Continuous Boundary Stitching

• Continuous boundary stitching is a method that unites adjacent representation segments, ensuring seamless transitions in geometry, appearance, and topology.
• It is applied in diverse fields such as NeRF compositing, deep image/video stitching, fluid–solid coupling, CAD modeling, trajectory planning, and microstructure synthesis.
• Key techniques involve explicit boundary representation, compatibility constraints, and propagation mechanisms to maintain both structural integrity and topological consistency.

Continuous boundary stitching denotes a family of constructions that join adjacent pieces of a representation so that the interface is continuous and the transition away from the interface remains compatible with the surrounding geometry, appearance, motion, or topology. In the cited literature, the term appears in settings as different as NeRF compositing, deep image and video stitching, fluid–solid coupling, CAD multi-patch modeling, trajectory generation, periodic microstructure synthesis, block-wise learned compression, and geometric inverse problems; across these settings, the common objective is to replace a visible seam, leak, gap, or discontinuity with a representation whose boundary conditions and interior behavior are jointly consistent (Gong et al., 2023, Kim et al., 2023, Panuelos et al., 3 Feb 2026, Martin et al., 2020, O'Mahoney et al., 12 May 2025, Ji et al., 2023, Zhang et al., 24 Feb 2025, Meyerson, 2020).

1. Scope and domain-specific meanings

Across the literature, “continuous boundary stitching” is not a single algorithm but a recurring structural problem. In radiance fields, it is the task of compositing multiple NeRFs into a single, continuous radiance field, with source appearance pinned at the contact region and target gradients preserved in the interior. In image stitching, it denotes hole-free and discontinuity-free transitions between overlapping and non-overlapping regions under a Dirichlet boundary condition. In fluid simulation, it denotes a continuous set of faces in a single mesh whose adjacency graph matches the connectivity of the physical fluid region around thin solids. In CAD, it denotes watertight or $G^1/G^2$ joining of trimmed spline patches. In trajectory planning, it denotes smooth connection of boundary states by recombining local segments rather than reproducing complete training trajectories. In periodic microstructure generation, it denotes continuation of grain boundaries across opposite faces. In block-wise learned compression, it denotes patch-wise inference whose stitched output is mathematically equivalent to whole-image inference (Gong et al., 2023, Kim et al., 2023, Panuelos et al., 3 Feb 2026, Martin et al., 2020, O'Mahoney et al., 12 May 2025, Ji et al., 2023, Zhang et al., 24 Feb 2025).

Domain	Boundary object	Continuity mechanism
NeRF composition (Gong et al., 2023)	3D contact region $\partial B_i$	Boundary color pinning + gradient preservation
Deep image stitching (Kim et al., 2023)	TPS control-grid boundary	Dirichlet zero displacement + recurrent residual warps
Video stitching (Nie et al., 8 May 2025)	Overlap region over time	Stitching trajectories + warp smoothing
Fluid–solid coupling (Panuelos et al., 3 Feb 2026)	Fluid–solid faces in stitched Voronoi mesh	Connectivity-preserving orphan reassignment + exact flux boundary conditions
CAD trimmed surfaces (Martin et al., 2020)	Shared patch boundaries	Base/ribbon blending with contact-order-controlled weights
Trajectory planning (O'Mahoney et al., 12 May 2025)	Start/goal and intermediate boundaries	Local inpainting-style flow model + trajectory splitting
Periodic microstructures (Ji et al., 2023)	Opposite sample faces	U-Net self-stitching through masked boundary reconstruction
Block-wise compression (Zhang et al., 24 Feb 2025)	Patch borders	Padding-free network + overlap bound from receptive field
Geodesic inverse problems (Meyerson, 2020)	Boundary-accessible geodesics	Quotient of geodesic intervals by intersection equivalence

Taken together, these usages suggest a cross-domain abstraction: a boundary must first be represented explicitly, then constrained by a compatibility condition, and finally extended into a neighborhood by a propagation or gluing mechanism. The exact representation varies—RGB radiance, TPS displacement, flux face, spline jet, latent trajectory, or quotient topology—but the role of the boundary is analogous.

2. Boundary conditions and formal interface constraints

The most direct formulations impose explicit boundary conditions. In SeamlessNeRF, the contact region for target field $F_i$ is

$$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$

and color continuity is enforced by

$$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$

This boundary pinning is combined with a gradient constraint in the target region, yielding

$$\mathcal{L} = \mathcal{L}_{\text{color}} + \lambda\,\mathcal{L}_{\text{grad}},$$

which functions as a Poisson-like blending objective in NeRF space (Gong et al., 2023).

REwarp expresses the boundary condition at the level of the warp parameterization rather than by a separate loss. The TPS control grid is $12\times 12$, all boundary control points are fixed to zero displacement, and the induced warp satisfies

$$\mathbf{F}(\mathbf{x}) = 0, \quad \mathbf{x}\in\partial\Omega.$$

Because TPS minimizes bending energy subject to these control constraints, the resulting warp is smooth ($C^2$), zero at the boundary, and smoothly approaches zero near the boundary (Kim et al., 2023).

In trimmed spline modeling, the boundary condition is encoded by a blend between a global base surface and local ribbons: $$a(\xi) := \frac {w(\xi)\, b(\xi) + \sum_{\ell=1}^L w_\ell(\xi)\, r_\ell(\kappa_\ell(\xi))} {w(\xi) + \sum_{\ell=1}^L w_\ell(\xi)}.$$ Here the ribbon associated with a boundary segment dominates near that segment, while the base surface dominates in the interior. Contact order $\partial B_i$0 corresponds to $\partial B_i$1, $\partial B_i$2, and $\partial B_i$3 boundary behavior under the stated ribbon and weight conditions (Martin et al., 2020).

In block-wise learned compression, the boundary condition is receptive-field-theoretic rather than geometric. For a padding-free network with receptive field $\partial B_i$4 and overall scale factor $\partial B_i$5, the required overlap is

$\partial B_i$6

Patches extracted with stride $\partial B_i$7 can then be stitched without block artefact, because every reconstructed pixel is taken from a valid region whose receptive field is fully available inside at least one patch (Zhang et al., 24 Feb 2025).

These formulations are heterogeneous, but each replaces heuristic seam hiding with an admissibility condition: pointwise equality, zero boundary displacement, ribbon dominance, or valid receptive-field support.

3. Propagation away from the interface

Boundary matching alone is insufficient in the cited methods. SeamlessNeRF states this explicitly: matching only the boundary values is insufficient for continuous stitching, so the method preserves the original target radiance gradients by finite differences,

$\partial B_i$8

The optimization then acts as a neural Poisson solver: boundary colors are re-anchored to the source, while the target’s internal texture, shading variation, and fine detail are preserved through gradient-domain constraint propagation. View dependence is handled by Closest Ray Sampling, so the view direction used in losses remains aligned with training rays (Gong et al., 2023).

REwarp uses a different propagation mechanism. A global homography provides coarse alignment, and a residual TPS warp refines the overlap. Both are learned recurrently: H-Cell predicts residual corner displacements for the homography, T-Cell predicts residual interior TPS control-point displacements, and both are accumulated across iterations. Because the boundary control points are fixed, the correction is “bulk” rather than “edge pulling”; continuity is preserved while alignment is improved incrementally (Kim et al., 2023).

In online video stitching, StabStitch++ makes the propagation explicitly temporal. It defines temporal meshes $\partial B_i$9, derives stitching motions $F_i$0 and $F_i$1, accumulates them into stitching trajectories $F_i$2 and $F_i$3, and smooths these trajectories with SmoothNet using $F_i$4, $F_i$5, and $F_i$6. The seam is therefore not merely aligned per frame; its motion is regularized as a spatiotemporal object (Nie et al., 8 May 2025).

In flow-based trajectory planning, propagation occurs in time rather than space. A UNet with local receptive field and inpainting-style conditioning is trained so that local temporal neighborhoods are plausible under varying boundary conditions. Action noise increases the density and diversity of overlapping local trajectory neighborhoods, while train-time and inference-time trajectory splitting introduce intermediate boundaries and enable local re-solve on halves of the trajectory. This yields a multi-scale stitching mechanism: first global endpoints are satisfied, then midpoint-constrained segments are refined locally (O'Mahoney et al., 12 May 2025).

A plausible unifying implication is that continuous boundary stitching requires a transport mechanism from the boundary into a neighborhood: gradients in radiance fields, recurrent TPS corrections in image space, smoothed mesh trajectories in video, or local temporal plausibility in trajectory space.

4. Exact topology, quotient constructions, and geometry-aware gluing

Some formulations treat continuity as a topological condition rather than a smoothness prior. In incompressible fluids coupled to thin deformables, clipping a Voronoi diagram by the solid surfaces creates valid and orphaned fluid fragments. The stitching algorithm reassigns orphan fragments to neighboring valid cells through existing fluid–fluid faces using a proxy geodesic distance, so no orphan crosses a solid barrier and every final fluid cell remains path-connected within the fluid domain. The adjacency graph of fluid cells thereby matches the connectivity of the physical fluid region, and the stitched mesh supports exact enforcement of normal velocity boundary conditions and direct pressure-force transfer on fluid–solid faces (Panuelos et al., 3 Feb 2026).

Geometric 4D Stitching treats the boundary as the image-space footprint of missing geometry. The information-addition mask is

$F_i$7

where projection holes and curtain masks both contribute to the missing region. Candidate content is generated only inside this mask, raw NVS depth is aligned to anchor depth by a patch-wise scale–shift field $F_i$8, and only geometrically consistent back-projected points are retained after render-disagreement distillation. The result is explicit 4D stitches added to the mesh rather than a radiance field that absorbs inconsistency into view dependence (Park et al., 11 May 2026).

At the most abstract end, stitching becomes an equivalence construction. For a collarable Riemannian manifold with boundary, “Stitching Data” defines the set

$F_i$9

and the equivalence relation

$$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$0

so that the quotient

$$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$1

is a stitched length space assembled from geodesic intervals. The paper proves that the resulting length structure is isomorphic to the manifold’s piecewise-geodesic length structure, and therefore that a stitching data determines the isometry class of a collarable manifold with boundary (Meyerson, 2020).

These works use different mathematical languages—adjacency graphs, back-projected 4D points, quotient spaces—but they share a strict requirement: the stitched representation must preserve which regions are connected, separated, or coincident.

5. Representative algorithmic realizations and evidence

The empirical literature evaluates continuity by seam visibility, topology correctness, stability over time, or explicit rate–distortion and planning metrics.

System	Reported evidence	Consequence for stitching
REwarp (Kim et al., 2023)	best average mPSNR $$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$2 dB on UDIS-D, zero failure rate $$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$3, and $$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$4 ms on RTX 3090	continuous, hole-free overlap boundaries
SeamlessNeRF (Gong et al., 2023)	works across Naïve NeRF, Instant-NGP, DirectVoxGO, and TensoRF; removing $$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$5 or $$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$6 degrades stitching	boundary pinning and gradient propagation are jointly necessary
Flow Planner (O'Mahoney et al., 12 May 2025)	stitching error $$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$7; Transformer with direct conditioning $$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$8; FP + split avoids obstacles up to four times as large	local receptive field and splitting improve continuous trajectory stitching
StabStitch++ (Nie et al., 8 May 2025)	$$\partial B_i \leftarrow \left\{\mathbf{x} \;\middle|\; S(\mathbf{x})=1 \ \text{and}\ \sigma_i(\boldsymbol{M}_i \mathbf{x}) > T_{\text{th}} \right\},$$9 ms per frame at $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$0	online spatial stitching and temporal stabilization are jointly feasible
Self-stitching microstructures (Ji et al., 2023)	mask-region accuracy exceeds $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$1; experiments use $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$2	periodic boundary repair can be learned from the structure itself
CPS compression (Zhang et al., 24 Feb 2025)	BD-rate $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$3 on Kodak and $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$4 on DIV8k relative to HEVC HM16.0; less than half of computing resources of existing models	seam-free block-wise compression with reduced memory/computation
Geometric 4D Stitching (Park et al., 11 May 2026)	under $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$5 minutes on a single NVIDIA RTX 5090 GPU per one-step scene expansion	explicit 4D stitching is practical for iterative expansion

The fluid literature uses different evidence. Instead of PSNR-like measures, the emphasis is on leakproofness, preservation of narrow passages, and qualitative scenarios such as the sealed bunny, the maze with thin passages, flexible heart valves, and parachutes with and without tears. The method’s correctness is expressed in terms of connectivity and discrete flux balance rather than image similarity (Panuelos et al., 3 Feb 2026).

The CAD and inverse-problem literatures are similarly structural. ABC-surfaces prove $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$6, $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$7, or $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$8 boundary behavior under stated conditions on ribbon consistency and contact order, while “Stitching Data” proves identifiability of the isometry class rather than reporting a benchmark metric (Martin et al., 2020, Meyerson, 2020).

6. Limitations, misconceptions, and current directions

A recurring misconception is that continuous boundary stitching is equivalent to local seam hiding. The cited work repeatedly rejects that view. SeamlessNeRF reports that removing $$\mathcal{L}_{\text{color}} = \sum_{\mathbf{x} \in \partial B_i} \left\| \mathbf{c}^M_i(\mathbf{x},\mathbf{d}) - \mathbf{c}_1(\mathbf{x},\mathbf{d}) \right\|_2^2.$$9 leads to oversmoothed or distorted textures, while using only gradient loss fails to align appearances; it also notes limitations from lighting/material mismatch, geometric misalignment, large style gaps, and compute cost (Gong et al., 2023). REwarp reports that very large parallax or complex depth variations can force unnatural bending elsewhere, and sequential multi-image stitching can accumulate distortions even when continuity is preserved (Kim et al., 2023).

Exactness-oriented methods face different bottlenecks. The stitched Voronoi fluid solver currently recomputes the Voronoi diagram from scratch each step, consuming about $$\mathcal{L} = \mathcal{L}_{\text{color}} + \lambda\,\mathcal{L}_{\text{grad}},$$0 of runtime, and the authors identify proper 3D visibility-constrained Voronoi construction, divergence-free interpolation on nonconvex cells, and monolithic implicit coupling as open directions (Panuelos et al., 3 Feb 2026). Geometric 4D Stitching depends on the quality of NVS and the depth estimator; extremely thin or complex occlusions can remain difficult, and generated unrevealed regions are not metrically accurate real-world geometry (Park et al., 11 May 2026).

Temporal and learning-based methods trade continuity against other desiderata. StabStitch++ notes challenges in low-texture or low-light overlaps, downsampling-induced loss of fine parallax cues, and residual instability under extreme motion (Nie et al., 8 May 2025). Flow Planner explicitly identifies a trade-off between action noise, which is needed for robust stitching, and smoothness, which must later be restored by filtering; it also observes that transformers with global receptive field do not support stitching well in this setting (O'Mahoney et al., 12 May 2025). CPS, by contrast, requires padding-free operations and overlap determined by receptive-field analysis; its guarantee is architectural rather than generic (Zhang et al., 24 Feb 2025).

The CAD and inverse-problem strands show that continuity can be exact but structurally constrained. ABC-surfaces can remain within standard CAD/CAM functionality and file formats, but the resulting NURBS degree can become high, and reentrant corners and more general topologies are left for future work (Martin et al., 2020). The geodesic “Stitching Data” theorems require collarability, and the delayed collision route additionally requires semi-nontrapping and the property that the manifold confirms intersections (Meyerson, 2020).

The cross-domain lesson is therefore precise rather than metaphorical: successful continuous boundary stitching requires more than matching interface values. It requires a representation in which the boundary condition is explicit, the interior extension mechanism is well posed, and topology or differential structure is preserved at the same time.