Mathematica - Fitting Penner's Skin Scattering

8/4/2019 Mathematica - Fitting Penner's Skin Scattering

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Coefficients and colors from d 'Eon and Luebke skin scattering work

ScatterCoeffs 0.0064, 0.0484, 0.187, 0.567, 1.99, 7.41 Sqrt2

ScatterColors 0.233, 0.455, 0.649, 0.1, 0.336, 0.344,

0.118, 0.198, 0, 0.113, 0.007, 0.007, 0.358, 0.004, 0, 0.078, 0, 0

GaussianFnvar_, x_ : 1 Sqrt2 Pi var Expx x 2 var

Skin scattering profile approximated a sum of multiple

gaussians. We don't want to work with threedimensional RGB

vectors so here one of the inputs selects which channel we want

ScatterFnx_, rgb_ :

ApplyPlus, TableGaussianFnvar, x , var, ScatterCoeffs ScatterColorsrgb

Out[1]= 0.00905097, 0.0684479, 0.264458, 0.801859, 2.81428, 10.4793Out[2]= 0.233, 0.455, 0.649, 0.1, 0.336, 0.344, 0.118, 0.198, 0,

0.113, 0.007, 0.007, 0.358, 0.004, 0, 0.078, 0, 0

In[5]:= This is the function defined with integrals, as in the paper

DiffuseLightΘ _, r_, rgb_ : NIntegrate

ClipCosΘ x , 0, 1 ScatterFn Abs2 r Sinx 2, rgb, x , Pi, Pi

DiffuseWeightr_, rgb_ : NIntegrate

ScatterFn Abs2 r Sinx 2, rgb, x , Pi, Pi

DiffuseIntegralΘ _, r_, rgb_ : DiffuseLightΘ, r, rgb DiffuseWeightr, rgb

This is a bit faster to evaluate,

it's a translation of the code used to generate the lookup table

DiffuseWeight2r_, rgb_ :

NSum ScatterFn Abs2 r Sinx 2, rgb, x , Pi, Pi, 2 Pi 20

DiffuseLight2Θ _, r_, rgb_ : NSum ClipCosΘ x , 0, 1

ScatterFn Abs2 r Sinx 2, rgb, x , Pi, Pi, 2 Pi 20

DiffuseIntegral2Θ _, r_, rgb_ : DiffuseLight2Θ, r, rgb DiffuseWeight2r, rgb

In[11]:= Let's theck that they are similar ...

DiffuseIntegral0.1, 8, 1




Out[11]= 0.979266

Out[12]= 0.990889

Out[13]= 0.990641

Out[14]= 0.994601

In[16]:= Let's generate a table of data to be used for the numerical fitting...

channel 1; 1red 2green 3 blue

dataThetaRVal FlattenTableΘ, r , DiffuseIntegralΘ, r , channel,

Θ, 0, Pi, 2 Pi 20, r, 0.25, 6, 1 20, 1;



an extended range that we will use to check how the fit extrapolates

dataIntervalExt FlattenTableΘ, r , Θ, 0, Pi, 2 Pi 20, r, 0, 12, 1 20, 1 ;

LineInterpa_,b_,x_,x0_,x1_ :

bClipxx0,0,1aClip1xx0x1x0,0,1

dataGraph ListPlot3DdataThetaRVal, InterpolationOrder 3

cosGraph ListPlot3D

x Firstx , Lastx , ClipCosFirstx , 0, 1 dataIntervalExt;

Plotting against the simple lambert cos lighting model,

of course that's not dependent on r ...

Show dataGraph, cosGraph

Out[18]=

Out[20]=

2 EricPenner_SkinScattering1.nb



In[24]:= Ok, let's have a peek at our function to understand its shape... We'll plot a

single slice, then a graph with three slices and the Lambert's clamped cos

plotIn2d PlotDiffuseIntegralx , 5, 1, x, 0, Pi

PlotDiffuseIntegralx , 1 , 1, DiffuseIntegralx, 3, 1,

DiffuseIntegralx , 6 , 1, Max Cosx , 0 , x, 0, Pi

Out[24]=

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

Out[25]=

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

EricPenner_SkinScattering1.nb 3



In[26]:= We have to find a function of cosΘ

which could look similar to the DiffuseIntegral

Show Plot Max Cosx 0.5 0.5, 0 ^ 3, x, 0, Pi, plotIn2d

Out[26]=

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0




In[27]:=

Let's find the fit now, first we have to define a parametric function,

the fitting process will optimize the parameters to minimize the error

piecewise linear models...

fitFunctionΘ _,r_ : Max CosΘ Mina0ra1,1 Max a2ra3,0,0

fitFunctionΘ _,r_ :

Max Max CosΘ Mina0ra1,1 Max a2ra3,0,0,

Max CosΘ Mina4ra5,1 Max a6ra7,0,0

nonlinear models...

fitFunctionΘ _,r_ : CosΘa0ra1 a2ra3^ 3

fitFunctionΘ _, r_ :

CosΘ a0 r a1 a2 r a3^ 3 Clipa4 r a5, 0, 1

Max CosΘ, 0 1 Clipa4 r a5, 0, 1

fitting NonlinearModelFitdataThetaRVal, fitFunctionΘ, r ,

a0, a1, a2, a3, a4, a5, a6, a7, Θ, r , Method NMinimize We need to use NMinimize for nonsmooth models

fitParams fitting"BestFitParameters"

Plus Abs fitting"FitResiduals" Print the sum of residuals

We plot the fit function over

dataIntervalExt to check that it extrapolates well

dataFit x Firstx , Lastx , fitFunctionFirstx , Lastx . fitParams

dataIntervalExt;

fitGraph ListPlot3DdataFit, InterpolationOrder 3,

ColorFunction "SouthwestColors", Mesh None;

Show fitGraph, cosGraph

Show fitGraph, dataGraph

Out[28]= FittedModel Clip 0.817767 0.11103 r, 0, 1 1131111

Out[29]= a0 0.0605281, a1 0.0903942, a2 0.0210583,

a3 0.689657, a4 0.11103, a5 0.817767, a6 7.15776, a7 5.89292Out[30]= 12.4611




Out[33]=

Out[34]=




lightVector Normalize1, 2, 0;

Lambert

SphericalPlot3D1, Θ, 0, Pi, Φ, 0, Pi,

ColorFunctionScaling False, False, False, False, False, False,

ColorFunction x, y, z, Θ, Φ, r Max x, y, z.lightVector, 0

Our function

radius 3;

SphericalPlot3D1, Θ, 0, Pi, Φ, 0, Pi,

ColorFunctionScaling False, False, False, False, False, False,

ColorFunction x, y, z, Θ, Φ, r

fitFunction ArcCosx, y, z.lightVector, radius . fitParams

Out[45]=




Out[47]=

todo...

Use NMinimize to find a fit with a custom error metric, i.e. perceptual,

weighting more the darker areas or we can fit the log of the

functions or we could use the Weights option of NonlinearModelFit

Constraint fit i.e. zero at r1,

Θ2Pi or better, equal to CosΘ for r going to

Find artisttunable parameters and use Manipulate to validate them...


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Mathematica - Fitting Penner's Skin Scattering

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