Regarding a conversion from measured reflectance values with Lambertian reflector target to BRDF values

[DavidJRitter904]

Describe the problem

A separate discussion started within Issue #167 about using measured reflectance values referenced to a Lambertian reflector target within a BRDF.

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Research done so far

If anything of the information we gathered on this topic is missing, please include them with a comment below!

BRDF definition

the ratio of radiance in the outgoing direction to the incident irradiance [2]

With the Assumptions below, we define a function $f_r$ that is the basic quantity that characterizes (geometrically) the reflecting properties of a surface called "Bidirectional Reflectance Distribution Function", abbreviated BRDF:

$$ BRDF: f_r(\theta_i , \phi_i ; \theta_r , \phi_r) = \frac{dL_r(\theta_i , \phi_i ; \theta_r , \phi_i ; E_i)} {dE_i(\theta_i , \phi_i)} $$ $$ dE_i = L_i(\theta_i , \phi_i) d\Omega_i $$

Since for a given pair of directions, the BRDF $f_r$ is a concentration of reflectance (per steradian) it may take on any value from zero to infinity.

BRDF properties

Positivity: $f_r(\theta_i , \phi_i ; \theta_r , \phi_r) \ge 0$
Energy conservation: $\forall\omega_i; \int_{\omega_r} \int_{\omega_i} f_r(\theta_i , \phi_i ; \theta_r , \phi_r) cos(\theta_r)d\Omega_i d\Omega_r \le 1$
Helmholtz reciprocity: $f_r(\theta_i , \phi_i ; \theta_r , \phi_r) = f_r(\theta_r , \phi_r ; \theta_i , \phi_i)$

Reflectance Factor

Continue by using the definition of the reflectance factor $R_F$ by Nicodemus et al.[1] which is the ratio of the radiant flux actually reflected by a sample surface to the radiant flux reflected by the ideal diffuse Lambertian reflector denoted as "id" ($\rho = 1$) in the same reflected-beam geometry, irradiated in exactly the same ways as the sample.

Combining equation (12) for the reflected flux by the surface element $dA$ into the solid angle $\omega_r$

$$ d\Phi_r = {dA\int_{\omega_r} \int_{\omega_i} f_r(\theta_i , \phi_i ; \theta_r , \phi_r) L_i(\theta_i , \phi_i)\ d\Omega_i d\Omega_r} $$

and equation (C8), the BRDF for an ideal standard diffuse reflector surface

$$ f_{r,id}(\theta_i , \phi_i ; \theta_r , \phi_r) = 1/\pi $$

including our assumptions, we arrive at the following reflectance factor $R_F$:

$$ R_F = \frac{d\Phi_r} {d\Phi_{r,id}} = \frac {dA\int_{\omega_r} \int_{\omega_i} f_r(\theta_i , \phi_i ; \theta_r , \phi_r) L_i(\theta_i , \phi_i)\ d\Omega_i d\Omega_r} {(dA/\pi)\int_{\omega_r} \int_{\omega_i} L_i(\theta_i , \phi_i)\ d\Omega_i d\Omega_r} = \pi f_r(\theta_i , \phi_i ; \theta_r , \phi_r)$$

This is in accordance with the definition of the Ocean Optics Web Book [4]:

and the derivation of a BRDF from a difusse reflectance: or albedo defined for a Lambertian surface within the Computer Vision book by Forsyth and Ponce, p.13, chapter 1.3.3 [2]:

$$ \rho_{brdf}=\frac {\rho_d} {\pi} $$

It should be noted, that for converting measurement values with respect to Lambertian reflectors / surfaces to a BRDF, the assumptions below need to be kept in mind. It is import to consider them and carefully evaluate the use of the converted BRDF values regarding the use case.
Therefore, the assumptions should be stated in the documentation where applicable to inform the user.

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Important notes

• $\Omega$: is the projected solid angle with unit [sr] to account for the foreshortening area.
• $\omega$: solid angle with unit [sr] - full hemispherical solid angle $\omega = 2\pi$
• Reflectivity and Reflectance are equal for half-space materials (not valid for thin layers or transmissive materials) [6]
• Radiant intensity follows Lambert's cosine law [7]: $I = I_0 * cos(\theta)$

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Assumptions

1. uniform and isotropic irradiance within full solid angle irradiance is constant
2. uniform and isotropic surface over a large enough area
3. BRDF used as a basic quantity to characterize geometrically reflecting properties of a surface
4. the radiance leaving a point on a surface is due only to radiance arriving at this point (although radiance may change directions at a point on a surface, we assume that it does not skip from point to point)
5. we assume that all light leaving a surface at a given wavelength is due to light arriving at that wavelength
6. we assume that the surfaces do not generate light internally and treat sources separately.
7. Helmholtz reciprocity of the BRDF: $f_r(\theta_i , \phi_i ; \theta_r , \phi_r) = f_r(\theta_r , \phi_r ; \theta_i , \phi_i)$
8. LiDAR measurements are considered independent of polarization

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Further researches on Reflectance & BRDF

• General overview of BRDF generation techniques: Link
• Data-driven BRDF generation from measurements: MERL database
• Different reflectance definitions/measures: Link

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Sources

1. Geometrical Considerations and Nomenclature for Reflectance – F.E. Nicodemus, J.C. Richmond, J.J. Hsia
2. Computer Vision - A modern Approach – Forsyth, Ponce
3. Ocean Optics Web Book
4. Ocean Optics Web Book - Reflectance Factor REEF
5. Ocean Optics Web Book - Lambertian BRDF
6. Wikipedia - Reflectance
7. Wikipedia - Lambert's cosine law

[lyndyRott]

one additional remark on the relation between surface reflectance $\rho$ and BRDF $f_r$, taken from (B 6) [Nicodemus, p.39]:

$$ f_r(\theta_i, \phi_i; \theta_r, \phi_r) = \frac{d\rho(\theta_i, \phi_i; \theta_r, \phi_r; \Omega_r)}{d\Omega_r} $$

quote:

$d\rho$ is not a measure of just the reflecting properties of the surface for the given pair of directions but it is also directly proportioned to the size of the element of projected solid angle of reflection (and collection) $d\Omega_r$. The invariant quantity for a particular element of reflecting surface $dA$, the one that characterizes its (geometrical) reflecting properties, is the bidirectional reflectance distribution function (BRDF) with the relations (B 6)

[ClemensLinnhoff]

Thank you very much for the collection!