Attribute-Aware Edge Collapse

• Attribute-aware edge collapse is a mesh simplification technique that integrates geometric data with per-vertex attributes (e.g., color, normals, texture) to maintain accuracy.
• It extends classical methods like Quadric Error Metrics by incorporating higher-dimensional quadrics and energy-based formulations to minimize attribute drift.
• Practical implementations leverage per-vertex quadric storage and optimized connectivity data structures to ensure the preservation of semantic boundaries and visual fidelity.

Attribute-aware edge collapse is a mesh simplification methodology that reduces the vertex, edge, and triangle count of a 3D mesh while preserving both geometric shape and salient per-vertex attributes such as color, normals, or texture coordinates. It extends classical edge collapse methods—including Quadric Error Metrics (QEM)—to operate directly on both spatial and attribute channels, thereby enhancing the preservation of semantic boundaries and feature fidelity during aggressive decimation operations (Kulkarni et al., 23 Dec 2025).

1. Foundations of Edge Collapse and Attribute Integration

The traditional edge collapse operation merges two mesh vertices into one, removing adjacent triangles and reconstructing connectivity. Classic simplification uses geometric error metrics, notably QEM, which quantifies the squared distance of vertices to their local approximating planes. Attribute-aware variants generalize this by incorporating additional vertex data into the error metric and optimization, folding per-vertex attributes (e.g., RGB values, normals, or texture coordinates) into the collapse cost function. The aim is to prevent attribute drift and preserve sharp boundaries in attribute channels during simplification (Kulkarni et al., 23 Dec 2025).

2. Higher-Dimensional Quadrics: QEM Generalization

The higher-dimensional QEM approach, following Garland & Heckbert (1998), represents each vertex as a single point in $\mathbb{R}^{3+d} = (x, y, z, a_1, ..., a_d)$, where the $a_i$ are scalar attributes. Each triangle face, comprised of three such points $p$, $q$, $r \in \mathbb{R}^{3+d}$, spans a 2-plane in this extended space. The squared distance from a candidate vertex $\tilde{v}$ to the plane $P$ is computed by:

• Establishing orthonormal basis vectors $e_1, e_2$ via Gram–Schmidt.
• Calculating the orthogonal component $u = w - (w\cdot e_1)e_1 - (w\cdot e_2)e_2$ for $w = \tilde{v} - p$.
• Defining the error as $$\epsilon_P(\tilde{v}) = \|w\|^2 - (w\cdot e_1)^2 - (w\cdot e_2)^2$$.

This error expands to the quadratic form:

$$\epsilon_P(\tilde{v}) = \tilde{v}^T H_P \tilde{v} + 2 c_P^T \tilde{v} + k_P,$$

with $$H_P = I - e_1 e_1^T - e_2 e_2^T \in \mathbb{R}^{(3+d)\times (3+d)}$$, $c_P = ((p\cdot e_1) e_1 + (p\cdot e_2) e_2 - p)$, and $k_P = p\cdot p - (p\cdot e_1)^2 - (p\cdot e_2)^2$. The total quadric at a vertex sums contributions over all incident faces: $H = \sum_i H_{P_i}$, $c = \sum_i c_{P_i}$, $k = \sum_i k_{P_i}$.

The optimal post-collapse vertex is $\tilde{v} = -H^{-1}c$ (if $H$ is invertible), yielding both the new 3D position and attribute values simultaneously. If $H$ is singular, fallback heuristics such as the midpoint or constraint selection are used (Kulkarni et al., 23 Dec 2025).

3. Hoppe’s Energy-Based Progressive Meshes

An alternate approach, originally by Hoppe (1996), leverages a global energy function encompassing geometric distance, edge “spring” regularization, attribute error, and explicit discontinuity penalties:

• $E_\text{dist}(M) = \sum_{x_i} \| x_i - p(x_i) \|^2$ for geometric fidelity.
• $$E_\text{spring}(M) = \kappa \sum_{(v_a,v_b)} \|v_a - v_b\|^2$$ to preserve edge lengths locally.
• $$E_\text{scalar}(M) = c_\text{scalar} \sum_{a_i} \| a_i - \hat{a}(p(x_i)) \|^2$$ for attribute preservation.
• $$D_\text{disc} = \text{numProject}(X', e)\cdot\|e\|^2$$ (with infinity penalty to prevent collapse of discontinuities).

The cost of an edge collapse is the net increase in total mesh energy: $$\Delta E = [E_\text{dist} + E_\text{spring} + E_\text{scalar} + D_\text{disc}]_{M^+} - [\cdots]_{M^-}$$, where $M^-, M^+$ are the pre- and post-collapse meshes. Optimization proceeds in two phases:

1. Minimizing $\Delta E_\text{spring} + \Delta E_\text{dist}$ for the positional component $v$, typically solved as a small quadratic problem in $v$.
2. Minimizing $\Delta E_\text{scalar}$ for the new per-vertex attribute vector $\bar{a}$, also quadratic.
3. Discontinuity costs are computed as projective penalties to preserve key seams.

After collapse, the surviving vertex’s spatial and attribute coordinates are set respectively to $v$ and $\bar{a}$ (Kulkarni et al., 23 Dec 2025).

4. Algorithmic Workflow and Data Structures

Both attribute-aware collapse strategies operate within a common infrastructure:

• Per-vertex quadric storage: Each vertex holds its current (possibly high-dimensional) quadric $(H, c, k)$.
• Connectivity representation: Typically via half-edge or corner-table data structures ensuring $O(\deg(v))$ incident face/edge retrieval.
• Mesh interface: Vertices expose both position and attributes as a composite vector.
• Edge selection: Edge collapse candidates are prioritized via a queue keyed on (cost, edge, optimal new $v$ and attributes).
• Collapse operation: After collapse, the endpoint quadrics merge, adjacent faces/edges are removed, and local quadric/energy updates propagate.

This architecture leverages the same edge selection, connectivity maintenance, and programmatic safeguards as in purely geometric collapse, with the extension that attribute channels are included in all relevant operations (Kulkarni et al., 23 Dec 2025).

5. Preservation of Attribute Discontinuities and Visual Outcomes

Attribute-aware edge collapse circumvents the characteristic “attribute bleeding” associated with geometric-only QEM, especially at color seams or material boundaries. Visual comparison demonstrates that standard QEM drifts colors across sharp boundaries, whereas embedding attributes into the quadric yields crisp preservation of color edges. No tabulated error metrics are cited, but qualitative results reported in the literature indicate substantial improvements in preserving visual and semantic features during decimation (Kulkarni et al., 23 Dec 2025).

6. Practical Implementations and Pseudocode Structures

Implementation incorporates the following reference structures:

class HighDimQEM : IConstraint EvaluateCost(ṽ): return ṽᵀ H ṽ + 2 cᵀ ṽ + k. class ProgressiveCost ComputeGeoQuadric(mesh, edge) → (H_geo, c_geo) ComputeAttrQuadric(mesh, edge) → (H_attr, c_attr) For each collapse: v = argmin vᵀ H_geo v + 2 c_geoᵀ v ā = argmin āᵀ H_attr ā + 2 c_attrᵀ ā cost = Q_geo(v) + Q_attr(ā) + D_disc

These structures allow each candidate vertex and collapse to be scored and solved via linear-algebraic methods, with fallback logic where system matrices are singular (Kulkarni et al., 23 Dec 2025).

7. Synthesis and Theoretical Significance

Attribute-aware edge collapse, as realized by higher-dimensional QEM and Hoppe’s energy-based progressive meshes, advances mesh simplification by embedding all per-vertex data into a unified minimization. Both formulations reduce to local quadratic forms and require only small, incremental data structure updates per collapse. The methodology leverages the established framework of geometric simplification, extending it to robustly preserve both spatial and attribute discontinuities, with demonstrable qualitative improvements in the fidelity of simplified meshes (Kulkarni et al., 23 Dec 2025).