Fast Harmonic Mappings EDEN FEDIDA HEFETZ BAR-ILAN UNIVERSITY, - PowerPoint PPT Presentation

Fast Harmonic Mappings EDEN FEDIDA HEFETZ BAR-ILAN UNIVERSITY, ISRAEL 1

Goal Find fast, locally injective harmonic mappings between shapes with low distortion. 1) Formulation of the optimization problem 2) Create custom-made solvers for the specific problem => Acceleration by orders of magnitude Performed on two types of harmonic mappings: 1) Planar shape deformation 2) Seamless parameterization 2

Fast Planar Harmonic Deformations with Alternating Tangential Projections EDEN FEDIDA HEFETZ, EDWARD CHIEN, OFIR WEBER BAR-ILAN UNIVERSITY, ISRAEL 3

The Mapping Problem 𝑔: 𝛻 → ℝ 2 • Desirable properties: • Locally-injective • Bounded conformal distortion • Bounded isometric distortion • Real-time 4

Previous Work • Cage based methods (barycentric coords): • Bounded distortion: [Hormann and Floater 2006] [Lipman 2012] [Joshi et al.2007] [Kovalsky at al. 2015] [Lipman et al. 2007] [Chen and Weber 2015] [Weber et al. 2011] [Levi and Weber 2016] [Weber et al. 2009] … … 5

Notations • Planar mapping: • Distortion measures: 𝑔: Ω → ℝ 2 • Jacobian: • Singular values of 𝐾 𝑔 𝐾 𝑔 = 𝑏 −𝑐 + 𝑑 𝑒 𝑐 𝑏 𝑒 −𝑑 Similarity Anti-similarity • Complex Wirtinger derivatives: 0 ≤ 𝜏 𝑐 ≤ 𝜏 𝑏 𝑔 𝑨 = 𝑏 + 𝑗𝑐 𝜏 𝑏 = 𝑔 𝑨 + 𝑔 ҧ 𝑨 𝑔 𝑨 = 𝑑 + 𝑗𝑒 𝜏 𝑐 = 𝑔 𝑨 − 𝑔 ҧ 𝑨 6

Bounded Distortion Harmonic Mappings • The BD space : ∀𝑨 ∈ 𝛻 𝜏 𝑏 −𝜏 𝑐 𝑔 ത 𝑨 𝑙 𝑨 = 𝜏 𝑏 +𝜏 𝑐 = 𝑨 ≤ 𝐷 𝑙 conformal 𝑔 𝜏 𝑏 z = 𝑔 𝑨 + 𝑔 ҧ 𝑨 ≤ 𝐷 𝑏 𝝊 = 𝐧𝐛𝐲 𝝉 𝒃 , 𝟐 isometric 𝜏 𝑐 z = 𝑔 𝑨 − 𝑔 ҧ 𝑨 ≥ 𝐷 𝑐 𝝉 𝒄 • Non-convex space 𝑫 𝒃 = 𝟑. 𝟔 𝑫 𝒃 = 𝟐𝟏 Source • Harmonic mapping enforce bounds only on 𝜖Ω [Chen and Weber 2015] 7

The ℒ 𝜉 Space [Levi and Weber 2016] • Change of variables: 𝝃 = 𝒈 𝒜 BD ℒ 𝜉 𝒎 = 𝒎𝒑𝒉 𝒈 𝒜 𝒈 𝒜 ℒ 𝜉 BD 𝒈 𝒜 = 𝒇 𝒎 𝒜 = 𝝃𝒇 𝒎 𝒈 ത • BD homeomorphic to ℒ 𝜉 8

The ℒ 𝜉 Space • Near convex space ∀𝑥 ∈ 𝜖𝛻 𝑙 𝑥 = 𝜉(𝑥) ≤ 𝐷 𝑙 𝜏 𝑏 w = 𝑓 𝑆𝑓(𝑚(𝑥)) (1 + 𝜉(𝑥) ) ≤ 𝐷 𝑏 𝜏 𝑐 w = 𝑓 𝑆𝑓 𝑚 𝑥 (1 − 𝜉(𝑥) ) ≥ 𝐷 𝑐 Convex 9

Discretization • Enforce distortion constraints on m densely sampled points n vertices • Use Cauchy complex barycentric coordinate : 𝑜 𝑜 𝑡 𝑘 , 𝑢 𝑘 ∈ ℂ 𝑚 𝑨 = ෍ 𝑡 𝑘 𝐷 𝑘 𝑨 & 𝜉 𝑨 = ෍ 𝑢 𝑘 𝐷 𝑘 𝑨 𝑘=1 𝑘=1 • Subspace of holomorphic functions • 4n-dimensional m sample points Affine 10

Our problem Convex Affine 4m-dimensional 4n-dimensional ∩ convex subspace of ℝ 4𝑛 Harmonic mapping Bounded distortion 11

Our problem • Input : 𝑚 and 𝜉 values from cage data • Find the closest point in the intersection of an affine space and a convex space A 𝑏 𝑗 B 12

Alternating Projections MAP ATP 𝐼 𝑗 13

Alternating Projections MAP ATP [Von Neumann 1950] Proof of convergence [Bauschke and Borwein 1993] 14

Large-Scale Bounded Distortion Mappings [Kovalsky et al. 2015] • Alternating Projections between an affine space and non-convex space • No convergence guarantees • Upon convergence, not necessarily locally injective • Only bounds the conformal distortion and not isometric 15

Gathering Input Data • Extract 𝑚 and 𝜉 values from cage data • Linear transformations 𝑓 𝑗 ⟼ ෝ 𝑓 𝑗 that preserves the unit normal 𝒇 𝒋−𝟐 ෟ 𝒇 𝒋−𝟐 𝒇 𝒋 𝒇 𝒋 ෝ 1 1 1 2 𝒇 𝒋+𝟐 ෟ 𝒇 𝒋+𝟐 16

Implementation • Local : • Project each sample point to the bounded distortion space • GPU kernel • Global : • Linear + fixed left hand side • GPU - Matrix-Vector products using cuBLAS 17

Results 18

Near-optimality of alternating projection methods source MOSEK ATP MAP 3 fps 35 fps 0.005 fps 19

Near-optimality of alternating projection methods source MAP ATP MOSEK 15 fps 140 fps 0.2 fps 20

Speedup × 𝟐𝟏 𝟒 ~170 × 𝟒 ∙ 𝟐𝟏 𝟒 ~30 21

𝐷 𝑏 = = 5 5 𝐷 𝑐 = = 0.2 𝝊 𝝊 = 𝐧𝐛𝐲 𝝉 𝒃 , 𝟐 𝝉 𝒄 22

Source Cauchy Coords [Kovalsky et al. 2015] ATP 23

Summary • Planar deformation • GPU accelerated – speedup of 3 × 10 3 • Guaranteed local injectivity and bounded distortion • General proof of convergence • Future Work: • Positional constraints • Extension to parametrization of surfaces 24

A Subspace Method for Fast Locally Injective Harmonic Mapping EDEN FEDIDA HEFETZ, EDWARD CHIEN, OFIR WEBER BAR-ILAN UNIVERSITY, ISRAEL 25

The Parametrization Problem • Given a 3-D triangular mesh S, find a map 𝑔 ∶ 𝑇 → Ω such that Ω ⊂ 𝑆 2 • Desirable properties: • Locally injectivity • Low distortion • Fast computation 26

Motivation • Texture mapping • Mesh correspondences • Remeshing 27/36

Credit: Hans-Christian Ebke Quad Remeshing Example 28

Previous Work • Linear methods • Nonconvex energy-based methods LSCM [Lévy et al. ‘ 02] CM [Schtengel et al.] Killing [Claici et al.] Angle-Based [Zayer et al. ‘ 07] SLIM [Rabinovich et al.] Conformal Flattening [Ben-Chen et al. ‘ 08] AQP [Kovalsky et al.] … … We aim for the speed of a linear method and the robustness of a nonconvex method 29

Tutte ’ s embedding • Discrete harmonic function Convex combination map Global bijection Convex boundary 30

Global Parametrization Mesh cut to a disk along a seam graph 𝐻 𝑡 1) 2) Resulting disk mapped to plane 3) Seam edge copies have isometric images 𝒘 𝒋 𝒄 𝒇 𝒋𝒌 𝒜 𝒌 𝒄 ) 𝒈(𝑓 𝑗𝑘 𝒘 𝒌 𝒜 𝒋 𝒃 ) 𝒈(𝒇 𝒋𝒌 𝒃 𝒜 𝒌 Seamless parametrization: The rotational part of the isometry is a rotation by some multiple of 𝜌/2 . 31

HGP [Bright et al. ‘ 17] Linear system: 𝜌𝑠𝑗𝑘 = 𝑓 𝑗 𝑏 𝑐 2 𝑔 𝑓 𝑗𝑘 1 . 𝑔 𝑓 𝑗𝑘 , 𝑓 ij ∈ 𝐻 𝑡 2. 3. 32

HGP [Bright et al. ‘ 17] Linear system: 𝜌𝑠𝑗𝑘 = 𝑓 𝑗 𝑏 𝑐 2 𝑔 𝑓 𝑗𝑘 1 . 𝑔 𝑓 𝑗𝑘 , 𝑓 ij ∈ 𝐻 𝑡 2. 3. 33

HGP [Bright et al. ‘ 17] Linear system: 𝜌𝑠𝑗𝑘 = 𝑓 𝑗 𝑏 𝑐 2 𝑔 𝑓 𝑗𝑘 1 . 𝑔 𝑓 𝑗𝑘 , 𝑓 ij ∈ 𝐻 𝑡 2. 3. 34

HGP [Bright et al. ‘ 17] HGP linear system Locally injective map Boundary and cone triangles are well-behaved i > f ҧ i f z , 𝑗 ∈ 𝑈 𝑑𝑐 𝑨 [Lipman 2012] Frame field from [Bommes et al. 2009] 35

Subspace construction Linear part of HGP: 𝑃(|𝐷|) Add interpolation constraints to complete the system dimension: 36

Subspace construction 0 𝐿𝑨 = 𝑑 𝑗𝑜𝑢 1 ⋯ 0 0 𝐿 −1 𝑑 𝑚𝑗𝑜 𝑑 𝑗𝑜𝑢 = 𝑨 , 𝑑 𝑚𝑗𝑜 = ⋮ 1 ⋮ 0 ⋯ 1 𝐶 𝑑 z = σ 𝑗=1 𝑑 𝑗𝑜𝑢 = 𝑨 𝑐 𝑗 𝑑 𝑗𝑜𝑢𝑗 𝑐 1 𝑐 2 ⋯ 𝑐 𝑑 For 𝑐 𝑗 , extract a column of 𝐿 −1 ! By HGP theorem, only need to extract entries near cones and boundaries! 37

Subspace construction -1 • Only need 𝑃( 𝐷 × 𝐷 ) elements of 𝐿 −1 • Use selective inverse from PARDISO [KLS13, VCKS17] • Detailed instructions for PARDISO use. => Linear subspace with dimension 𝑃( 𝐷 ) Affine 38

Boundary and cone triangles 1 , 𝑗 ∈ 𝑈 𝑑𝑐 (2) (2) 1 Convex 39

Our problem Convex Affine Dimension: 2 𝑈 𝐷𝐶 Dimension: 𝑃( 𝐷 ) ∩ 𝐼 𝑗 Harmonic mapping Locally injective 40

Projected Newton • Symmetric Dirichlet energy is optimized • Nonconvex, project Hessians to the PSD cone • [Chen & Weber 2017] 41

Results! • Comparable quality to HGP results • Fairly robust results: 66 of 77 successful on a benchmark • One order of magnitude faster than HGP; comparable to 1-2 linear solves (next slide) 42

Timing Results 100,000.00 10,000.00 Runtime (s) 1,000.00 100.00 10.00 1.00 0.10 0 20,000 40,000 60,000 80,000 100,000 120,000 140,000 160,000 # Triangles Subspace Method HGP [Chien et al. 2016] 43

Algorithm Parts Timing 4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 1 2 3 4 Series1 Series2 Series3 44

Summary • Surface parametrization • Speedup by order of magnitude • Locally injective • Analysis of linear system from HGP • Future Work: • Higher genus • Convexication frames 45

Conclusion ▪ Two methods have been accelerated: ▪ Input: flat 2D manifold ▪ Input: curved 2D manifold ▪ Continuous functions ▪ Triangular mesh discretization ▪ Convex characterization of the ▪ Non-convex space space. ▪ May not be feasible ▪ Solution is guaranteed 46

The End 47