Sinefitting : Robust Curvature Estimator On Surface Triangulation J - - PowerPoint PPT Presentation

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Sinefitting : Robust Curvature Estimator On Surface Triangulation J - - PowerPoint PPT Presentation

Aug 16, 2023 264 likes •566 views

Motivation Problematic Related work Objectif Sinefitting Results Conclusion Sinefitting : Robust Curvature Estimator On Surface Triangulation J er ome Charton, Stefka Gueorguieva, Pascal Desbarats LaBRI Universit e Bordeaux 1

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion

Sinefitting : Robust Curvature Estimator On Surface Triangulation

J´ erˆ

  • me Charton, Stefka Gueorguieva, Pascal Desbarats

LaBRI Universit´ e Bordeaux 1

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion

To obtain a surface variation descriptor on unstructured data.

1

Point Cloud

1Acquilon scanner of Kreon

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Presentation

Curvature estimation methods generally divided into two parts: Normal estimation Curvature tensor estimation itself

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Presentation

Curvature estimation methods generally divided into two parts: Normal estimation Curvature tensor estimation itself For the evaluation of the curvature estimators we use 3 criterions : Pointwise Convergence Precision Robustness

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Theoretical base

Notations 1/2 (Neighborhood & plane section)

  • ni

Ci P Pi Ci is the normal section containing Pi P: target Pi: neighbors

  • N: normal of the surface at P
  • Ti: tangent of Ci at P
  • ni: normal of Ci at P

ki: curvature of Ci at P

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Theoretical base

Notations 2/2 (Principal directions & curvatures)

rc pi p

  • Tmax
  • Tmin
  • Ti θi

kmax&kmin: maximal and mininal curvatures KH: Mean curvature: KH = (kmax + kmin)/2 KG: Gaussian curvature: KG = kmax ∗ kmin

  • Tmax&

Tmin: respectively kmax&kmin directions θi: angle between

  • Tmax and

Ti

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Theoretical base

Euler theorem

ki = kmaxcos2(θi) + kminsin2(θi) (1)

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Theoretical base

Meusnier theorem

2

ki = k.cos(β) (2) Where β is the angle between n and N and k is the curvature of C at the point P

2Illustration extracted from Chen and Schmitt book [CS92]

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Classification

We can classify curvature estimators in three classes: Averaging methods (Meyer’s et al. method [MMB02] (SDA)) Surface fitting methods (Mc Ivor’s et al. method [MW97] (SQFA)) Curve fitting methods (Chen’s, Taubin’s and Langer’s methods)

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Averaging methods

Meyer et al. [MMB02] (SDA)

3

Angle weighted area or Vorono¨ ı area around vertex P in grey This method just computes K and H

3Illustration extracted from Bac et al. [BDM05]

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Averaging methods

Meyer et al. [MMB02] (SDA)

Weakness of this method:

3

3Illustration extracted from Bac et al. [BDM05]

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Surface fitting methods

Mc Ivor et al. [MW97]: Simple Quadratic Fitting (SQFA)

Consists in solving an equation like eq.(3) by using the spatial coordinates of each Pi z = ax2 + by2 + cxy (3) Is an overdetermined system usually solved by least squares. Researched values are obtained by using the coefficients.

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Surface fitting methods

Mc Ivor et al. [MW97]: Simple Quadratic Fitting (SQFA)

Weakness of this method: Highly sensitive to the distrubution of the neighborhood.

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Curve fitting methods

Chen & Smith [CS92]

4

1 Find the most opposite triplets 2 Compute k for each circle fitted over each choosen triplet 3 Use the Meusnier theorem to evaluate the ki 4 Finally, fit a transformed equation of the Euler theorem.

4Illustration extracted from Chen and Schmitt book [CS92]

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Curve fitting methods

Chen & Smith [CS92]

Weakness of this method: Theoretical curvature Chen KG estimation Local instabilities at saddle point and low curvature.

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Curve fitting methods

Taubin [TFA95] and Langer [LBS07]

Firstly: both methods compute ki as ki ≈ 2

Nt( PPi) || PPi||2

Secondly: Taubin gives a matricial system representation of the curvature tensor. Whereas Langer evaluate the curvature as two integrals modeling KH and KG.

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Curve fitting methods

Taubin [TFA95] and Langer [LBS07]

Weakness of this methods: Taubin KG estimation Langer KG estimation Taubin is imprecise and Langer has occasional errors

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion

All this curvature estimators present dysfunctions Can we find a new curvature estimator less sensitive to neighborhood geometry ?

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Algorithm

The SineFitting algorithm is composed of two steps

1 Evaluation of ki as in Taubin and Langer algorithms by circle

fitting.

2 Fitting a transformed equation of Euler theorem as in Chen

algorithm but without using Meunsier theorem. (Recall Euler equation) ki = kmaxcos2(θi) + kminsin2(θi)

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion ki evaluations

  • N

P ψ Pi ψ O H M Known data cos ψ = ||

PH|| || PPi|| = || MPi|| || OPi|| ; ...; ||

OPi|| = ri = | 1

  • 2. ||

PPi||2

  • PPi.

N |; ki = 2.

  • PPi.

N || PPi||2

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Sinewave fitting

  • Tmax is unknown, so θi cannot be directly computed.

Let ϕ an angle such that θi = αi + ϕ, where αi = ∠( T0, Ti) Euler equation is rewritten as: ki = kmax cos2(αi + ϕ) + kmin sin2(αi + ϕ) ...

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Sinewave fitting

  • Tmax is unknown, so θi cannot be directly computed.

Let ϕ an angle such that θi = αi + ϕ, where αi = ∠( T0, Ti) Euler equation is rewritten as: ki = kmax cos2(αi + ϕ) + kmin sin2(αi + ϕ) ... ki = a cos(2αi) + b sin(2αi) + c where (if a > 0 for example), ϕ = −

tan−1( b

a )

2

, kmax = c + √ a2 + b2, kmin = c − √ a2 + b2

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Experimentation

Hamann’s discretization surface for robustness [Ham91] Different convergent discretisation methods of mathematical

  • surfaces. Called NeighborDealers

P P0 P1 P2 P3 P4 P5 P P2 P3 P1 P4 P0 P5 P P0 P1 P2 P3 P4 P5 P P0 P1 P2 P3 P4 P5

Nreg(P) Nirreg(P) NregδDist(P) NregδAngle(P)

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Pointwise convergence 17 / 21

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Precision

SDA Taubin Chen SineFitting SQFA Langer

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion Robustness 19 / 21

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion

Conclusion: According to the performed tests, Sinefitting is not always the most accurate method, but is far more stable. It is easy to implement. Perspectives: Test robustness on noised data following perturbations of Gatzke [GG06]. Experiment on point cloud. Future work: We will try to use the same intuition for the normal estimator.

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion

Thank you for listening jerome.charton@labri.fr Experiment platform: http://smithdr.labri.fr/ All results are available on: http://dept-info.labri.fr/∼ charton/curvature analysis/

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Motivation Problematic Related work Objectif Sinefitting Results Conclusion

  • A. Bac, M. Daniel, and J-L. Maltret.

3d modeling and segmentation with discrete curvatures. Journal of Medical Informatics and Technology, 9:13–24, 2005.

  • X. Chen and F. Schmitt.

Intrinsic Surface Properties from Surface Triangulation. T´ el´ ecom Paris, D. ´ Ecole Nationale Sup´ erieure des T´ el´ ecommunications, 1992. Timothy D. Gatzke and Cindy M. Grimm. Estimating curvature on triangular meshes. International Journal of Shape Modeling, 12(1), 2006. Bernd Hamann. Visualization and Modeling of Contours of Trivariate Functions. between January and May 1991. Torsten Langer, Alexander Belyaev, and Hans-Peter Seidel. Exact and interpolatory quadratures for curvature tensor estimation.

  • Comput. Aided Geom. Des., 24(8-9):443–463, November 2007.

Peter Schr¨ uder Mark Meyer, Mathieu Desbrun and Alan H. Barr. Discrete differential-geometry operators for triangulated 2-manifolds. VisMath, 2002. Alan M. McIvor and Peter T. Waltenberg. Recognition of simple curved surfaces from 3d surface data, 1997. Gabriel Taubin, Surface From, and A Polyhedral Approximation. Estimating the tensor of curvature of a surface from a polyhedral approximation, 1995. 21 / 21