Green Coordinates T obias G. Pfeiffer Freie Universitt Berlin AG - PowerPoint PPT Presentation

Green Coordinates T obias G. Pfeiffer Freie Universität Berlin AG Mathematical Geometry Processing November 6, 2008

Outline Introduction Barycentric Coordinates Problems with Existing Methods Green Coordinates About Idea Derivation Extension to the Outside of the Cage Motivation Problems When Extending Coordinates Solution Implementation , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 2 / 27

Contents Introduction Barycentric Coordinates Problems with Existing Methods Green Coordinates About Idea Derivation Extension to the Outside of the Cage Motivation Problems When Extending Coordinates Solution Implementation , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 3 / 27

What are Barycentric Coordinates ◮ Idea: Spatial coordinates of a point are represented as linear combination of the vertices of an ambient cage. ◮ x ∈ R d point, v i ∈ R d vertices of a cage P ; find φ i ( x ) so that: � x = φ i ( x ) · v i i ∈ I V ◮ Motivation: 1. Interpolate function values given on the boundary: � f ( x ) := φ i ( x ) · f ( v i ) i 2. Move the cage vertices and see how the internal points move along: � F ( · , P ′ ) : x �→ x ′ := φ i ( x ) · v ′ i i We look only at (2.) here; special case of (1.), with f being the transformation applied to P . , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 4 / 27

Problems with Existing Methods ◮ Linear combinations of cage vertices must lead to affine-invariant transformations, not shape-preserving . ◮ Shape-preserving ◮ Close to rotations with isotropic scale ◮ Infinitesimal circles are mapped to infinitesimal ellipsoids with bounded axis ratio (quasi-conformal) ◮ Affine-invariant ◮ Affine transformation applied to cage results in same transformation applied to geometry ⇒ problems with shearing and anisotropic scale ◮ Especially: Changes in only one direction do not affect the other directions , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 5 / 27

Problems with Existing Methods Original, affine-invariant transformation Solution: Green Coordinates , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 6 / 27

Contents Introduction Barycentric Coordinates Problems with Existing Methods Green Coordinates About Idea Derivation Extension to the Outside of the Cage Motivation Problems When Extending Coordinates Solution Implementation , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 7 / 27

Facts about Green Coordinates ◮ Paper from Y . Lipman, D. Levin, D. Cohen-Or, presented on SIGGRAPH 2008 ◮ Can be used with piecewise smooth boundaries in any dimension ◮ Cages must not be necessarily simply connected ◮ Yields conformal transformations in 2D, quasi-conformal transformations in higher dimensions , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 8 / 27

Idea of Green Coordinates ◮ T ake not only vertices of cage, but also face orientation (= normals) into account. ◮ P a cage, v i ∈ R d vertices ( i ∈ I V ), t j faces with normals n j ∈ R d ( j ∈ I T ) � � x = φ i ( x ) · v i + ψ j ( x ) · n j i ∈ I V j ∈ I T ◮ With cage change P �→ P ′ , transformation is then given by � � F ( · , P ′ ) : x �→ x ′ = φ i ( x ) · v ′ ψ j ( x ) · s j · n ′ i + j i ∈ I V j ∈ I T ◮ s j scaling factors, chosen appropriately to obtain desired properties , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 9 / 27

Example Transformation Original, transformation induced from Green Coordinates , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 10 / 27

Derivation of Green Coordinates Theorem (Green’s Third Identity) Let Ω ⊂ R d with a smooth boundary, G a a fundamental solution of the Laplace equation (i. e. ∆ G a ( x ) = δ a , x ). If u : Ω → R is twice continuously differentiable, then for all a ∈ Ω , the following equality holds: � � � � ∂ G a ∂ u u ( a ) = u ( x ) · ( x ) − G a ( x ) · ( x ) d σ x + G a ( x ) · ∆ u ( x ) dx ∂ n ∂ n ∂ Ω Ω � �� � vanishes if u harmonic Those functions G a in R d have the form � 1 2 π log � a − x � d = 2 G a ( x ) = 1 ( 2 − d ) ω d � a − x � 2 − d d ≥ 3 (with ω d volume of the d -unit sphere). , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 11 / 27

Derivation of Green Coordinates Treat coordinate functions u = ( x , y , z ) : Ω → R 3 as special harmonic functions (in each component): � � � ∂ G a u ( a ) = a = x · ( x ) − G a ( x ) · n ( x ) d σ x ∂ n ∂ Ω Remark 1 Let d = 2 ⇒ G a ( x ) = 2 π log � a − x � . Compare the above representation to Cauchy’s integral formula: � 1 1 a = · z d σ z 2 π i z − a ∂ D In 2D, Green and Complex Coordinates (Gotsman) are equivalent! , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 12 / 27

Derivation of Green Coordinates ◮ normal n j constant on each triangle t j � ◮ for x ∈ t j , x = v k ∈ V ( t j ) Γ k ( x ) · v k (real barycentric coordinates; Γ k piecewise linear hat function with Γ k ( v i ) = δ ik ) � � Rearrange and for x = i ∈ I V φ i ( x ) · v i + j ∈ I T ψ j ( x ) · n j , one obtains: � ∂ G a φ i ( a ) = Γ i ( x ) · ( x ) d σ x i ∈ I V ∂ n x ∈ AdjFaces ( v i ) � ψ j ( a ) = − G a ( x ) d σ x j ∈ I T x ∈ t j , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 13 / 27

Desired Properties For the transformation � � F ( x , P ′ ) = φ i ( x ) · v ′ ψ j ( x ) · s j · n ′ i + j , i ∈ I V j ∈ I T the scaling factors s j (depending on source and target cage!) are still to be defined to ensure the following properties: 1. Linear reproduction: x = F ( x , P ) 2. Translation invariance: F ( x , P + v ) = x + v 3. Rotation and scale invariance: F ( x , TP ) = Tx for T an affine transformation consisting of rotation with isotropic scale 4. Shape preservation: x �→ F ( x , P ′ ) is conformal ( d = 2) or quasi-conformal ( d ≥ 3) 5. Smoothness: ϕ i , ψ j should be smooth , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 14 / 27

Scaling Factors � � � � � t ′ ◮ In 2D, choose s j = � / � t j � . j ◮ In 3D, choose �� 1 � 2 − 2 ( u ′ · v ′ )( u · v ) + � u ′ � � 2 � � � v ′ � � � 2 � � � 2 , � � v � u 8area ( t j ) where u , v , u ′ , v ′ span the old and new triangles t j , t ′ j . ◮ If t j = t ′ j , then s j = 1. (necessary for linear reproduction) ◮ Conformality for d = 2 is proven in T echnical Report yet to be published. ◮ Quasi-Conformality for d ≥ 3: ◮ distortion measured by quotient of singular values of DF ◮ experimentally found distortion bounded by constant ≤ 6 (Mean-Value Coordinates and Harmonic Coordinates yield unbounded distortion proportional to cage distortion) , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 15 / 27

Some Images I Deformations using Green, Mean-Value, Harmonic Coordinates , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 16 / 27

Some Images II Deformation using a non-simply connected cage , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 17 / 27

Contents Introduction Barycentric Coordinates Problems with Existing Methods Green Coordinates About Idea Derivation Extension to the Outside of the Cage Motivation Problems When Extending Coordinates Solution Implementation , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 18 / 27

Partial Cages: Motivation ◮ Sometimes only part of a geometry should be deformed. ◮ Large cages are harder to construct and increase computation time. ◮ Requirements: ◮ Smooth transition where geometry crosses “exit face”. ◮ Diminishing influence of cage movement outside the cage. , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 19 / 27

Problems ◮ Green’s Identity only holds inside the cage, i. e. for x ∈ P in . ◮ Coordinate functions: � ◮ Normal weights ψ j ( a ) = − x ∈ t j G a ( x ) d σ x are smooth across ∂ Ω : � x ∈ AdjFaces ( v i ) Γ i ( x ) · ∂ G a ◮ Vertex weights φ i ( a ) = ∂ n ( x ) d σ x are discontinuous across adjacent faces of v i : ◮ F ( x , P ) = 0 if x ∈ P ext , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 20 / 27

Solution ◮ Goal: Find analytic (complex-analytic in d = 2, real-analytic in d ≥ 3) continuations of φ i across a fixed face t r . ◮ Let I r ⊂ I V be the index set of vertices spanning t r . ◮ Define ˜ ψ r and ˜ φ i ( i ∈ I r ) such that: ◮ linear reproduction holds: � � � φ i ( x ) v i + ˜ ˜ ψ r ( x ) n r = x − φ i ( x ) v i − ψ j ( x ) n j i ∈ I r i ∈ I V \ I r j � = r ◮ translation invariance holds: � � ˜ φ i ( x ) = 1 − φ i ( x ) i ∈ I r i ∈ I V \ I r This yields an (invertible!) linear equation system that can be φ i ( x ) and ˜ used to compute ˜ ψ j ( x ) . ψ j ( x ) = ψ j ( x ) if x ∈ P in (by construction) φ i ( x ) = φ i ( x ) and ˜ ◮ ˜ , T obias Pfeiffer, FU Berlin, AG GEOM: Green Coordinates, November 6, 2008 21 / 27