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Sparse Randomized Approximation of Normal Cycles

Allen Paul, Neill Campbell, Tony Shardlow

TL;DR

This paper addresses the computational bottleneck of the curvature-aware normal-cycle shape metric by extending SparseNystromCurrVar to compress normal-cycles via a Nystrom RKHS framework with ridge leverage sampling. It derives a Dirac-delta, real-valued embedding for discrete normal-cycles, enabling aggressive compression with exponential error decay and $O(mn)$ costs for downstream distance computations. The approach yields substantial speedups (often 9–20x) with negligible loss in registration quality in large-scale LDDMM problems, demonstrating practicality for datasets with $10^5$–$10^6$ elements. Overall, the work makes curvature-aware, correspondence-free shape analysis scalable to massive geometry processing tasks while retaining theoretical guarantees on approximation error.

Abstract

We develop a compression algorithm for the Normal-Cycles representations of shape, using the Nystrom approximation in Reproducing Kernel Hilbert Spaces and Ridge Leverage Score sampling. Our method has theoretical guarantees on the rate of convergence of the compression error, and the obtained approximations are shown to be useful for down-line tasks such as nonlinear shape registration in the Large Deformation Metric Mapping (LDDMM) framework, even for very high compression ratios. The performance of our algorithm is demonstrated on large-scale shape data from modern geometry processing datasets, and is shown to be fast and scalable with rapid error decay.

Sparse Randomized Approximation of Normal Cycles

TL;DR

This paper addresses the computational bottleneck of the curvature-aware normal-cycle shape metric by extending SparseNystromCurrVar to compress normal-cycles via a Nystrom RKHS framework with ridge leverage sampling. It derives a Dirac-delta, real-valued embedding for discrete normal-cycles, enabling aggressive compression with exponential error decay and costs for downstream distance computations. The approach yields substantial speedups (often 9–20x) with negligible loss in registration quality in large-scale LDDMM problems, demonstrating practicality for datasets with – elements. Overall, the work makes curvature-aware, correspondence-free shape analysis scalable to massive geometry processing tasks while retaining theoretical guarantees on approximation error.

Abstract

We develop a compression algorithm for the Normal-Cycles representations of shape, using the Nystrom approximation in Reproducing Kernel Hilbert Spaces and Ridge Leverage Score sampling. Our method has theoretical guarantees on the rate of convergence of the compression error, and the obtained approximations are shown to be useful for down-line tasks such as nonlinear shape registration in the Large Deformation Metric Mapping (LDDMM) framework, even for very high compression ratios. The performance of our algorithm is demonstrated on large-scale shape data from modern geometry processing datasets, and is shown to be fast and scalable with rapid error decay.

Paper Structure

This paper contains 30 sections, 3 theorems, 90 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Existing Work
  4. Outline
  5. Background
  6. Currents
  7. Dual space metrics for Currents
  8. Normal Cycles
  9. Unit Normal Bundle and Normal Cycle of a smooth submanifold
  10. Normal Cycle of a union of smooth sets
  11. Dual Space Metric - Normal Cycles
  12. Decomposition and compression of Normal-Cycles
  13. Curves
  14. Surfaces
  15. Planar component
  16. ...and 15 more sections

Key Result

Theorem 1

NCpaper Let $\mathcal{T},\mathcal{T}'$ be two triangulated surfaces. The inner product between the associated discrete Normal Cycles can be computed as, where $\partial \mathcal{T}$ denotes the boundary of the surface $\mathcal{T}$ and similarly for $\mathcal{T}'$. In the above, $n_{e},m_{e}$ denotes the number of unique edges in each triangulation, and $f_{i},g_{j}$ the unique edges respectively

Figures (4)

  • Figure 1: Left: Cat (14410 triangles) Middle: Head (31620 triangles) Right: Flamingo (52895 triangles)
  • Figure 2: Numerical curves comparing RKHS error decay (black) of RLS compression of normal-cycles, to theoretical trace bound (red) and Uniformly sampled compression (blue), on cat (left), head (centre) and flamingo (right) surfaces.
  • Figure 3: Top left: spherical template. Top right: target mesh. Bottom left: Matching with full metrics taking $2$ hours and $42$ minutes with $d_{H} = 0.0442$. Bottom Right: Matching with $97\%$ compression of template and target taking only $17$ minutes with $d_{H} = 0.0298$.
  • Figure 4: Top left: spherical template. Top right: target mesh. Bottom left: an example matching with full normal-cycles, from spherical template to target without compression, taking $5$ hours and $37$ minutes with $d_{H} = 0.1621$. Bottom right: the same example matching but with $99\%$ compression taking only $17$ minutes with $d_{H} = 0.1662$.

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 1
  • Theorem 2: NCThesis
  • Theorem 3
  • Definition A.1
  • Definition A.2
  • Definition B.1