Motivation Problematic Related work Objectif Sinefitting Results Conclusion Sinefitting : Robust Curvature Estimator On Surface Triangulation J´ erˆ ome Charton, Stefka Gueorguieva, Pascal Desbarats LaBRI Universit´ e Bordeaux 1
Motivation Problematic Related work Objectif Sinefitting Results Conclusion To obtain a surface variation descriptor on unstructured data. Point Cloud 1 1 Acquilon scanner of Kreon 1 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Presentation Curvature estimation methods generally divided into two parts: Normal estimation Curvature tensor estimation itself 2 / 21
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 2 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Theoretical base Notations 1/2 (Neighborhood & plane section) P C i � n i P i C i is the normal section containing P i � P : target T i : tangent of C i at P P i : neighbors n i : normal of C i at P � � N : normal of the surface at P k i : curvature of C i at P 3 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Theoretical base Notations 2/2 (Principal directions & curvatures) r c p i � � T i θ i T min p � T max k max & k min : maximal and mininal curvatures K H : Mean curvature: K H = ( k max + k min ) / 2 K G : Gaussian curvature: K G = k max ∗ k min T max & � � T min : respectively k max & k min directions T max and � � θ i : angle between T i 4 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Theoretical base Euler theorem k i = k max cos 2 ( θ i ) + k min sin 2 ( θ i ) (1) 5 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Theoretical base Meusnier theorem 2 k i = k . cos ( β ) (2) n and � Where β is the angle between � N and k is the curvature of C at the point P 2 Illustration extracted from Chen and Schmitt book [CS92] 6 / 21
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) 7 / 21
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 3 Illustration extracted from Bac et al. [BDM05] 8 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Averaging methods Meyer et al. [MMB02] ( SDA ) Weakness of this method: 3 3 Illustration extracted from Bac et al. [BDM05] 8 / 21
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 P i z = ax 2 + by 2 + cxy (3) Is an overdetermined system usually solved by least squares. Researched values are obtained by using the coefficients. 9 / 21
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. 9 / 21
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 k i 4 Finally, fit a transformed equation of the Euler theorem. 4 Illustration extracted from Chen and Schmitt book [CS92] 10 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Curve fitting methods Chen & Smith [CS92] Weakness of this method: Theoretical curvature Chen K G estimation Local instabilities at saddle point and low curvature. 10 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Curve fitting methods Taubin [TFA95] and Langer [LBS07] Firstly: both methods compute k i as k i ≈ 2 � N t ( � PP i ) || � PP i || 2 Secondly: Taubin gives a matricial system representation of the curvature tensor. Whereas Langer evaluate the curvature as two integrals modeling K H and K G . 11 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Curve fitting methods Taubin [TFA95] and Langer [LBS07] Weakness of this methods: Taubin K G estimation Langer K G estimation Taubin is imprecise and Langer has occasional errors 11 / 21
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 ? 12 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Algorithm The SineFitting algorithm is composed of two steps 1 Evaluation of k i 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) k i = k max cos 2 ( θ i ) + k min sin 2 ( θ i ) 13 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion k i evaluations � N P M ψ P i H ψ Known data O cos ψ = || � PP i || = || � 2 . || � PP i || 2 PP i .� � PH || MP i || OP i || ; ...; || � OP i || = r i = | 1 N N | ; k i = 2 . || � || � PP i .� � || � PP i || 2 14 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Sinewave fitting � T max is unknown, so θ i cannot be directly computed. Let ϕ an angle such that θ i = α i + ϕ , where α i = ∠ ( � T 0 , � T i ) Euler equation is rewritten as: k i = k max cos 2 ( α i + ϕ ) + k min sin 2 ( α i + ϕ ) ... 15 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Sinewave fitting � T max is unknown, so θ i cannot be directly computed. Let ϕ an angle such that θ i = α i + ϕ , where α i = ∠ ( � T 0 , � T i ) Euler equation is rewritten as: k i = k max cos 2 ( α i + ϕ ) + k min sin 2 ( α i + ϕ ) ... k i = a cos(2 α i ) + b sin(2 α i ) + c where (if a > 0 for example), √ √ tan − 1 ( b a ) a 2 + b 2 , k min = c − a 2 + b 2 ϕ = − , k max = c + 2 15 / 21
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 5 P 5 P 5 P 5 P 0 P 4 P 0 P 4 P 0 P 4 P 0 P P 4 P P P P 1 P 3 P 1 P 3 P 1 P 1 P 3 P 3 P 2 P 2 P 2 P 2 N reg ( P ) N irreg ( P ) N reg δ Dist ( P ) N reg δ Angle ( P ) 16 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Pointwise convergence 17 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Precision SDA Taubin Chen SineFitting SQFA Langer 18 / 21
Motivation Problematic Related work Objectif Sinefitting Results Conclusion Robustness 19 / 21
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. 20 / 21
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/ 21 / 21
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