Edge Collapse Algorithm

• Edge collapse is a local operation that contracts an edge to simplify meshes while preserving key geometric and topological features.
• It employs cost functions like Quadric Error Metrics and attribute-aware methods to balance geometric error and structure preservation.
• The algorithm enhances efficiency in mesh decimation and persistent homology by maintaining topological invariance with optimized local operations.

Edge collapse refers to a class of local operations central to mesh simplification, topological filtration reduction, and persistent homology computations. In its archetypal form, the edge collapse operation contracts an edge $(v_1, v_2)$ into a single vertex, locally modifying the mesh’s or graph’s connectivity while introducing a quantifiable, preferably minimal, geometric and/or topological error. This framework admits both geometric realizations (in mesh processing and LOD pipelines) and homotopical/topological instantiations (in TDA, e.g., flag-filtration reduction). Recent developments generalize and refine edge collapse algorithms for improved attribute preservation, quad-dominance, parallelizability, and explicit persistent homology guarantees (Glisse et al., 2022, Knodt, 2024, Kulkarni et al., 23 Dec 2025).

1. Formalism and Key Definitions

Let $K$ be an abstract simplicial complex on vertex set $V$. For flag complexes generated by a 1-skeleton graph $G=(V,E)$, the edge collapse is defined as the removal of an edge $e=[ab]\in E$ (and all simplices containing $e$) provided there exists a vertex $v\in V\setminus\{a,b\}$ satisfying the domination property:

$$N_G[e] := (N_G(a) \cap N_G(b)) \cup \{a,b\} \subseteq N_G(v) \cup \{v\}$$

Here, $N_G(x)$ denotes the neighborhood of $x$ in $G$. In mesh geometry, given a triangular or quad mesh $(\mathcal{V},\mathcal{E},\mathcal{F})$, collapsing $e=(v_0,v_1)$ entails merging $v_0$ and $v_1$ into a new vertex $v^*$, updating topological and attribute information, and deleting faces incident to $e$.

For persistent homology, a filtration is a sequence $K_1\subseteq K_2\subseteq \cdots \subseteq K_m$. Edge collapse, under suitable domination or contractibility conditions, yields a simplified filtration $K'_1\subseteq K'_2\subseteq \cdots$ such that the persistence diagrams of the original and simplified filtrations are isomorphic (Glisse et al., 2022).

Cost functions for geometric edge collapse are most commonly based on Quadric Error Metrics (QEM), which associate to each vertex a 4×4 matrix encoding the sum of squared distances to adjacent planes; mesh energy-based and constrained methods (e.g., Lindström–Turk, Hoppe) further extend this to volume, boundary, and attribute preservation (Kulkarni et al., 23 Dec 2025). For quad-dominant decimation, per-edge dihedral-angle weighted quadrics and a partial ordering (recency-based tie-breaking among ε-equivalent costs) are vital (Knodt, 2024).

2. Algorithmic Operations and Topological Guarantees

For flag-filtration reduction, three edge-collapse primitives guarantee barcode equivalence (Glisse et al., 2022):

• Swap: Exchanging the order of simultaneously-presented edges with equal filtration values does not alter persistence.
• Shift: Deferring the addition of a dominated edge to a later step preserves persistent homology.
• Trim: Removing the last dominated edge entirely, if it makes no homological distinction, is safe.

These primitives compose into algorithms that maintain isomorphism of persistent homologies by star-contraction arguments. In the zigzag filtration setting, additional primitives—zigzag-shift and cancellation—handle forward and backward arrow modifications, with analogous correctness proofs.

In mesh simplification, the collapse operation is always validated by a series of manifoldness, orientation (triangle-flip/normal check), and boundary merging checks to permit only admissible local modifications (Kulkarni et al., 23 Dec 2025).

3. Cost Functions, Attribute Preservation, and Partial Ordering

The collapse cost $c_e$ for an edge $e=(v_0,v_1)$ is computed as follows:

Classical QEM:

$$E(\mathbf{x}) = \mathbf{x}^T H \mathbf{x} + 2\mathbf{c}^T \mathbf{x} + k,\quad Q = Q_{v_0} + Q_{v_1}$$

Optimal position $v^* = -H^{-1}\mathbf{c}$ if $H$ invertible; else, midpoint fallback.

Quad-dominant Extension:

Each edge receives a tangent-space quadric enforcing regular spacing; dihedral-angle weighting penalizes collapse of sharp edges. Equivalence bands of collapse costs $|a-b|<\varepsilon_{abs}$ are processed in arbitrary order, with a recency heuristic to control quad-chord preservation (Knodt, 2024).

Constrained/Attribute-aware:

Lindström–Turk constraints introduce linear conditions for volume and boundary preservation, occasionally requiring principal-plane solutions or fallback strategies. Garland & Heckbert’s attribute-lifting QEM treats $(x,y,z,attributes)$ in an augmented space (Kulkarni et al., 23 Dec 2025).

Cost computation encompasses geometric error, attribute error, and (for skinned/animated meshes) joint-influence changes, by upweighting relevant edges. For memoryless QEM (Knodt, 2024), the cost is the introduced, not absolute, error:

$$\text{cost}(e) = \min_x [(Q_{v_0} + Q_{v_1})(x)] - (Q_{v_0}(v_0) + Q_{v_1}(v_1))$$

4. Complete Algorithmic Workflows

A robust edge collapse pipeline includes the following steps (Kulkarni et al., 23 Dec 2025, Knodt, 2024):

1. Initialization:
   • Construct mesh connectivity (half-edge/corner-table).
   • Compute all edge costs and optimal collapse positions.
   • Insert all edges into a priority queue (min-heap by cost, possibly with secondary partial-order heap for quad meshes).
2. Simplification Loop:
   • Extract edge with smallest (or ε-minimal) cost.
   • Validate collapse (manifold, normal, boundary conditions).
   • Perform edge collapse; update mesh, attributes, recency counters (for quads), and local costs in heap(s).
3. Termination:
   • Continue until the target face count, error threshold, or quad-preservation goals are met.

Pseudo-code and detailed constraint assembly for QEM, Lindström–Turk, Hoppe–PM energy, and attribute-aware variants are standard (Kulkarni et al., 23 Dec 2025). All attribute mixing and quadric updates after each collapse comply with these formulations.

5. Specialized Techniques for Persistent Homology and Filtration Reduction

The edge collapse algorithm in persistent homology applications operates over the 1-skeleton of flag (clique) complexes within a filtration, entirely determined by the edge array and their sorted filtration times. The swap/shift/trim paradigm allows aggressive simplification—dominated edges can be shifted or deleted, and event order can be flexibly manipulated without affecting the barcode (Glisse et al., 2022). Zigzag filtrations introduce bidirectional event handling via zigzag-shift and event cancellation.

Complexity for flag-filtration reduction is $O(n_e k^2)$ (practically, $O(n_e k)$) where $k$ is maximum degree; for general mesh decimation, O($E \log E$) due to priority-queue operations and validation. Empirically, simplification of very large Rips graphs (hundreds of thousands of edges) via backward collapse achieves several orders of magnitude reduction in edge count with dramatic persistent homology speedups (e.g., Torus3D $k=2$: $75s\to6.4s$) (Glisse et al., 2022).

Approximate collapse (ε-interleaving in topological TDA or soft error thresholds in geometric meshes) allows further edge removal, trading negligible error in the persistence/barcode or geometric metric for efficiency gains.

6. Experimental and Practical Outcomes

In mesh processing, single-edge collapse with dihedral-angle weighted edge quadrics, ε-approximate cost partial ordering, and recency bias consistently achieves high quad-preservation ratios without visible degradation:

• Median quad-preservation at 50% reduction: 0.95
• Median Chamfer: $1.8\times10^{-4}$ vs standard QEM’s $2.9\times10^{-4}$ MeshLab’s classical QEM achieves slightly lower Hausdorff at the expense of catastrophic quad loss (Knodt, 2024).

For animated/skinned mesh decimation, QEM with per-edge joint-influence weights outperforms commercial-style QEM and multi-pose optimizers on both Chamfer and Hausdorff distance for the majority of real-world test subjects.

Persistent homology benchmarks show extreme speedups post-collapse, e.g., backward collapse on complete graphs ($900$ vertices: $404{,}550 \to 5{,}980$ edges in $0.4s$, $43s$ forward), with negligible loss in barcode accuracy if ε-interleaving is permitted (Glisse et al., 2022).

Parallelization (thread-pool, divide-and-conquer) is effective for both mesh and filtration reduction, displaying near-linear speedup on commodity hardware for large instances.

7. Safeguards, Implementation Considerations, and Research Frontiers

Rigorous safeguards are central: triangle-flip checks (for orientation), two-neighbor manifoldness, boundary-hole validation, fallback strategies for ill-conditioned linear systems (midpoint/centroid), and boundary/attribute seam preservation are standard best practices (Kulkarni et al., 23 Dec 2025). Efficient data structures—half-edge, corner-table—enable $O(1)$ local updates; priority queues (with efficient lazy invalidation) maintain schedule order. Attribute-aware and region‐weighted edge collapses enable high-quality outputs in practical downstream uses, such as LOD generation for skinned models or real-time rendering assets.

This suggests that the ongoing research trend is toward simultaneously optimizing geometric, attribute, and topological error within a unified collapse scheme that is adaptable, parallelism-friendly, and tailored (via local quadrics/weights) to application-specific regions of interest. Edge collapse remains a fundamental primitive, with new theoretical work (e.g., persistent homology safety via star-contractions and extended zigzag operations), hybrid geometric/topological cost functions, and domain‐specific constraints shaping current methodology (Glisse et al., 2022, Knodt, 2024, Kulkarni et al., 23 Dec 2025).