Sampling Analysis Cengiz ztireli using Correlations Gurprit Singh - PowerPoint PPT Presentation

Sampling Analysis Cengiz Öztireli using Correlations Gurprit Singh for Monte Carlo Rendering

Point Patterns in Computer Graphics Random distributions of points with characteristics Fundamental for many applications in graphics

Imaging

Patterns of Nature

Dynamic Structures

Simulations

Geometry Processing

Fabrication

Non-photorealistic Rendering

Rendering – Computing Integrals

Estimating Integrals with Points Sample and sum the sampled values of an integrand 1 Z I := f ( x ) d x |D| D n ˆ X I := w i f ( x i ) I =1 bias P [ˆ I ] = I − E P [ˆ I ] var P [ˆ I ] = E P [ˆ I 2 ] − ( E P [ˆ I ]) 2

Stochastic Point Processes Formal characterization of point patterns

Stochastic Point Processes Formal characterization of point patterns Point Process

Stochastic Point Processes Examples of point processes Natural Process Manuel Process

General Point Processes Infinite point processes Observation window

General Point Processes Assign a random variable to each set B B B N p B q “ 3 N p B q “ 5 N p B q “ 2

General Point Processes Joint probabilities define the point process B 1 B 1 B 1 B 2 B 2 B 2 p N p B 1 q ,N p B 2 q

Point Process Statistics Correlations as probabilities % ( n ) ( x 1 , · · · , x n ) dV 1 · · · dV n = p ( x 1 , · · · , x n q Product density Small volumes Points in space dV i x i

Point Process Statistics First order product density % (1) ( x ) = � ( x ) Expected number of points around x Measures local density x

Point Process Statistics First order product density ) = � ( x )

Point Process Statistics First order product density ) = � ( x ) Constant

Point Process Statistics Second order product density % (2) ( x , y ) = % ( x , y ) Expected number of points around x & y y Measures the joint probability p ( x , y ) x

Point Process Statistics Higher order product density? z Expected number of points around x , y , z y Not necessary: second order dogma x

Point Process Statistics Higher order not necessary: second order dogma % (1) ( x ) = � ( x ) % (2) ( x , y ) = % ( x , y ) y x x

Point Process Statistics Summary: 1 st & 2 nd order correlations sufficient % (1) ( x ) = � ( x ) % (2) ( x , y ) = % ( x , y ) y x x

Point Process Statistics Example: homogenous Poisson point process a.k.a. random sampling p ( x ) = p p ( x , y ) = p ( x ) p ( y ) p ( x , y ) = % ( x , y ) dV x dV y λ ( x ) dV = p λ ( x ) = λ y = p ( x ) p ( y ) = ) = � ( x ) dV x � ( y ) dV y % ( x , y ) = � ( x ) � ( y ) = � 2

Point Process Statistics Summary: 1 st & 2 nd order correlations sufficient % (1) ( x ) = � ( x ) % (2) ( x , y ) = % ( x , y ) y x x

Stationary Point Processes Stationary Isotropic (translation invariant) (translation & rotation invariant)

Stationary Point Processes Stationary (translation invariant) λ ( x ) = λ % ( x , y ) = % ( x − y ) = λ 2 g ( x − y ) Pair Correlation Function (PCF) DoF reduced from 2d to d

Stationary Point Processes Isotropic point process (translation & rotation invariant) λ ( x ) = λ g ( x − y ) = g ( || x − y || ) g p r q PCF r

Estimating Correlations Campbell’s Theorem Z hX i f ( x i ) = R d f ( x ) λ ( x ) d x E P 2 3 Z 4X 5 = f ( x i , x j ) R d ⇥ R d f ( x , y ) % ( x , y ) d x d y E P ( x i I 6 = j

Estimating Correlations First order λ ( x ) = " X # hX i I D ( x i ) = E P 1 E P x i ∈ D # Z Z = λ d x = λ d x = λ |D| D D Point distribution P P k N k ( D ) ˆ λ = Number of point K |D| distributions

Estimating Correlations Second order stationary - pair correlation function (PCF) 2 3 4X δ ( r − ( x i − x j )) E P 5 ( x i I 6 = j Z = R d × R d � ( r − ( x − y )) % ( x − y ) d x d y Z = λ 2 R d × R d δ ( r − ( x − y )) g ( x − y ) d x d y = λ 2 g ( r )

Estimating Correlations Second order stationary - pair correlation function (PCF) 1 X X g ( r ) = ˆ δ ( r − ( x i − x j )) K λ 2 P k x i , x j 2 P k ,i 6 = j Finite domains: 1 X X g ( r ) = ˆ δ ( r − ( x i − x j )) K λ 2 a I D ( r ) P k x i , x j 2 P k ,i 6 = j

Estimating Correlations Second order stationary - pair correlation function (PCF) Point Distribution Pair Correlation Function

Estimating Correlations Second order isotropic - pair correlation function (PCF) 1 X g ( r ) = ˆ k ( r � k x i � x j k ) λ 2 r d � 1 |S d | i 6 = j Volume of the unit Kernel hypercube in d dimensions e.g. Gaussian

Pair Correlation Function g ( r ) = ˆ 1 g ( r ) = ˆ

Pair Correlation Function g ( r ) = ˆ 1 g ( r ) = ˆ g ( r ) = ˆ

Pair Correlation Function g ( r ) = ˆ 1 g ( r ) = ˆ

Spectral Statistics Power spectrum Fourier transform of PCF

Spectral Statistics Points PCF Power spectrum Points

Spectral Statistics Power spectrum Radial average Radial anisotropy

Statistics for Stationary Processes Summary Stationary: Spatial (PCF) & spectral (power spectrum) PCF Power spectrum Isotropic: radial averages