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Fast Modeling and Rendering of
Participating Media
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Abstract
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.
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Publications
“Legendre
Fluids: A Unified Framework for Analytic Reduced Space Modeling and
Rendering of
Participating Media”
Mohit Gupta, SG Narasimhan,
Eurographics/ ACM SIGGRAPH
Symposium on Computer Animation 2007
[PDF] [LowResolution
PDF]
“Legendre
polynomials Triple Product Integral and lower-degree approximation of
polynomials
using Chebyshev
polynomials”
Mohit Gupta, SG Narasimhan,
Tech.
Report CMU-RI-TR-07-22, Robotics
Institute, Carnegie
Mellon University,
May, 2007
[PDF]
Videos
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Confetti
Added to Christmas Video: (Apple Quicktime 7.0).
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Mist
and Snow:
(Apple Quicktime 7.0).
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SCA
2007 Video (with audio):
This video is a compilation of the main results of this project.
(Apple Quicktime 7.0).
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Pictures
(click on thumbnails to enlarge images)
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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).
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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.
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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.
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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.
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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.
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Typical
computational speed-ups achieved for 3D simulation and rendering in
Legendre domain as compared to the spatial domain.
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