What is the purpose of the projected solid angle $dA^\perp$ term in the definition of Radiance?

Question

In the definition of Radiance, which according to PBRT is defined as

flux per unit solid angle $d\omega$ per unit projected area $dA^\perp$

$$L_i=\frac{d\Phi}{d\omega\ dA^\perp}$$

My original assumption was that the $dA^\perp$ term is used to weaken the irradiance contribution due to the incident angle (which I assume is just Lambert's cosine law). However, in the rendering equation, we've already explicitly apply Lambert's law with the $\cos\theta$ term:
$$\int_\Omega L_i(\omega_{i}) \, f_r(\omega_{i}\rightarrow \omega_{o}) \, \cos\theta_{i} \, d\omega_{i}$$

I assume we arn't applying Lambert's law twice, so what exactly is this $dA^\perp$ term?

Answer 1

Your main idea is more or less correct. The cosine hidden in the projected area measure $dA^\perp = dA\cos(θ)$ compensates the weakening of irradiance due to incident angle (the Lambert's cosine law). This makes radiance independent from the incident angle. My guess is that the main motivation was to make it more practical to work with.

The cosine in the rendering equation comes from the definition of BRDF:

$$ f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{\mathrm{d}E\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\cos\theta_{i}\mathrm{d}\omega_{i}} $$

Which can be rewritten as

$$ \mathrm{d}L_{o}\left(\omega_{o}\right) =f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Which can be integrated over the hemisphere the get the radiance reflected in a given direction $\omega_o$

$$ \int_\Omega f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Voila, the rendering equation!

Conclusion: Althought both cosines come from the same fundamental priciple, they serve different purposes.

Answer 2

Since you seem to want an explanation in terms of irradiance, consider both the radiance and irradiance definitions: $$E = \frac{d\Phi}{dA}, \quad L = \frac{d^2\Phi}{d\omega dA^{\perp}}$$ We can reformulate the radiance definition as: $$d^2\Phi = L \cos\theta d\omega dA$$ Integrating both sides over the solid angle $d\omega$ (in the sense of Lebesgue-Stieltjes) yields: $$d\Phi = dA \int_{\Omega}{L\cos\theta d\omega}$$ And thus: $$E = \frac{d\Phi}{dA} = \int_{\Omega}{L\cos\theta d\omega}$$ So you can think of irradiance as the integral of radiance with respect to the solid angle measure, or vice versa, you can think of radiance as the derivative of irradiance with respect to the solid angle measure. The other answers also provide the link to the rendering equation. So no, you do not have the cosine two times. Though if you were to rewrite the rendering equation in its area formulation you do get another cosine term (which is due to the projection of a differential area patch onto the hemisphere). This uses the relationship: $$d\omega = \frac{\cos\theta_y}{r^2}dA(y)$$ Then one can rewrite the rendering equation as: $$L_o(x_0 \rightarrow x_{-1}) = \int_{S}{f(x_1 \rightarrow x_0 \rightarrow x_{-1}) L(x_1 \rightarrow x_0) \cos\theta_{x_0} \frac{\cos\theta_{x_1}}{||x_1-x_0||^2} V(x_0,x_1) \,dA(x_1)}$$ Where the visibility term $V(x_0,x_1)$ appears since it was originally implicit in the solid angle formulation. This is because you cannot get radiance on a straight line between point $x_0$ and $x_1$ if there is an occluder in-between, that's automatically taken care of in the solid angle formulation by using $L_i(x_0,\omega = x_0 \rightarrow x_1)$, which gives us the radiance arriving at $x_0$ from direction $\omega$, note that this considers the first point from which this radiance arrives, and not specifically $x_1$. Formally $L_i(x_0, \omega) = L(r(x_0,\omega), -\omega))$, note the raycasting operator $r$ in which the visibility term is implicitly hidden, since $r(x_0,x_0 \rightarrow x_1) \ne x_1$ if there is an occluder inbetween.

Answer 3

The two cosine terms basically refer to the foreshortening effect which says:

The irradiance falling onto a patch must be modified by the cosine of the angle of the patch to the source.

Now from the definition of the irradiance:

irradiance is flux per unit area, therefore:

Reordering it gives:

which is the definition of radiance, with the same cosine term that was in definition of irradiance(integrating irradiance on the hemisphere and multiplying it by BRDF gives rendering equation).

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So both cosine terms refer to one effect but I think you are right that there are two cosine terms in rendering equation. The first which is visible in equation compensates for the weakening of irradiance on the receiver patch, and the second cosine implicit in radiance definition corresponds to the radiance of the source patch subtended foreshortened surface area. In path tracing this radiance (Li) either comes from a direct light with a known radiance value or an indirect which is unknown and will be solved recursively. But if you were to calculate the source radiance you need to multiply both terms too.