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

Fast Harmonic Mappings

EDEN FEDIDA HEFETZ BAR-ILAN UNIVERSITY, ISRAEL

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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

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Fast Planar Harmonic Deformations with Alternating Tangential Projections

EDEN FEDIDA HEFETZ, EDWARD CHIEN, OFIR WEBER BAR-ILAN UNIVERSITY, ISRAEL

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The Mapping Problem

𝑔: 𝛻 → ℝ2

• Desirable properties:
• Locally-injective
• Bounded conformal distortion
• Bounded isometric distortion
• Real-time

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Previous Work

• Cage based methods (barycentric coords):

[Hormann and Floater 2006] [Joshi et al.2007] [Lipman et al. 2007] [Weber et al. 2011] [Weber et al. 2009] …

• Bounded distortion:

[Lipman 2012] [Kovalsky at al. 2015] [Chen and Weber 2015] [Levi and Weber 2016] …

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Notations

• Distortion measures:
• Singular values of 𝐾𝑔

𝜏𝑏= 𝑔

𝑨 + 𝑔 ҧ 𝑨

𝜏𝑐= 𝑔

𝑨 − 𝑔 ҧ 𝑨

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𝑔: Ω → ℝ2

• Planar mapping:
• Jacobian:

𝐾𝑔 = 𝑏 −𝑐 𝑐 𝑏 + 𝑑 𝑒 𝑒 −𝑑

• Complex Wirtinger derivatives:

𝑔

𝑨 = 𝑏 + 𝑗𝑐

𝑔𝑨 = 𝑑 + 𝑗𝑒 0 ≤ 𝜏𝑐 ≤ 𝜏𝑏

Similarity Anti-similarity

𝑫𝒃 = 𝟐𝟏

Bounded Distortion Harmonic Mappings

• The BD space: ∀𝑨 ∈ 𝛻

𝑙 𝑨 =

𝜏𝑏−𝜏𝑐 𝜏𝑏+𝜏𝑐 = 𝑔ത

𝑨

𝑔

𝑨 ≤ 𝐷𝑙

𝜏𝑏 z = 𝑔

𝑨 + 𝑔 ҧ 𝑨 ≤ 𝐷𝑏

𝜏𝑐 z = 𝑔

𝑨 − 𝑔 ҧ 𝑨 ≥ 𝐷𝑐

• Non-convex space
• Harmonic mapping enforce bounds only on 𝜖Ω

[Chen and Weber 2015]

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conformal isometric Source 𝑫𝒃 = 𝟑. 𝟔

𝝊 = 𝐧𝐛𝐲 𝝉𝒃, 𝟐 𝝉𝒄

The ℒ𝜉 Space

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𝒎 = 𝒎𝒑𝒉 𝒈𝒜 𝝃 = 𝒈𝒜 𝒈𝒜 𝒈𝒜 = 𝒇𝒎 𝒈ത

𝒜 = 𝝃𝒇𝒎

BD ℒ𝜉 ℒ𝜉 BD

[Levi and Weber 2016]

• Change of variables:
• BD homeomorphic to ℒ𝜉

The ℒ𝜉 Space

𝑙 𝑥 = 𝜉(𝑥) ≤ 𝐷𝑙 𝜏𝑏 w = 𝑓𝑆𝑓(𝑚(𝑥))(1 + 𝜉(𝑥) ) ≤ 𝐷𝑏 𝜏𝑐 w = 𝑓𝑆𝑓 𝑚 𝑥 (1 − 𝜉(𝑥) ) ≥ 𝐷𝑐

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• Near convex space

∀𝑥 ∈ 𝜖𝛻

Convex

Discretization

• Enforce distortion constraints on m densely sampled points
• Use Cauchy complex barycentric coordinate :

𝑚 𝑨 = ෍

𝑘=1 𝑜

𝑡

𝑘𝐷 𝑘 𝑨

& 𝜉 𝑨 = ෍

𝑘=1 𝑜

𝑢𝑘𝐷

𝑘 𝑨

• Subspace of holomorphic functions
• 4n-dimensional

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n vertices m sample points

Affine

𝑡

𝑘, 𝑢𝑘 ∈ ℂ

Our problem

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Bounded distortion Convex

∩

Harmonic mapping Affine convex subspace

• f ℝ4𝑛

4n-dimensional 4m-dimensional

Our problem

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• Input: 𝑚 and 𝜉 values from cage data
• Find the closest point in the intersection of an affine space and a convex

space

B A

𝑏𝑗

Alternating Projections

MAP ATP

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𝐼𝑗

Alternating Projections

MAP ATP

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Proof of convergence [Bauschke and Borwein 1993] [Von Neumann 1950]

Large-Scale Bounded Distortion Mappings

• 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

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[Kovalsky et al. 2015]

Gathering Input Data

• Extract 𝑚 and 𝜉 values from cage data
• Linear transformations 𝑓𝑗 ⟼ ෝ

𝑓𝑗 that preserves the unit normal

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𝒇𝒋 ෝ 𝒇𝒋

1 1 1 2

𝒇𝒋−𝟐 𝒇𝒋+𝟐 ෟ 𝒇𝒋−𝟐 ෟ 𝒇𝒋+𝟐

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

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Results

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ATP MAP

Near-optimality of alternating projection methods

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MOSEK source 35 fps 3 fps 0.005 fps

Near-optimality of alternating projection methods

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MOSEK source MAP ATP 140 fps 15 fps 0.2 fps

Speedup

× 𝟐𝟏𝟒 × 𝟒 ∙ 𝟐𝟏𝟒

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~30 ~170

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𝝊 = 𝐧𝐛𝐲 𝝉𝒃, 𝟐 𝝉𝒄

𝝊

𝐷𝑏 = = 5 5 𝐷𝑐 = = 0.2

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[Kovalsky et al. 2015] Cauchy Coords ATP Source

Summary

• Planar deformation
• GPU accelerated – speedup of 3 × 103
• Guaranteed local injectivity and bounded distortion
• General proof of convergence
• Future Work:
• Positional constraints
• Extension to parametrization of surfaces

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A Subspace Method for Fast Locally Injective Harmonic Mapping

EDEN FEDIDA HEFETZ, EDWARD CHIEN, OFIR WEBER BAR-ILAN UNIVERSITY, ISRAEL

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The Parametrization Problem

• Given a 3-D triangular mesh S, find a map 𝑔 ∶ 𝑇 → Ω such that Ω ⊂ 𝑆2

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• Desirable properties:
• Locally injectivity
• Low distortion
• Fast computation

Motivation

• Texture mapping
• Mesh correspondences
• Remeshing

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Credit: Hans-Christian Ebke

Quad Remeshing Example

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Previous Work

• Linear methods

LSCM [Lévy et al. ‘02] Angle-Based [Zayer et al. ‘07] Conformal Flattening [Ben-Chen et al. ‘08] …

• Nonconvex energy-based methods

CM [Schtengel et al.] Killing [Claici et al.] SLIM [Rabinovich et al.] AQP [Kovalsky et al.] …

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We aim for the speed of a linear method and the robustness of a nonconvex method

Tutte’s embedding

• Discrete harmonic function

Convex combination map Global bijection Convex boundary

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Global Parametrization

1) Mesh cut to a disk along a seam graph 𝐻𝑡 2) Resulting disk mapped to plane 3) Seam edge copies have isometric images Seamless parametrization: The rotational part of the isometry is a rotation by some multiple of 𝜌/2.

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𝒇𝒋𝒌 𝒈(𝒇𝒋𝒌

𝒃)

𝒈(𝑓𝑗𝑘

𝒄)

𝒘𝒌 𝒜𝒋 𝒘𝒋 𝒜𝒌

𝒃

𝒜𝒌

𝒄

HGP

Linear system:

• 1. 𝑔 𝑓𝑗𝑘

𝑏

= 𝑓𝑗

𝑐

, 𝑓ij ∈ 𝐻𝑡 2. 3.

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[Bright et al. ‘17]

HGP

Linear system:

• 1. 𝑔 𝑓𝑗𝑘

𝑏

= 𝑓𝑗

𝑐

, 𝑓ij ∈ 𝐻𝑡 2. 3.

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[Bright et al. ‘17]

HGP

Linear system:

• 1. 𝑔 𝑓𝑗𝑘

𝑏

= 𝑓𝑗

𝑐

, 𝑓ij ∈ 𝐻𝑡 2. 3.

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[Bright et al. ‘17]

fz

i > f ҧ 𝑨 i

, 𝑗 ∈ 𝑈𝑑𝑐

HGP

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[Bright et al. ‘17] HGP linear system Locally injective map Boundary and cone triangles are well-behaved [Lipman 2012] Frame field from [Bommes et al. 2009]

Subspace construction

Linear part of HGP: Add interpolation constraints to complete the system dimension:

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𝑃(|𝐷|)

Subspace construction

𝐿𝑨 =

𝑑𝑗𝑜𝑢

𝐿−1

𝑑𝑚𝑗𝑜 𝑑𝑗𝑜𝑢 = 𝑨

, 𝑑𝑚𝑗𝑜 =

1 ⋯ ⋮ 1 ⋮ ⋯ 1

𝑐1 𝑐2 ⋯ 𝑐𝑑 𝑑𝑗𝑜𝑢 = 𝑨 z = σ𝑗=1

𝑑

𝑐𝑗𝑑𝑗𝑜𝑢𝑗

For 𝑐𝑗, extract a column of 𝐿−1! By HGP theorem, only need to extract entries near cones and boundaries!

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𝐶

Subspace construction

• 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 )

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• 1

Affine

Boundary and cone triangles

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Convex

, 𝑗 ∈ 𝑈𝑑𝑐

1 (2) 1 (2)

Our problem

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Locally injective Convex

∩

Harmonic mapping Affine

Dimension: 𝑃( 𝐷 ) Dimension: 2 𝑈𝐷𝐶

𝐼𝑗

Projected Newton

• Symmetric Dirichlet energy is optimized
• Nonconvex, project Hessians to the PSD

cone

• [Chen & Weber 2017]

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Results!

• Comparable quality to HGP results
• One order of magnitude faster

than HGP; comparable to 1-2 linear solves (next slide)

• Fairly robust results: 66 of 77

successful on a benchmark

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Timing Results

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0.10 1.00 10.00 100.00 1,000.00 10,000.00 100,000.00 20,000 40,000 60,000 80,000 100,000 120,000 140,000 160,000

Runtime (s) # Triangles

Subspace Method HGP [Chien et al. 2016]

Algorithm Parts Timing

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0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 1 2 3 4

Series1 Series2 Series3

Summary

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

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Conclusion

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

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▪ Two methods have been accelerated:

The End

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