Multi-directional Geodesic Neural Networks via Equivariant Convolution

• Curved surfaces lack a direct convolution analogue that satisfies all the properties of Euclidean convolution.Source: §1 · p. 1
• Extrinsic mappings and projections can distort surfaces or apply only to restricted topological classes.Source: §1 · p. 1
• Intrinsic methods avoid embedding changes but face unresolved tangent-plane orientation ambiguity when mapping kernels onto surfaces.Source: §1 · p. 1
• Prior remedies either use unstable canonical directions or angular max pooling, which loses relative kernel orientation across layers.Source: §1 · p. 2
• Directional convolution preserves responses across rotations through aligned layers and postpones the maximal response operation until the final layer, improving training and test accuracy.The method uses directional functions so relative responses at different surface points remain comparable across layers.Source: §1 · p. 2

2.1 Extrinsic and Volumetric Techniques

• Extrinsic and volumetric methods include rendered views, planar mappings, volumetric convolutions, and point-cloud networks, but depend on shape embedding and can be sensitive to pose changes.Source: §2.1 · p. 2

2.2 Intrinsic and Graph-based Techniques

• Intrinsic and graph-based methods define convolution through spectral, graph, planar-embedding, or Laplace and Dirac operators directly on shape data.Source: §2.2 · p. 2
• Local parameterization methods convolve learned kernels in tangent planes, but face directional ambiguity, unstable principal-curvature directions, or angular max pooling.Source: §2.2 · p. 2

2.3 Contribution

• The paper maintains directional ambiguity across deep-network layers without choosing a canonical direction or retaining only maximal responses.Source: §2.3 · p. 2
• Directional functions can be convolved into new directional functions, allowing ambiguity to be resolved by taking the maximum response only once at the final layer.Source: §2.3 · p. 2
• The approach reports improved accuracy and robustness across applications and extends prior architectures with mesh-simplification pooling and residual-learning blocks.Source: §2.3 · p. 2

3 CONVOLUTION OVER MANIFOLDS

• Surface convolution is formulated intrinsically on oriented 2D manifolds, using exponential maps and tangent-plane coordinate alignments to compare signals with templates.The discussion assumes embedded, boundaryless surfaces for exposition, while the practical implementation supports boundaries.Source: §3 · p. 3

3.1 Convolution of Directional Functions

• Directional functions extend surface signals by associating each point with every unit tangent direction, allowing ordinary signals to be lifted without changing their values.A lifted signal ignores its directional argument, while general directional functions retain angular dependence.Source: §3.1 · p. 3
• Directional convolution uses a completed exponential map with parallel transport, producing another directional function rather than collapsing responses across orientations.The completed map sends a tangent-plane vector to a surface point and the transported unit direction there; regular signals are handled through lifting.Source: §3.1 · pp. 3–4

3.2 Directional vs. Geodesic Convolution

• Directional convolution preserves geodesic paths and relative angular information, whereas repeated geodesic convolution spreads a point response over a ball and loses directional detail.For a shifted Dirac kernel, directional propagation reaches a circle at exact distance 2t, while geodesic propagation covers the disc of radius 2t.Source: §3.2 · p. 4, Figure 3 · p. 4, Figure 4 · p. 4

3.3 Directional Convolution in Angular Coordinates

• Angular-coordinate implementations remain equivariant to arbitrary reference-direction changes, with circularly permuted inputs producing correspondingly permuted outputs.The remaining rotational ambiguity is deferred to the final layer, where angular max pooling produces a regular signal.Source: §3.3 · p. 5

3.4 Multi-Directional Convolutional Neural Networks

• Directional convolution layers can be stacked in neural networks because their outputs remain directional functions, with angular max pooling applied only at the final layer.The resulting architectures are called Multi Directional Geodesic Convolutional Neural Networks, or MDGCNNs.Source: §3.4 · p. 5

4.1 Template Functions

• Template functions are discretized on uniform radial and angular polar samples, stored as trainable matrices over the resulting window.The window is bounded by radius R and sampled at Nρ radial and Nθ angular locations.Source: §4.1 · p. 6

4.2 Angular Functions

• Angular functions are represented at mesh vertices across discrete angular bins relative to an arbitrary reference direction in each tangent plane.For Nv vertices, the representation has size (Nv, Nθ).Source: §4.2 · p. 6

4.3 Exponential Map

• Geodesic polar coordinates sample neighborhoods by distance and angle, while barycentric interpolation handles off-vertex samples and improves resilience to mesh-structure changes.The exponential-map representation uses a tensor E whose fixed window sample generically has three nonzero barycentric weights.Source: §4.3 · p. 6

4.4 Parallel Transport

• Parallel transport aligns angular samples between tangent planes through angle differences and interpolation, yielding a discrete tensor for spatial and angular transport.The transported angular exponential map is interpolated in both spatial and angular domains.Source: §4.4 · p. 6
• Directional convolution is implemented by matrix multiplication after aligning template and geodesic angular coordinates, with rotated templates handled by circular angular permutations.A learned or fixed central-point matrix can be added because the window convolution otherwise excludes the central signal value.Source: §4.4 · p. 6

4.5 Spatial Pooling

• Mesh pooling transfers signals to a simplified mesh produced by quadratic edge collapse, while directional pooling also rotates and interpolates angular bins using normal-induced offsets.Un-pooling transfers values back by selecting the closest collapsed vertex.Source: §4.5 · pp. 6–7

5 GPC AND PARALLEL TRANSPORT

• An extended fast-marching procedure propagates geodesic distance, polar angle, and reference-direction angle differences until a prescribed radius threshold Rmax.The method uses local transfer and update rules while selecting candidate vertices by a priority criterion.Source: §5 · p. 7
• Angles are transported across mesh edges by aligning edge directions in neighboring GPCs, measuring their relative offset, and adding that offset during propagation.This produces an estimate of parallel transport between source and target reference directions.Source: §5 · p. 7
• Figure 7 encodes GPC angular coordinates by color and illustrates transport of one center direction, alongside the discretized radial and angular bins.The sphere example separates the geometric transport visualization from the discrete sampling illustration.Source: Figure 7 · p. 7, §5 · p. 7

6.1 Architecture

• MDGCNN uses ResNet stacks with mesh down-sampling, angular max pooling, and global averaging for classification, while U-ResNet adds decoding and shortcut connections for pointwise segmentation.The encode block follows the ResNet design, whereas decoding replaces pooling with un-pooling to restore spatial resolution.Source: §6.1 · pp. 7–8, Figure 9 · p. 8, Figure 8 · p. 8

6.2 Experiments

• MDGCNN is evaluated against GCNN and other methods across CIFAR-10 classification, human-body segmentation, and FAUST shape matching.Experiments use matched architectures for MDGCNN and GCNN where possible, with ADAM optimization and varied geometric inputs.Source: §6.2 · p. 8, §6.3 · p. 8, §6.4 · p. 9
• MDGCNN is slower per epoch than GCNN because directional alignment and interpolation increase implementation cost, but its faster convergence partly compensates for this difference.On the sphere-based image classification experiment, one epoch took 10 min 13 s for MDGCNN versus 7 min 13 s for GCNN.Source: §6.2 · p. 8, Figure 14 · p. 9

6.3 Image classification

• MDGCNN achieves higher CIFAR-10 scores than GCNN and compares favorably with PointCNN across grid and sphere domains despite spherical stretching and irregular meshing.Table 1 reports classification accuracy for the different methods and domains.Source: §6.3 · pp. 8–9, Table 1 · p. 8, Figure 11 · p. 8

6.4 Shape segmentation

• On human-body segmentation, MDGCNN learns rigid-motion-invariant features from 3D coordinates through augmentation, producing smoother results and outperforming GCNN in that input setting.MDGCNN also shows more stable training than GCNN, while performance is close to WKS-based results and comparable to the Toric Cover method.Source: Figure 13 · p. 9, Table 2 · p. 9, Table 3 · p. 9

6.5 Shape matching

• On FAUST correspondence, MDGCNN maintains precision after remeshing whereas GCNN degrades, while both methods behave similarly on the original shared-connectivity meshes.The comparison measures geodesic error directly and includes SHOT-descriptor nearest neighbors as a baseline.Source: §6.5 · pp. 10–11, Figure 15 · p. 10, Figure 16 · p. 10

6.6 Limitations & Future Work

• The method is limited by intrinsic local coordinates, which can separate nearby embedded regions, mishandle thin or merged parts, and cause feathers to be segmented as separate tails.The paper illustrates these failure cases on bird shapes and notes successful segmentation when such limiting cases are absent.Source: §6.6 · p. 11, Figure 18 · p. 11
• A single fixed local-window scale can reduce generalization when object-part proportions differ, while angular-bin interpolation introduces errors that grow with depth, especially with few bins.The authors suggest multi-scale modules and Fourier representations as future alternatives, while noting that Fourier representations complicate nonlinear operations and angular pooling.Source: §6.6 · p. 11

7 CONCLUSION

• The method defines convolution on curved surfaces without global or canonical coordinates by aligning local systems and learning consistent filters across domains.Directional functions generalize real-valued signals and are convolved with learned template functions.Source: §7 · p. 11
• Directional convolution composes across network layers without losing directional information, preserving multiple orientations instead of discarding them prematurely.This maintains the directional ambiguity through intermediate layers rather than resolving it with only a maximal response at each layer.Source: §7 · pp. 11–12
• The approach compares favorably with direct analogues, produces smoother and more robust results, and competes with newer techniques even when using 3D coordinates as weak input signals.The conclusion presents this pattern across the paper’s evaluated geometric learning tasks.Source: §7 · p. 12
• The paper suggests that multi-directional convolution could generalize to other data representations with ambiguity by allowing neurons to interpret and communicate inputs under different alignments.This is presented as a broader possible direction rather than an experimentally established result.Source: §7 · p. 12

8.1 Details on the implementation of convolution

• For triangle meshes, geodesic polar coordinates around each vertex define convolution windows, with points inside triangles sampled using barycentric coordinates.Source: §8.1 · p. 13
• Mesh signals are pulled back to window systems, while template functions are stored in ab-polar kernel tensors for geodesic convolution.Source: §8.1 · pp. 13–14
• Parallel transport is represented by angular offsets and discretized transport tensors, with lower and upper angular indices enabling interpolation within directional sectors.Source: §8.1 · p. 14

8.2 A Stronger Notion of Directional Convolution

• Directional convolution transports tangent directions along geodesics, producing a local radial vector field that is rotation invariant and simplifies implementation.Source: §8.2 · p. 14
• On the plane, the resulting directional function stores ordinary convolution responses for rotated kernels across directions, linking directional CNNs to maximal rotated-kernel responses.Source: §8.2 · p. 14