Is warp-area sampling method easier to understand and implement in path space?

[LiuLinyun]

I'm reading your paper Unbiased Warped-Area Sampling for Differentiable Rendering.

I found the key idea is constructing the warp field $\vec{\mathcal{V}}_{\theta}(\mathbf{x})$ satisifying the two conditions.

In your paper, it is designed and implemented as $\vec{\mathcal{V}}_{\theta}^{(\mathrm{filtered})}(\omega)$ in solid angle space.

Can it be desinged and implemented as $\vec{\mathcal{V}}_\theta(\mathbf{x})=\frac{\partial} {\partial\theta}\mathbf{x}$ , where $\mathbf{x}$ is in mesh triangles and interploated by its three vertices $(\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3)$ , and then solve rendering equation in path space using veach's path integral formulation ?

[LiuLinyun]

I did not have a correct understanding of it, since only interpolated by triangle's vertices' $\mathcal{V}_{\theta}$ causes discontinuous. But I should think how could it be implemented in path space.