The varifold representation of non-oriented shapes for diffeomorphic registration

04/22/2013
by   Nicolas Charon, et al.
0

In this paper, we address the problem of orientation that naturally arises when representing shapes like curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient to model a wide variety of shapes. However, in such approaches, orientation of shapes is a fundamental issue that can lead to several drawbacks in treating certain kind of datasets. More specifically, problems occur with structures like acute pikes because of canceling effects of currents or with data that consists in many disconnected pieces like fiber bundles for which currents require a consistent orientation of all pieces. As a promising alternative to currents, varifolds, introduced in the context of geometric measure theory by F. Almgren, allow the representation of any non-oriented manifold (more generally any non-oriented rectifiable set). In particular, we explain how varifolds can encode numerically non-oriented objects both from the discrete and continuous point of view. We show various ways to build a Hilbert space structure on the set of varifolds based on the theory of reproducing kernels. We show that, unlike the currents' setting, these metrics are consistent with shape volume (theorem 4.1) and we derive a formula for the variation of metric with respect to the shape (theorem 4.2). Finally, we propose a generalization to non-oriented shapes of registration algorithms in the context of Large Deformations Metric Mapping (LDDMM), which we detail with a few examples in the last part of the paper.

READ FULL TEXT

page 5

page 26

page 27

research
12/07/2016

Sparse Geometric Representation Through Local Shape Probing

We propose a new shape analysis approach based on the non-local analysis...
research
06/15/2012

Functional Currents : a new mathematical tool to model and analyse functional shapes

This paper introduces the concept of functional current as a mathematica...
research
03/30/2020

PointGMM: a Neural GMM Network for Point Clouds

Point clouds are a popular representation for 3D shapes. However, they e...
research
07/22/2022

Deep learning of diffeomorphisms for optimal reparametrizations of shapes

In shape analysis, one of the fundamental problems is to align curves or...
research
10/18/2010

Revisiting Complex Moments For 2D Shape Representation and Image Normalization

When comparing 2D shapes, a key issue is their normalization. Translatio...
research
07/31/2020

Orientation-Preserving Vectorized Distance Between Curves

We introduce an orientation-preserving landmark-based distance for conti...
research
02/09/2018

Shapes Characterization on Address Event Representation Using Histograms of Oriented Events and an Extended LBP Approach

Address Event Representation is a thriving technology that could change ...

Please sign up or login with your details

Forgot password? Click here to reset