Reading 5 – The Laplacian

The Laplace-Beltrami operator plays a fundamental role in geometry and physics; it is also a key object in digital geometry processing, take the role (in some sense) of the fast Fourier transform from traditional signal processing. Your next reading assignment will be to gain a better understanding of the Laplacian, both from a formal as well as intuitive geometric viewpoint. (If you haven’t seen vector calculus in a while, this may also be a good moment to brush up on things like the divergence, gradient, and curl.)

Some potential starting points:

  • Laplace operator – the obligatory Wikipedia link
  • Laplace-Beltrami – a generalized version of the Laplacian used in geometry; a discrete version of this operator is what we will effectively use to develop a number of algorithms throughout the remainder of the term.
  • Laplace-Beltrami: The Swiss Army Knife of Geometry Processing – some slides that give a nice visual overview of the Laplacian, its relationship with geometry, its discretization, and its applications in digital geometry processing.
  • Chapter 6 – The Laplacian – from our course notes; you will see a version of this chapter again very soon, since it forms the basis for your next assignment.
  • A discrete Laplace-Beltrami operator for simplicial surfaces – for folks who want to dig a bit deeper, this paper goes into some of the deeper questions about how to define a Laplace operator on a discrete surface, while capturing important features of the smooth operator.
  • Discrete Laplace operators: No free lunch – another “go deeper” paper; this one shows that (under certain assumptions) it’s actually impossible to preserve all the properties you might like to have.

Submission: As usual, please send an email to kmcrane@cs.cmu.edu and nsharp@cs.cmu.edu no later than 10:00 AM on Tuesday, February 9th including the string DDGSpring2016 in your subject line.  Your email for readings should always include:

  1. a short (2-3 sentence) summary of what you read, and
  2. at least one question about something you found confusing / interesting / incomplete / not addressed.

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