pith. sign in

arxiv: 2606.18069 · v1 · pith:PDAQVCREnew · submitted 2026-06-16 · 💻 cs.GR · cs.CG· cs.CV

Blended Chart Surfaces: A Seamless Explicit Representation for Smooth Surface Fitting

Romy Williamson , Niloy Mitra This is my paper

Pith reviewed 2026-06-26 21:49 UTC · model grok-4.3

classification 💻 cs.GR cs.CGcs.CV
keywords blended chart surfacesexplicit surface representationsmooth surface fittingpolynomial mapsone-ring blendingproxy meshdifferentiable surfacesgeometry processing
0
0 comments X

The pith

Blended Chart Surfaces jointly optimize per-vertex polynomial maps on a proxy mesh and fuse them with one-ring coordinate blending to produce globally smooth explicit surfaces without parametrizations or seams.

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

The paper introduces Blended Chart Surfaces as an explicit surface representation that stays smooth by construction while remaining compact and network-free. It begins with a coarse proxy mesh that supplies the topology and rough shape, then fits a polynomial map at each vertex to match an implicit target shape. Neighboring maps are combined through a smooth blending scheme based on one-ring coordinates so that the result has no visible seams and supports direct evaluation of derivatives. This construction decouples the topology and coarse geometry carried by the proxy from the finer geometric details carried by the patches. A sympathetic reader would care because the approach gives reliable access to normals, curvatures, and surface energies inside modern differentiable pipelines while avoiding the seam artifacts or iso-surfacing steps common in other representations.

Core claim

Given a coarse proxy mesh that encodes intended topology and approximate geometry, Blended Chart Surfaces optimize a polynomial map at each proxy vertex to fit an implicit target, then fuse neighboring maps with a smooth one-ring coordinate blending scheme. The resulting surface is globally smooth and fully differentiable by construction, equivariant to rigid motions and scaling of the proxy, and directly supplies differential quantities without requiring an input parametrization or post-processing.

What carries the argument

The one-ring coordinate blending scheme that fuses neighboring per-vertex polynomial maps to guarantee continuity and differentiability across chart boundaries.

If this is right

  • The surface remains globally smooth and C1-continuous across all patch boundaries by construction.
  • Differential quantities such as normals and surface energies become directly accessible without extra computation.
  • The representation supports arbitrary surface topologies supplied by the proxy mesh.
  • No input parametrization is required because the optimization fits directly to the implicit target.
  • The construction is equivariant under rigid motions and uniform scaling of the proxy mesh.

Where Pith is reading between the lines

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

  • The decoupling of topology from detail could let modelers edit coarse connectivity independently of fine geometry in interactive design tools.
  • Because the method is network-free and uses an off-the-shelf optimizer, it may integrate more readily into existing geometry-processing codebases than neural alternatives.
  • Higher-degree polynomials per chart could be substituted to increase local expressivity while retaining the same blending framework.

Load-bearing premise

The proxy mesh must correctly encode the intended surface topology and lie close enough to the target geometry that the independent per-vertex polynomial optimizations converge to a single consistent surface.

What would settle it

Visible jumps in position or first derivatives when the surface is evaluated along paths that cross from one polynomial patch into a neighboring patch would show that the blending scheme fails to produce global smoothness.

Figures

Figures reproduced from arXiv: 2606.18069 by Niloy Mitra, Romy Williamson.

Figure 1
Figure 1. Figure 1: We propose Blended Chart Surfaces, a coarse proxy mesh– guided, network-free, explicit surface representation formed by com￾posing local polynomial maps. The model is compact, faithfully captures surface geometry, and is fully differentiable for optimiza￾tion in modern learning pipelines. Here, |V| = 250 and dpoly = 2. submitted to arXiv (2026) arXiv:2606.18069v1 [cs.GR] 16 Jun 2026 [PITH_FULL_IMAGE:figur… view at source ↗
Figure 2
Figure 2. Figure 2: Motivation. (Top-left) Implicit fields (stored on a grid or as a neural field) require high effective resolution (here iso￾surfaced from a 103 grid) to avoid visible discretization artifacts when extracted with Marching Cubes. (Bottom-left) Mesh-based displacement fields parametrized by an MLP (e.g. [SLR24]) require fewer parameters but retain the tangent discontinuities present in the proxy. (Right) Our B… view at source ↗
Figure 3
Figure 3. Figure 3: Curve case. Each coarse proxy vertex is assigned a poly￾nomial vertex function. Within each edge, the vertex functions are combined using blending functions (partition-of-unity weights), pro￾ducing a local map that evaluates curve points (and derivatives) smoothly on that base domain. See text for details. τA,B(t) represents the same position in the coordinates of domain B. Although these transition maps a… view at source ↗
Figure 4
Figure 4. Figure 4: blended chart curve fitting. (Left to right) We start from the target implicit field (closed-curve SDF) and a coarse proxy polygon (V,E). At each coarse vertex, we optimize a local curvelet parametrized by maps mi (in this example, degree-5 polynomials) together with their associated local frame (i.e, rotation Ri , translation vi , scale si). These optimized curvelets are then repositioned to form the vert… view at source ↗
Figure 6
Figure 6. Figure 6: Choices of blending functions on an equilateral trian￾gle and their induced vertex-centered ‘hat’ functions. (Top) Weight distributions over the equilateral triangle: barycentric interpola￾tion and three other variants based on radial distance (exponential, trigonometric, and smooth transition). See Section 4 for the defi￾nitions. (Bottom) The corresponding blending function around a valence-6 vertex, illu… view at source ↗
Figure 5
Figure 5. Figure 5: Computing one-ring coordinates. Our one-ring coor￾dinates are based on the idea that if we affinely transform every triangle in a small part of a mesh into a unit equilateral trian￾gle, without changing the connectivity, then the one-rings in this transformed mesh-part would all look like (be isometric to) cones— including hyperbolic cones, where η > 6, planar parts, where η = 6, elliptic points, where η <… view at source ↗
Figure 7
Figure 7. Figure 7: Discovering a surface parametrization. Blended Chart Surfaces implicitly induce a seamless correspondence between the proxy and the fitted surface via the one-ring coordinates and blended local maps. Left (Bob): We visualize the optimized correspondence by pulling back a procedural color pattern defined on the coarse proxy mesh onto the Blended Chart Surface. Right (Fertility): We illustrate how a color fi… view at source ↗
Figure 8
Figure 8. Figure 8: Result gallery. BCS on different targets (Igea, Bob, and a twisted torus). For each model, we show the coarse proxy mesh, the optimized but unblended local polynomial patches (illustrating the patch structure prior to blending; quadratic polynomials used in these examples), the resulting Blended Chart Surface, and the corresponding error map (color bar at right; models are normalized to a unit-width boundi… view at source ↗
Figure 9
Figure 9. Figure 9: Effect of the blending function on continuity and surface quality (see [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Mean Curvature colormaps computed using auto￾matic differentiation on a BCS with ‘smooth transition’ blending and a BCS with trigonometric blending, and discrete mean cur￾vature computed on a high-resolution mesh of an Interpolating Spline [DFK∗ 25], using the same coarse net and the same class of vertex functions (quadratic polynomials). Access to surface properties. Blended Chart Surfaces support sta￾bl… view at source ↗
Figure 11
Figure 11. Figure 11: Elastic energy. We displace one proxy vertex of a Blended Chart Surfaces torus while keeping the vertex functions fixed. We compute the deformation gradient from the Jacobians of the deformed and undeformed tori, and evaluate elastic energy via the Green–Lagrange strain (rubber-like material parameters). Red indicates high energy density and white indicates zero. We then reoptimize the vertex functions wi… view at source ↗
Figure 12
Figure 12. Figure 12: Effect of the coarse proxy mesh on fitting quality. We fit Blended Chart Surfaces to a ‘rippled’ target with strongly varying curvature using proxy meshes of increasing resolution (|V|=100,150,250). The results are largely robust to changes in proxy connectivity, reflecting the locality of our one-ring coordinates and PoU blending; however, at very low resolution, the local optimization can stall, leading… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison with Djuren et al. [DFK∗ 25]. Whilst our Blended Chart Surface are constructed similarly to the Vertex￾Centric Interpolating Splines (by blending together vertex functions, in this case polynomials), the key difference is that we jointly op￾timize vertex functions, guided by a target implicit shape, while Vertex-Centric Interpolating Splines use individually optimal vertex functions to best int… view at source ↗
Figure 14
Figure 14. Figure 14: Levels of detail controlled by polynomial degree. We fit a Blended Chart Surface using cubic vertex polynomials and visualize coarser models by truncating coefficients to obtain constant (degree 0), linear (degree 1), and quadratic (degree 2) variants, respectively. Increasing degree improves local approximation quality. 18|V|. For most of the examples in this paper, the proxy mesh has around 250 vertices… view at source ↗
read the original abstract

A surface representation suitable for geometry processing should be compact and explicit, provide global smoothness guarantees, support a wide range of surface topologies, and offer reliable access to differential quantities such as normals and surface energies, while remaining compatible with modern differentiable optimization. Existing neural representations typically sacrifice one or more of these properties: implicit fields typically require iso-surfacing for downstream use, while explicit neural maps are constrained by canonical-domain parametrizations or exhibit seam artifacts between local charts. We introduce Blended Chart Surfaces, a compact, network-free, explicit representation that is smooth by construction and anchored to user-provided topology. Given a coarse proxy mesh encoding the intended surface topology and approximate geometry, Blended Chart Surfaces jointly optimize for a polynomial map at each proxy vertex using an off-the-shelf optimizer to fit to an implicit target shape, avoiding the need for an input parametrization. Neighboring maps are fused using a smooth 'one-ring coordinate' blending scheme, decoupling topology and coarse geometry (carried by the proxy) from geometric details (carried by the local patches). The surface is globally smooth, fully differentiable, and enables stable evaluation of derivatives, making differential quantities and surface energies directly accessible. Additionally, our construction is equivariant to rigid motions and scaling of the proxy mesh. We evaluate Blended Chart Surfaces on various topologies and geometric complexity, and compare against explicit alternatives including interpolating-function baselines and mesh-displacement MLPs. Across these, Blended Chart Surfaces achieve a favorable trade-off among compactness, simplicity, access to differential quantities, and expressivity while remaining smooth across patch boundaries.

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 introduces Blended Chart Surfaces, a compact explicit surface representation anchored to a user-provided coarse proxy mesh. Polynomial maps are jointly optimized per proxy vertex via an off-the-shelf optimizer to fit an implicit target shape; neighboring maps are then fused via a smooth one-ring coordinate blending scheme. The construction is claimed to be globally smooth and differentiable by design, equivariant to rigid motions and scaling, and to decouple topology/coarse geometry (from the proxy) from fine details (from the patches), while avoiding input parametrizations and providing direct access to differential quantities.

Significance. If the central claims hold, the representation offers a network-free explicit alternative that guarantees smoothness across charts and supports stable derivative evaluation, which would be useful for geometry-processing tasks requiring differentiable energies or optimization. The decoupling of topology from details and the avoidance of seam artifacts are potentially valuable strengths.

major comments (2)
  1. [Abstract / construction description] The central smoothness-by-construction claim depends on the optimized per-vertex polynomials producing consistent values and derivatives under the one-ring blending weights. No analysis, convergence criteria, or regularization terms are supplied to guarantee that the joint optimization reaches such an alignment when the proxy mesh deviates from the target; the abstract only states that the proxy supplies 'approximate geometry.'
  2. [Abstract / evaluation description] The weakest assumption—that the proxy mesh encodes correct topology and lies sufficiently close for off-the-shelf optimization to converge without topology changes or additional regularization—is load-bearing for the 'globally consistent smooth surface' claim, yet no supporting experiments, failure cases, or sensitivity analysis appear.
minor comments (2)
  1. [Abstract] The abstract mentions comparisons to 'interpolating-function baselines and mesh-displacement MLPs' but does not specify the exact baselines, metrics, or quantitative results; these details are needed to assess the claimed favorable trade-off.
  2. [Abstract] The free parameters (polynomial degree per chart, blending function parameters) are listed but their selection procedure and sensitivity are not discussed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract / construction description] The central smoothness-by-construction claim depends on the optimized per-vertex polynomials producing consistent values and derivatives under the one-ring blending weights. No analysis, convergence criteria, or regularization terms are supplied to guarantee that the joint optimization reaches such an alignment when the proxy mesh deviates from the target; the abstract only states that the proxy supplies 'approximate geometry.'

    Authors: The joint optimization minimizes a global fitting objective across all per-vertex polynomials simultaneously. Because the one-ring blending weights are smooth and the loss is evaluated over the entire surface, this process encourages alignment of values and derivatives in overlap regions. We agree that the manuscript does not supply formal convergence analysis or additional regularization terms beyond the base fitting loss. In revision we will expand the method section with a description of the loss, optimizer settings, and empirical checks for boundary consistency to better substantiate the smoothness claim. revision: yes

  2. Referee: [Abstract / evaluation description] The weakest assumption—that the proxy mesh encodes correct topology and lies sufficiently close for off-the-shelf optimization to converge without topology changes or additional regularization—is load-bearing for the 'globally consistent smooth surface' claim, yet no supporting experiments, failure cases, or sensitivity analysis appear.

    Authors: The manuscript states that the proxy supplies approximate geometry and correct topology, and the reported experiments use user-provided proxies that satisfy this condition. We acknowledge the absence of explicit sensitivity tests or failure-case analysis. In the revision we will add a new subsection with experiments that vary proxy quality (e.g., small perturbations) and discuss observed failure modes when the proxy deviates substantially or carries incorrect topology. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper describes an optimization procedure that fits per-vertex polynomial maps to an external implicit target shape using an off-the-shelf optimizer, followed by a one-ring coordinate blending construction to enforce smoothness. No load-bearing step reduces by definition, by fitted-input renaming, or by self-citation chain to the inputs themselves; the smoothness guarantee is produced by the explicit blending weights rather than presupposed, and the fitting objective is independent of the final surface representation. The proxy-mesh assumption is an empirical precondition for convergence, not a circular definitional step.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The construction relies on the existence of a suitable proxy mesh, the ability of low-degree polynomials to capture local geometry, and the mathematical properties of the one-ring blending to guarantee global smoothness; these are not derived from first principles in the abstract.

free parameters (2)
  • polynomial degree per chart
    Degree is not specified in abstract but must be chosen to balance expressivity and smoothness.
  • blending function parameters
    One-ring coordinate blending requires definition of weights or falloff that are not detailed.
axioms (2)
  • domain assumption A user-provided coarse proxy mesh correctly encodes topology and approximate geometry.
    Invoked when stating that the proxy carries topology and coarse geometry while patches carry details.
  • domain assumption Polynomial maps can be optimized independently per vertex and then blended without introducing discontinuities.
    Central to the claim of global smoothness by construction.
invented entities (1)
  • one-ring coordinate blending scheme no independent evidence
    purpose: To fuse neighboring polynomial maps smoothly without seams.
    New blending mechanism introduced to decouple topology from details.

pith-pipeline@v0.9.1-grok · 5818 in / 1536 out tokens · 18616 ms · 2026-06-26T21:49:45.893596+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

115 extracted references · 7 canonical work pages

  1. [1]

    J. M. Buhmann and D. W. Fellner and M. Held and J. Ketterer and J. Puzicha , TITLE =. 1998 , PAGES =. doi:10.1111/1467-8659.00269 , NOTE =

    work page doi:10.1111/1467-8659.00269 1998

  2. [2]

    and Helmberg, Christoph , TITLE =

    Fellner, Dieter W. and Helmberg, Christoph , TITLE =. 1993 , PAGES =

    1993

  3. [3]

    Kobbelt and M

    L. Kobbelt and M. Stamminger and H.-P. Seidel , title =. doi:10.1111/1467-8659.16.3conferenceissue.36 , note =

    work page doi:10.1111/1467-8659.16.3conferenceissue.36

  4. [4]

    Lafortune and Sing-Choong Foo and Kenneth E

    Eric P. Lafortune and Sing-Choong Foo and Kenneth E. Torrance and Donald P. Greenberg , title =. Proc. SIGGRAPH '97 , volume = 31, pages =

  5. [5]

    : Blended barycentric coordinates

    Anisimov D., Panozzo D., Hormann K. : Blended barycentric coordinates. Computer Aided Geometric Design 52-53 (2017), 205--216

    2017

  6. [6]

    : Reconstruction and representation of 3d objects with radial basis functions

    Carr J., Beatson R., Cherrie J., Mitchell T., Fright W., McCallum B., Evans T. : Reconstruction and representation of 3d objects with radial basis functions. In Proc. Annual Conference on Computer Graphics and Interactive Techniques (2001)

    2001

  7. [7]

    : Digital geometry processing with discrete exterior calculus

    Crane K., de Goes F., Desbrun M., Schröder P. : Digital geometry processing with discrete exterior calculus. In ACM SIGGRAPH 2013 courses (New York, NY, USA, 2013), SIGGRAPH '13, ACM

    2013

  8. [8]

    : Algebraic smooth occluding contours

    Capouellez R., Dai J., Hertzmann A., Zorin D. : Algebraic smooth occluding contours. In Proc. SIGGRAPH (2023)

    2023

  9. [10]

    : Learning implicit fields for generative shape modeling

    Chen Z., Zhang H. : Learning implicit fields for generative shape modeling. In Proc. IEEE/CVF Conf. on Computer Vision & Pattern Recognition (June 2019), pp. 5939--5948

    2019

  10. [11]

    : Better patch stitching for parametric surface reconstruction

    Deng Z., Bednarik J., Salzmann M., Fua P. : Better patch stitching for parametric surface reconstruction. In Proc. 3DV (11 2020), pp. 593--602

    2020

  11. [12]

    : Interpolating splines over triangulated surfaces by blending vertex-centric local geometries

    Djuren T., Finnendahl U., Kohlbrenner M., Worchel M., Alexa M. : Interpolating splines over triangulated surfaces by blending vertex-centric local geometries. Computers and Graphics 131 (2025), 104316

    2025

  12. [13]

    : Discrete differential geometry - an applied introduction

    Desbrun M., Polthier K., Schröder P., Stern A. : Discrete differential geometry - an applied introduction. In ACM SIGGRAPH course notes (2006)

    2006

  13. [14]

    J., Wimmer M

    Erler P., Guerrero P., Ohrhallinger S., Mitra N. J., Wimmer M. : Points2Surf : Learning implicit surfaces from point clouds. In Proc. Euro. Conf. on Computer Vision (2020)

    2020

  14. [15]

    : A generalized blending scheme for arbitrary order of continuity

    Fang X. : A generalized blending scheme for arbitrary order of continuity. Preprint (2023)

    2023

  15. [16]

    G., Russell B., Aubry M

    Groueix T., Fisher M., Kim V. G., Russell B., Aubry M. : Atlasnet: A papier-m\^ach\'e approach to learning 3d surface generation. In Proc. IEEE/CVF Conf. on Computer Vision & Pattern Recognition (2018)

    2018

  16. [17]

    M., Hughes J

    Grimm C. M., Hughes J. F. : Modeling surfaces of arbitrary topology using manifolds. In Proc. Annual Conference on Computer Graphics and Interactive Techniques (1995), p. 359–368

    1995

  17. [18]

    J., Riesenfeld R

    Gordon W. J., Riesenfeld R. F. : B-spline curves and surfaces. In Computer Aided Geometric Design. Academic Press, 1974, pp. 95--126

    1974

  18. [19]

    : Parametric pseudo-manifolds

    Gallier J., Xu D., Siqueira M. : Parametric pseudo-manifolds. Differential Geometry and its Applications 30 (12 2012), 702--736. https://doi.org/10.1016/j.difgeo.2012.09.002 doi:10.1016/j.difgeo.2012.09.002

    work page doi:10.1016/j.difgeo.2012.09.002 2012

  19. [20]

    : Functional networks for b-spline surface reconstruction

    Iglesias A., Echevarr \' a G., G \'a lvez A. : Functional networks for b-spline surface reconstruction. Future Generation Computer Systems 20, 8 (2004), 1337--1353

    2004

  20. [21]

    : G ^2 interpolating spline with local maximum curvature

    Jiang B., Chen R. : G ^2 interpolating spline with local maximum curvature. ACM Trans. on Graphics 44, 6 (2025), 1--13

    2025

  21. [22]

    : Dual contouring of hermite data

    Ju T., Losasso F., Schaefer S., Warren J. : Dual contouring of hermite data. In Proc. Annual Conference on Computer Graphics and Interactive Techniques (July 2002), vol. 21, p. 339–346

    2002

  22. [23]

    : Adam: A method for stochastic optimization

    Kingma D., Ba J. : Adam: A method for stochastic optimization. International Conference on Learning Representations (2015)

    2015

  23. [24]

    : Poisson Surface Reconstruction

    Kazhdan M., Bolitho M., Hoppe H. : Poisson Surface Reconstruction . In Symposium on Geometry Processing (2006), Sheffer A., Polthier K., (Eds.)

    2006

  24. [25]

    : Provably good moving least squares

    Kolluri R. : Provably good moving least squares. ACM Trans. Algorithms 4, 2 (May 2008)

    2008

  25. [26]

    : Relu fields: The little non-linearity that could

    Karnewar A., Ritschel T., Wang O., Mitra N. : Relu fields: The little non-linearity that could. In Proc. SIGGRAPH (2022)

    2022

  26. [27]

    E., Cline H

    Lorensen W. E., Cline H. E. : Marching cubes: A high resolution 3d surface construction algorithm. In Proc. Annual Conference on Computer Graphics and Interactive Techniques (1987), p. 163–169

    1987

  27. [28]

    Lee J. M. : Introduction to Smooth Manifolds, 2 ed., vol. 218 of Graduate Texts in Mathematics. Springer, 2013

    2013

  28. [30]

    D., Kim V

    Liu H. D., Kim V. G., Chaudhuri S., Aigerman N., Jacobson A. : Neural subdivision. CoRR abs/2005.01819 (2020)

    arXiv 2005

  29. [31]

    : Mash: Masked anchored spherical distances for 3d shape representation and generation

    Li C., Xin Y., Zhou X., Shamir A., Zhang H., Liu L., Hu R. : Mash: Masked anchored spherical distances for 3d shape representation and generation. Proc. SIGGRAPH (2025), 11 pages

    2025

  30. [32]

    Morreale L., Aigerman N., Kim V., Mitra N. J. : Neural surface maps. In Proc. IEEE/CVF Conf. on Computer Vision & Pattern Recognition (2021)

    2021

  31. [33]

    : Instant neural graphics primitives with a multiresolution hash encoding

    M\" u ller T., Evans A., Schied C., Keller A. : Instant neural graphics primitives with a multiresolution hash encoding. ACM Trans. on Graphics 41, 4 (July 2022)

    2022

  32. [34]

    : Occupancy networks: Learning 3d reconstruction in function space

    Mescheder L., Oechsle M., Niemeyer M., Nowozin S., Geiger A. : Occupancy networks: Learning 3d reconstruction in function space. In Proc. IEEE/CVF Conf. on Computer Vision & Pattern Recognition (2019), pp. 4460--4470

    2019

  33. [35]

    : Polyhedral control-net splines for analysis

    Mishra B., Peters J. : Polyhedral control-net splines for analysis. Computers & Mathematics with Applications 151 (2023), 215--221

    2023

  34. [36]

    P., Tancik M., Barron J

    Mildenhall B., Srinivasan P. P., Tancik M., Barron J. T., Ramamoorthi R., Ng R. : Nerf: representing scenes as neural radiance fields for view synthesis. Commun. ACM 65, 1 (Dec. 2021), 99–106

    2021

  35. [37]

    Munkres J. R. : Elements of Algebraic Topology. Addison-Wesley Publishing Company, 1984

    1984

  36. [38]

    : Smooth Manifolds and Observables, vol

    Nestruev J. : Smooth Manifolds and Observables, vol. 220 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003

    2003

  37. [39]

    : Multi-level partition of unity implicits

    Ohtake Y., Belyaev A., Alexa M., Turk G., Seidel H.-P. : Multi-level partition of unity implicits. In ACM Trans. Graph. (July 2003), vol. 22, p. 463–470

    2003

  38. [40]

    : Applications of Lie Groups to Differential Equations

    Olver P. : Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Springer New York, 1993. URL: https://books.google.co.uk/books?id=sI2bAxgLMXYC

    1993

  39. [41]

    : Smooth free-form surfaces over irregular meshes generalizing quadratic splines

    Peters J. : Smooth free-form surfaces over irregular meshes generalizing quadratic splines. Computer Aided Geometric Design 10, 3 (1993), 347--361

    1993

  40. [42]

    J., Florence P., Straub J., Newcombe R., Lovegrove S

    Park J. J., Florence P., Straub J., Newcombe R., Lovegrove S. : Deepsdf: Learning continuous signed distance functions for shape representation. In Proc. IEEE/CVF Conf. on Computer Vision & Pattern Recognition (2019), pp. 165--174

    2019

  41. [44]

    : Polyhedral design with blended n -sided interpolants

    Salvi P. : Polyhedral design with blended n -sided interpolants. http://arxiv.org/abs/2601.19322 arXiv:2601.19322

    arXiv

  42. [45]

    : Neural geometry fields for meshes

    Sivaram V., Li T.-M., Ramamoorthi R. : Neural geometry fields for meshes. In Proc. SIGGRAPH (2024)

    2024

  43. [46]

    Sitzmann V., Martel J. N. P., Bergman A. W., Lindell D. B., Wetzstein G. : Implicit neural representations with periodic activation functions. In Proc. Conf. on Neural Information Processing Systems (2020)

    2020

  44. [47]

    Saupe D., Vrani \'c D. V. : 3d model retrieval with spherical harmonics and moments. In Joint Pattern Recognition Symposium (2001), Springer, pp. 392--397

    2001

  45. [48]

    : Neural geometric level of detail: Real-time rendering with implicit 3D shapes

    Takikawa T., Litalien J., Yin K., Kreis K., Loop C., Nowrouzezahrai D., Jacobson A., McGuire M., Fidler S. : Neural geometric level of detail: Real-time rendering with implicit 3D shapes. In Proc. IEEE/CVF Conf. on Computer Vision & Pattern Recognition (2021)

    2021

  46. [49]

    P., Mildenhall B., Fridovich-Keil S., Raghavan N., Singhal U., Ramamoorthi R., Barron J

    Tancik M., Srinivasan P. P., Mildenhall B., Fridovich-Keil S., Raghavan N., Singhal U., Ramamoorthi R., Barron J. T., Ng R. : Fourier features let networks learn high frequency functions in low dimensional domains. Proc. Conf. on Neural Information Processing Systems (2020)

    2020

  47. [50]

    Williamson R., Mitra N. J. : Neural geometry processing via spherical neural surfaces. Proc. Eurographics (2025)

    2025

  48. [51]

    : Subdivision Methods for Geometric Design: A Constructive Approach

    Warren J., Weimer H. : Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann, 2002

    2002

  49. [52]

    : Neupps: Neural piecewise parametric surfaces

    Yang L., Liang Y., Li X., Zhang C., Lin G., Lin C., Sheffer A., Schaefer S., Keyser J., Wang W. : Neupps: Neural piecewise parametric surfaces. vol. 45

  50. [53]

    : A simple manifold-based construction of surfaces of arbitrary smoothness

    Ying L., Zorin D. : A simple manifold-based construction of surfaces of arbitrary smoothness. 271–275

  51. [54]

    : 3dshape2vecset: A 3d shape representation for neural fields and generative diffusion models

    Zhang B., Tang J., Nie ner M., Wonka P. : 3dshape2vecset: A 3d shape representation for neural fields and generative diffusion models. ACM Trans. Graph. 42, 4 (jul 2023)

    2023

  52. [55]

    CAD-Assistant: Tool-Augmented VLLMs as Generic

    Dimitrios Mallis and Ahmet Serdar Karadeniz and Sebastian Cavada and Danila Rukhovich and Niki Maria Foteinopoulou and Kseniya Cherenkova and Anis Kacem and Djamila Aouada , year =. CAD-Assistant: Tool-Augmented VLLMs as Generic. 2412.13810 , archiveprefix =

    arXiv

  53. [56]

    Deepsdf: Learning continuous signed distance functions for shape representation , author=

  54. [57]

    2019 , pages =

    Chen, Zhiqin and Zhang, Hao , title =. 2019 , pages =

    2019

  55. [58]

    MASH: Masked Anchored SpHerical Distances for 3D Shape Representation and Generation , author =

  56. [59]

    Occupancy Networks: Learning 3D Reconstruction in Function Space , author=

  57. [60]

    2024 , booktitle = SIGGRAPH, articleno =

    Sivaram, Venkataram and Li, Tzu-Mao and Ramamoorthi, Ravi , title =. 2024 , booktitle = SIGGRAPH, articleno =

    2024

  58. [61]

    Neural Geometry Processing via Spherical Neural Surfaces , author =

  59. [62]

    Neural Surface Maps , author=

  60. [63]

    AtlasNet: A Papier-M\^ach\'e Approach to Learning 3D Surface Generation , author=

  61. [64]

    and Hughes, John F

    Grimm, Cindy M. and Hughes, John F. , title =. 1995 , booktitle = SIGGRAPH_old, pages =

    1995

  62. [65]

    Neural Implicit Surface Evolution , author=

  63. [66]

    Exploring differential geometry in neural implicits , volume =

    Tiago Novello and Guilherme Schardong and Luiz Schirmer and Vinícius. Exploring differential geometry in neural implicits , volume =. Computers and Graphics , pages =

  64. [67]

    Shape Reconstruction by Learning Differentiable Surface Representations , author=

  65. [68]

    2013 , publisher =

    Introduction to Smooth Manifolds , author =. 2013 , publisher =

    2013

  66. [69]

    1984 , publisher =

    Elements of Algebraic Topology , author =. 1984 , publisher =

    1984

  67. [70]

    2017 , publisher=

    Scalable locally injective mappings , author=. 2017 , publisher=

    2017

  68. [71]

    SSD-C: Smooth Signed Distance Colored Surface Reconstruction

    Calakli, Fatih and Taubin, Gabriel. SSD-C: Smooth Signed Distance Colored Surface Reconstruction. Expanding the Frontiers of Visual Analytics and Visualization. 2012

    2012

  69. [72]

    1992 , booktitle = SIGGRAPH_old, pages =

    Hoppe, Hugues and DeRose, Tony and Duchamp, Tom and McDonald, John and Stuetzle, Werner , title =. 1992 , booktitle = SIGGRAPH_old, pages =

    1992

  70. [73]

    Symposium on Geometry Processing , editor =

    Kazhdan, Michael and Bolitho, Matthew and Hoppe, Hugues , year =. Symposium on Geometry Processing , editor =

  71. [74]

    2022 , issue_date =

    Lin, Siyou and Xiao, Dong and Shi, Zuoqiang and Wang, Bin , title =. 2022 , issue_date =. doi:10.1145/3554730 , journal =

    work page doi:10.1145/3554730 2022

  72. [75]

    2024 , journal =

    Jia-Mu Sun and Tong Wu and Lin Gao , title =. 2024 , journal =

    2024

  73. [76]

    Learning Implicit Fields for Generative Shape Modeling , author=

  74. [77]

    2019 , volume=

    Mescheder, Lars and Oechsle, Michael and Niemeyer, Michael and Nowozin, Sebastian and Geiger, Andreas , booktitle= CVPR, title=. 2019 , volume=

    2019

  75. [78]

    2020 , organization=

    Convolutional occupancy networks , author=. 2020 , organization=

    2020

  76. [79]

    1974 , author =

    B-Spline Curves and Surfaces , booktitle =. 1974 , author =

    1974

  77. [80]

    Future Generation Computer Systems , volume=

    Functional networks for B-spline surface reconstruction , author=. Future Generation Computer Systems , volume=. 2004 , publisher=

    2004

  78. [81]

    Joint Pattern Recognition Symposium , pages=

    3D model retrieval with spherical harmonics and moments , author=. Joint Pattern Recognition Symposium , pages=. 2001 , organization=

    2001

  79. [82]

    and Cline, Harvey E

    Lorensen, William E. and Cline, Harvey E. , title =. 1987 , booktitle = SIGGRAPH_old, pages =

    1987

  80. [83]

    2002 , volume =

    Ju, Tao and Losasso, Frank and Schaefer, Scott and Warren, Joe , title =. 2002 , volume =

    2002

Showing first 80 references.