CIS 7000

NeurCross

A Self-Supervised Neural Approach for Representing Cross

Fields in Quad Mesh Generation

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QIUJIE DONG, HUIBIAO WEN, RUI XU, XIAOKANG YU, JIARAN ZHOU, SHUANGMIN CHEN, WENPING WANG

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Shandong University, Qingdao University,Texas A&M University

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Carlos López Garcés

MS Computer Graphics

At a Glance

Problem

• Learn cross field of curvature of triangle mesh surface to obtain quad mesh

Outline

• Key concepts
• Challenges and goals of quadrangulation
• Solution
• Surface fitting module
• Cross field prediction module
• Results
• Limitations

Mesh Quadrangulation

Quadrangulation

• Conversion of triangle-based 3D mesh to quad-based
• Applications in CAD, character animation, physics simulations

Curvature

Curvature at a point

• How fast the surface bends in different directions
• Infinitely many directions in the tangent plane at a point, if smooth

Example:

• Let df(X) be unit tangent direction
• Plane defined by df(X) and N intersects surface at a curve
• Curvature 𝜅n in direction X

Principal Curvature Directions

Principal directions

• Directions in tangent plane where curvature is minimum and maximum
• Orthogonal

Principal curvatures

• Curvatures 𝜅1 and 𝜅2 of principal directions

Cross Fields

Cross field

• Direction field of pairs of orthogonal vectors at each point on a surface
• Ideally, aligned with principal directions

1-direction: Vector Field

2-direction: �Line Field

4-direction: Cross Field

Quadrangulation and Cross Fields

• Cross fields define the orientation and alignment of the quads
• Quad mesh edges should align with principal directions (among other features)
• 2D (u,v) parameterization should align with cross field (for extraction)

Singularities

• Vertices where local connectivity is not the usual 4-way “valence”
• Where principal directions converge or diverge
• Unavoidable on most curved, closed surfaces

Positive

5-valent vertex

Negative

3-valent vertex

Challenges and Goals of Quadrangulation

Minimize singularities

• Singularities introduce local distortions
• Make mesh subdivision, UV mapping, and texture alignment more difficult

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Place them strategically

• Peaks and valleys of curvature
• Sharp corners
• Less visible regions
• Away from regions of high deformation (deformation simulation)

Challenges and Goals (continued)

Maintain fidelity with the original triangular surface

• Minimize area and angle distortion

1000 vertices

5000 vertices

Severe loss of fidelity

Some loss of fidelity

Looks almost like the triangular mesh

Good job, despite few vertices

Challenges and Goals (continued)

Resistance of cross field to noise and fine geometric details

• Cross field should align with principal curvature directions despite noise

Resistant

Not Resistant

Solution Overview

NeurCross

• Self-supervised neural representation of the cross field

2 modules

• SIREN-based module for fitting a Signed Distance Function (SDF)
• U-Net based module for predicting the cross field

Loss function

• Ensure faithful SDF
• Enforce alignment of cross field with principal curvature directions
• Promote spatial coherence of cross field

Surface Fitting Module

SIREN-based module

• Goal: learn a neural SDF whose zero level set corresponds to input triangle mesh
• Computes Hessian matrix needed by Cross Field Prediction module
• From SDF
• Uses SIREN neural network as neural representation of SDF

Surface Fitting Module (continued)

Signed Distance Functions (SDFs)

• Shortest Euclidean signed distance from a given point 𝐱 in space to the surface

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• Implicit definition of surface as zero level set:
• ∇𝑓(𝐱) points to closest point on surface
• |∇𝑓(𝐱)|= 1
• Differentiable

Surface Fitting Module (continued)

SIRENs: SInusoidal REpresentation Networks

• Effective at fitting continuous, high-frequency signals and their derivatives
• 3D shapes, SDFs, images, audio waveforms
• Key idea: periodic/sinusoidal activation functions
• Learn mapping of 3D coordinates (point cloud) to scalar values (signed distance)
• Solve Eikonal boundary value problem:
• PDE |∇𝑓(𝐱)|= 1 with boundary condition 𝑓(𝐱) = 0 on the surface
• Learned neural SDF is continuous and differentiable

Surface Fitting Module (continued)

Self-supervised fitting process

1. Extract triangle centroids 𝒑 and normals np: defines sample set 𝓟
2. Define Gaussian distribution at each 𝒑 based on 50 nearest neighbor centroids
3. Sample distribution to obtain set 𝛀, a thin shell space enclosing 𝓟
4. Use this data to optimize SIREN-based SDF using 3 different losses

Surface Fitting Module (continued)

SDF fitting is subject to 3 conditions, enforced by loss functions

• Eikonal condition: enforce unit gradient
• 𝓟, sample points on input triangle mesh (triangle centroids)
• 𝛀, randomly sampled points in thin-shell space enclosing 𝓟

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Surface Fitting Module (continued)

• Dirichlet condition: enforce 𝒑 on triangle mesh is in zero level set, 𝑓(𝒑) = 0
• 𝙌, uniformly drawn from 𝒑’s bounding box [-0.5,0.5]³
• External to 𝓟 (not on surface)
• Encourage 𝑓(𝒒) ≠ 0

e^-x

Surface Fitting Module (continued)

• Alignment condition: orient/align SDF with surface based on surface normal
• Hessian 𝐇𝒙 of SDF at point 𝒙 describes how ∇𝑓(𝒙) changes near 𝒙
• Eigenvalues/eigenvectors of Hessian 𝐇𝒙:
• Eigenvalue 0: no ∇𝑓(𝒙) change, no curvature along eigenvector, i.e. normal n𝒙
• n𝒙 is in the null space of 𝐇𝒙:
• Other eigenvalues/eigenvectors of Hessian projected onto tangent plane: principal curvatures 𝜅1 and 𝜅2 and directions
• Align 𝐇𝒙 with normal vectors np from input triangles (𝓟)

Cross Field Prediction Module

U-Net-based module

• Goal: infer cross field vectors 𝛼𝒑 and 𝛽𝒑 that align with principal directions

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where 𝜇𝒑 and 𝑣𝒑 define a fixed coordinate system on the triangle

• Boils down to predicting rotation angle 𝛳𝒑 for each input triangle 𝒑
• 𝛼𝒑 and 𝛽𝒑 result from rotating frame 𝜇𝒑 and 𝑣𝒑
• Uses U-Net neural network for prediction of 𝛳𝒑

Cross Field Prediction Module (continued)

Alignment of cross field vectors 𝛼𝒑 and 𝛽𝒑 with principal curvature directions

• Hessian 𝐇𝒑 of surface at point 𝒑 approximated from SDF
• Eigenvectors are mapped to a multiple of themselves by a matrix
• Eigenvectors of 𝐇𝒑 are principal directions
• If 𝛼𝒑 and 𝛽𝒑 are aligned with 𝐇𝒑𝛼𝒑 and 𝐇𝒑𝛽𝒑, then they are principal directions
• Collinearity condition:
• Alignment loss:

Cross Field Prediction Module (continued)

Coherence of adjacent crosses:

• 2 unit vector pairs (𝛼1, 𝛽1) and (𝛼2, 𝛽2) align if 𝛼1 and 𝛼2 or 𝛽1 and 𝛽2 are collinear
• It can be proven that that’s the case if the following equals 2:

• Coherence loss:

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where rotations are for each of the 3 adjacent vertices q

Cross Field Prediction Module (continued)

U-Net, CNN

• For predicting rotation angle 𝛳𝒑
• Good for predicting point-wise, dense, structured outputs
• Residual connections to mitigate vanishing gradients problem
• Preserves fine-grained geometric details, like local curvature

Total Loss

• SDF and cross field are updated simultaneously during optimization
• Total loss combines losses from the 2 modules

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• Annealing factor 𝜏 on SDF alignment loss decreases its influence
• 1 initially (first 20% iterations), then decreases linearly, then 0
• SDF and surface orientation alignment is likely solved early in the process

Final Steps

Quad Mesh Extraction from Cross Field

1. Global-seamless parameterization, borrowed from libigl
• 3D point to 2D (u,v) mapping
• Cross field aligns with integer isolines, which define boundaries of quads
2. Extract quad mesh from parameterization using libQEx

Example Parameterization

Example Parameterization

Integer Isolines

=

Integer UV coords

NeurCross

Initial

cross field

Learned

cross field

Surface fitting

Cross field orientation prediction

Total loss

Neural SDF

Triangle centroids

Learned principal directions

Learned

rotation angle

Output

quad mesh

Hessian

Results

Evaluation metrics

1. Area distortion (Area): standard deviation of areas of quads
2. Angle distortion (Angle)
3. Number of singularities (# of Sings)
4. Chamfer distance (CD): quantifies similarity between 2 surfaces (sets of points)

Methods compared to

• MIQ, Instant Meshes (IM), Quadriflow, Dual Marching Cubes (DMC)
• None are neural methods

Datasets

• ShapeNet and Thingi10k

Quantitative Evaluation

ShapeNet

Thingi10k

Best

2nd Best

Evaluation: Alignment

Alignment with geometric features, including following curvature

Evaluation: Complexity

Handles complex geometry well

• High genus (holes, handles), thin shells, and non-orientable rings

Evaluation: Fidelity

High fidelity at low and high resolution

Fidelity: Hausdorff Distance

Evaluation: Number of Singularities

Limitations

Resulting quad mesh doesn’t retain feature lines

• Feature lines are prominent geometric features, like sharp edges or ridges
• Cause: no optimization constraints for aligning quadrangulation with feature lines

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Limitations

Zig-zagging open boundaries

• Cause: no optimization constraints at open boundaries

Input

NeurCross

Conclusions and Future Work

Conclusions

• First self-supervised approach for cross fields
• Good approximation accuracy and regularity of cross field
• Good placement of singular points

Future work

• Address limitations:
• Add alignment constraints for feature lines
• Add alignment constraints at open boundary locations

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