ccetaw/taichi-sggx

Simple implementation of SGGX microflake model for validation use. Written in the framework of Taichi lang.

SGGX Microflake Model

This repo contains a simplest implementation of volume rendering including

• heterogeneous(homogeneous) volume rendering
• SGGX volume rendering, please refer to The SGGX Microflake Distribution

As it's for validation use only, only necessary components are included:

• perspective camera, with simple box reconstruction filter
• environment map
• voxel based volume representation
• transform could be applied on above components
• single scattering model, path tracing

To run this repo, installing taichi and numpy will be sufficient. Use

python main.py

to render.

Camera

We follow the OpenGL camera convention that the camera space is right-handed and the z-axis faces backwards. For convenience, we consider a image plane at focus and don't consider near and far clip. Please refer to OpenGL Transformation

Environment Map

We use an hdr image as environment map. The environment is considered placed so far away that rays with the same direction result in the same color. For more details, please refer to PBR book.

Heterogeneous Volume

BoundingBox	Homogeneous	Heterogeneous

We consider scattering, absorptive and non-emissive volume. For basics of volume rendering, please refer to PBR book.

For simplicity, the volume is confined in an AABB and the data is stored in voxels. Each voxel stores 2 3D vectors, standing for extinction coefficient and albedo. Trilinear interpolation is applied for points sampled in the volume. Also we consider uniform phase function, as we don't focus on heterogeneous volume.

With path tracing, we have

BoundingBox	Homogeneous	Heterogeneous

SGGX Microflake

The SGGX microflake model is implemented as in the supplement material of originial paper.

BoundingBox	SGGX

Renderer

The overall rendering equation is

$$ \begin{align}L(\mathbf{x}, \omega) &= T_r(\mathbf{x}, \mathbf{x}_{\infty}(\omega, \mathbf{x}))L_e(\omega) \\ &+ \int_0^{z} T_r(\mathbf{x}, \mathbf{x}_t)\sigma_t(\mathbf{x}_t)\alpha(\mathbf{x}_t)\int_{S^2} f_p(\mathbf{x}_t,\omega,\omega')L_e(\omega') d\omega'dt \end{align} $$

The radiance at point $\mathbf{x}$ from direction $\omega$ is composed of 2 terms. THe first term is the radiance from the background, with $T_r(\mathbf{x},\mathbf{y})$ being the transmittance between points $\mathbf{x}$ and $\mathbf{x}$, $L_e(\omega)$ being the radiance from the environment. As we mentioned, the environment radiance only depends on direction. And $\mathbf{x}_\infty(\omega)$ is the point at infinity in direction $\omega$ from point $\mathbf{x}$.

The second term is the accumulated scattering radiance. $\sigma_t$ is the extincition coefficient, $\alpha$ is the albedo. In SGGX rendering, these two terms may depend on directions. The inner integration is the radiance from all directions scattered to direction $\omega$. $f_p$ is the phase function.