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Geometry and Shape Analysis in Biological Sciences (8-9 June 2017)
Tutorial 3: Comparing Biological Shapes (1)
Organizers
Overview
In a chapter titled "The Comparison of Related Forms", D'Arcy Thompson explored how differences in the forms of related animals can be described by means of simple mathematical transformations This inspired the development of several shape comparison techniques, with the goal of defining a map, or correspondence, between two shapes that can be used to measure their similarity. This is a challenging problem, as the space of all possible maps is extremely large and difficult to characterize mathematically. In this tutorial and its follow-up, we will cover this problem, focusing first on the discrete representation of shapes in biology, and later on the theory of conformal mapping and its applications to study biological shapes. While three-dimensional (3D) data representing a shape come in many forms, we concentrate on the important case where the surface of the shape is available and described with a discrete triangular mesh. Mathematically, these objects are 2-dimensional Riemannian manifolds in the smooth case, and piecewise-flat surfaces in the discrete setting.
Slides of the presentation
A simple tool to compute 2D Delaunay triangulations
Visualizing shapes: a simple 3D mesh viewer
Gallery:
Resources
Books and Review articles
Useful links
- Visualization programs
- Reading, manipulating and performing operations on meshes
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