DDISH-GI: Dynamic Distributed Spherical Harmonics Global Illumination

Abstract

We propose a real-time hybrid rendering algorithm that off-loads computationally complex rendering of indirect lighting from mobile client devices to dedicated ray tracing hardware on the server with a hybrid real-time computer graphics rendering algorithm. Spherical harmonics (SH) light probes are updated with path tracing on the server side, and the final frame is rendered with a fast rasterization-based pipeline that uses the light probes to approximate high quality indirect diffuse lighting and glossy specular reflections. That is, the rendering workload can be split to multiple devices across the network with a small bandwidth usage. It also benefits multi-user and multi-view scenarios by separating indirect lighting computation from camera positioning. Compared to simply streaming fully remotely rendered frames, the approach is more robust to network interruptions and latency. Furthermore, we propose a specular approximation for GGX materials via zonal harmonics (ZH). This alleviates the need to implement more computationally complex algorithms, such as screen space reflections, which was suggested in the state-of-the-art dynamic diffuse global illumination (DDGI) method. We show that the image quality of the proposed method is similar to that of DDGI, with a 23 times more compact data structure.

Appendices

Appendix A GGX Approximation with ZH Lobes

ZH is a subset of SH that consists of the SH functions that are rotationally symmetric with respect to the z-axis. In particular, if we assume that our physically-based material model has rotationally symmetric BRDF lobes, we can represent those lobes more compactly and efficiently with the ZH functions instead of requiring a full SH representation. We refer the reader to [24] and [25] for general details on SH and ZH, respectively.

For the physically-based material model in our renderer, we chose to use the common GGX material model [30], also known as Trowbridge-Reitz distribution [28], to support the glTF 2.0 specification [27] as closely as possible. The GGX BSDF has parameters that allows it to be used for a large variety of different real-world materials with a good degree of realism [30]. It is quite fast to evaluate, which is why it has seen a lot of use in the real-time rendering industry.

SH has been previously used as an approximation for arbitrary BRDFs [13]. Hence, in addition to using an SH approximation for path traced indirect lighting, we also investigate using ZH for GGX-based rotationally invariant BRDF material approximations. The indirect lighting from the SH probes is convolved with the GGX material approximation for the final lighting contribution. As shown in [24], the convolution between a rotationally symmetric function g and some function f – in our case the ZH GGX basis and the SH light probe basis, respectively – has projection coefficients satisfying

$$\begin{aligned} \left( g *f \right) ^m_l = \sqrt{\frac{4\pi }{2l+1}} g^0_l f^m_l, \end{aligned}$$

(1)

where the left-hand side of the equation denotes the projection coefficients of the basis of g convolved with f, m is the order of the basis function coefficient, and l is the degree of the basis function coefficient. Thus, a convolution of SH over the ZH is simple because it only amounts to scaling each degree of the SH basis with the respective ZH. On the other hand, the SH-over-SH convolution would require several multiplications added up for each coefficient.

Utilizing SH probes and SH approximation for surface materials was proposed in [6] for specular lighting. Their method uses several texture lookups and has to rotate the SH approximation, making it computationally challenging. As the SH probes are already an approximation for the indirect lighting components with smoothed expressiveness at lower degree coefficients, using a sharper convolution lobe for the material, like in [6], is wasteful and only highlights the low-frequency nature of the SH probes. Furthermore, our novel contribution is applying the ZH GGX approximation only for indirect glossy highlights from the SH probes. For the final indirect lighting contribution, we decided to adopt a less complex approach utilizing the split sum approximation published in [12]:

$$\begin{aligned} \int _{\varOmega }\frac{L_i(\boldsymbol{\omega }_i)f(\boldsymbol{\omega }_i,\boldsymbol{\omega }_o)\boldsymbol{n}\cdot \boldsymbol{\omega }_i)}{p(\boldsymbol{\omega }_i,\boldsymbol{\omega }_o)}d\boldsymbol{\omega _i} \approx \left( \int _{\varOmega }L_i(\boldsymbol{\omega }_i)d\boldsymbol{\omega _i}\right) \left( \int _{\varOmega }\frac{f(\boldsymbol{\omega }_i,\boldsymbol{\omega }_o)\boldsymbol{n}\cdot \boldsymbol{\omega }_i}{p(\boldsymbol{d}_k,\boldsymbol{v})}d\boldsymbol{\omega _i}\right) , \end{aligned}$$

(2)

where $L_i(\boldsymbol{\omega }_i$ is the incoming radiance from the direction $\boldsymbol{\omega }_i$, $f(\boldsymbol{\omega }_i,\boldsymbol{\omega }_o)$ is the BRDF from direction $\boldsymbol{\omega }_i$ to the direction $\boldsymbol{\omega }_o$, $\boldsymbol{n}\cdot \boldsymbol{\omega }_i$ is the angle between $\boldsymbol{\omega }_i$ and the surface normal $\boldsymbol{n}$, and $p(\boldsymbol{\omega }_i,\boldsymbol{\omega }_o)$ is the probability of sampling from $\boldsymbol{\omega }_i$ to $\boldsymbol{\omega }_o$ based on the BRDF. In our case, the left side integral of the product is calculated as the convolution between the nearest interpolated SH probes and our rotationally symmetric ZH approximation of the GGX specular lobe (similarly as in [12]). This convolution produces a new rotationally symmetric ZH function with coefficients calculated by Eq. 1. Furthermore, the right side integral of the split sum is evaluated as an environment BRDF which is encoded in a 2D 2-channel texture varying by surface roughness on one axis and $\boldsymbol{n}\cdot \boldsymbol{\omega }_i$ on the other.

Fig. 3.

Values of the precomputed ZH coefficients for approximating GGX specular lobes.

In order to use the ZH basis to approximate GGX materials with different roughnesses, we numerically integrated 1024 samples on the 0 to 1 roughness interval and fit a ZH basis function onto each GGX lobe value. The coefficients of the ZH basis are plotted as a function of the material roughness in Fig. 3. The figure shows how the specularity intensifies on smoother materials with large coefficients on the left, and dampens down to diffuse rough materials with small coefficients on the higher degrees. We experimented with different basis degrees from L0 to L4 and found that the L2 basis was a good trade-off between coefficient compactness and approximation quality. The fit is exactly correct to the Trowbridge-Reitz GGX lobe at the impulse direction of the surface normal.

Even though the split sum approximation doesn’t take into account the skewness of the GGX lobe at grazing angles, it is an accepted trade-off in the industry and works well enough together with the SH-probes. In [12], the authors used Eq. 2 to compute the separate integrals in advance for environment maps on cubemap bases, whereas our novel contribution is applying this to an SH light probe basis for glossy specular highlights with the GGX lobe further approximated by a ZH basis. The split sum method constrains the materials’ BRDF lobes to be axially symmetric, which means that in our implementation of the SH probe basis, we only consider the rotationally symmetric basis which is exactly the ZH basis. This provides us with the aforementioned faster SH-over-ZH convolution. In each degree of the SH basis, the ZH is unique and so, we can refer to the different degrees from L0 to L4 uniquely.

As discussed earlier, our SH probe approximations are only for the indirect lighting component produced by path tracing and do not consider effects from direct lights. In order to support specular highlights from direct lighting, we approximate direct lights as almost singular points and directly sample the BRDF, which is less accurate for larger lights. However, it serves as a decent approximation as long as the surface is not perfectly smooth and has some roughness present.

Appendix B Comparison Images

Fig. 4.

Comparison images from Sponza.

Fig. 5.

Comparison images from Breakfast Room.

Fig. 6.

Comparison images from Sibenik Cathedral.

We present comparison images between DDGI, the proposed method DDISH-GI, and a 16384 spp path traced reference from Sponza (Fig. 4), Breakfast Room (Fig. 5), and Sibenik Cathedral (Fig. 6). We observe that DDGI exhibits overly spread out indirect lighting in occluded areas whereas the proposed method is closer to the reference in those situations, as seen behind the benches of the Sibenik Cathedral. Both probe-based methods have challenges with high frequency detail in indirect lighting effects, such as in the corners of the Breakfast Room, due to the limited spatial resolution of the probes.