How to use the fitted data

Overview
Fitted parameters
1. BSDF parameters
2. Spatial distribution parameters
Values outside the control points

Overview

In http://www-evasion.imag.fr/Membres/Antoine.Bouthors/research/phd/thesis/tables/ you will find the following files:

Matlab files (fit_*.mat) containing the fitted parameters for the simulation data, see below

Text files (Mie*.txt) containing the phase function used.

Mie0.txt represents the chopped peak of the Mie phase function, i.e., P_peak(Θ) (equation A.1 in the thesis)
MiePF3.txt represents the chopped Mie phase function, i.e., P_chopped(Θ) in RGB format (equation A.3 in the thesis)

I am not sure whether these functions are normalized or not. I advise you to re-generate them using MiePlot, with the parameters described in appendix A.

The other text files are simply interpolated values of the fitted parameters contained in the matlab files. I won't describe them here, I suggest you regenerate these as you see fit from the matlab files (see below)

The files fit_o*.mat contain the fitted parameters for the BSDF, i.e., the equations in §5.4 of the thesis.
The files fitXZ8_o*.mat contain the fitted parameters for the moments of the spatial distribution, i.e., the equations in §5.3 of the thesis.
The numbers in the file names corresponds to the orders of scattering that where fitted (under the form n+1). fit_o1.mat contains the fitted parameters for order 2. fitXZ8_o6-8.mat contains the fitted parameters for the orders [7,9]. fit_o15p.mat contains the fitted parameters for the orders [16,∞], etc.

BSDF parameters

When you open the matlab files for the BSDF parameters (fit_o*.mat), you will find the following variables in the workspace. All the variables follow the same "table of control points" format.

Peak value

The variable 'a' contains the fitted values for A₁(n, z₁, θ) (used in equation 5.48 in the thesis). It the I3D paper, it corresponds to A(t, μ_V). It is a table of control points. Each row of the table is a control point. Each control point has 3 values:

z₁ (in meters)
θ (under the form -cos(θ) I believe, i.e., μ_V in the I3D paper)
A₁

For example if you open fit_o2.mat, and read [50,-0.6,0.039801] in the 3rd row, that means A₁( n=3, z₁₌50m, θ=cos^-1(0.6) ) = 0.039801. And so on.

Note there is a discontinuity around θ=90°. That is normal and expected.

The variable 'x' contains the fitted values for A₂(n, z₁, θ') (used in equation 5.48 in the thesis). It the I3D paper, it corresponds to X(t, μ_L). Each control point has 3 values:

For example if you open fit_o2.mat, and read [50,0.2,1.904] in the 3rd row, that means A₂( n=3, z₁₌50m, θ'=cos^-1(0.2) ) = 1.904. And so on.

Note that A₂ is normalized so that A₂(n, z₁, 0) = 1, as explained in the thesis. Also, θ' ranges from 0° to 90° (whereas θ ranges from 0° to 180°), which is normal since the slab can only be lit from above.

Peak location

The variable 'b1' contains the fitted values for B₁(n, z₁, θ) (used in equation 5.49 in the thesis). It the I3D paper, it corresponds to B₁(t, μ_V). Each control point has 3 values:

z₁ (in meters)
θ (under the form -cos(θ) I believe, i.e., μ_V in the I3D paper)
B₁

For example if you open fit_o2.mat, and read [50,-0.6,9.2099] in the 3rd row, that means B₁( n=3, z₁₌50m, θ=cos^-1(0.6) ) = 9.2099. And so on.

The variable 'b2' contains the fitted values for B₂(n, z₁, θ') (used in equation 5.49 in the thesis). It the I3D paper, it corresponds to B₂(t, μ_L). Each control point has 3 values:

z₁ (in meters)
θ' (under the form cos(θ') I believe, i.e., μ_L in the I3D paper)
B₂

For example if you open fit_o2.mat, and read [50,0.2,26.066] in the 3rd row, that means B₂( n=3, z₁₌50m, θ=cos^-1(0.2) ) = 26.066. And so on.

Broadness

The variable 'c' contains the fitted values for C(n, z₁, θ) (used in equation 5.50 in the thesis). It the I3D paper, it corresponds to C(t, μ_V). Each control point has 3 values:

z₁ (in meters)
θ (under the form -cos(θ) I believe, i.e., μ_V in the I3D paper)
C

For example if you open fit_o2.mat, and read [50,-0.6,18.529] in the 3rd row, that means C( n=3, z₁₌50m, θ=cos^-1(0.6) ) = 18.529. And so on.

Logarithmic behavior

The variable 'd' contains the fitted values for D(n, z₁, θ) (used in equation 5.51 in the thesis). It the I3D paper, it corresponds to D(t, μ_V). Each control point has 3 values:

z₁ (in meters)
θ (under the form -cos(θ) I believe, i.e., μ_V in the I3D paper)
D

For example if you open fit_o2.mat, and read [50,-0.6,631.07] in the 3rd row, that means D( n=3, z₁₌50m, θ=cos^-1(0.6) ) = 631.07. And so on.

Anisotropy

The variable 'p' contains the fitted values for P(n, Θ) (used in equation 5.52 in the thesis). It the I3D paper, it corresponds to P(θ). Each control point has 3 values:

z₁ (in meters)
Θ (under the form cos(Θ) I believe)
P

Note there is an inconsistency here: although in the thesis and paper we say that P does not depend on z₁, we have z₁ as a parameter in the tables. This is simply due to the fact that the fitting was done per slab thickness. If you observe the data, you will note that the values for P are actually almost identical for a given Θ, regardless of z₁. This is a nice validation of our assumption that P does not depend on z₁. In consequence, to read the value for P, you can either read the values of P for an arbitrary z₁, or read P for all values of z₁ and take the average. I think we did the latter.

For example if you open fit_o2.mat, and read

[50, -0.5, 2.2514] in the 3rd row,
[100, -0.5, 2.0483] in the 12th row,
[200, -0.5, 2.0575] in the 21st row,
[250, 0.5, 2.0711] in the 30th row,
[300, -0.5, 2.0132] in the 39th row,
[400, -0.5, 2.0722] in the 48th row and
[500, -0.5, 2.0974] in the 57th row,

that means P( n=3, Θ=cos^-1(0.5) ) = 2.0873. And so on.

Finally, note that there is no data for P(n=[16,∞], Θ). This is normal: these orders are isotropic, therefore P(n=[16,∞], Θ)=1 everywhere, as explained in the thesis.

Spatial distribution parameters

Notes

I would also like to remind you that z' is the viewpoint depth: z'=z₁-z (thesis p. 114).

Finally, bear in mind that P(n) (equation 5.26) is not at all the same function than P(n, Θ) (equation 5.52). Similarly, g in equations 5.11 and 5.17 is not the assymetry parameter of the phase function. This is just the result of a poor naming convention.

First moment

Matlab variable	Thesis function equivalent	Control points	I3D paper function equivalent
b	O(n, μ, μ') (eq.5.24)	Column 1&2: -μ (i.e., μ_Vin the I3D paper), Column 3: μ', Column 4: O	N(μ_V, μ_L)
c	I(n, μ) (eq.5.19)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: I	I(μ_V)
d	M(n, μ) (eq.5.23)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: M	M(μ_V)
e	E(n, μ) (eq.5.15)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: E	E(μ_V)
j	G(n, μ) (eq.5.20)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: G	G(μ_V)
k	F(n, μ) (eq.5.16)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: F	F(μ_V)
l	H(n, μ) (eq.5.18)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: H	H(μ_V)
u	K(n, μ) (eq.5.21)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: K	K(μ_V)
v	L(n, μ) (eq.5.22)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: L	L(μ_V)
w	J(n, μ) (eq.5.20)	Column 1: -μ (i.e., μ_Vin the I3D paper), Column 2: J	J(μ_V)

Second moment

The equation 5.25 in the thesis should be

σ(n, z₁, z, θ, ψ, θ') = p + q z₁ log(1 + rz') + s log(1 + tz₁), n=[16,∞]

σ(n, z₁, z, θ, ψ, θ') = S_a + S_b log(1 + S_cz'), n<16

With these equations, the data for n=[16,∞] is not present in the matlab files. Here are the values:

P(n=[16,∞]) = 171.0
Q(n=[16,∞]) = 0.1645
R(n=[16,∞]) = 0.002321
S(n=[16,∞]) = 504.6956
T(n=[16,∞]) = 0.00096153

For n<16, the parameters S_a, S_b, S_c are defined by the scalar variables sa, sb, and sc in the matlab files.

Values outside the control points

The parameters have been fitted only for the given control points. To compute the value of a parameter outside the control points, you have to interpolate. The choice of the interpolation method is important, as it will give different results. We used the linspace() matlab function to sample uniformly the domain, then interpolated using interp1() for 1D data and griddata() for 2D data. Although it is tempting to use a cubic or spline interpolation method to get smooth variations, these are bad choices as they result in overshoots. Thus we interpolated all the data using the linear method, which gave satisfactory results. Care must be taken to interpolate finely enough so that the discontinuities are not smoothed out.