Efficient participating media rendering with differentiable regularization

Fig. 1 For high-order scattering-dominant media, the light path is extremely long, especially when the albedo is high because the attenuation after each scattering is minor; when the albedo is low, the light path will be shortened as the attenuation becomes quicker. Thus, low-albedo media will have less noise.

Fig. 2 Source image (left) and denoised gradient image (right) for the Lucy scene. In this example, the extinction coefficient ${\sigma }_{\mathrm{t}}$ is $10$ , target albedo ${\alpha }_{\mathrm{t}}$ is $0.97$ , source albedo is $0.93\times {\alpha }_{\mathrm{t}}$ , and sample rate is 512.

Fig. 3 Comparison of gradient images without (top right) and with (top left) denoising and their predicted results (bottom).

Fig. 4 Radiance curves (top) and gradient curves (bottom) as functions of the albedo for three pixels with different extinction coefficients (shown in different colors). The shapes of these curves inspired us the design of our prediction function ( Eq. (11) ).

Fig. 5 Comparison among our method, temporal denoised path tracing with equal time, path tracing with equal time, and the reference. The difference between path tracing and the reference and that between our method and the reference are shown in the last two difference maps.

Fig. 6 Comparison between our method and volumetric path tracing (equal-time) on media with varying extinction coefficients ${\sigma }_t\in \{5,10,15,20,50\}$ . Our method consistently produces better results than path tracing consistently. The sample rate of the source scene is set as 256.

Fig. 7 Radiance variation curves on a slice of pixels (locations shown as the red line in the left image) of the ground truth (GT), predicted results of linear, log-linear, our prediction functions, and source image. The source ${\alpha }_{\mathrm{s}}=0.95\times {\alpha }_{\mathrm{t}}$ and target ${\alpha }_{\mathrm{t}}=0.95$ . Our prediction function is the closest to the GT.

Fig. 8 Impact of the second exponential term in our prediction function. Here, we compare our prediction function with the second exponential term (orange curve), our prediction function without the second exponential term (yellow curve), and the reference radiance curve (blue curve).

Fig. 9 Error (MSE) curve as a function of source albedo ${\alpha }_{\mathrm{s}}$ on the Lucy scene. Here, ${\alpha }_{\mathrm{s}}$ ranges from 0.1 to 0.94 $(0.1-0.97\times$ ${\alpha }_{\mathrm{t}})$ . The sample rate of the source scene is set as 512. The errors are measured on cropped images $(30\times 30)$ . The error and time cost curve (rightmost) are on a small range of the source albedo, $0.9-0.95\times {\alpha }_{\mathrm{t}}$ . The images corresponding to the orange and green dots are shown at the bottom.

Fig. 10 Predicted results (top) from different sources (bottom) on the Candle scene. When the source albedo is far from the target albedo, there is obvious color bias after the prediction.

Fig. 11 MSE between our method or path tracing with the reference (rendered with 33,768 spp) over varying rendering time on the Cube scene. The error of our method decreases until approximately 16 s and then remains almost constant. The error before 16 s comes from both variance and bias. After the result is converged, the error comes from the bias. Therefore, our method is suitable for rendering with a low time budget.

Fig. 12 Failure case: comparison between our method and path tracing on rendering the low-albedo media. The source ${\alpha }_{\mathrm{s}}=0.9\times {\alpha }_{\mathrm{t}}$ and target ${\alpha }_{\mathrm{t}}=0.5$ . The extinction coefficient is $10$ .