Confusion of deriving the Jacobian of the reflection transformation in Walter et. al's 2007 paper

Question

When deriving the Jacobian of the reflection transformation in Walter et. al's 2007 paper:

First we have a macrosurface BSDF which is:

$f_{s}(\mathbf{i}, \mathbf{o}, \mathbf{n})=\int\left|\frac{\mathbf{i} \cdot \mathbf{m}}{\mathbf{i} \cdot \mathbf{n}}\right| f_{s}^{m}(\mathbf{i}, \mathbf{o}, \mathbf{m})\left|\frac{\mathbf{o} \cdot \mathbf{m}}{\mathbf{o} \cdot \mathbf{n}}\right| G(\mathbf{i}, \mathbf{o}, \mathbf{m}) D(\mathbf{m}) d \omega_{m}$

Then we need an equation for $f_s^m$, the microsurface BSDF. And the paper gives:

$f_{s}^{m}(\mathbf{i}, \mathbf{o}, \mathbf{m})=\rho \frac{\delta_{\omega_{o}}(\mathbf{s}, \mathbf{o})}{|\mathbf{o} \cdot \mathbf{m}|}$

Then they rewrite it to fit in the macrosurface BSDF by changing the associated measure of the Dirac delta function:

$f_{s}^{m}(\mathbf{i}, \mathbf{o}, \mathbf{m})=\rho(\mathbf{i}, \mathbf{m}) \frac{\delta_{\omega_{m}}(\mathbf{h}(\mathbf{i}, \mathbf{o}), \mathbf{m})}{|\mathbf{o} \cdot \mathbf{m}|}\left\|\frac{\partial \omega_{\mathbf{h}}}{\partial \omega_{\mathbf{0}}}\right\|$.

And my question is, why should the jacobian be a $\left\|\frac{\partial \omega_{\mathbf{h}}}{\partial \omega_{\mathbf{o}}}\right\|$ instead of a $\left\|\frac{\partial \omega_{\mathbf{m}}}{\partial \omega_{\mathbf{o}}}\right\|$? I noticed that the associated mesure changed from $\omega_{\mathbf{o}}$ to $\omega_{\mathbf{m}}$, instead of $\omega_{\mathbf{h}}$.

Answer 1

There is a functional relationship between $\mathbf{h}$ and $\mathbf{o}$ (and also $\mathbf{i}$) whose Jacobian is being taken. But there's no prior functional relationship between $\mathbf{m}$ and $\mathbf{o}$; they are unrelated variables, until $\mathbf{m}$ is constrained to equal $\mathbf{h}$ by the delta function $\delta(\mathbf{h}, \mathbf{m})$.

So, really you could write the Jacobian either way, but in my mind it makes a little more sense to write it as $\| \partial \omega_\mathbf{h} / \partial \omega_\mathbf{o} \|$ as that's the underlying functional relationship.