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Sampling Analysis Cengiz ztireli using Correlations Gurprit Singh - - PowerPoint PPT Presentation
Sampling Analysis Cengiz ztireli using Correlations Gurprit Singh - - PowerPoint PPT Presentation
Sampling Analysis Cengiz ztireli 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
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Imaging
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Patterns of Nature
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Dynamic Structures
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Simulations
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Geometry Processing
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Fabrication
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Non-photorealistic Rendering
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Rendering – Computing Integrals
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Estimating Integrals with Points
Sample and sum the sampled values of an integrand I := 1 |D| Z
D
f(x)dx ˆ I :=
n
X
I=1
wif(xi) biasP[ˆ I] = I − EP[ˆ I] varP[ˆ I] = EP[ˆ I2] − (EP[ˆ I])2
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Stochastic Point Processes
Formal characterization of point patterns
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Stochastic Point Processes
Formal characterization of point patterns
Point Process
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Stochastic Point Processes
Examples of point processes
Natural Process Manuel Process
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General Point Processes
Infinite point processes
Observation window
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General Point Processes
Assign a random variable to each set
B
NpBq “ 3
B
NpBq “ 5
B
NpBq “ 2
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General Point Processes
Joint probabilities define the point process
B1 B1 B1 B2 B2 B2
pNpB1q,NpB2q
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Point Process Statistics
Correlations as probabilities
Product density Small volumes Points in space
xi dVi
%(n)(x1, · · · , xn)dV1 · · · dVn = p(x1, · · · , xnq
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Point Process Statistics
First order product density
x
Expected number of points around x Measures local density
%(1)(x) = (x)
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Point Process Statistics
First order product density
) = (x)
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Point Process Statistics
First order product density
) = (x)
Constant
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Expected number of points around x & y Measures the joint probability p(x, y)
x y %(2)(x, y) = %(x, y)
Point Process Statistics
Second order product density
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Expected number of points around x, y, z
x y
Point Process Statistics
Higher order product density?
z
Not necessary: second order dogma
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Point Process Statistics
Higher order not necessary: second order dogma
%(2)(x, y) = %(x, y) %(1)(x) = (x) x x y
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Point Process Statistics
Summary: 1st & 2nd order correlations sufficient
%(2)(x, y) = %(x, y) %(1)(x) = (x) x x y
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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)dVxdVy λ(x)dV = p λ(x) = λ
y = p(x)p(y) =
) = (x)dVx(y)dVy %(x, y) = (x)(y) = 2
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Point Process Statistics
Summary: 1st & 2nd order correlations sufficient
%(2)(x, y) = %(x, y) %(1)(x) = (x) x x y
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Stationary Point Processes
Stationary (translation invariant) Isotropic (translation & rotation invariant)
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Stationary Point Processes
Stationary (translation invariant)
λ(x) = λ %(x, y) = %(x − y) = λ2g(x − y)
Pair Correlation Function (PCF) DoF reduced from 2d to d
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Stationary Point Processes
Isotropic point process (translation & rotation invariant)
λ(x) = λ g(x − y) = g(||x − y||)
PCF
r
gprq
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EP 2 4X
I6=j
f(xi, xj) 3 5 = Z
Rd⇥Rd f(x, y)%(x, y)dxdy
(xi
Estimating Correlations
Campbell’s Theorem
EP hX f(xi) i = Z
Rd f(x)λ(x)dx
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Estimating Correlations
First order EP hX ID(xi) i = EP " X
xi∈D
1 # # = Z
D
λdx = λ Z
D
dx = λ|D| ˆ λ = P
Pk Nk(D)
K|D|
λ(x) =
Point distribution Number of point distributions
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Estimating Correlations
Second order stationary - pair correlation function (PCF)
EP 2 4X
I6=j
δ(r − (xi − xj)) 3 5 = Z
Rd×Rd (r − (x − y))%(x − y)dxdy
= λ2 Z
Rd×Rd δ(r − (x − y))g(x − y)dxdy = λ2g(r)
(xi
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ˆ g(r) = 1 Kλ2aID(r) X
Pk
X
xi,xj2Pk,i6=j
δ(r − (xi − xj))
Estimating Correlations
Second order stationary - pair correlation function (PCF) Finite domains: ˆ g(r) = 1 Kλ2 X
Pk
X
xi,xj2Pk,i6=j
δ(r − (xi − xj))
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Estimating Correlations
Second order stationary - pair correlation function (PCF)
Pair Correlation Function Point Distribution
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ˆ g(r) = 1 λ2rd1|Sd| X
i6=j
k(r kxi xjk) Estimating Correlations
Second order isotropic - pair correlation function (PCF)
Kernel e.g. Gaussian Volume of the unit hypercube in d dimensions
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Pair Correlation Function
1
ˆ g(r) = ˆ g(r) =
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Pair Correlation Function ˆ g(r) =
1
ˆ g(r) = ˆ g(r) =
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Pair Correlation Function
1
ˆ g(r) = ˆ g(r) =
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Spectral Statistics
Power spectrum Fourier transform
- f PCF
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Spectral Statistics
PCF Power spectrum Points Points
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Spectral Statistics
Power spectrum Radial average Radial anisotropy
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