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Stochastic Point Processes Stochastic Point Processes Cengiz Öztireli, 2015 Distributions in Nature [Filckr user Hans Dekker] Sampling • Conversion from continuous to discrete • Integration in rendering [Filckr user Josh Pesavento ] Point Distributions • All kinds of patterns Studying Point Distributions • How can we analyze patterns? Periodogram Density [Heck et al. 2013] [Faal 2011] Point Processes • Formal characterization of point patterns Point Processes • Definition Point Process Point Processes • Infinite point processes Observation window Point Process Statistics • Correlations as probabilities %(n)(x , , x )dV dV = p(x , , x N) 1 ··· n 1 ··· n 1 ··· n| Product density Small volumes Points in space Point process xi Point Process Statistics • Correlations as probabilities – First order product density %(1)(x)=λ(x) Expected number of points around x x Measures local density Point Process Statistics • Correlations as probabilities – First order product density %(1)(x)=λ(x) Point Process Statistics • Correlations as probabilities – First order product density %(1)(x)=λ(x) Constant Point Process Statistics • Correlations as probabilities – Second order product density %(2)(x, y)=%(x, y) Expected number of points around x & y y x Measures the joint probability p(x, y) Point Process Statistics • Correlations as probabilities – Higher order? %(3)(x, y, z) z Expected number of points around x y z y The “second order dogma” in physics x Point Process Statistics • Summary: – First and second order product densities %(1)(x)=λ(x) %(2)(x, y)=%(x, y) y x x Point Process Statistics • Example: homogenous Poisson process – a.k.a random sampling p(x)=p p(x, y)=p(x)p(y) 2 λ(x)dV = p p(x, y)=%(x, y)dVxdVy = p(x)p(y)=λ(x)dVxλ(y)dVy = λ 2 λ(xp)=(x, yλ)=%(x, y)dVxdVy = p(x)p(y)=λ(x)dVxλ(y)dVy = λ 2 p(x, y)=%(x, y)dVxdVy = p(x)p(y)=λ(x)dVxλ(y)dVy = λ %(x, y)=λ(x)λ(y)=λ2 Stationary Processes Stationary Isotropic (translation invariant) (translation & rotation invariant) [Zhou et al. 2012] Stationary Processes • Stationary processes λ(x)=λ %(x, y)=%(x y) − = λ2g(x y) − Pair Correlation Function (PCF) DoF reduced from d2 to d! Stationary Processes • Pair Correlation Functions= λ2 g(x y) − No points close to each other For isotropic processes: radially symmetric PCF Stationary Processes • PCF is related to the periodogram g(h) = P (!) F{ } g(h) =gP(h()!)= P (!) F{ }F{ } Isotropic Processes • Rotation invariant λ(x)=λ g(x y)=g( x y ) − || − || PCF Isotropic Processes • Smooth estimator of the PCF 1 g(r) d 1 2 k(r d(xi, xj)) ⇡ @Vd r λ − − i=j | | X6 Volume of the unit Intensity Kernel, e.g. Distance hypercube in d dimensions Gaussian measure Isotropic Processes • Estimated PCFs Random Stratified Poisson Disc 1 Isotropic Processes • Estimated PCFs [Balzer [Schlömer Regular et al. 2009] et al. 2011] 1 Isotropic Processes • Estimated PCFs Clustering Clustering Clustering 1 Reconstructing Point Patterns [Ozreli and Gross 2012] • Least squares fitting of point patterns Sample Points Target PCF PCF of sample points Gradient descend on the @E xk+1 = xk t sample point locations i i k − @xi Reconstructing Point Patterns [Ozreli and Gross 2012] • Least squares fitting of point patterns Reconstructing Point Patterns • General isotropic patterns – [Wachtel et al. 2014] – [Heck et al. 2013] – [Zhou et al. 2012] – [Oztireli and Gross 2012] General Point Processes • What about: general point patterns [Faal 2011] General Point Processes • Idea 1: Use a tailored distance 1 g(r) d 1 2 k(r d(xi, xj)) ⇡ @Vd r λ − − i=j | | X6 PCF [Faal 2011] General Point Processes • Idea 1: Use a tailored distance d(x, y)=(x y)T M(x)(x y) − − [Li et al. 2010] General Point Processes • Idea 1: Use a tailored distance [Chen et al. 2013] General Point Processes • Idea 2: Change an initial point set Example: thinning according to a density Start with an isotropic sampling Remove each point with probability 1 - p(x) General Point Processes • Idea 2: Change an initial point set Example: thinning according to a density Start with an isotropic sampling Remove each point with probability 1 - p(x) λ (x)=p(x)λ %(x, y)=g( x y )λ(x)λ(y) || − || General Point Processes • Idea 3: Define the probability of a sampling – Fall back to classical statistics – So far: infinite point processes – Now: finite with n number of points – Define the probability of a configuration: f(x , , x ) 1 ··· n General Point Processes • Idea 3: Define the probability of a sampling f(x , , x )=f f(x , , x )=f 1 ··· n 1 1 ··· n 2 General Point Processes • Idea 3: Define the probability of a sampling – Generation of point patterns - MCMC Accept with probability P (X0)M(X X0) min 1, | P (X)M(X X) ✓ 0| ◆ Proposal distribution X X0 General Point Processes • Idea 4: General correlations %(1)(x)=λ(x) %(2)(x, y)=%(x, y) y x x Further Generalizations • Marked point processes Point locations + marks PCF for class 1 g1(r) PCF for class 2 g2(r) Cross PCF for class 1 & 2 g12(r) d(x1, x2) x1 d(x1, xx22) i j i 2 Ci1 ji 2 C2 Further Generalizations • Marked point processes – Discrete vs. continuous marks [Wei 2010] Further Generalizations • Space-time point processes [Ma et al. 2013] Conclusions • Stochastic point processes – Theoretical foundations – Explains general point patterns – Unified analysis and synthesis – Generalizes to measure spaces Thanks Cengiz Öztireli .
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