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A Signal-Processing Framework for
Forward and Inverse Rendering

Ravi Ramamoorthi


Pat Hanrahan

June 01, 2001

Notes by Ken Turkowski

  • Forward Rendering
    • Interactive Rendering
    • Reflection/Environment/Reflectance Maps
    • Complex Illumination
  • Inverse Rendering
    • Measure realistic material models and lighting from real photographs
  • Object Recognition
  • Reflection as Convolution
  • Efficient Rendering: Environment Maps
  • Lighting Variability in Object Recognition
  • Deconvolution, Inverse Rendering

The results of this paper stem from the fundamental observation that

          distant lights as the source of illumination,
          the radiance integral reduces to a convolution.

Convolution is efficiently implemented as multiplication in the frequency domain.

Spherical harmonics are used as the basis for illumination functions defined on the sphere. By representing these in Cartesian rather than spherical coordinates, the spherical harmonics are represented as simple polynomials, rather than trancendental functions, thus simplifying the computations enormously.

        Light is the signal,
        the BRDF (bi-directional reflection distribution function)
        is the filter, and reflection on a curved surface is convolution.

Inverse rendering is deconvolution.

The BRDF of a mirror ball is an impulse response. The environment lighting can be determined directly.

Lambertian reflection falls off quickly, with periodic zeros in the spherical harmonic coefficients. This can be represented with only 9 parameters, and can be well approximated with a quadratic polynomial.

The diffuse component is localized in frequency space, and the specular component is localized in angular space. This observation leads to an efficient dual algorithm for global illumination, where

       diffuse reflection is computed in frequency space, and
       specular reflection is computed with ray-tracing

From a practical viewpoint, the method works well even for most local lighting.

Inverse rendering results:

Illumination can be recovered from the reflected image of a specular surface, but not from a diffuse surface. Attempting to do so from a diffuse surface yields division by numbers very close to zero, yielding very noisy and mostly bogus results.

Light fields (a.k.a. Lumigraphs) can be factored to extract both the BRDF and illumination from an image simultaneously.

Inverse rendering via deconvolution or other methods is (obviously) an inverse problem, which can be ill-conditioned. Success is highly dependent on the characteristics of the source image, and any a priori knowledge of the scene (i.e. characteristics of the illumination or BRDF; geometry is assumed to be known).

The BRDF can be robustly and quickly computed with knowledge of the illumination (and geometry) upon a diffuse surface.


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