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I am reading through Kajiya - Ray Tracing Volume Densities paper. And I've already got stuck into section 2. I wonder if there's a mistake in that equation.

I'll quote the relevant bit

The quantity to be calculated in a scattering problem is the energy per unit solid angle per unit area $$ dE = I(x,\omega)\sin\vartheta d\omega d\sigma $$ This quantity is called the intesity of radiation at a point $x$ in the direction of the solid angle $d\omega$. The scattering equation can be derived by considering a differential cylindrical volume $dV = d\sigma ds$, where $d\sigma$ is the cross section of the cylinder and $ds$ is the length. If we follow a pencil of radiation along the length of the cylinder, we find the difference in intensity between the two ends is given by $$ dI = -absorbed + emitted = -k\rho ds d\sigma d\omega + j \rho ds d\sigma d\omega \;\;\; (2.1) $$ where $\rho$ is the density of the matter in the volume element $k$ is the absorbption coefficient...

I believe there's a mistake in the absorbed intensity expression, there's a reference quoted at the very beginning of the section where this formulation is taken from (the reference is Chandrasekhar - Radiative Transfer). I cannot to manage what later the paper defines as scattering equation

$$ \frac{1}{k\rho} s\cdot\nabla_xI(x,s) - I(x,s) + \frac{1}{4\pi} \int_{\left\lVert s \right\rVert = 1} p(s,\overline{s})I(x,\overline{s}) d\overline{s} $$

I believe in the reference (Chandrasekhar) I've found what was used for Kajiya's derivation, I'll put an image just in case not everyone can see it from the preview:

enter image description here

In Chandrasekhar's formula 44 there's a $I_\nu$ factor, which should correspond to $I(x,s)$ in Kajiya's. And formula (48) in Chandrasekhar is exactly the same as Kajiya's scattering equation, except that the former takes into account in his formulation time and wavelength. But apart from this they're the same.

So I suspect there's a typo in Kajiya's. I think in general absorption is formulate as some form of exponatial decay of the form $f'(x) = -\alpha f(x)$, which is used in Chandrasekhar. In summary I think equation (2.1) from Kajiya should be written as

$$ dI d\sigma d\omega = -k\rho I ds d\sigma d\omega + j \rho ds d\sigma d\omega $$

Can you confirm if my suspicion is correct? Thank you

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I believe your suspicion is correct,otherwise dimensions won't match.

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  • $\begingroup$ Thank you for checking $\endgroup$ Mar 14, 2023 at 2:11

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