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arxiv: 2606.22880 · v1 · pith:QQLNI6GSnew · submitted 2026-06-22 · 💻 cs.GR

DJM: Compact Base Meshes for Displacement Mapping using Triangle Jacobians

Congyi Zhang , Nicholas Vining , Yanhong Lin , Alireza Khatami , Ziyu Sun , Xiaohu Guo , Wenping Wang , Alla Sheffer This is my paper

Pith reviewed 2026-06-26 06:39 UTC · model grok-4.3

classification 💻 cs.GR
keywords displacement mappingbase mesh simplificationJacobian metricQEMbijective mappingmicromeshmesh parameterization
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The pith

A Jacobian lower-bound constraint during QEM simplification produces smaller base meshes that still support bijective, low-distortion displacements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a Displacement Jacobian Metric to guide construction of coarse base meshes for displacement mapping of detailed surfaces. It modifies quadric-error-metric simplification so that every intermediate mesh maintains a minimum pointwise Jacobian value for the displacement function between input and base. This constraint, combined with explicit storage of the input-to-base mapping and a robust inverse-barycentric solver, replaces unreliable ray casting. The resulting base meshes achieve higher reconstruction accuracy at any given triangle count than prior methods. The approach is demonstrated on micromesh rendering and neural displacement encoding.

Core claim

The central claim is that enforcing a lower bound on the pointwise Jacobian of the displacement mapping throughout the simplification process yields base meshes whose supported displacements remain bijective and low-distortion while using fewer triangles than meshes produced by unconstrained QEM or other existing schemes.

What carries the argument

The Displacement Jacobian Metric (DJM), which supplies a pointwise lower bound on the Jacobian determinant of the displacement function and is inserted directly into the QEM edge-collapse cost to reject collapses that would violate bijectivity or distortion limits.

If this is right

  • Base meshes can be reduced in size while preserving reconstruction accuracy.
  • Explicit input-to-base correspondences replace ray-mesh intersections for all subsequent displacement computations.
  • The same base meshes support both traditional micromesh rasterization and neural displacement encoders with less error.
  • The method inherits the linear complexity of QEM while adding only the Jacobian check and mapping bookkeeping.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Jacobian-tracking idea could be applied to other coarsening tasks that require guaranteed bijective mappings, such as progressive parameterization.
  • Because the mapping is stored explicitly, the approach may extend to dynamic or animated base meshes without recomputing correspondences from scratch.

Load-bearing premise

Maintaining a lower bound on the Jacobian at every step of simplification is sufficient to ensure the final base mesh admits a bijective low-distortion displacement for the original detailed surface.

What would settle it

An input mesh for which the DJM-simplified base, after displacement encoding, produces visible folds, overlaps, or high local distortion in the reconstructed surface would show the Jacobian bound is insufficient.

Figures

Figures reproduced from arXiv: 2606.22880 by Alireza Khatami, Alla Sheffer, Congyi Zhang, Nicholas Vining, Wenping Wang, Xiaohu Guo, Yanhong Lin, Ziyu Sun.

Figure 1
Figure 1. Figure 1: We compactly represent input shapes (a) as a combination of a coarse base-mesh (d, left) and a displacement map enabling accurate reconstruction (d, [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representing shapes (a) as a base mesh (b, arrows visualize corner [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ray casting-based methods are not stable for displacement calcula [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: DJM base mesh construction workflow. Our base mesh construction method loosely follows the standard QEM workflow [Garland and Heckbert 1997], but seeks to balance the classical QEM geometric collapse metric versus mapping dis￾tortion and triangle shape. One naive approach to address these additional desiderata is to run QEM as-is, but reject any collapse operation that introduces base mesh triangles that v… view at source ↗
Figure 5
Figure 5. Figure 5: We optimize the sampling pattern on a per base-mesh triangle basis [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Visual comparison of micromeshes constructed by Maggiordomo et al. [2023], QEM [Garland and Heckbert 1997], and our method. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visual comparison with NGF [Sivaram et al. 2024]. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Visual comparison with Pentapati et al.[Pentapati et al. 2025]. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

Representing complex geometry as a displacement function defined over a coarse base mesh enables compact storage and accelerated rendering. The core challenge in converting detailed triangle meshes into this representation is computing base meshes that have as few triangles as possible, while also supporting displacement functions that accurately approximate the input. Accurate approximation requires the supported displacement functions to bijectively map the input surface onto the base with low parametric distortion. We observe that this distortion can be measured by evaluating the pointwise Jacobian of the displacement functions. Our new DJM (Displacement Jacobian Metric)-based base-mesh construction method uses the Jacobian of the displacement functions to guide base mesh computation, enabling us to outperform prior approaches in terms of accuracy to size trade-off. We achieve this goal by proposing a variant of the QEM-based simplification scheme that constrains the displacement mapping between the input and the base to be bijective and low distortion (defined as satisfying a lower bound on the mapping Jacobian). When evaluating and encoding the displacement maps, we avoid unreliable ray-mesh intersections by explicitly storing the mapping between the input mesh and the base throughout the construction process, and use this mapping within a robust inverse barycentric displacement solver to obtain dense base-to-mesh correspondences to assist all computations. We demonstrate DJM to outperform alternative schemes in terms of reconstruction accuracy to size trade-off, and demonstrate its robustness and usability for micromesh-based rendering and neural encoding.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims to introduce DJM, a Displacement Jacobian Metric for constructing compact base meshes supporting displacement maps. It modifies QEM simplification to enforce a lower bound on the pointwise Jacobian of the displacement function (defining this as ensuring bijectivity and low distortion), stores explicit input-to-base mappings to enable a robust inverse barycentric solver, and reports improved accuracy-to-size trade-offs over prior methods for micromesh rendering and neural encoding.

Significance. If the construction reliably yields globally bijective low-distortion displacements with superior trade-offs, the approach could improve compact geometry representations in graphics pipelines. The explicit mapping storage is a practical engineering contribution that sidesteps ray-intersection unreliability.

major comments (1)
  1. [Abstract] Abstract (paragraph on the QEM variant): the central claim defines bijectivity and low distortion solely via a lower bound on the mapping Jacobian enforced during edge collapses. However, det(J) > 0 only guarantees local invertibility; global injectivity on a closed surface requires additional conditions (properness, no global folds). The manuscript provides no invariant, propagation argument, or post-simplification injectivity check showing that the per-collapse constraint yields a globally bijective final displacement map.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the distinction between local and global bijectivity. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on the QEM variant): the central claim defines bijectivity and low distortion solely via a lower bound on the mapping Jacobian enforced during edge collapses. However, det(J) > 0 only guarantees local invertibility; global injectivity on a closed surface requires additional conditions (properness, no global folds). The manuscript provides no invariant, propagation argument, or post-simplification injectivity check showing that the per-collapse constraint yields a globally bijective final displacement map.

    Authors: We agree that a strictly positive Jacobian determinant supplies only a local invertibility guarantee and does not by itself preclude global folds or non-injectivity on a closed surface. The manuscript defines the target property via the per-collapse Jacobian lower bound and relies on the incremental construction (starting from the identity map) together with explicit input-to-base correspondence storage to maintain usable mappings in practice. No formal propagation invariant or post-process global injectivity test is supplied. We will revise the abstract and the relevant method section to state explicitly that the Jacobian constraint enforces local bijectivity and low distortion, and we will add a short discussion of the distinction together with empirical checks (e.g., absence of foldovers detected via the inverse barycentric solver and visual inspection of rendered displacements) on the evaluated models. These changes will temper the claim while preserving the engineering contribution of the explicit mapping storage. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic enforcement of Jacobian bound is independent construction

full rationale

The paper defines low-distortion as a lower bound on the pointwise Jacobian and enforces this bound inside a QEM variant. This is a direct algorithmic constraint rather than a self-definition, fitted prediction, or reduction to prior self-citation. No equations equate the final bijectivity claim to the input by construction, and the accuracy-to-size improvement is presented as an empirical outcome of the procedure. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review prevents exhaustive extraction; the method rests on the domain assumption that Jacobian lower bounds control bijectivity and distortion, plus the standard math definition of the Jacobian.

axioms (1)
  • domain assumption The pointwise Jacobian of the displacement function measures parametric distortion of the mapping from input surface to base mesh.
    Invoked in the abstract as the basis for the new metric and the simplification constraint.
invented entities (1)
  • DJM (Displacement Jacobian Metric) no independent evidence
    purpose: To quantify and constrain distortion during base-mesh simplification.
    New metric introduced by the paper to guide the QEM variant.

pith-pipeline@v0.9.1-grok · 5804 in / 1294 out tokens · 23346 ms · 2026-06-26T06:39:40.523523+00:00 · methodology

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