Fast Modeling and Rendering of Scattering

Legendre domain 3D fluid simulation and rendering: In this example, we have 3000 snow flakes being carried by a wind field (middle). We then add mist to the scene (right). Notice how further objects appearing brighter due to the air-light effect, and distant snow-flakes becoming invisible as the mist density is increased. For the complete video, see below.

In this paper, we present a unified framework for reduced space modeling and rendering of dynamic and non-homogenous participating media, like snow, smoke, dust and fog. The key idea is to represent the 3D spatial variation of the density, velocity and intensity fields of the media using the same analytic basis. In many situations, natural effects such as mist, outdoor smoke and dust are smooth (low frequency) phenomena, and can be compactly represented by a small number of coefficients of a Legendre polynomial basis. We derive analytic expressions for the derivative and integral operators in the Legendre coefficient space, as well as the triple product integrals of Legendre polynomials. These mathematical results allow us to solve both the Navier-Stokes equations for fluid flow and light transport equations for single scattering efficiently in the reduced Legendre space. Since our technique does not depend on volume grid resolution, we can achieve computational speedups as compared to spatial domain methods while having low memory and pre-computation requirements as compared to data-driven approaches. Also, analytic definition of derivatives and integral operators in the Legendre domain avoids the approximation errors inherent in spatial domain finite difference methods. We demonstrate many interesting visual effects resulting from particles immersed in fluids as well as volumetric scattering in non-homogenous and dynamic participating media, such as fog and mist.


Legendre Fluids: A Unified Framework for Analytic Reduced Space Modeling and Rendering of Participating Media

Mohit Gupta, Srinivasa Narasimhan

ACM Symposium on Computer Animation 2007

Legendre polynomials Triple Product Integral and Chebyshev Polynomials

Mohit Gupta, Srinivasa Narasimhan

Technical Report, Robotics Institute, CMU

Scene and Viewing Geometry:

The participating medium is illuminated by a distant light source and is viewed by an orthographic camera. Under the single scattering assumption, the intensity field within the medium volume can be split into two sets of light rays: the pre-scattering (direct transmission) intensity field Ed(x,t) and post-scattering intensity field Es(x,t) (red rays).

2D Legendre domain Simulation results:

Evolution of density and velocity fields for different number of Legendre coefficients. More coefficients allow for higher frequencies and vorticities in the density and velocity fields.

3D Legendre domain simulation and advection of optical properties:

Vertically upwards impulse applied to a vase shaped smoke density field. Also, we advect the scattering albedos of the media along with the densities and velocities to create the effect of mixing of different media.

Rendering of Non-homogenous participating media:

Rendering non-homogenous media under the single scattering model. Mist is added to a clear weather scene (Images courtesy Google Earth). Notice how distant objects appear brighter due to the air-light effect.

Snapshots from a fly-through of Swiss Alps with Non-homogenous and dynamic fog added (Images courtesy Google Earth). Complete fly-through is included with the supplemental video.

Typical computational speed-ups achieved for 3D simulation and rendering in Legendre domain as compared to the spatial domain.