Mean and median shapes using concepts from geometric measure theory

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Even though shapes and shape problems are ubiquitous in science and engineering, there is in our opinion, no completely satisfactory framework for dealing with shapes. In this work, we take the perspective that shapes in R n are often best represented as k-currents in R n . While we discuss what this means in more detail in this thesis, for the purposes of this introduction, they can be thought of as countable unions of pieces of oriented C 1 k-submanifolds in R n . We usually require their total k-dimensional measure to be finite. Moreover, there might be a function associated with this set that assigns a weight or density to each point of the set. There are very few other works that take this viewpoint. Using the flat norm from Geometric Measure Theory, we present a new shape mean and median, and then focus on the shape median that can be computed with ease due to a formulation as a linear program. We discuss existence and the lack of uniqueness, shortest paths on shape manifolds and the exploration of shape manifolds defined by point clouds in shape spaces. We also prove a few theorems concerning the properties of shape medians, both with and without the minimal mass regularization that we suggest as a selection principle. We show computed examples and provide code for readers to use in their own explorations of shape data. The result of our experiment shows that we should carefully explore shape space or current space to have a meaningful median shape. Though meaningful does not always have to match theoretical results. Since it is an optimization problem, we can get an empty median for some large [Lambda]s which does not agree with how we perceive a median especially when input currents have nonzero lengths. So we recommend to use smaller [Lambda]s and working on to find theoretical bound for the scaling factors or mass regularization coefficients. The main goal we wanted to achieve was to find a way to formulate a flatnorm-based median shape definition which can find a median over input currents. The goal has been pursued both from mathematical and computational perspectives. We did define mean and median shapes and implemented Mass regularized Simplicial Median Shape (MRSMS) which finds median shapes in simplicial settings. Furthermore, we raised many open questions along with several solutions such as adding a mass regularization term to narrow down median shapes closer to unique property and deforming shapes to one another by adding weights of convex combination to their corresponding flat norm difference from the median shape.

Mean and median shapes using concepts from geometric measure theory

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