This spring I am going to follow along with the posted video lectures from Justin Solomon's Shape Analysis course from spring 2017. Justin video taped most of his lectures and posted the videos on youtube along with his slides on his course website. The videos can be found here and the course website is here. The specific name and number for the MIT course is 6.838 Shape Analysis.
I plan to make a 'blog post' of my notes for each lecture. This is really just to help me keep tract of the information I learn and to force me to take notes. It may also help others learn more about shape analysis, but no guarantees! Click on the category SA Lecture Notes for all the lectures.
Video for lecture 1
Differential Geometry is the study of smooth manifolds.
A manifold is a topological space that locally resembles Euclidian space near each point.
A surface is a 2D manifold, meaning that at any and all points on the surface, locally it looks like a Euclidean plane. Here is a useful way to explain it from WolframMathWorld: To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly "flat" on small scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem.
Differential Geometry 'Toolbox':
- Geodesic distance - distance along the shape
Flows and Vector fields
- spectral geometry - hit a shape with a hammer, hear the set of frequencies, based on the set of frequencies can you tell what the shape looks like? turns out yes, because it is really hard to find two objects that 'sound' the same.
Many notions of shape:
can be a triangular mesh, graph, point cloud, pairwise distance matrix,
really can be nearly anything with a notion of proximity/distance/curvature.
Triangular mesh: collection of triangles, approximation of a smooth surface
where is the curvature on a triangular mesh? the triangles are technically flat...
Discrete Differential Geometry:
asks the question: can i develop a discrete theory of diff. geometry from the ground up based on triangles and angles, parallel to differential geometry
two important things you want:
Structure Preservation: keeping properties from the continuos abstraction exactly true in a discretization
Convergence: increasing approximation quality as a discretization is refined
but you cannot have both perfectly. Have to pick and choose which are needed most for your purposes.
I will be using this blog space as a repository of notes on various articles and lectures related to my research. Click on the categories below to organize the posts by topic.