INSTITUTO POLITÉCNICO NACIONAL CENTRO DE INVESTIGACION EN COMPUTACION Laboratorio de Ciberseguridad Fusión de Sensores Dr. Ponciano Jorge Escamilla Ambrosio pescamilla@cic.ipn.mx http://www.cic.ipn.mx/~pescamilla/
CIC Sensor Fu Sensor Fusion sion Pr Probabil obabilit ity y rev review iew Based on Hugh Durrant-Whyte, Introduction to Sensor Data Fusion 2 http://www.acfr.usyd.edu.au/teaching/graduate/Fusion/index.html
CIC Sensor Fu Sensor Fusion sion Pr Probabil obabilit ity y rev review iew 3
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CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average One of the simplest and most intuitive general methods of sensor data fusion is to take a weighted average of redundant information provided by a group of sensors and use this as the fused value. Formally, the weighted average of N sensor measurements x i with weights 0 ≤ w i ≤ 1 is, 8
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average Formally, the weighted average of N sensor measurements x i with weights 0 ≤ w i ≤ 1 is: where 𝑗 𝑥 𝑗 = 1 and w i = 0 if x i is not within some specified thresholds. How to obtain the weights w i ? 9
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average Example 10
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average 11
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average If w 1 = w 2 = 1/2 then: 12
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average What if we have heterogeneous senosrs? 13
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average 14
CIC Sensor Fu Sensor Fusion sion Weighted average Weighted average If the x i , i = 1, 2,…, n measurements are assumed to be independent normally distributed random 2 ) , then a variables, with distribution 𝑂( 𝑦 𝑗 , 𝜏 𝑗 linear weighted mean aggregation model combining these random variables into one random variable x f is given by: 𝑦 𝑔 = 𝛾 1 𝑦 1 + 𝛾 1 𝑦 1 + ⋯ + 𝛾 𝑜 𝑦 𝑜 with variance: 15
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average Where 𝛾 𝑗 is a positive weighting factor calculated by: 16
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average 17
CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average 18
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CIC Sensor Fu Sensor Fusion sion Weighted Weighted average average w 1 w 2 20
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CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method method Conditional probability definition: If we multiply both sides of the definition of P(A|B) by P(B) we obtain: P(A ∩ B) = P(A|B) P(B) Similarly, if we multiply both sides of the definition of P(B|A) by P(A) we obtain: P(B ∩ A) = P(B|A) P(A) 23
CIC Sensor Fu Sensor Fusion sion Bayesian method Bayesian method Because P(A ∩ B) = P(B ∩ A), Bayes’ rule: 𝑄 𝐵 𝐶 = 𝑄 𝐶 𝐵 𝑄(𝐵) 𝑄(𝐶) 𝑄 𝐶 𝐵 𝑄(𝐵) 𝑄 𝐵 𝐶 = 𝑗 𝑄(𝐶|𝐵 𝑗 ) 𝑄(𝐵 𝑗 ) where: P(A|B) = a posteriori probability. P(B|A) = likelihood function of A. P(A) = a priori probability of A. 24
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method method Commonly, Bayes’ rule is thought of in terms of updating the belief about a hypothesis A in the light of new evidence B. Thus, the posterior belief P(A|B) is calculated by multiplying the prior belief P(A) by the likelihood P(B|A) that B will occur if A is true. 25
CIC Sensor Fu Sensor Fusion sion Bayesian method Bayesian method Replacing A with x and B with z , we obtain the following relation: 𝑄 𝐴 𝐲 𝑄(𝐲) 𝑄 𝐲 𝐴 = 𝑗 𝑄(𝐴|𝐲 𝑗 ) 𝑄(𝐲 𝑗 ) The items x and z are regarded as random variables. x is a state or parameter of the system. z is the sensor measurements. 26
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method method The Bayes’ theorem is interpreted as the computation of the posterior probability P( x | z ), given the prior probability of the state or parameter (P( x )), and the observation probability (P( z | x )): the value of x that maximizes the term ( x |data). 27
CIC Sensor Fu Sensor Fusion sion Bayesian method Bayesian method The term P( z | x ) assumes the role of a sensor model in the following way: (1) First build a sensor model: fix x = x and then ask what probability density function (pdf) on z results. (2) Use the sensor model: observe z = z and then ask what the pdf on x is. (3) Practically P( z | x ) is constructed as a function of both variables (or a matrix in discrete form). (4) For each fixed value of x , a distribution in z is defined. Therefore as x varies, a family of distributions in z is created. 28
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 1 29
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 1 30
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 2a 2a 31
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 2a 2a 32
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 2b 2b 33
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: method: Data fusion Data fusion 34
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: method: Data fusion Data fusion 35
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: method: Data fusion Data fusion 36
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Data method: Data fusion fusion 37
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Data method: Data fusion fusion 38
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Data method: Data fusion fusion 39
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Data method: Data fusion fusion 40
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Data method: Data fusion fusion 41
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Data method: Data fusion fusion 42
CIC Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Data method: Data fusion fusion 43
CIC Sensor Sensor Fusion Fusion Recursive Recursive Bayes Bayes Updating Updating 44
CIC Sensor Fu Sensor Fusion sion Recursive Recursive Bayes Bayes Updating Updating 45
CIC Sensor Fu Sensor Fusion sion Recu Recursi rsive ve Bayes Bayes Updati Updating: Exampl ng: Example 46
CIC Sensor Fu Sensor Fusion sion Recu Recursi rsive ve Bayes Bayes Updati Updating: Example ng: Example 47
CIC Sensor Fu Sensor Fusion sion Recu Recursi rsive ve Bayes Bayes Updati Updating: Example ng: Example 48
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CIC Generalised Generalised Bayesian Bayesian Filter Filtering: ing: Problem Statement Problem Statement 52
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CIC Generalised Generalised Bayesian Bayesian Filt Filteri ering ng 56
CIC Generalised Generalised Bayesian Bayesian Filt Filteri ering ng 57
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