A working group about Shapes

• The original slides are here.
• What for ?
  • Consideration of time : development, growth (time evolutions)
  • Human perception; psychology; psychophysics
  • Geometry (interaction of molecules due to their configuration)
• What ?
  • I : What do we mean by shapes ?
    • Shapes could be just a list of points, a set of key-points (landmarks)
    • The shape might not exist in reality (or the choice of its limits might be arbitrary if several possibilities of definition) but be a practical way of modeling problems
  • II : Representation
    • A shape can be seen as an infinite number of points (e.g. continuous curve as a set of points)
    • Distinction between mathematical models and use in practice (a model could be continuous and need discretization when implemented)
    • Related : level-set representation do not imply a discretization as a mathematical model (remove the word "approximation")
    • Topology : genus = shape for a particular choice of a wide class of invariance
    • Shape as a set of parameters for a particular model
  • III: families
    • Fractals
    • Tubular shapes (as a skeleton + a ball)
    • Trees, graphs (e.g. models of skeletons)
    • Meshes
    • Minimal surfaces
    • Shape occultation : parts of a shape, or several shapes overlapping = the same family ?
    • Parameterized model
• Pairwise shape comparison
  • Similarity vs. distances (do we need distances in the mathematical sense [with triangular inequality, etc.])
  • The distance depends obviously on the space of shapes chosen
  • For shapes that are 0-level of some function, consider e.g. Bernstein polynomials, distance can be based on how much the graph of the function varies based on the (polynomial) basis coefficients
  • Based on eigenvalues of a kernel defined on the shape (the heat kernel typically), as in Bronstein & Bronstein
  • About optimization : gradient descent + stochastics
• Where from ?
  • Shapes can come from mathematical formulas (e.g. the solution of a problem)
• Shape statistics
  • The way of computing statistics depends on the family of shapes studied and their representation
  • Parameterized models : statistics on the model parameters