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)NPUT ATA2ANGE 3CAN#! 4OMOGRAPHY2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS!NALYSIS OF SURFACE QUALITY3URFACE SMOOTHING FOR NOISE REMOVALDifferential Geometry0ARAMETERIZATIONMark Pauly3IMPLIFICATION FOR COMPLEXITY REDUCTION2EMESHING FOR IMPROVING MESH QUALITY&REEFORM AND MULTIRESOLUTION MODELING
Outline Differential Geometry– curvature– fundamental forms– Laplace-Beltrami operator Discretization Visual Inspection of Mesh QualityMark Pauly2
Differential Geometry Continuous surface x(u, v)x(u, v) y(u, v) , (u, v) IR2z(u, v) Normal vectorn (xu xv )/"xu xv "– assuming regular parameterization, i.e.xu xv " 0Mark Pauly3
Differential Geometry Normal Curvaturexu xvn "xu xv "nxutxvpxvxu sin φt cos φ!xu !!xv !Mark Pauly4
Differential Geometry Normal Curvaturexu xvn "xu xv "ntcpxvxu sin φt cos φ!xu !!xv !Mark Pauly5
Differential Geometry Principal Curvatures– maximum curvature κ1 max κn (φ)φ– minimum curvature κ2 min κn (φ)φ Euler Theorem: κn (t̄) κn (φ) κ1 cos2 φ κ2 sin2 φ Mean Curvature1κ1 κ2 H 22π!2πκn (φ)dφ0 Gaussian Curvature K κ1 · κ2Mark Pauly6
Differential Geometry Normal curvature is defined as curvature of thenormal curve c x(u, v) at a point p c Can be expressed in terms of fundamental formsas22Tea 2f ab gbt̄ II t̄ κn (t̄) TEa2 2F ab Gb2t̄ I t̄ntcpMark Paulyt axu bxv7
Differential Geometry First fundamental formI !EFFG": !xTu xuxTu xvxTu xvxTv xv"xTuu nxTuv nxTuv nxTvv n" Second fundamental formII !effg": !Mark Pauly8
Differential Geometry I and II allow to measure– length, angles, area, curvature– arc elementds2 Edu2 2F dudv Gdv 2– area element!dA EG F 2 dudvMark Pauly9
Differential Geometry Intrinsic geometry: Properties of the surface thatonly depend on the first fundamental form– length– angles– Gaussian curvature (Theorema Egregium)6πr 3C(r)K limr 0πr3Mark Pauly10
Differential Geometry A point x on the surface is called– elliptic, if K 0– parabolic, if K 0– hyperbolic, if K 0– umbilical, if κ1 κ2 Developable surfaceK 0Mark Pauly11
Laplace OperatorgradientoperatorLaplaceoperator f div f 2nd partialderivatives! 2fifunction inEuclidean spacedivergenceoperatorMark Pauly x2iCartesiancoordinates12
Laplace-Beltrami Operator Extension of Laplace to functions on manifoldsgradientoperatorLaplaceBeltrami S f divS S ffunction onmanifold SdivergenceoperatorMark Pauly13
Laplace-Beltrami Operator Extension of Laplace to functions on ure S x divS S x 2HncoordinatefunctiondivergenceoperatorMark Paulysurfacenormal14
Outline Differential Geometry– curvature– fundamental forms– Laplace-Beltrami operator Discretization Visual Inspection of Mesh QualityMark Pauly15
Discrete Differential Operators Assumption: Meshes are piecewise linearapproximations of smooth surfaces Approach: Approximate differential properties atpoint x as spatial average over local meshneighborhood N(x), where typically– x mesh vertex– N(x) n-ring neighborhood or local geodesic ballMark Pauly16
Discrete Laplace-Beltrami Uniform discretization1 uni f (v) : N1 (v) !(f (vi ) f (v))vi N1 (v)– depends only on connectivity simple andefficient– bad approximation for irregular triangulationsMark Pauly17
Discrete Laplace-Beltrami Cotangent formula2 S f (v) : A(v)!(cot αi cot βi ) (f (vi ) f (v))vi N1 (v)vvviA(v)vαiβiviMark Paulyvi18
Discrete Laplace-Beltrami Cotangent formula2 S f (v) : A(v)!(cot αi cot βi ) (f (vi ) f (v))vi N1 (v) Problems– negative weights– depends on triangulationMark Pauly19
Discrete Curvatures Mean curvatureH ! S x! Gaussian curvatureG (2π !Aθj )/Ajθj Principal curvatures!κ1 H H 2 G!κ2 H H 2 GMark Pauly20
Links & Literature P. Alliez: Estimating Curvature Tensorson Triangle Meshes (source code)– iez/demos/curvature/ Wardetzky, Mathur, Kaelberer,Grinspun: Discrete Laplace Operators:No free lunch, SGP 2007principal directionsMark Pauly21
Outline Differential Geometry– curvature– fundamental forms– Laplace-Beltrami operator Discretization Visual Inspection of Mesh QualityMark Pauly22
Mesh Quality Smoothness– continuous differentiability of a surface (Ck) Fairness– aesthetic measure of “well-shapedness”– principle of simplest shape– fairness measures from physical models!#2 "#2! " κ1 κ222κ1 κ2 dA dA t1 t2SSstrain energyvariation of curvatureMark Pauly23
Mesh Quality!Sκ21 κ22dA! "Sstrain energy κ1 t1#2 " κ2 t2#2dAvariation of curvatureMark Pauly24
Mesh Quality Visual inspection of “sensitive” attributes– Specular shadingMark Pauly25
Mesh Quality Visual inspection of “sensitive” attributes– Specular shading– Reflection linesMark Pauly26
Mesh Quality Visual inspection of “sensitive” attributes– Specular shading– Reflection lines differentiability one order lower than surface can be efficiently computed using graphics hardwareC0C1Mark PaulyC227
Mesh Quality Visual inspection of “sensitive” attributes– Specular shading– Reflection lines– Curvature Mean curvatureMark Pauly28
Mesh Quality Visual inspection of “sensitive” attributes– Specular shading– Reflection lines– Curvature Mean curvature Gauss curvatureMark Pauly29
Mesh Quality Criteria Smoothness– Low geometric noiseMark Pauly30
Mesh Quality Criteria Smoothness– Low geometric noise Adaptive tessellation– Low complexityMark Pauly31
Mesh Quality Criteria Smoothness– Low geometric noise Adaptive tessellation– Low complexity Triangle shape– Numerical robustnessMark Pauly32
Triangle Shape Analysis Circum radius / shortest edger1r2r1 e1e2e2r2e1 Needles and capsNeedleCapMark Pauly33
Mesh Quality Criteria Smoothness– Low geometric noise Adaptive tessellation– Low complexity Triangle shape– Numerical robustness Feature preservation– Low normal noiseMark Pauly34
Normal Noise AnalysisMark Pauly35
Mesh Optimization Smoothness Mesh smoothing Adaptive tessellation Mesh decimation Triangle shape Repair, remeshingMark Pauly36
Mark Pauly Outline Differential Geometry – curvature – fundamental forms – Laplace-Beltrami oper
