Flood Lighting |
Polarized Light Striping |
Comparison |
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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.
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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,
August 2007.
[PDF]
[Low Resolution 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]
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Videos
(Video Result Playlist)
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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
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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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