Source-linked AI summary
The varifold representation of non-oriented shapes for diffeomorphic registration
Nicolas Charon, Alain Trouvé
TL;DR
Currents are effective for shape analysis but require orientation, creating cancellation and consistency problems for non-oriented or disconnected shapes. The paper develops varifold representations and reproducing-kernel Hilbert metrics, derives their shape variation, and applies them to non-oriented registration. The approach avoids requiring shape orientations, although the paper notes limitations related to kernel injectivity and emphasizes synthetic examples.
Problem
Current-based shape representations depend on orientation, creating cancellation and consistency problems for non-oriented, disconnected, or sharply structured shapes.
Method
The paper represents shapes as distributions of unoriented tangent spaces and constructs varifold Hilbert metrics using reproducing kernels on ambient space and Grassmann manifolds.
Results
The paper derives shape-variation formulas and adapts LDDMM registration to non-oriented shapes, with examples demonstrating the framework.
Takeaways & Limitations
The approach does not require shape orientation, making it relevant to datasets that are difficult or impossible to orient consistently.
Takeaways & Limitations
The paper states that its numerical results are mainly focused on synthetic examples and notes that some varifold kernel constructions may be non-injective.
Abstract
from arXiv · showhide
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.
1. Introduction
Computational anatomy models shape variability through deformations of ambient space, making a relevant residual shape-difference measure fundamental. The paper focuses on replacing orientation-dependent currents with varifolds to represent shapes without orientation.
- Computational anatomy studies shape variability among subjects as deformations of the ambient space.
- A central requirement for computational-anatomy algorithms is a relevant measure of residual shape difference.
- Currents create an orientation dilemma, motivating a representation completely free of shape orientation.
- The paper adapts varifolds, an alternative concept from geometric measure theory, for this purpose.
2. Representation of shapes with currents : strengths and limitations
Currents embed oriented shapes into functional spaces and support kernel-based computations, but their dependence on orientation causes cancellation, orientation requirements, and difficulties for disconnected or sharp structures. These limitations motivate orientation-free shape representations.
- Currents embed oriented submanifolds and rectifiable subsets into a common functional space by integrating differential forms.
- Kernel norms of rectifiable currents are bounded by Hausdorff measure, but orientation prevents the converse domination.
- Comparing shapes as currents requires orientations that are consistent with each other.
- Oppositely oriented nearby parts can cancel under kernel metrics, causing meaningful shape structures to disappear from the representation.
- Fiber bundles with N connected components have 2^N possible orientations, making consistent orientation difficult and potentially producing unsatisfactory matching results.
- Sharp spines or tails can be annihilated in current-based Hilbert representations even when their Hausdorff measure remains substantial.
3. Varifolds
Varifolds represent non-oriented shapes by combining positions with unoriented tangent spaces, supporting both continuous and discrete constructions. Their transport under diffeomorphisms is compatible with shape transport, providing a basis for computational registration.
- Computational role: The section concludes that varifolds provide continuous and computational representations of non-oriented shapes, while an appropriate metric remains needed for large-deformation matching.The representation is intended for later adaptation of large deformation matching to varifolds.
- Grassmannian representation: Varifolds represent tangent spaces through the Grassmann manifold G_d(E), the set of all d-dimensional subspaces of E.The Grassmann manifold also has a compact Riemannian-manifold structure and can be viewed through orthogonal projections.
- Varifold definition: A d-dimensional varifold is a finite Borel measure on E × G_d(E), pairing spatial position with an unoriented tangent subspace.Its Dirac elements record a position and a non-oriented subspace that serves as a tangent space.
- Rectifiable sets: A non-oriented rectifiable set X is represented by measuring points together with their tangent spaces, yielding the varifold μ_X.This construction uses Hausdorff measure on the set of position–tangent-space pairs.
- Discrete representation: Meshed curves and surfaces become finite sums of Dirac varifolds carrying cell centers, unoriented tangent spaces, and local cell volumes.Reversing the vertex order of a curve edge leaves the equations unchanged, consistent with non-orientation.
- Transport by deformation: Varifold transport by a diffeomorphism applies the differential to tangent spaces and weights them by the corresponding d-dimensional Jacobian.The resulting transport commutes with representing the transformed rectifiable set as a varifold.
4. Kernel metrics on varifolds
The paper constructs RKHS metrics for varifolds by combining kernels on the ambient space with kernels on the Grassmann manifold. It analyzes when these metrics separate shapes, preserve volume-related information, remain sensitive to tangent directions, and support shape variation calculations.
- Kernel construction: Tensor products of kernels on E and Gd(E) provide a generic, computationally effective construction of RKHS metrics for varifolds.Grassmannian kernels can be induced through projection-matrix embeddings or principal angles, yielding a broad class of positive kernels.
- Kernel construction: Cauchy-Binet kernels need not be C0-universal, so their dual applications may fail to be injective.This non-injectivity means the associated Hilbert norm does not necessarily define a true distance on all varifolds.
- Metric separation: Under the stated conditions, the resulting metrics distinguish finite unions of compact submanifolds, while Gaussian Grassmannian kernels recover injectivity on all varifolds.A C0-universal kernel on E tensored with the specified Gaussian kernel on Gd(E) is itself C0-universal.
- Large-scale behavior: At large ambient scale, varifold norms compare distributions of non-oriented tangent spaces rather than collapsing closed shapes to boundary-sensitive representations.This makes the Grassmannian kernel decisive when multiple directions cross near one another; Gaussian kernels are presented as more effective for fibers and tree-like structures.
- Volume consistency: Theorem 4.1 establishes consistency between varifold kernel norms and shape volume, preventing the pathological cancellation behavior highlighted for currents.The paper connects this property to avoiding artificial mass elimination during registration or template estimation.
- Metric variation: Theorem 4.2 gives the variation of varifold metrics under a flow, with interior variation depending only on the normal component of the deformation field.The formula also contains boundary terms, while the overall variation is controlled by the vector field rather than its derivatives.
5. Large deformation matching of unoriented shapes : an algorithm
The paper adapts LDDMM registration to unoriented shapes by using varifold-based attachment terms within the existing deformation framework. Experiments show advantages over currents when shapes contain narrow branches, disconnected components, inconsistent orientations, or fine surface details.
- Algorithmic framework: Varifold kernels provide distances between unoriented shapes that can serve as attachment terms in LDDMM matching.The framework is developed for curves and surfaces in 3D Euclidean space.
- Algorithmic framework: The registration objective balances deformation-path length against the distance between the deformed source and target through a trade-off parameter.The optimization can be handled using gradient descent or, almost straightforwardly, geodesic shooting.
- Optimization: The optimal vector field is parameterized by momenta, reducing the optimization to a finite-dimensional control problem with a backward integration scheme for the covariable.The resulting descent direction is formed from the difference between the covariable and the momenta.
- Algorithmic framework: LDDMM retains the same deformation dynamics for oriented and unoriented objects; only the attachment quantity and its gradient must be specified.This allows varifold data attachment to be inserted into established LDDMM optimization schemes.
- Experiments: Gaussian varifold registration recovers narrow branches that currents nearly ignore, using the same parameters in the 2D curve example.Changing the spatial scale σe provides a second varifold registration comparison.
- Experiments: Varifolds handle inconsistently oriented disconnected components and avoid several current-based mismatches in tree-like structures and surface registrations.The reported surface examples include thin envelopes, a sphere-to-bunny match, ears, head details, and unwanted membranes produced by currents.
6. Conclusion
The paper adapts varifolds to represent unoriented shapes, constructs kernel-based Hilbert metrics, and extends LDDMM registration. The framework avoids orientation-driven volume loss and supports registration without consistently orienting shapes.
- Varifolds represent shapes as spatial distributions of unoriented tangent spaces modeled on the Grassmann manifold.
- Reproducing-kernel constructions provide Hilbert structures and numerically usable metrics for varifold representations.
- The paper adapts LDDMM registration to unoriented shapes, including examples involving trees and surface matching.
- The metrics do not artificially eliminate shape volume because of orientation, unlike current-based representations.
Appendix A. A variation formula for varifold metrics
The appendix derives the variation of a varifold metric when a smooth compact submanifold is deformed by a vector field. The derivation separates positional, tangent-space, and volume-change contributions and concludes with theorem 4.2.
- For a smooth compact orientable submanifold X, the variation of µX(ω) is computed under a diffeomorphic deformation generated by a compactly supported vector field.
- Combining the positional, tangent-space, volume, and boundary terms yields the stated variation formula and proves theorem 4.2.
- Differentiating the deformed integral yields contributions from spatial position, tangent-space direction, and tangent-volume change.
- The volume-change contribution is decomposed using tangential and normal components of the deformation field, with mean curvature appearing in the normal term.
- The tangential contribution is rewritten as a boundary integral using the divergence theorem and the outward boundary normal.
Appendix B. Discrete computations for varifold LDDMM
The appendix makes varifold LDDMM registration computationally explicit for discrete curves and surfaces. It evaluates kernel-based attachment distances from mesh representations and differentiates them with respect to source-shape points.
- Varifold LDDMM uses an attachment distance between the deformed source and target, computed in an RKHS from spatial and Grassmann-manifold kernels.
- Discrete unoriented shapes are represented from vertices and unoriented simplices, whose varifold measures are assembled from cell centers, sizes, and tangent directions.
- Distance evaluation reduces to repeated evaluations of the spatial kernel ke and tangent-space kernel kt once the shape representations are constructed.
- The gradient is obtained by differentiating the attachment with respect to cell centers, sizes, and tangent directions, then propagating derivatives to the shape vertices.
- For curves and triangulated surfaces in R3, tangent directions and derivative formulas are computed from the corresponding geometric point and mesh equations.