Macrofacet Theory for Gaussian Process Statistical Surfaces

TL;DR

The approach converts Gaussian process implicit surfaces into classic exponential media to render surfaces, volumes and in-betweens without realization to enable efficient rendering with performance improvement compared to realization-based approaches, while bridging microfacet models and Gaussian processes theoretically.

Abstract

We present macrofacet theory, taking microfacet theory from micro-space to macro-space by stretching a surface to a volume to make it have microfacet characteristic in marco-space. In this way, we have a macroscopic microfacet formulation that uses a classic exponential participating medium. Meanwhile, we observe that traditional microfacet models are equivalent to Gaussian processes in definition but ignore the correlation along the geometric normal of macro-surface. We extend microfacet theory so that macrofacet can handle this problem and represent Gaussian process implicit surfaces in a statistical way. As a result, our approach converts Gaussian process implicit surfaces into classic exponential media to render surfaces, volumes and in-betweens without realization. These enable efficient rendering with performance improvement compared to realization-based approaches, while bridging microfacet models and Gaussian processes theoretically. Moreover, our approach is easy to implement and friendly for artists.

Figures (13)

• Figure 1: Extension of a macro-surface. The top row illustrates the micro-scale extension of macrofacet, while the bottom row illustrates the macro-scale extension of macrofacet. Originally, the object is a surface at macro-scale and microfacet at micro-scale. We stretch the macro-surface upwards and downwards by the length of $3\sigma$. In this way, the macro-surface becomes a shell volume. At the same time, microfacets become microflakes floating inside the shell.
• Figure 2: Comparison of transmittance between macrofacets and GPISes. The figure shows the transmittance of a ray intersecting macrofacet and GPIS planes at a $45^\circ$ incident angle, where $-\omega_o\cdot\omega_g=\sqrt{2}/2$. The transmittance of Beckmann macrofacet (green dashed line) matches the one of fully anisotropic GPIS (orange line) very well. The transmittance of generalized macrofacet (red dashed line) matches the one of isotropic GPIS (blue line) well in the first half, but has a slight difference in the latter half, probably due to the de-correlated assumption.
• Figure 3: GPISes with different correlations on the $z$-axis. These two figures show SDFs in $yz$-slices of realizations of (a) a strongly anisotropic GPIS with $l_z=10$ and (b) an isotropic one with $l_z=1$. The red lines are the zero level sets, indeed the realizations of GPISes. We can see that (a) the strongly anisotropic GPIS is a height field, while (b) the isotropic one is no longer a height field, having normals pointing downward.
• Figure 4: The equivalence between the normalized factor in vNDF and projected area. We compare the normalized factor computed by numerical integration (orange, red and brown lines) and the projected area $\sigma(\omega_o)$ in Equation \ref{['eqn:generalized_sigma_t']} (blue, green and purple lines) with different correlation on the $z$-axis $l_z$. The normalized factor matches projected area well whatever $l_z$ is.
• Figure 5: We compare our vNDF with the one generated by GPIS at different incident angles. The first and fourth columns are vNDF generated by GPIS. The second and fifth columns are our vNDF. The third and sixth columns are MSE between our vNDF and the one generated by GPIS. From left to right, from top to bottom, incident angles are $0^\circ$, $30^\circ$, $45^\circ$ and $60^\circ$, respectively. The roughness on all axes are $1.0$. MSE shows that our vNDF matches the reference accurately.