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arxiv: 1907.10250 · v1 · pith:JUPTRX6Dnew · submitted 2019-07-24 · 💻 cs.CV · cs.CG· cs.LG

Learning Embedding of 3D models with Quadric Loss

Nitin Agarwal , Sung-Eui Yoon , M Gopi This is my paper

Pith reviewed 2026-05-24 17:11 UTC · model grok-4.3

classification 💻 cs.CV cs.CGcs.LG
keywords 3D reconstructionquadric lossChamfer losssharp featurespoint cloudmesh reconstructiondeep learningsurface fitting
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The pith

Quadric loss combined with Chamfer loss improves reconstruction of sharp features over either loss alone or other point-surface alternatives.

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

The paper introduces quadric loss as a new point-surface loss function designed to minimize quadric error between reconstructed points and the input surface, with the goal of better preserving edges and corners that matter for human perception of 3D shapes. Quadric matrices are precomputed once, making the loss efficient and fully differentiable for training point-based or mesh-based networks. Experiments show that adding quadric loss to Chamfer loss produces higher-quality results than using Chamfer loss by itself or compared with other common point-surface losses. A reader would care because current reconstruction networks often smooth out or miss these sharp features, limiting their use in graphics and vision tasks.

Core claim

The paper claims that quadric loss, which minimizes the quadric error between the reconstructed points and the input surface using precomputed local quadric matrices, when combined with Chamfer loss, achieves better reconstruction results as compared to any one of them or other point-surface loss functions.

What carries the argument

Quadric loss: a differentiable point-surface loss computed from precomputed local quadric matrices that measures quadric error to the input surface.

If this is right

  • Reconstructed 3D models preserve edges and corners more accurately than with Chamfer loss alone.
  • Training remains efficient because quadric matrices are computed once before optimization.
  • The loss works with both point-based and mesh-based network architectures.
  • Overall reconstruction quality exceeds that obtained from other point-surface loss functions.

Where Pith is reading between the lines

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

  • The same precomputed quadric approach might be adapted to other geometric tasks such as surface denoising or registration.
  • Performance could be tested on inputs with varying noise levels or missing regions to check robustness.
  • The method might reduce the need for post-processing steps that restore sharp features in current pipelines.

Load-bearing premise

Minimizing quadric error on precomputed local quadrics will reliably improve perceptual quality of sharp features across diverse input surfaces and network architectures without introducing new artifacts.

What would settle it

An experiment on a test set of models with prominent edges and corners where the combined quadric-plus-Chamfer loss shows equal or worse edge-preservation metrics than Chamfer loss alone.

Figures

Figures reproduced from arXiv: 1907.10250 by M Gopi, Nitin Agarwal, Sung-Eui Yoon.

Figure 1
Figure 1. Figure 1: (a) Input point cloud reconstructed using an auto-encoder network with (b) Cham [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Computation of point-surface losses: Let the reconstructed point [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometric Interpretation of quadric loss: Quadric loss is an [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Effect of Point-Surface loss: Reconstruction results (2500 points) on example 3D [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reconstruction results of 3D models from the test set. To obtain a mesh from the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

Sharp features such as edges and corners play an important role in the perception of 3D models. In order to capture them better, we propose quadric loss, a point-surface loss function, which minimizes the quadric error between the reconstructed points and the input surface. Computation of Quadric loss is easy, efficient since the quadric matrices can be computed apriori, and is fully differentiable, making quadric loss suitable for training point and mesh based architectures. Through extensive experiments we show the merits and demerits of quadric loss. When combined with Chamfer loss, quadric loss achieves better reconstruction results as compared to any one of them or other point-surface loss functions.

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

2 major / 2 minor

Summary. The paper proposes quadric loss, a differentiable point-surface loss for 3D reconstruction networks that minimizes quadric error using precomputed 4x4 matrices Q_i for each input point. It is presented as easy to compute and suitable for point- and mesh-based architectures; when combined with Chamfer loss the authors claim it yields better reconstruction of sharp features than either loss alone or other point-surface losses, supported by extensive experiments.

Significance. If the empirical superiority holds under the stated conditions, quadric loss would supply a lightweight, precomputable regularizer that improves edge and corner fidelity without new parameters or heavy overhead. The precomputation of quadrics and full differentiability are concrete engineering strengths that could be adopted in existing pipelines.

major comments (2)
  1. [Section 3] Section 3: the quadric loss is defined directly as ∑ p^T Q_i p for reconstructed points p. The manuscript provides no analysis or bound on the approximation error incurred when p falls outside the local neighborhood used to fit Q_i; if this locality assumption is violated the combined loss can introduce new artifacts rather than sharpen features, directly undermining the central claim.
  2. [Experiments] Experiments (abstract and results sections): the headline assertion that quadric+Chamfer outperforms Chamfer alone or other point-surface losses is presented without reference to specific quantitative tables, ablation rows, metrics (e.g., Hausdorff, normal consistency on sharp edges), or error bars. This absence makes the superiority claim unverifiable from the given text and load-bearing for acceptance.
minor comments (2)
  1. [Abstract] Abstract: 'apriori' should be written as 'a priori'.
  2. [Section 3] Notation: the manuscript should explicitly state whether the same Q_i matrices are reused for all reconstructed points or whether nearest-neighbor assignment is performed at each iteration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will incorporate revisions to strengthen the presentation and verifiability of our claims.

read point-by-point responses

  1. Referee: [Section 3] Section 3: the quadric loss is defined directly as ∑ p^T Q_i p for reconstructed points p. The manuscript provides no analysis or bound on the approximation error incurred when p falls outside the local neighborhood used to fit Q_i; if this locality assumption is violated the combined loss can introduce new artifacts rather than sharpen features, directly undermining the central claim.

    Authors: We agree that the manuscript would benefit from an explicit discussion of the locality assumption. In the revision we will add a paragraph to Section 3 deriving a simple error bound based on the quadratic form and the radius of the neighborhood used to compute each Q_i, together with a short argument that training dynamics keep reconstructed points sufficiently close to the surface for the bound to remain useful. This directly addresses the possibility of new artifacts. revision: yes

  2. Referee: [Experiments] Experiments (abstract and results sections): the headline assertion that quadric+Chamfer outperforms Chamfer alone or other point-surface losses is presented without reference to specific quantitative tables, ablation rows, metrics (e.g., Hausdorff, normal consistency on sharp edges), or error bars. This absence makes the superiority claim unverifiable from the given text and load-bearing for acceptance.

    Authors: We accept that the superiority claim requires explicit quantitative support. The revised manuscript will add a results table reporting Chamfer distance, Hausdorff distance, and normal consistency restricted to sharp edges, with rows for the combined loss, each loss alone, and the other point-surface baselines. Standard deviations across three independent training runs will be included as error bars, and the table will be referenced from both the abstract and the experimental section. revision: yes

Circularity Check

0 steps flagged

No circularity: quadric loss is a direct definition of a new point-surface term

full rationale

The paper defines quadric loss directly as the sum of p^T Q_i p over reconstructed points p, with Q_i precomputed once from input neighborhoods via the standard quadric-error formulation. This is an explicit new loss function, not obtained by fitting parameters to data and then relabeling the fit as a prediction, nor by any self-citation chain that would make the central claim equivalent to its own inputs. Effectiveness is asserted via empirical comparison with Chamfer loss and other baselines; no equation reduces the proposed loss to a fitted quantity or to a prior result by the same authors. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that local quadric approximations capture the geometric features worth preserving and that precomputed matrices remain valid during training.

axioms (1)
  • domain assumption Quadric matrices can be computed a priori from the input surface
    Stated directly in the abstract as the source of efficiency.
invented entities (1)
  • Quadric loss no independent evidence
    purpose: Point-surface loss that penalizes deviation from local quadric surfaces
    Newly introduced term; no independent evidence supplied beyond the claim of better reconstruction.

pith-pipeline@v0.9.0 · 5645 in / 1200 out tokens · 20983 ms · 2026-05-24T17:11:28.889270+00:00 · methodology

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