Markov Point Process for Multiple Object Detection

Transcription

1 Markov Point Process for Multiple Object Detection Xavier Descombes INRIA Sophia Antipolis Méditerranée 24/06/2013

2 Outline Motivation The different issues : - objects - reference measure - prior - data term - optimization Some results 2

3 Motivation : from context to geometry 1) High resolution data : the object geometry is an important source of information 2) The pixel scale does not contain the main information 3

4 Motivation : from context to geometry 1) Consider prior information (Bayesian approach, Markov Random Fields, interactions) 2) Embed geometric information (graph of objects) 3 ) Modeling the scene structure (interactions between objects, unknown number of objects) 4) Need algorithms for simulating, optimizing the models Marked point processes 4

5 Motivation : a scene as a collection of objects 5

6 Motivation : a scene as a collection of objects 6

7 Motivation : a scene as a collection of objects 7

8 The configuration space «Simple» parametric shapes : S = { s = ( x,m),x K,m M} x,r ( ) x,a,b,θ ( ) x,l,l,θ ( ) x,l,θ ( ) Ω n = {{ s 1,,s n },s i S} Ω 0 = Ω = n Ω n 8

9 Usually the Poisson measure : The reference measure π ν ( B) ν ( χ ) π ν ( B) = e 1[ 0/ B ] + n n= 0 n! π ν n ( B) = 1[ { x1,,x n } B ] ν ( dx1 ) ν ( dx n ) ν ν Intensity measure: uniform or not ν ( A) λ( x ) dx A NDVI MAP 9

10 The density The model is defined by a density (usually un-normalised) w.r.t the reference measure: f : Ω 0, [ [, f x ( )dπ ν x ( ) Ω < Mimicking the Bayesian approach: f (x) = g(x)h I (x) Prior Data (I) term 10

11 Overlap penalization (pairwise interaction): Alignment : θ g(x) = g(x) = Overlap penalization : g(x) = i~ j ϕ( x i, x ) j ϕ( x i, x j, x ) k i~ j ~k i ϕ x i,x j, j ~ i And many more ϕ x i,x j The prior min s i, s j ( ) = Φ s i s j ( ) ϕ( x i, x j, x ) k = Φ( π θ ) i, j,k ( ) ϕ( x i,x j, j ~ i) = Φ max j ~i s i s j min s i, s j ( ) 11

12 Bayesian approach: likelihood The data term h Y x l background y i ( ) = l ( object y ) i i inside objects i outside objects ( ) Distance between interior and exterior h Y Geometrical consistency objects ( x) = dissimilarity( red pixels, blue pixels) h Y objects ( x) = exp U( ω) And many more Examples : Bhattacharya distance, Statistical test, U( ω) = 1 ω U d ω ( ω,y) =ψ U( ω),t Y( u),n( u) du Y( u) 2 + ε 2 ( ) 12

13 Optimization : RJMCMC initialize the temperature T and the configuration x (empty set) Choose a proposition kernel Q m (x,.) with probability p m (x), or let the configuration unchanged probability 1-Σ m p m (x). Sample x according to the chosen kernel Compute the acceptation ratio : With probability α = min(1,r m ) set x t+1 = x, else reject the proposition : x t+1 = x. Some perturbation kernels (proposal) Adding an object Removing an object Modifying an object (translation, rotation, dilation) Merging/Splitting objects 13

14 Optimization : RJMCMC Pros : Generality Choice for kernels Convergence to the global optimum Cons : Rejection Simulated annealing scheme (parameters setting) Kernels usually involve one or two objects 14

15 Optimization :Multiple births and deaths Goals : Avoid rejection Consider several objects at once Idea : Extend Langevin s dynamics (Stochastic Differential Equation : diffusion process) 15

16 Optimization :Multiple births and deaths Consider : f ( x) = exp βe( x) A birth intensity consisting in adding an object u to the configuration x : b( x,u)du = zdu if u D( x), where D x ( ) = B v ( ε) v x K A death intensity consisting in removing an object u from the configuration x : ( ) = expβ E( x) - E(x/u) d x,u [ ] if u x, Detailed balance condition holds 16

17 Optimization :Multiple births and deaths A New Approximation Process Markov Chain : T β, δ ( m ), m = t δ = 0,1,2, ( discretization of time) Birth transition : Death transition : x n +1 = x 1 x 2, x 1 x n, x 1 x 2 = x 2 : Poisson law (with intensity z) distributed q u,δ = zδδu, if x x { u } 1 zδδu, if x x (no birth in Δy) p u,δ = δ exp E x 1+ δ exp E x 1 1+ δ exp E x ( ) E( x /u) ( ) E( x /u) [ ] [ ] ( ) E( x /u) [ ], if x x /u, if x x (x survives) 17

18 Optimization :Multiple births and deaths A New Algorithm 1) Precomputing of the data term / birth map 2) Repeat : 2.1) Birth: For each pixel, add an object with probability : δb E d ( u) ( ) 2.2) Sort the objects with respect to their data term value 2.3) Death: For each object u taken in the list order, remove it with probability : δ exp β E x ( ) E( x /u) ( ) E( x /u) [ ( )] ( ) 1+ δ exp β E x [ ] 18

19 Optimization :Multiple births and deaths Pros : Cons : No rejection in the birth step Birth does not depends on the temperature Convergence to the global optimum Only births and deaths kernels 19

20 Optimization :Multiple births and cut Idea : Combine multiples births and deaths with graphcut techniques 20

21 Optimization :Multiple births and cut Pros : No rejection in the birth step No cooling schedule Cons : Only births and deaths No proof of convergence 21

22 Result :Trees counting 22

23 Result :Trees counting 23

24 Result :Trees counting Results : 24

25 Result : Flamingos counting Ariana/INRIA 25

26 Result : Flamingos counting Tour du Valat 26

27 Result : Flamingos counting Ariana/INRIA 27

28 Result : Cells counting 28

29 Result : Cells counting 29

30 Result : Vesicule (co)localization 30

31 Conclusion Pros : General framework / Numerous application Embed strong geometric constraint Future work : parallelism parameter estimation open source software 31