the following topics:


    Chapter 1: Particle filter


    Chapter 2: Importance sampling


    Chapter 3: Point process


    Chapter 4: Fokker-Planck equation


    Chapter 5: Wiener's lemma


    Chapter 6: Klein-Kramers equation


    Chapter 7: Mean-field particle methods


    Chapter 8: Dirichlet kernel


    Chapter 9: Generalized Pareto distribution


    Chapter 10: Superprocess


    (II) Answering the public top questions about particle filter.


    (III) Real world examples for the usage of particle filter in many fields.


    Who this book is for


    Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Particle Filter.

    LanguageEnglish
    PublisherOne Billion Knowledgeable
    Release dateMay 13, 2024
    Particle Filter: Exploring Particle Filters in Computer Vision

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      Book preview

      Particle Filter - Fouad Sabry

      Chapter 2: Importance sampling

      To assess the characteristics of a distribution of interest using just samples drawn from another distribution, the Monte Carlo technique of importance sampling is employed. Importance sampling is similar to umbrella sampling in computational physics, and its inception in statistics is typically credited to a study by Teun Kloek and Herman K. van Dijk in 1978. Sampling from this alternate distribution, making inferences based on such samples, or both, may all be referred to by this word, depending on the context.

      Let {\displaystyle X\colon \Omega \to \mathbb {R} } be a random variable in some probability space (\Omega ,{\mathcal {F}},P) .

      We want to calculate the probability distribution of X given P, marked with the EX;P symbol].

      If we have statistically independent random samples x_{1},\ldots ,x_{n} , produced in accordance with P, Hence, a practical calculation of EX;P] is

      {\displaystyle {\widehat {\mathbf {E} }}_{n}[X;P]={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\quad \mathrm {where} \;x_{i}\sim P(X)}

      and the accuracy of this approximation is proportional to the dispersion of X:

      {\displaystyle \operatorname {var} [{\widehat {\mathbf {E} }}_{n};P]={\frac {\operatorname {var} [X;P]}{n}}.}

      The primary idea behind importance sampling is to reduce the uncertainty in the estimation of EX;P] by drawing samples from a separate distribution for the states, or when it is difficult to sample from P.

      This is accomplished by first choosing a random variable L\geq 0 such that EL;P] = 1 and that P-almost everywhere L(\omega )\neq 0 .

      With the variable L we define a probability {\displaystyle P^{(L)}} that satisfies

      {\mathbf {E}}[X;P]={\mathbf {E}}\left[{\frac {X}{L}};P^{{(L)}}\right].

      The variable X/L will thus be sampled under P(L) to estimate EX;P] as above and this estimation is improved when

      \operatorname {var}\left[{\frac {X}{L}};P^{{(L)}}\right]<\operatorname {var}[X;P]

      .

      When X is of constant sign over Ω, the best variable L would clearly be L^{*}={\frac {X}{{\mathbf {E}}[X;P]}}\geq 0 , so that X/L* is the searched constant EX;P] and a single sample under P(L*) suffices to give its value.

      We are unable to go that route, unfortunately, since EX;P] is the value we need to find! This theoretical best-case L*, however, sheds light on the function of importance

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