SQEM: Spherical Quadratic Error Metric

• SQEM is a metric that extends traditional QEM to spherical domains by quantifying tangency deviations for accurate fitting of spheres or patches.
• It is applied in medial mesh computation and surface approximation, employing least-squares optimization to maintain intrinsic geometric structures.
• By leveraging spherically quasi-convex optimization, SQEM ensures global optimality and preserves structural fidelity in complex mesh processing.

The Spherical Quadratic Error Metric (SQEM) is a class of error metrics that extend the classical Quadratic Error Metric (QEM) to contexts where measurements or constraints are imposed in a spherical, curved, or otherwise non-Euclidean domain. SQEM plays a pivotal role in geometric optimization, medial axis transform computation, shape approximation, and mesh processing, where the preservation of spherical or intrinsic geometric relationships is crucial for accuracy and structural fidelity.

1. Mathematical Formulation

SQEM is described by the aggregation of squared tangency deviations between a geometric object (e.g., medial sphere, patch approximant) and an associated surface, typically across multiple sample points. The general form, as introduced in the context of medial mesh optimization (Wang et al., 12 Oct 2025), is

$$\mathrm{SQEM}_i = \sum_{x_{ik} \in \Omega_i} \left( (p_{ik} - c_i)^\top n_{ik} - r_i \right)^2,$$

where for each sample point $x_{ik}$ in the restricted power cell $\Omega_i$ of a medial sphere $m_i$ (with center $c_i$, radius $r_i$), $p_{ik}$ is the projection of $x_{ik}$ to the shape boundary, and $n_{ik}$ the local surface normal at $p_{ik}$. The inner term computes the deviation from the ideal tangency between the sphere and the surface, i.e., the difference between the distance from $c_i$ to the tangent plane at $p_{ik}$ and $r_i$.

The quadratic structure of SQEM admits a least-squares solution: $\min_{m_i} \|A m_i - b\|^2,$ with $m_i = (c_i, r_i) \in \mathbb{R}^4$ and $(A, b)$ constructed from the sampling process. The metric thus succinctly encodes alignment of spheres (or other primitives) to curved surfaces in a manner sensitive to both position and orientation.

2. Spherical Quadratic Error Metrics in Optimization

Characterizing when a quadratic function $q_A(x) = \langle Ax, x \rangle$ is spherically quasi-convex on a spherically convex set $\mathcal{C}$ is critical to using SQEM-type objectives in global optimization (Ferreira et al., 2018). The key equivalence is: $$q_A \text{ is spherically quasi-convex} \iff \varphi_A(x) = \frac{\langle Ax, x \rangle}{\|x\|^2}$$ is quasi-convex (i.e., its sublevel sets are convex). For $\mathcal{C}$ defined as an intersection of the unit sphere $S^{n-1}$ and a proper cone $\mathcal{K}$, necessary and sufficient conditions involve both spectral properties of $A$ and the sign structure of its eigenvectors. For example, for the positive orthant $\mathcal{K} = \mathbb{R}^n_+$, if $A$ is a $Z$-matrix and $(\lambda_2 I - A)$ is copositive, then $q_A$ is spherically quasi-convex.

The significance of this is that any strict local minimum of a spherically quasi-convex quadratic objective is also a global minimum. This property underpins the robust use of SQEM in optimization algorithms, especially when used to fit spheres, patches, or other primitives under spherical constraints.

3. SQEM in Medial Mesh Computation

The structure-aware optimization framework for computing the medial axis transform (MAT) utilizes SQEM to regulate the movement and placement of medial spheres (Wang et al., 12 Oct 2025). Each medial sphere $m_i$ is optimized not solely for even spatial coverage (particle repulsion), but subject to the local tangency constraints encoded by the SQEM. During optimization:

• The singular value decomposition of matrix $A$ in the SQEM least-squares system determines the null space (i.e., directions in which the sphere can move without altering critical tangency conditions).
• Fully constrained spheres (large singular values) are held fixed, whereas underconstrained ones have their gradient projected onto the null space, enforcing movements that do not break local structural alignment with the medial locus.

SQEM thereby provides a rigorous geometric safety net, ensuring sphere placement fidelity and preventing degradations in mesh quality, particularly at seams, sheets, or junctions. Empirically, this constraint leads to medial meshes with improved topological correctness and better triangle quality compared to MATFP and MATTopo schemes, as validated by the Medial Structure Error Ratio (Wang et al., 12 Oct 2025).

4. Relation to Intrinsic and Extrinsic Error Metrics

Classical mesh decimation methods, such as QEM, accumulate error by summing squared Euclidean distances in the ambient space. The intrinsic error metric (ICE) framework (Liu et al., 2023) generalizes this concept by accumulating mass-weighted drift in tangent spaces, allowing error to be tracked in terms of intrinsic (geodesic) properties rather than purely extrinsic geometry. ICE tracks curvature drift via complex-valued tangent vectors, offering guarantees on mesh quality through intrinsic Delaunay retriangulation. Such intrinsic strategies decouple mesh resolution from the size of numerical systems used by PDE solvers.

A plausible implication is that SQEM emerges naturally as the spherical (extrinsic or intrinsic) analogue to QEM and ICE, encoding errors in quadratic form but adapted to curved domains, and measuring deviation either as tangency errors or curvature drift weighted by local geometric structure.

5. Application to Spherical Surface Approximation

In polynomial surface approximation, particularly in computer-aided geometric design, spherical surfaces are approximated by tensor product quadratic Bézier patches (Vavpetič et al., 2023). Error metrics analogous to SQEM are employed—specifically, simplified radial errors $f(u,v) = \|p(u,v)\|_2^2 - 1$—to facilitate optimization over the patch control parameters. The optimization seeks the minimum maximal radial deviation over the patch, subject to geometric interpolation constraints and, where necessary, smoothness constraints (e.g., $G^1$).

Results show that imposing $G^1$ continuity increases the radial error compared to $G^0$ (only positional) continuity, demonstrating the trade-off between smoothness and geometric fidelity. In complex geometries (e.g., spherical rectangles instead of squares), multiple or even infinitely many optimal approximants may exist for the same aspect ratio.

6. Structural and Algorithmic Implications

The integration of SQEM into particle-based optimization or geometric design algorithms confers several advantages:

• Global Optimality: When the underlying quadratic forms are spherically quasi-convex, all local minima are global, ensuring predictable optimization landscapes (Ferreira et al., 2018).
• Structure Preservation: During medial mesh computation, SQEM projection restricts movement to directions that maintain medial structure, minimizing mesh artifacts (Wang et al., 12 Oct 2025).
• Algorithmic Tractability: The quadratic form enables efficient solution via linear least squares; structural awareness is enforced by controlling gradient directions via the null space analysis of $A$.
• Flexibility: For surface simplification or approximation, SQEM-generalized metrics adapt to both extrinsic (radial, tangency) and intrinsic (curvature, geodesic) errors, supporting application-specific fidelity criteria (Liu et al., 2023, Vavpetič et al., 2023).

These characteristics make SQEM a foundational tool for high-accuracy geometric modeling, medial axis computation, and mesh processing in both theoretical and practical settings.

7. Comparative Overview and Future Directions

Context/Method	SQEM Role	Key Advantage
Medial Mesh (MATStruct)	Medial sphere optimization	Structural awareness, mesh quality
Surface Approximation	Patch radial error metric	Geometric fidelity, $G^n$ trade-offs
Intrinsic Simplification	Potential intrinsic error metric	Element quality, decoupled matrices

The continued development of SQEM and its variants is expected to further improve the robustness of algorithms for mesh simplification, medial shape representation, and geometric optimization, particularly as more applications prioritize intrinsic geometric structure and the global properties of the error landscape.

Recent work clarifies that the adoption of spherically quasi-convex quadratic forms is crucial when deploying SQEM for global optimization or mesh quality certification, particularly in high-dimensional or highly constrained settings (Ferreira et al., 2018). Future research may address more general intrinsic error metrics on curved spaces, tighter theoretical guarantees for structure-aware optimization projections, and scalable implementations for industrial geometry processing workflows.