Geometric Queries on Tessellation-Free Displaced Surfaces

1 Introduction

• Tessellating displaced surfaces into dense triangle meshes can incur heavy memory costs, despite displacement maps encoding fine detail compactly.Recent tessellation-free methods support direct ray tracing, but broader geometric queries remain needed.Source: §1 · p. 1
• A local affine height plane and conservative residual retain regional height variation, supplying slope estimates and enclosing the height function where scalar height intervals alone are insufficient.The residual bounds height error throughout the region.Source: §1 · p. 2
• The hierarchy shares height planes and residual bounds across triangles using the same displacement map, supporting enclosures for intersection and distance queries and area estimates for sampling.Additional slope bounds support specular-path search; query-specific data are added only where needed.Source: §1 · p. 2
• The paper develops query-specific traversal, pruning, and sampling algorithms for ray intersection, specular light paths, surface sampling, and closest-point queries.The algorithms use piecewise-linear ray approximation, height-and-slope bounds, regional area estimates, and world-space distance lower bounds, respectively.Source: §1 · p. 2

Displacement mapping and tessellation-free ray tracing

• Direct displaced-surface ray tracing uses scalar height hierarchies, tightened texture/world bounds, rational texture-space rays, or fixed-step projective marching; this work instead preserves local height variation for broader queries.Prior approaches include TFDM [Thonat et al. 2021], RMIP [Thonat et al. 2023], Ogaki [2023], and projective displacement mapping [Hoetzlein 2025].Source: §2 · p. 2

Bounding methods for geometric queries

• The hierarchy combines affine height models with bounded residuals and query-specific bounds, extending displaced-surface queries to area, normals, and operations beyond ray intersection.Related continuous-surface methods use affine arithmetic, Taylor models, or range analysis; prior displacement bounds include normal bounds with min–max mipmaps.Source: §2 · p. 2

Specular light transport

• Mesh-based specular-path methods use manifold solving, hierarchical pruning, or polynomial constraints, while this hierarchy supports specular next-event estimation without pre-tessellation.The cited methods expect ordinary triangle meshes or require significant extensions for tessellation-free displaced surfaces.Source: §2 · p. 3

Surface and light sampling

• Prior surface and light sampling methods use mesh areas, all-hit ray queries, hierarchical product sampling, or light trees; this work estimates displaced-region area and combines it with emission and receiver-dependent weights.The resulting importance sampling targets displaced emitters without pre-tessellation.Source: §2 · p. 3

Closest-point queries and surface PDEs

• Closest-point methods for meshes or continuous representations differ from this hierarchy’s distance lower bounds for direct displaced-surface queries, which support surface projections in Projected Walk on Spheres.The application uses repeated closest-point projections for surface PDE solving [Sugimoto et al. 2024].Source: §2 · p. 3

3.1 Displaced surface

• A displaced triangle mesh combines a base triangle, interpolated displacement directions, texture coordinates, and a scalar height map that determines displacement.The shell map relates the texture-space height graph to the displaced face and surrounding shell.Source: §3.1 · pp. 3–4
• Shell space is the world-space volume swept by triangles at fixed heights over an interval containing displacement heights; its canonical coordinates are (α, β, h).The shell map sends a height interval over a texture region to a world-space region containing the corresponding surface patch.Source: §3.1 · pp. 3–4

3.2 Geometric queries and their requirements

• Ray intersection uses regional height bounds to reject regions, while specular-path search also needs position and normal bounds to prune regions before local constraint solving.Specular connections must satisfy reflection or refraction constraints, and the normal bounds must account for base geometry and the slope field.Source: §3.2 · p. 4
• Surface sampling needs regional area estimates because texture-space density generally differs from surface-area density, which depends on height and its derivatives.The pointwise Jacobian converts texture-space density to area density; regional estimates guide sample allocation.Source: §3.2 · p. 4
• Closest-point search combines surface candidates as distance upper bounds with enclosure-based lower bounds to discard regions that cannot improve the best distance.The paper demonstrates these queries for surface projection in Projected Walk on Spheres [Sugimoto et al. 2024].Source: §3.2 · p. 4
• The sought representation retains local height and slope information alongside approximation-error bounds, since searches need exclusion bounds and sampling needs placement estimates.A texture-space hierarchy lets queries reject regions or assign sample weights without examining every texel.Source: §3.2 · p. 4

4.1 Node representation

• The hierarchy is shared across triangles using the same height map, and each query uses the node region clipped to the triangle’s texture footprint.Each node covers a rectangular region in a regular texture-space quad-tree.Source: §4.1 · p. 5
• Each texture-space quad-tree node stores an affine height plane and conservative residual, retaining local variation while bounding height error across its region.Unlike scalar intervals used in prior methods [Thonat et al. 2021, 2023], the plane includes an offset and two slopes; higher-order models would require more coefficients and involved bound calculations.Source: §4 · p. 4, §4.1 · p. 5
• The plane-following height band can remain narrow across a large height range, but the residual bounds height error rather than derivative error and the world-space image need not be planar.Stored slopes need not equal the height function’s derivatives at the region center.Source: §4.1 · p. 5

4 First-order hierarchy

• The resulting records support ray rejection, distance lower bounds, regional area estimates, and—when augmented with slope bounds—specular-path search.The paper develops query-specific traversal, pruning, and sampling strategies for these uses.Source: §4.1 · p. 5, §4.2 · p. 6

4.2 Hierarchy construction

• Leaf records use the constant and linear terms of bilinear height interpolation as the plane, with the remaining cross term bounded by the cell half-widths.The residual bound also holds for piecewise-linear interpolation over the cell’s two triangles.Source: §4.2 · p. 5
• Parent records combine four child records by averaging slopes, evaluating child planes at their corners, expanding by child residuals, and enclosing the resulting range with a midpoint and half-width.Because the representation is affine, corner evaluations and child residuals enclose the full child height band; this construction is repeated to the coarsest level.Source: §4.2 · pp. 5–6

5.1 Piecewise-linear ray approximation

• The mapped ray is partitioned at singularities, barycentric turning points, and triangle boundaries before each retained interval is subdivided according to its midpoint deviation.Subdivision uses equal height increments within each event interval; the 0.06-texel threshold sets the approximation target but does not bound maximum deviation.Source: §5.1 · p. 7

5.2 Hierarchical DDA traversal

• Linear segments enable incremental DDA grid traversal and endpoint tests against node height bands, replacing repeated nonlinear ray–box tests.Subtracting a node’s plane makes the relative height affine, so endpoint values determine whether it lies outside the residual band; padding accounts for sideways curve deviation.Source: §5 · p. 6, §5.2 · p. 7

5.3 Results

• Across 228 mesh–displacement map pairs, our method achieves a 3.404× average speedup over NRT, with LPIPS at most 0.004 across nine scenes and no visible discrepancies.It passes quantitative checks for hit positions, normals, and image differences against NRT.Source: §5.3 · p. 8, Table 1 · p. 9
• Pre-tessellation at one micro-quad per texel is estimated to require 544 MB, about 53× the method’s 10.3 MB geometric-mean footprint.The 10.3 MB includes the height map, retained hierarchy levels, per-triangle records, and prism BVH; the comparison covers 228 pairs.Source: §5.3 · pp. 8–9
• The first-order representation retains efficient ray tracing while supporting broader queries, with similar performance on tested curved bases and about 14% faster rendering on two planar-base scenes.The overall renderer speedups also include traversal and leaf-intersection changes.Source: §5.3 · p. 9

6 Specular next-event estimation

• Specular paths are challenging to sample with standard path tracing [Veach 1997], and their optical constraints place them on a low-dimensional path-space manifold.The section addresses specular-path queries that tessellation-free methods had largely overlooked.Source: §6 · p. 9

6.1 Conservative first-order pruning of specular paths

• The pruning test bounds possible n × m values over a node, projects the bound to 2D, and uses separating axes to test whether it contains zero.The projected test is conservative and cheaper than the full 3D test; Figure 8 illustrates the node bound and a case where the origin prevents pruning.Source: §6.1 · p. 10, Figure 8 · p. 10

6.2 Hierarchical specular next-event estimation query

• Top-down traversal prunes child nodes, then runs Newton’s method from each accepted leaf’s texel center and accumulates the solutions’ contributions.The leaf-center initialization is usually close, and the constraint is nearly linear on a small smooth patch.Source: §6.2 · p. 10
• The leaf solver assumes one solution per texel, so a caustic fold can hide a second solution and cause slight energy loss.The authors say this case is rare, becomes less likely with smaller texels, and caused no noticeable darkening or artifact in their tests.Source: §6.2 · p. 10

6.3 Results

• In equal-time caustic renders, our method converges significantly faster than SMS; Figure 1 also shows displaced caustic emitters fulfilling the scene’s artistic direction.SMS can produce excessive noise without good initialization in its unbiased variant, or lose energy in its biased variant.Source: §6.3 · pp. 10–11
• The current implementation handles specular paths with a single specular vertex, not consecutive specular interactions.The authors identify extending the method to consecutive interactions as future work.Source: §6.3 · p. 11

7.1 Area-guided hierarchical sampling

• Hierarchical descent selects texture-space children in proportion to estimated displaced area, yielding unbiased surface samples despite using estimated areas.The method weights each node by its estimated area and samples uniformly within the selected leaf footprint.Source: §7.1 · p. 11

7.2 Weighted area sampling for displaced emitters

• Emission weighting makes the sampling PDF approximately proportional to emission luminance, and receiver-aware weighting further allocates samples to nearby sources.The receiver-aware strategy reduces variance around strong emitters; the E+R variant achieves more than 10× variance reduction without loss of energy against SMS variants.Source: Figure 9 · p. 12, §7.2 · p. 13, §7.3 · p. 13

7.3 Results

• E+R achieves sampling efficiency close to a light BVH without requiring heavy pre-tessellation.The comparison uses equal-time rendering; the light BVH achieves lower errors at a heavy memory cost, while E+R has comparable errors under equal-sample conditions.Source: §7.3 · p. 13

8.1 Initialization; 8.3 Refinement and termination

• An initial evaluated surface point supplies a valid distance upper bound, and valid bounds let termination return a candidate within ε_cp of the minimum distance.The search retains bounds from closed regions and completed leaf searches when deciding whether to stop.Source: §8.1 · p. 14, §8.3 · p. 15

8.2 Distance bounds and traversal

• The search visits regions in increasing distance-lower-bound order and prunes regions that cannot improve the best candidate within the requested tolerance.Height-plane approximations and deviation bounds provide conservative lower bounds; leaf regions can be subdivided and refined as needed.Source: §8.2 · p. 15, §8.3 · p. 15, Figure 11 · p. 15

8.4 Results

• First-order bounds yield a 3.06× geometric-mean speedup over scalar min–max prism bounds across all test surfaces.Traversal and the leaf solver are fixed in this ablation, isolating the regional bounds' contribution.Source: §8.4 · pp. 15–16, Figure 13 · p. 16
• At least 55.4× lower query memory and 7.48× faster queries for FCPW on the 36 surfaces meeting the 10^-4 target summarize the pre-tessellation comparison.Across 56 surfaces, 20 did not meet the target within the tessellation budget; the memory comparison counts geometry and acceleration data retained after preprocessing.Source: §8.4 · p. 15, Figure 12 · p. 16
• For Projected Walk on Spheres, query memory falls by factors of 12.52 and 91 at distance-error targets of 10^-4 and 10^-5, respectively.In a separate complete-walk ablation, the method is 3.19× faster than the min–max AABB variant.Source: §8.4 · p. 16

• The hierarchy supports caustics rendering, displaced-emitter sampling, and closest-point queries for surface PDE solvers while retaining efficient direct ray tracing.Source: §9 · p. 16
• High-fidelity graphics and geometry workloads can demand so much geometry that it cannot fit into memory.Source: §9 · p. 16
• The paper presents tessellation-free displaced surfaces as a promising high-fidelity, general-purpose graphics primitive.Source: §9 · p. 16