<p>[I will try to keep this answer updated as I learn more]</p>
<p>The best reference on the foundations of light transport seems to be Arvo's <a href="http://www0.cs.ucl.ac.uk/research/vr/Projects/VLF/vlfpapers/sampling/Arvo_J__Transfer_Equations_in_Global_Illumination__SIGGRAPH_course_notes_1993.pdf" rel="nofollow noreferrer">Transfer Equations in Global Illumination</a> (~30 pages)</p>
<p>Some things he doesn't cover:</p>
<ul>
<li>Argument for symmetry of BRDF (see Veach)</li>
<li>Transfer in media with continuously varying refractive index</li>
</ul>
<p>The state of the art model for Light Transport seems to be <a href="http://isgwww.cs.uni-magdeburg.de/graphics/projects/dissertation/index.html" rel="nofollow noreferrer">Christian Lessig's PhD thesis</a> (~600 pages). The theory is developed in terms of differentiable manifolds and Hilbert spaces and is much more technical than Arvo's note.</p>
<p>Lessig carries out translation from the mathematical theory to algorithms (e.g. path tracing, photon mapping) in Section 4.3.</p>

CC BY-SA 4.0
2024-08-20T22:54:53.343Z
I don't know whether you've already seen this, but I like very much Arvo's Theorem 4 (pg 24) of his 1995 thesis, where he derives invariance of radiance along lines, as it were, "just" by keeping track of a particular set of photons as time elapses" https://www.cs.cornell.edu/courses/cs667/2005sp/readings/ArvoThesis.pdf
Simon
CC BY-SA 4.0
2021-01-22T18:23:59.100Z
2021-01-22T20:16:31.343Z
2021-01-22T20:16:31.343Z
user357269
<p>This is almost verbatim from Veach. Let the trajectory space be <span class="math-container">$\mathbb{P} = \mathbb{R} \times \mathcal{M} \times \mathcal{S}^2 \times \mathbb{R}^+$</span>, corresponding respectively to: time, set of points of all scene surfaces, unit sphere (directions), wavelength. Let <span class="math-container">$\mathcal{P}$</span> be the set of measurable subsets of <span class="math-container">$\mathbb{P}$</span> and we have a measure <span class="math-container">$\rho$</span>. Let <span class="math-container">$\rho = l \times A \times \sigma^{\perp} \times l^+$</span>. Where <span class="math-container">$l, l^+$</span> are the Lebesgue measures on <span class="math-container">$\mathbb{R}$</span> and <span class="math-container">$\mathbb{R}^+$</span>, <span class="math-container">$A$</span> is the Lebesgue measure on <span class="math-container">$\mathcal{M}$</span>, and <span class="math-container">$\sigma^{\perp}$</span> is the projected solid angle measure <span class="math-container">$(d\sigma^{\perp}(\omega) = |\cos\theta|\sin\theta\,d\theta\,d\phi$</span>). Then given the radiant energy measure <span class="math-container">$Q(\mathcal{T}, \mathcal{D}, \Omega, \Lambda) \in [0,\infty]$</span> (<span class="math-container">$\mathcal{T} \times \mathcal{D} \times \Omega \times \Lambda \in \mathcal{P}$</span>), we can define the density <span class="math-container">$L_{\lambda}$</span> of <span class="math-container">$Q$</span> with respect to the measure <span class="math-container">$\rho$</span> as:</p>
<p><span class="math-container">\begin{equation}
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int_{\Lambda}\int_{\Omega}\int_{\mathcal{D}}\int_{\mathcal{T}}L_{\lambda}(t, x, \omega,\lambda)d\rho(t,x,\omega,\lambda).
\end{equation}</span></p>
<p>Then the density <span class="math-container">$L_{\lambda}$</span> is the Radon-Nikodym derivative of <span class="math-container">$Q$</span> w.r.t. <span class="math-container">$\rho$</span>:</p>
<p><span class="math-container">\begin{equation}
\frac{dQ}{d\rho}(t, x, \omega,\lambda) := L_{\lambda}(t, x, \omega,\lambda).
\end{equation}</span></p>
<p>We can do a similar thing with respect to the appropriate measures:</p>
<p>Radiant flux:
<span class="math-container">\begin{gather}
\rho_1 = l \\
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int_{\mathcal{T}}\Phi(t, \mathcal{D}, \Omega, \mathcal{\Lambda})\,d\rho_1(t) \\
\frac{dQ}{d\rho_1}(t, \mathcal{D}, \Omega, \mathcal{\Lambda}) := \Phi(t, \mathcal{D}, \mathcal{\Omega}, \mathcal{\Lambda}).
\end{gather}</span></p>
<p>Irradiance:
<span class="math-container">\begin{gather}
\rho_2 = l\times A \\
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int_{\mathcal{D}}\int_{\mathcal{T}}E(t, x, \Omega, \mathcal{\Lambda})\,d\rho_2(t,x) \\
\frac{dQ}{d\rho_2}(t, x, \Omega, \mathcal{\Lambda}) := E(t, x, \Omega, \mathcal{\Lambda}).
\end{gather}</span></p>
<p>Intensity:
<span class="math-container">\begin{gather}
\rho_3 = l\times \sigma \\
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int_{\Omega}\int_{\mathcal{T}}I(t, \mathcal{D}, \omega, \mathcal{\Lambda})\,d\rho_3(t,\omega) \\
\frac{dQ}{d\rho_3}(t, \mathcal{D}, \omega, \mathcal{\Lambda}) := I(t, \mathcal{D}, \omega, \mathcal{\Lambda}).
\end{gather}</span></p>
<p>Radiance:
<span class="math-container">\begin{gather}
\rho_4 = l\times A \times \sigma^{\perp} \\
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int_{\Omega}\int_{\mathcal{D}}\int_{\mathcal{T}}L(t, x, \omega, \mathcal{\Lambda})\,d\rho_4(t,x,\omega) \\
\frac{dQ}{d\rho_4}(t, x, \omega, \mathcal{\Lambda}) := L(t, x, \omega, \mathcal{\Lambda}).
\end{gather}</span></p>

CC BY-SA 4.0
2021-01-21T11:10:00.730Z
Just a comment: $\rho_4$ isn't really a product of measures since the $\sigma^\perp$ measure depends on the surface orientation. It's some kind of twisted product, see https://en.wikipedia.org/wiki/Markov_kernel#Semidirect_product or https://en.wikipedia.org/wiki/Disintegration_theorem
user357269
CC BY-SA 4.0
2021-01-21T12:29:05.490Z
@user357269 The measure $\sigma^{\perp}_x(\Omega)$ is defined on page 77, eq (3.1) in Veach's thesis as $\sigma^{\perp}_x(\Omega) = \int_{\Omega}|\omega \cdot N_x|\,d\sigma(\omega)$. So yes, $\sigma^{\perp}$ depends on $x$, but that isn't really an issue.
lightxbulb
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2021-01-06T18:03:12.803Z
2021-01-06T18:39:54.513Z
2021-01-06T18:39:54.513Z
lightxbulb
<p>I'm new to learning about ray tracing, but have been fairly confused by the mathematical foundations of it all. Specifically of radiometry/radiative transfer. I've internet searched a lot and skimmed a ton of references, but still very confused.</p>
<p>Here is a list of some of the things I have had to struggle a lot to understand:</p>
<ol>
<li><p>Irradiance is a function of 3 variables <span class="math-container">$E(p,n, \mathrm{in}/\mathrm{out})$</span> and it's not really the derivative of anything with respect to anything else (in the function sense)</p>
</li>
<li><p>If <span class="math-container">$p$</span> is a point on an actual surface in a scene, irradiance doesn't make sense in directions that aren't the surface normal, except perhaps as limit.</p>
</li>
<li><p>Radiance is not meaningfully the derivative of irradiance (Wikipedia says it's the derivative w.r.t. étendue, which is a great way to get more confused).</p>
</li>
<li><p>Radiance can be defined without ever using the words &quot;projected area&quot;, &quot;solid angle&quot; or &quot;subtend&quot;.</p>
</li>
<li><p>You don't need to draw pictures with infinitesimals to think about radiance. I see infinitesimals in pictures a lot, e.g. <a href="https://en.wikipedia.org/wiki/Lambert%27s_cosine_law#Details_of_equal_brightness_effect" rel="nofollow noreferrer">Figures 1 and 2 Wikipedia's Lambert's cosine law article</a>.</p>
</li>
<li><p>Radiance is preserved along paths (in vacuum). Is this an axiom of the theory or a theorem?</p>
</li>
</ol>
<p>I've  found the following approach more insightful than the usual presentations:</p>
<ol>
<li><p>If you have an oriented surface, you can define the flux entering it <span class="math-container">$\Phi(S)$</span>.</p>
</li>
<li><p>Irradiance is something that's integrated into flux:
<span class="math-container">$$ \Phi(S) = \int_SE(p, n_p, \mathrm{in}) \mathrm{d}S(p)$$</span></p>
</li>
<li><p>For two surfaces, we can define direct flux transport between them: the power that exist <span class="math-container">$S_1$</span> and directly enters <span class="math-container">$S_2$</span>, <span class="math-container">$\Phi(S_1, S_2)$</span>.</p>
</li>
<li><p>&quot;Flux transport density&quot; (not sure if correct term) is something that's integrated into flux transport. It is the function of 4 variables <span class="math-container">$K(p_\text{source}, n_\text{source}, p_\text{target}, n_\text{target})$</span>
<span class="math-container">$$ \Phi(S_1,S_2) = \int_{S_2} \int_{S_1} K(p, n_p, q, n_q) \mathrm{d}S_1(p) \mathrm{d}S_2(q)$$</span></p>
</li>
<li><p>&quot;Light hull principle&quot; consider two oriented disks <span class="math-container">$D_p = D(p, n_p, r_p)$</span> and <span class="math-container">$D_q = D(q, n_q, r_q)$</span> that are sufficiently small.</p>
<ol>
<li><p>If <span class="math-container">$(q-p) \cdot n_p \le 0$</span> then <span class="math-container">$\Phi(D_p, D_q) = 0$</span></p>
</li>
<li><p>Taking the limit <span class="math-container">$r_p \to 0$</span> shows that <span class="math-container">$(q-p) \cdot n_p$</span> implies <span class="math-container">$K(p, n_p, q, n_q) = 0$</span></p>
</li>
<li><p>If <span class="math-container">$(p-q) \cdot n_q \le 0$</span> then <span class="math-container">$\Phi(D_p, D_q) = 0$</span></p>
</li>
<li><p>Taking the limit <span class="math-container">$r_q \to 0$</span> shows that <span class="math-container">$(p-q) \cdot n_q$</span> implies <span class="math-container">$K(p, n_p, q, n_q) = 0$</span></p>
</li>
<li><p>Let's assume <span class="math-container">$(q-p) \cdot n_p &gt; 0$</span> and <span class="math-container">$(p-q) \cdot n_q &gt; 0$</span> going forward. Maybe it's simplier to write <span class="math-container">$|\cos(\theta_p)| &lt; \pi/2$</span>, <span class="math-container">$|\cos(\theta_q)| &lt; \pi/2$</span></p>
</li>
<li><p>Let's form the &quot;light hull&quot; of <span class="math-container">$D_p$</span> and <span class="math-container">$D_q$</span>. This is a funny truncated cone like shape defined by <span class="math-container">$$H(D_p, D_q) = \{ t p' + (1-t) q' | p' \in D_p, q' \in D_q, t \in [0, 1] \}$$</span> <a href="https://i.sstatic.net/qTJlU.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qTJlU.png" alt="enter image description here" /></a> It may later make sense to allow  <span class="math-container">$t$</span> to go negative/greater than 1, extending the hull a bit.</p>
</li>
<li><p>Let <span class="math-container">$a$</span> be a point on the segment <span class="math-container">$pq$</span>. Intersecting the light hull with a plane through <span class="math-container">$a$</span> gives as circular shaped section:
<span class="math-container">$$H(a, n_a, D_p, D_q) = \{ b | (b - a) \cdot n_a = 0 \} \cap H(D_p, D_q) $$</span></p>
</li>
<li><p>In free space we have <span class="math-container">$$\Phi(D_p, D_q) = \Phi(D_p, H(a, n_a, D_p, D_q)) = \Phi(H(a, n_a, D_p, D_q), D_q)$$</span> I call this the <em>light hull principle</em> and is basically just geometry saying &quot;if I put a transparent screen in the light's way, all the light will go through it&quot;.</p>
</li>
<li><p>Using the light hull principle, we see that we can replace <span class="math-container">$D_q$</span> with a similar disk-ish shape that's orthogonal to <span class="math-container">$pq$</span>: <span class="math-container">$$ \Phi(D_p, D_q) = \Phi(D_p, H(q, p-q, D_p, D_q))$$</span> Let's call this shape <span class="math-container">$D_q'$</span>.
<a href="https://i.sstatic.net/or5th.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/or5th.png" alt="enter image description here" /></a>
For small <span class="math-container">$r_p^2$</span>, <span class="math-container">$r_q^2$</span>, the left hand side is
<span class="math-container">$$ \Phi(D_p, D_q) \approx K(p, n_p, q, n_q) \pi^2 r_p^2 r_q^2$$</span>
Or more formally
<span class="math-container">$$ \frac{\partial^4 \Phi(D_p, D_q) }{(\partial r_p)^2 (\partial r_q)^2} \bigg|_{r_p = r_q = 0} = 4 \pi^2 K(p, n_p, q, n_q) $$</span>
I don't want to go into too much detail here, but it's possible to prove for the right hand side that <span class="math-container">$$ \frac{\partial^4 \Phi(D_p, H(q, p-q, D_p, D_q)) }{(\partial r_p)^2 (\partial r_q)^2} \bigg|_{r_p = r_q = 0} = 4 \pi^2 \cos(\theta_q) K\left(p, n_p, q, \frac{p-q}{||p-q||} \right) $$</span>
It follows that <span class="math-container">$$ K(p, n_p, q, n_q) = \cos(\theta_q) K\left(p, n_p, q, \frac{p-q}{||p-q||} \right)$$</span></p>
</li>
<li><p>Repeating the same argument at <span class="math-container">$p$</span> instead of <span class="math-container">$q$</span>, we get
<span class="math-container">$$ K(p, n_p, q, n_q) = \cos(\theta_p) \cos(\theta_q) K\left(p, \frac{q-p}{||p-q||}, q, \frac{p-q}{||p-q||} \right)$$</span></p>
</li>
<li><p>Assuming now that the two disks are directly facing each other, i.e. <span class="math-container">$n_p = \frac{q-p}{||p-q||} = -n_q$</span>, we can apply the light hull principle and take a limit/derivative as <span class="math-container">$r_p, r_q \to 0$</span> to show the inverse square law
<span class="math-container">$$ K(p, n_p, p+r n_p, -n_p) = \frac{1}{r^2} K(p, n_p, p+n_p, - n_p)$$</span></p>
</li>
</ol>
</li>
<li><p>Let <span class="math-container">$r = ||p-q||$</span> and <span class="math-container">$n = \frac{q-p}{r}$</span>. We now know that
<span class="math-container">$$ \begin{aligned}
K(p, n_p, q, n_q) &amp;= K(p, n, p+n, -n) \frac{\cos(\theta_p) \cos(\theta_q)}{r^2} \\
&amp;=: L(p,n,\mathrm{out}) \frac{\cos(\theta_p) \cos(\theta_q)}{r^2} \\
&amp;=: L(q,-n,\mathrm{in}) \frac{\cos(\theta_p) \cos(\theta_q)}{r^2}
\end{aligned}
$$</span>
Where <span class="math-container">$L$</span> is the defined by the equation above. We call <span class="math-container">$L$</span> radiance.</p>
</li>
<li><p>Taking the limit <span class="math-container">$p \to q$</span> shows that <span class="math-container">$L(q, n, \mathrm{out}) = L(q, -n, \mathrm{in}) = L(p, n, \mathrm{out})$</span>. Therefore radiance is constant along rays (and <span class="math-container">$K$</span> is translation invariant).</p>
</li>
</ol>
<p>The significance of this is that we have defined radiance and proved it invariant by using a simple geometric principle on finite (non-infinitesimal) surfaces and taking appropriate limits. It's a lot more work than saying irradiance over projected surface area times solid angle, but to me much more intuitive.</p>
<p>The light hull principle is an axiom/key &quot;physical&quot; fact, and everything else is a theorem.</p>
<p>My question is; is there a reference that deals with radiative transfer in general in a similar way as I've just dealt with radiance?</p>
<p>The closest thing I could find is <a href="https://graphics.stanford.edu/papers/veach_thesis/" rel="nofollow noreferrer">Veach's PhD thesis</a> where he defines some interesting concepts using measure theory in Chapter 3 and its appendix. The idea is that a scene is a system of <span class="math-container">$N$</span> non-interacting photons moving around. Radiometric quantities can then be defined by a snapshot of this system in time. I think it's a really cool idea and would love to see more of this. To me these are the true foundations. Sadly, I don't think Veach develops them in enough detail. (And I also don't see how a finite number of photons as Veach models them could create equilibrium in the sense that irradiance is independent of time)</p>
<p>Also, in Chapter 6, he presents a novel argument for the symmetry of the BRDF that's much more satisfying than &quot;physical objects have symmetric BRDFs, now moving on...&quot; that I've seen everywhere else.</p>
<p>I also found <a href="https://arxiv.org/pdf/1206.3301.pdf" rel="nofollow noreferrer">this paper by Lessig and Castro</a> that says:</p>
<blockquote>
<p>Radiative transfer describes the transport of electromagnetic energy
in macroscopicenvironments, classically when polarization effects are
neglected [37]. The theory originates in work by Bouguer [6, 7] and
Lambert [22] in the 18thcentury where light intensity and its
measurement were first studied systematically, cf. Fig. 1. In the
19thand early 20thcentury the theory was then extended to include
transport and scattering effects [26, 9, 42, 43]. To this day, however,
radiative transfer is a phenomenological theory with a mathematical
formulation that still employs the concepts introduced by Lambert in
the 18thcentury—and this despite the importance of the theory in a
multitude of fields, such as medical imaging, remote sensing,computer
graphics, atmospheric science, climate modelling, and astrophysics.</p>
</blockquote>
<p>I don't know any physics so the paper is way over my head, but maybe it's the holy grail?</p>
<p>Please flood me with references!</p>

CC BY-SA 4.0
2021-01-06T16:45:41.230Z
"Sadly, I don't think Veach develops them in enough detail." - what details are you missing? You can define every quantity $\Phi$, $E$, $B$, $L$ as a density of your energy measurement function $Q$. If you want something even more involved: https://arxiv.org/pdf/1205.4447.pdf
lightxbulb
CC BY-SA 4.0
2021-01-06T17:06:23.237Z
@lightxbulb thanks very much, this seems like a great reference! Re: Veach, he defines these cool measures and goes on to not use them? He doesn't use them do define radiance, for instance. And his definition of irradiance is informal, almost "in passing". Lack of equilibrium is also a problem...
user357269
CC BY-SA 4.0
2021-01-06T17:11:56.663Z
He gives an example of spectral radiance $L_{\lambda}$ as the Radon-Nikodym derivative of $Q$ with respect to the measure $\rho$. In a similar way you can define radiance $L$ as the Radon-Nikodym derivative of $Q$ w.r.t. a different measure (one that does not include $\lambda$). The definition for $\Phi, E, B, I$ is analogous, simply as Radon-Nikodym derivatives w.r.t. different measures. I do not know what equilibrium you are talking about.
lightxbulb
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2021-01-06T18:45:45.777Z
@lightxbulb I need to read it again more carefully, will get back to you
user357269
CC BY-SA 4.0
2021-01-06T18:46:48.300Z
@lightxbulb Re: equilibrium, I'm reading pbrt, which contains this paragraph: "Steady state: Light in the environment is assumed to have reached equilibrium, so its radiance distribution isn’t changing over time. This happens nearly instantaneously with light in realistic scenes, so it is not a limitation in practice."
user357269
CC BY-SA 4.0
2021-01-06T19:07:38.893Z
Yes, in the rendering equation you assume that light travels instantaneously. $L$ is the solution of the equation if you assume such instantaneous travel. I believe that Veach discusses this in the very first chapter, where he states the assumptions on which the model is based, which ignore wave effects, quantum effects, and even some geometric optics effects.
lightxbulb
CC BY-SA 4.0
2021-01-09T09:24:32.047Z
Did this answer your questions or do you have more?
lightxbulb
CC BY-SA 4.0
2021-01-20T17:15:27.610Z
@lightxbulb Just digested the arxiv paper: the big issue there is that the authors don't explain how radiance is related to radiative transfer (in fact they emphasize that it isn't). Lessig's general formulation of radiative transfer (in his thesis and the paper I quote) is too technical for me to grasp at this point
user357269
CC BY-SA 4.0
2021-01-20T19:34:03.053Z
"in fact they emphasize that it isn't" which part are you referring to? I still don't really understand what you expect as an answer. I see no issue with the radiance as a Radon-Nikodym derivative in Veach's thesis. What's your issue with it?
lightxbulb
CC BY-SA 4.0
2021-01-21T11:06:18.033Z
My issue is the same: they don't explain why radiance is preserved along straight lines. This should somehow follow from geometric optics. Veach's Radon-Nikodym derivative and Lessig's 2-form define radiance in the context of measurements, but not in the context of transport. See Remark 1 in Geometry of Radiative Transfer (linked above). Unless I missed it, Veach just takes preservation as given?

When you look at the path of light in a volume with continously varying refractive index, the concept of radiance is still meaningful (measurements) but is not preserved along the path (transport)
user357269
CC BY-SA 4.0
2021-01-21T12:39:26.533Z
Yes this follows from geometric optics, it is called conservation of energy. It is taken to be true since at the end of the day you are modeling a physical setting where this is true (considering the assumptions made). "When you look at the path of light in a volume with continuously varying refractive index" - Veach mentions that he does not model this setting, since it is computationally a lot more expensive, see page 76 in his thesis, and also chapter 1. Transport in Veach is handled through the propagation and scattering operators.
lightxbulb
CC BY-SA 4.0
2021-01-21T13:33:57.853Z
Right, that's the sort of thing that's missing from my ray tracing books - some argument, possibly based on conservation of energy - explaining why radiance is conserved. I am interested in references where this is justified. (Veach doesn't seem to be one of them). And I think it's interesting (if not directly practical) that something more general that Lessig calls "light energy density" is conserved by reflections/refractions too.
user357269
CC BY-SA 4.0
2021-01-21T14:26:38.650Z
As mentioned, it's simply energy conservation: https://physics.stackexchange.com/questions/177775/why-does-radiance-remain-constant-along-rays-of-light-through-empty-space
if you have assumed vacuum. Veach cites Nicodemus regarding that, and some other books.
lightxbulb
CC BY-SA 4.0
2021-01-21T16:55:11.877Z
Thanks! The answer https://physics.stackexchange.com/a/547998/284109 was actually the basis of my derivation of conservation of radiance above. I wanted to remove the infinitesimals. Just looked at Nicodemus' paper "Radiance", will digest
user357269
CC BY-SA 4.0
2021-01-21T20:42:43.853Z
You don't need to get rid of differentials. Also considering there's already established notation and theory, I would suggest to stick to it. Anyways, I hope everything should be clear at this point?
lightxbulb
<raytracing><3d>
Mathematical Foundations of Radiative Transfer
user357269
# Mathematical Foundations of Radiative Transfer

(Asked by: username_0 on 2021-01-06)

I'm new to learning about ray tracing, but have been fairly confused by the mathematical foundations of it all. Specifically of radiometry/radiative transfer. I've internet searched a lot and skimmed a ton of references, but still very confused.

Here is a list of some of the things I have had to struggle a lot to understand:

1. Irradiance is a function of 3 variables $E(p,n, \mathrm{in}/\mathrm{out})$ and it's not really the derivative of anything with respect to anything else (in the function sense)
2. If $p$ is a point on an actual surface in a scene, irradiance doesn't make sense in directions that aren't the surface normal, except perhaps as limit.
3. Radiance is not meaningfully the derivative of irradiance (Wikipedia says it's the derivative w.r.t. étendue, which is a great way to get more confused).
4. Radiance can be defined without ever using the words "projected area", "solid angle" or "subtend".
5. You don't need to draw pictures with infinitesimals to think about radiance. I see infinitesimals in pictures a lot, e.g. [Figures 1 and 2 Wikipedia's Lambert's cosine law article](https://en.wikipedia.org/wiki/Lambert%27s_cosine_law#Details_of_equal_brightness_effect).
6. Radiance is preserved along paths (in vacuum). Is this an axiom of the theory or a theorem?

I've found the following approach more insightful than the usual presentations:

1. If you have an oriented surface, you can define the flux entering it $\Phi(S)$.
2. Irradiance is something that's integrated into flux:
   $$ \Phi(S) = \int\_SE(p, n\_p, \mathrm{in}) \mathrm{d}S(p)$$
3. For two surfaces, we can define direct flux transport between them: the power that exist $S\_1$ and directly enters $S\_2$, $\Phi(S\_1, S\_2)$.
4. "Flux transport density" (not sure if correct term) is something that's integrated into flux transport. It is the function of 4 variables $K(p\_\text{source}, n\_\text{source}, p\_\text{target}, n\_\text{target})$
   $$ \Phi(S\_1,S\_2) = \int\_{S\_2} \int\_{S\_1} K(p, n\_p, q, n\_q) \mathrm{d}S\_1(p) \mathrm{d}S\_2(q)$$
5. "Light hull principle" consider two oriented disks $D\_p = D(p, n\_p, r\_p)$ and $D\_q = D(q, n\_q, r\_q)$ that are sufficiently small.

   1. If $(q-p) \cdot n\_p \le 0$ then $\Phi(D\_p, D\_q) = 0$
   2. Taking the limit $r\_p \to 0$ shows that $(q-p) \cdot n\_p$ implies $K(p, n\_p, q, n\_q) = 0$
   3. If $(p-q) \cdot n\_q \le 0$ then $\Phi(D\_p, D\_q) = 0$
   4. Taking the limit $r\_q \to 0$ shows that $(p-q) \cdot n\_q$ implies $K(p, n\_p, q, n\_q) = 0$
   5. Let's assume $(q-p) \cdot n\_p > 0$ and $(p-q) \cdot n\_q > 0$ going forward. Maybe it's simplier to write $|\cos(\theta\_p)| < \pi/2$, $|\cos(\theta\_q)| < \pi/2$
   6. Let's form the "light hull" of $D\_p$ and $D\_q$. This is a funny truncated cone like shape defined by $$H(D\_p, D\_q) = \{ t p' + (1-t) q' | p' \in D\_p, q' \in D\_q, t \in [0, 1] \}$$ [![enter image description here](https://i.sstatic.net/qTJlU.png)](https://i.sstatic.net/qTJlU.png) It may later make sense to allow $t$ to go negative/greater than 1, extending the hull a bit.
   7. Let $a$ be a point on the segment $pq$. Intersecting the light hull with a plane through $a$ gives as circular shaped section:
      $$H(a, n\_a, D\_p, D\_q) = \{ b | (b - a) \cdot n\_a = 0 \} \cap H(D\_p, D\_q) $$
   8. In free space we have $$\Phi(D\_p, D\_q) = \Phi(D\_p, H(a, n\_a, D\_p, D\_q)) = \Phi(H(a, n\_a, D\_p, D\_q), D\_q)$$ I call this the *light hull principle* and is basically just geometry saying "if I put a transparent screen in the light's way, all the light will go through it".
   9. Using the light hull principle, we see that we can replace $D\_q$ with a similar disk-ish shape that's orthogonal to $pq$: $$ \Phi(D\_p, D\_q) = \Phi(D\_p, H(q, p-q, D\_p, D\_q))$$ Let's call this shape $D\_q'$.
      [![enter image description here](https://i.sstatic.net/or5th.png)](https://i.sstatic.net/or5th.png)
      For small $r\_p^2$, $r\_q^2$, the left hand side is
      $$ \Phi(D\_p, D\_q) \approx K(p, n\_p, q, n\_q) \pi^2 r\_p^2 r\_q^2$$
      Or more formally
      $$ \frac{\partial^4 \Phi(D\_p, D\_q) }{(\partial r\_p)^2 (\partial r\_q)^2} \bigg|\_{r\_p = r\_q = 0} = 4 \pi^2 K(p, n\_p, q, n\_q) $$
      I don't want to go into too much detail here, but it's possible to prove for the right hand side that $$ \frac{\partial^4 \Phi(D\_p, H(q, p-q, D\_p, D\_q)) }{(\partial r\_p)^2 (\partial r\_q)^2} \bigg|\_{r\_p = r\_q = 0} = 4 \pi^2 \cos(\theta\_q) K\left(p, n\_p, q, \frac{p-q}{||p-q||} \right) $$
      It follows that $$ K(p, n\_p, q, n\_q) = \cos(\theta\_q) K\left(p, n\_p, q, \frac{p-q}{||p-q||} \right)$$
   10. Repeating the same argument at $p$ instead of $q$, we get
       $$ K(p, n\_p, q, n\_q) = \cos(\theta\_p) \cos(\theta\_q) K\left(p, \frac{q-p}{||p-q||}, q, \frac{p-q}{||p-q||} \right)$$
   11. Assuming now that the two disks are directly facing each other, i.e. $n\_p = \frac{q-p}{||p-q||} = -n\_q$, we can apply the light hull principle and take a limit/derivative as $r\_p, r\_q \to 0$ to show the inverse square law
       $$ K(p, n\_p, p+r n\_p, -n\_p) = \frac{1}{r^2} K(p, n\_p, p+n\_p, - n\_p)$$
6. Let $r = ||p-q||$ and $n = \frac{q-p}{r}$. We now know that
   $$ \begin{aligned}
   K(p, n\_p, q, n\_q) &= K(p, n, p+n, -n) \frac{\cos(\theta\_p) \cos(\theta\_q)}{r^2} \\
   &=: L(p,n,\mathrm{out}) \frac{\cos(\theta\_p) \cos(\theta\_q)}{r^2} \\
   &=: L(q,-n,\mathrm{in}) \frac{\cos(\theta\_p) \cos(\theta\_q)}{r^2}
   \end{aligned}
   $$
   Where $L$ is the defined by the equation above. We call $L$ radiance.
7. Taking the limit $p \to q$ shows that $L(q, n, \mathrm{out}) = L(q, -n, \mathrm{in}) = L(p, n, \mathrm{out})$. Therefore radiance is constant along rays (and $K$ is translation invariant).

The significance of this is that we have defined radiance and proved it invariant by using a simple geometric principle on finite (non-infinitesimal) surfaces and taking appropriate limits. It's a lot more work than saying irradiance over projected surface area times solid angle, but to me much more intuitive.

The light hull principle is an axiom/key "physical" fact, and everything else is a theorem.

My question is; is there a reference that deals with radiative transfer in general in a similar way as I've just dealt with radiance?

The closest thing I could find is [Veach's PhD thesis](https://graphics.stanford.edu/papers/veach_thesis/) where he defines some interesting concepts using measure theory in Chapter 3 and its appendix. The idea is that a scene is a system of $N$ non-interacting photons moving around. Radiometric quantities can then be defined by a snapshot of this system in time. I think it's a really cool idea and would love to see more of this. To me these are the true foundations. Sadly, I don't think Veach develops them in enough detail. (And I also don't see how a finite number of photons as Veach models them could create equilibrium in the sense that irradiance is independent of time)

Also, in Chapter 6, he presents a novel argument for the symmetry of the BRDF that's much more satisfying than "physical objects have symmetric BRDFs, now moving on..." that I've seen everywhere else.

I also found [this paper by Lessig and Castro](https://arxiv.org/pdf/1206.3301.pdf) that says:

> Radiative transfer describes the transport of electromagnetic energy
> in macroscopicenvironments, classically when polarization effects are
> neglected [37]. The theory originates in work by Bouguer [6, 7] and
> Lambert [22] in the 18thcentury where light intensity and its
> measurement were first studied systematically, cf. Fig. 1. In the
> 19thand early 20thcentury the theory was then extended to include
> transport and scattering effects [26, 9, 42, 43]. To this day, however,
> radiative transfer is a phenomenological theory with a mathematical
> formulation that still employs the concepts introduced by Lambert in
> the 18thcentury—and this despite the importance of the theory in a
> multitude of fields, such as medical imaging, remote sensing,computer
> graphics, atmospheric science, climate modelling, and astrophysics.

I don't know any physics so the paper is way over my head, but maybe it's the holy grail?

Please flood me with references!

**Tags:** raytracing, 3d

> **Comments:**
> **username_1**: "Sadly, I don't think Veach develops them in enough detail." - what details are you missing? You can define every quantity $\Phi$, $E$, $B$, $L$ as a density of your energy measurement function $Q$. If you want something even more involved: https://arxiv.org/pdf/1205.4447.pdf  
> **username_0**: @username_1 thanks very much, this seems like a great reference! Re: Veach, he defines these cool measures and goes on to not use them? He doesn't use them do define radiance, for instance. And his definition of irradiance is informal, almost "in passing". Lack of equilibrium is also a problem...  
> **username_1**: He gives an example of spectral radiance $L_{\lambda}$ as the Radon-Nikodym derivative of $Q$ with respect to the measure $\rho$. In a similar way you can define radiance $L$ as the Radon-Nikodym derivative of $Q$ w.r.t. a different measure (one that does not include $\lambda$). The definition for $\Phi, E, B, I$ is analogous, simply as Radon-Nikodym derivatives w.r.t. different measures. I do not know what equilibrium you are talking about.  
> **username_0**: @username_1 I need to read it again more carefully, will get back to you  
> **username_0**: @username_1 Re: equilibrium, I'm reading pbrt, which contains this paragraph: "Steady state: Light in the environment is assumed to have reached equilibrium, so its radiance distribution isn’t changing over time. This happens nearly instantaneously with light in realistic scenes, so it is not a limitation in practice."  
> **username_1**: Yes, in the rendering equation you assume that light travels instantaneously. $L$ is the solution of the equation if you assume such instantaneous travel. I believe that Veach discusses this in the very first chapter, where he states the assumptions on which the model is based, which ignore wave effects, quantum effects, and even some geometric optics effects.  
> **username_1**: Did this answer your questions or do you have more?  
> **username_0**: @username_1 Just digested the arxiv paper: the big issue there is that the authors don't explain how radiance is related to radiative transfer (in fact they emphasize that it isn't). Lessig's general formulation of radiative transfer (in his thesis and the paper I quote) is too technical for me to grasp at this point  
> **username_1**: "in fact they emphasize that it isn't" which part are you referring to? I still don't really understand what you expect as an answer. I see no issue with the radiance as a Radon-Nikodym derivative in Veach's thesis. What's your issue with it?  
> **username_0**: My issue is the same: they don't explain why radiance is preserved along straight lines. This should somehow follow from geometric optics. Veach's Radon-Nikodym derivative and Lessig's 2-form define radiance in the context of measurements, but not in the context of transport. See Remark 1 in Geometry of Radiative Transfer (linked above). Unless I missed it, Veach just takes preservation as given?  When you look at the path of light in a volume with continously varying refractive index, the concept of radiance is still meaningful (measurements) but is not preserved along the path (transport)  
> **username_1**: Yes this follows from geometric optics, it is called conservation of energy. It is taken to be true since at the end of the day you are modeling a physical setting where this is true (considering the assumptions made). "When you look at the path of light in a volume with continuously varying refractive index" - Veach mentions that he does not model this setting, since it is computationally a lot more expensive, see page 76 in his thesis, and also chapter 1. Transport in Veach is handled through the propagation and scattering operators.  
> **username_0**: Right, that's the sort of thing that's missing from my ray tracing books - some argument, possibly based on conservation of energy - explaining why radiance is conserved. I am interested in references where this is justified. (Veach doesn't seem to be one of them). And I think it's interesting (if not directly practical) that something more general that Lessig calls "light energy density" is conserved by reflections/refractions too.  
> **username_1**: As mentioned, it's simply energy conservation: https://physics.stackexchange.com/questions/177775/why-does-radiance-remain-constant-along-rays-of-light-through-empty-space if you have assumed vacuum. Veach cites Nicodemus regarding that, and some other books.  
> **username_0**: Thanks! The answer https://physics.stackexchange.com/a/547998/284109 was actually the basis of my derivation of conservation of radiance above. I wanted to remove the infinitesimals. Just looked at Nicodemus' paper "Radiance", will digest  
> **username_1**: You don't need to get rid of differentials. Also considering there's already established notation and theory, I would suggest to stick to it. Anyways, I hope everything should be clear at this point?  

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## 2 answers

### Answer by username_0 on 2021-01-22. Score: 1

[I will try to keep this answer updated as I learn more]

The best reference on the foundations of light transport seems to be Arvo's [Transfer Equations in Global Illumination](http://www0.cs.ucl.ac.uk/research/vr/Projects/VLF/vlfpapers/sampling/Arvo_J__Transfer_Equations_in_Global_Illumination__SIGGRAPH_course_notes_1993.pdf) (~30 pages)

Some things he doesn't cover:

* Argument for symmetry of BRDF (see Veach)
* Transfer in media with continuously varying refractive index

The state of the art model for Light Transport seems to be [Christian Lessig's PhD thesis](http://isgwww.cs.uni-magdeburg.de/graphics/projects/dissertation/index.html) (~600 pages). The theory is developed in terms of differentiable manifolds and Hilbert spaces and is much more technical than Arvo's note.

Lessig carries out translation from the mathematical theory to algorithms (e.g. path tracing, photon mapping) in Section 4.3.

> **Comments:**
> **username_2**: I don't know whether you've already seen this, but I like very much Arvo's Theorem 4 (pg 24) of his 1995 thesis, where he derives invariance of radiance along lines, as it were, "just" by keeping track of a particular set of photons as time elapses" https://www.cs.cornell.edu/courses/cs667/2005sp/readings/ArvoThesis.pdf  

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### Answer by username_1 on 2021-01-06. Score: 1

This is almost verbatim from Veach. Let the trajectory space be $\mathbb{P} = \mathbb{R} \times \mathcal{M} \times \mathcal{S}^2 \times \mathbb{R}^+$, corresponding respectively to: time, set of points of all scene surfaces, unit sphere (directions), wavelength. Let $\mathcal{P}$ be the set of measurable subsets of $\mathbb{P}$ and we have a measure $\rho$. Let $\rho = l \times A \times \sigma^{\perp} \times l^+$. Where $l, l^+$ are the Lebesgue measures on $\mathbb{R}$ and $\mathbb{R}^+$, $A$ is the Lebesgue measure on $\mathcal{M}$, and $\sigma^{\perp}$ is the projected solid angle measure $(d\sigma^{\perp}(\omega) = |\cos\theta|\sin\theta\,d\theta\,d\phi$). Then given the radiant energy measure $Q(\mathcal{T}, \mathcal{D}, \Omega, \Lambda) \in [0,\infty]$ ($\mathcal{T} \times \mathcal{D} \times \Omega \times \Lambda \in \mathcal{P}$), we can define the density $L\_{\lambda}$ of $Q$ with respect to the measure $\rho$ as:

\begin{equation}
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int\_{\Lambda}\int\_{\Omega}\int\_{\mathcal{D}}\int\_{\mathcal{T}}L\_{\lambda}(t, x, \omega,\lambda)d\rho(t,x,\omega,\lambda).
\end{equation}

Then the density $L\_{\lambda}$ is the Radon-Nikodym derivative of $Q$ w.r.t. $\rho$:

\begin{equation}
\frac{dQ}{d\rho}(t, x, \omega,\lambda) := L\_{\lambda}(t, x, \omega,\lambda).
\end{equation}

We can do a similar thing with respect to the appropriate measures:

Radiant flux:
\begin{gather}
\rho\_1 = l \\
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int\_{\mathcal{T}}\Phi(t, \mathcal{D}, \Omega, \mathcal{\Lambda})\,d\rho\_1(t) \\
\frac{dQ}{d\rho\_1}(t, \mathcal{D}, \Omega, \mathcal{\Lambda}) := \Phi(t, \mathcal{D}, \mathcal{\Omega}, \mathcal{\Lambda}).
\end{gather}

Irradiance:
\begin{gather}
\rho\_2 = l\times A \\
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int\_{\mathcal{D}}\int\_{\mathcal{T}}E(t, x, \Omega, \mathcal{\Lambda})\,d\rho\_2(t,x) \\
\frac{dQ}{d\rho\_2}(t, x, \Omega, \mathcal{\Lambda}) := E(t, x, \Omega, \mathcal{\Lambda}).
\end{gather}

Intensity:
\begin{gather}
\rho\_3 = l\times \sigma \\
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int\_{\Omega}\int\_{\mathcal{T}}I(t, \mathcal{D}, \omega, \mathcal{\Lambda})\,d\rho\_3(t,\omega) \\
\frac{dQ}{d\rho\_3}(t, \mathcal{D}, \omega, \mathcal{\Lambda}) := I(t, \mathcal{D}, \omega, \mathcal{\Lambda}).
\end{gather}

Radiance:
\begin{gather}
\rho\_4 = l\times A \times \sigma^{\perp} \\
Q(\mathcal{T}, \mathcal{D}, \Omega, \mathcal{\Lambda}) = \int\_{\Omega}\int\_{\mathcal{D}}\int\_{\mathcal{T}}L(t, x, \omega, \mathcal{\Lambda})\,d\rho\_4(t,x,\omega) \\
\frac{dQ}{d\rho\_4}(t, x, \omega, \mathcal{\Lambda}) := L(t, x, \omega, \mathcal{\Lambda}).
\end{gather}

> **Comments:**
> **username_0**: Just a comment: $\rho_4$ isn't really a product of measures since the $\sigma^\perp$ measure depends on the surface orientation. It's some kind of twisted product, see https://en.wikipedia.org/wiki/Markov_kernel#Semidirect_product or https://en.wikipedia.org/wiki/Disintegration_theorem  
> **username_1**: @username_0 The measure $\sigma^{\perp}_x(\Omega)$ is defined on page 77, eq (3.1) in Veach's thesis as $\sigma^{\perp}_x(\Omega) = \int_{\Omega}|\omega \cdot N_x|\,d\sigma(\omega)$. So yes, $\sigma^{\perp}$ depends on $x$, but that isn't really an issue.  

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