Measure (or Jacobian) conversion for direction sampling

Question

If in the path tracing use case, I have some sampling strategies to sample a scattering direction in a 2D space ($\theta, \phi$). Of course, I need to convert it to a 3D direction vector (with length being 1). My question is, is the PDF of this 3D direction vector the same as that of $(\theta, \phi)$? If I recall correctly, there should be a Jacobian term for the conversion from spherical coords to 3D Cartesian space, so the PDF is probably different?

If the sampling PDF for $\theta$ is $p_1(\theta), \theta \in [0, \pi)$ (volumetric scattering requires sampling on the entire sphere), and PDF for sampling $\phi$ is $p_2(\phi), \phi \in [0, 2\pi)$.

• What will be the PDF of the 3D direction vector $p_d(\omega)$?
• What about the case that $\theta, \phi$ is sampled from a 2D joint distribution $p(\theta, \phi)$ (like, sampling from precomputed tabulations), is there any difference from the first case?
• And if I am gonna evaluate the MC integral here, is the following correct? $$ \text{scattering throughput}=f(\omega, \omega') \times C / p_d(\omega), C=\begin{cases} \sigma_s, \text{volumetric}\\ \omega^T \omega',\text{surface} \end{cases} $$

Answer 1

If your map $\omega:[0,2\pi) \times [0,\pi] \to \mathbb{S}^2\subset \mathbb{R}^3$ is the standard spherical coordinates map:

$$\omega = (\cos\phi \sin\theta, \sin\phi \sin\theta, \cos\theta),$$

then the Jacobian is:

$$J = \frac{\partial \omega}{\partial (\phi, \theta)} = \begin{bmatrix} \uparrow & \uparrow \\\frac{\partial\omega}{\partial \phi} & \frac{\partial\omega} {\partial\theta} \\ \downarrow & \downarrow \end{bmatrix} = \begin{bmatrix} -\sin\phi \sin\theta & \cos\phi\cos\theta \\ \cos\phi\sin\theta & \sin\phi \cos\theta \\ 0 & \sin\theta\end{bmatrix}.$$

The area measure on the sphere is $d\sigma(\omega) = \sqrt{\det J^T J} \,d\lambda(\phi)d\lambda(\theta) = |\sin\theta|\,d\lambda(\phi)d\lambda(\theta)$. Here $\lambda$ is just the Lebesgue measure and it is often omitted. Now assume the probability density function for picking $(\phi,\theta)$ is $p_{\phi,\theta}$. Taking into account $d\sigma(\omega) = |\sin\theta|\,d\lambda(\phi)d\lambda(\theta)$ we have: $$p_{\phi,\theta}(\phi,\theta) \,d\lambda(\phi) d\lambda(\theta) = p_{\sigma}(\omega)\,d\sigma(\omega) \implies p_{\sigma}(\omega) = \frac{p_{\phi,\theta}(\phi,\theta)}{|\sin\theta|}.$$

Note that translations and rotations do not change the term $\sqrt{J^T J}$ since the derivative of a translation is zero, and a rotation is orthogonal $R^TR=I$ and thus $\sqrt{J^TR^TRJ} = \sqrt{J^TJ} = |\sin\theta|$. I am mentioning this since typically you integrate over the sphere not w.r.t. the standard coordinate axes, but some frame oriented around the normal, which can be implemented as $\omega' = R\omega$.

If $p_{\phi,\theta}$ is separable you can write it as $p_{\phi,\theta}(\phi,\theta) = p_{\phi}(\phi)p_{\theta}(\theta)$.

The change of variables in an integral over the unit sphere happens as follows:

\begin{align} I &=\int_{\mathbb{S}^2} L_i(x, \omega_i) f(\omega_o, x, \omega_i) (\omega_i \cdot n)\,d\sigma(\omega_i) \\ &= \int_{0}^{2\pi}\int_0^{\pi} L_i(x,\omega_i(\phi,\theta))f(\omega_o,x, \omega_i(\phi,\theta)) (\omega_i(\phi,\theta) \cdot n)|\sin\theta|\,d\lambda(\theta) d\lambda(\phi) \\ &= \int_0^1\int_0^1 L_i(x,\omega_i(u,v))f(\omega_o,x, \omega_i(u,v)) (\omega_i(u,v) \cdot n)\underbrace{|\sin\theta|\left|\det \frac{\partial(\phi,\theta)}{\partial (u,v)}\right|}_{1/p_{\sigma}(\omega_i)}\,d\lambda(u) d\lambda(v) \\ &= \int_0^1\int_0^1 \frac{L_i(x,\omega_i(u_j,v_j))f(\omega_o,x, \omega_i(u,v)) (\omega_i(u,v) \cdot n)}{p_{\sigma}(\omega_i(u,v))}\,d\lambda(u) d\lambda(v) \\ &\approx \frac{1}{N} \sum_{j=1}^N\frac{L_i(x,\omega_i(u_j,v_j))f(\omega_o,x, \omega_i(u_j,v_j)) (\omega_i(u_j,v_j) \cdot n)}{p_{\sigma}(\omega_i(u_j,v_j))}, \quad (u_j,v_j) \sim U[0,1]^2. \end{align}