A Practical Introduction to
Frequency Analysis of Light Transport

Laurent Belcour - Unity Technologies


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  • About this course
    • Result of 11 years of research so far
    • Started with Durand et al. [2005]

  • About the lecture
    • Not targetting completness or details
    • Give a taste of different possibilities

Noise: An Issue in Physically Based Rendering

  • PBR rely a lot on sampling
    • Plagued by noise
    • Hard to know a stopping criterion

  • Frequency Analysis can solve part of it!


  • Examples of today's lecture's applications
    • Adaptive sampling
    • Upsampling
    • Density estimation
    • Anti-aliasing

Frequency Analysis

  • In the Signal Processing toolbox
    • Study of signals and their transformations
    • Application to digitalization, sampling, numerical integration

Example: Nyquist Rate

Distance between samples $ d < \dfrac{1}{2 B_w} $

The Fourier Transform

The Fourier Transform

Rendering has a lot to do with Blur

For Example: Defocus Blur

Our Goal

  • Use the Fourier Transform
    • Predict the smoothness in rendering
    • Use it to adapt and optimize rendering

  • However
    • We cannot fully render the image
    • We only need a conservative prediction

  • But without limitations
    • Be able to treat defocus, motion blur, ...
    • Work for offline and real-time contexts

Fourier Transform of Rendering?

Fourier Transform of Rendering?

  • Need to go to the fundementals
    • Rendering Equation [Kajiya 1986]
    • Even a little step further ...

  • Break the Rendering Equation into pieces
    • Decompose shading, defocus, motion, ...
    • To perform and compose the analysis

  • First a quick recap of the Rendering Equation


  1. Introduction
  2. Frequency Operators
    • Travel & Occlusion
    • Reflection & Refraction
    • Scattering & Absorption
  3. Applications
    • Adaptive sampling & denoising
    • Upsampling
    • Density estimation
    • Antialiasing
  4. Conclusion

Break: Questions?

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