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

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 $$\omega_{\mathbf{h}}$$.

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.