Overlap-Aware Tiling Methods

• Overlap-aware tiling is a framework that defines and manages tile overlaps using mathematical constraints and algorithmic strategies to enable precise partitioning and controlled assembly.
• It employs methods such as sliding-window extraction, weighted blending, and graph-based selection to aggregate overlapping data while mitigating boundary artifacts and runaway growth.
• The approach is practically applied in remote sensing, high-resolution imaging, and object detection, delivering measurable improvements in accuracy, speed, and artifact-free outputs.

Overlap-aware tiling encompasses a spectrum of mathematical, algorithmic, and applied strategies in which the design, placement, and aggregation of tiles must explicitly control, exploit, or avoid overlaps between tile instances. Overlap is crucial both as a mathematical constraint (e.g., ensuring partitions without redundant covering) and as a tool to enhance inference, computation, termination proofs, or aperiodicity in physical, computational, and geometric systems. The topic spans discrete, combinatorial tiling theory, hierarchical assembly models, computational algorithms for segmentation and image processing, neural optimization frameworks, symbolic dynamics, and geometric substitution systems.

1. Mathematical Foundations and Definitions

Overlap-aware tiling involves various formal models depending on the domain:

• Discrete Tiling: Tiles (usually geometric shapes or labeled squares) are placed to cover a domain such as $\mathbb{Z}^2$ or $\mathbb{R}^d$, where overlaps are either forbidden (to ensure partitioning) or managed (to resolve ambiguities, as in overlapping substitutions) (Akiyama et al., 2024).
• Hierarchical/TAS Models: In hierarchical tile assembly systems (TAS), an assembly is a partial function $\alpha: \mathbb{Z}^2 \rightharpoonup T$ where $T$ is the set of tile types. Overlap is formalized via translations: $\alpha$ and $\alpha+\vec v$ overlap if their domains intersect; if they overlap consistently (identical tile types in the overlap), unbounded growth is possible (Chen et al., 2014).
• Substitution Systems: Overlapping substitutions employ an expansive map $\phi$ and substitution rules $\rho$ that can generate patches whose interiors can overlap. Consistency is defined by the absence of contradictory overlaps after iteration, quantified using the substitution matrix $M$ with potentially non-integer entries reflecting weighted overlaps (Akiyama et al., 2024).
• Segmentation and Image Processing: The input (e.g., an image $I$) is partitioned into overlapping tiles, each processed independently. Overlaps guarantee coverage of features near tile borders, mitigating boundary artifacts. Mathematical aggregation of predictions from overlapping regions uses weighted averages (Jia et al., 10 Jul 2025, Kwon et al., 7 Oct 2025).
• String Rewriting: In string rewriting systems, tiles are $k$-length factors over an extended alphabet. Overlap closures manage the concatenation and interaction between tiles under rewrite rules, which is critical for local and relative termination proofs (Geser et al., 2020).

2. Algorithms and Aggregation Mechanisms

Overlap-aware tiling algorithms are designed for computational efficiency, precision, and avoidance of artefacts or unintended growth:

• Sliding-window Overlapping: Tiles are extracted with prescribed stride $s$ and overlap $o$, e.g., for a tile of size $s$ and overlap ratio $o$, the stride is $t_s = \lceil s(1-o) \rceil$ (Jia et al., 10 Jul 2025, Kwon et al., 7 Oct 2025). Hierarchical schemes deploy multiple scales to enable both fine and coarse context aggregation (as in HOTA).
• Fusion/Aggregation: Outputs from all overlapping tiles are fused by computing, for pixel $(x,y)$,

$$P_{\mathrm{final}}(x,y) = \frac{1}{\sum_i w_i(x,y)} \sum_i w_i(x,y)\,P_i(x,y)$$

where $w_i(x,y)$ are tile-specific weights and $P_i(x,y)$ are local predictions (Jia et al., 10 Jul 2025).

• Weighted Blending: For image super-resolution and high-resolution editing, overlapping regions are blended using separable hat-windows or similar weighting schemes. In MobilePicasso’s ACPT, tiles are merged via

$$\hat{I}(x,y) = \frac{\sum_k w_k(x,y)\,I_k(x,y)}{\sum_k w_k(x,y)}$$

with weights defined to smooth overlaps and eliminate seams (Kwon et al., 7 Oct 2025).

• Automaton Completion in String Rewriting: Sparse tiling and overlap closures are constructed via automaton completion, iteratively adding states and transitions to encode all admissible overlaps and ensure termination or relative termination (Geser et al., 2020).
• Graph-based Selection: In neural optimization—such as TilinGNN—nodes represent candidates and edges encode overlaps and contacts. Two-branch GNNs process neighbor and overlap connections independently, and training loss terms penalize overlaps while rewarding coverage (Xu et al., 2020).

3. Termination, Consistency, and Theoretical Guarantees

Overlap, when unmanaged, can cause undesirable outcomes such as infinite growth, non-uniqueness, or contradictory assemblies:

• Runaway Growth in Hierarchical TAS: If an assembly admits a nontrivial translation that overlaps itself consistently, arbitrarily large assemblies can be produced, violating the design goal of finite structures. Detection algorithms iterate over all possible translations within bounded radii to check for consistency in overlapped regions (Chen et al., 2014).
• Consistency in Overlapping Substitutions: The open set condition (OSC), linear GIFS conditions, and graph-directed IFS constructions provide sufficient criteria to guarantee global consistency in overlapping substitution tilings (Akiyama et al., 2024). Perron-Frobenius theory determines patch frequencies in weighted substitutions and ensures expansion constants are algebraic integers under mild conditions.
• Relative Termination in String Rewriting: Overlap closure is essential for transforming relative string rewriting problems into automata amenable to termination analysis. Theorems guarantee that termination or relative termination holds iff the sparse tiling reflects all possible overlaps correctly (Geser et al., 2020).
• Aperiodicity via Overlapping IFS Tops: By applying inverse compositions along top addresses in overlapping IFS, one obtains tilings (often aperiodic) that resolve ambiguities in overlaps and guarantee full coverage by stabilized limit shapes (Barnsley et al., 16 Apr 2025).

4. Practical Applications and Empirical Results

Overlap-aware tiling strategies see extensive use in computational and applied contexts:

Context	Overlap Function	Quantitative Impact
Remote sensing segmentation	Multi-scale overlap	IoU improved by 4–6 pp, boundary error <0.5 m (Jia et al., 10 Jul 2025)
High-res image processing	ACPT, blending	99.7% PSNR recovery, 3.12× latency reduction (Kwon et al., 7 Oct 2025)
GPU pipelines (PolyMage-GPU)	Warp-overlapped, hybrid	1.65×–2.25× speedup vs Halide (Jangda et al., 2019)
Dense object detection	Overlap-induced group evidence	Recall gain 0.685→0.778, F1=0.669, 0.095s/img (Xiao, 13 Sep 2025)
Tiling optimization (TilinGNN)	Graph overlap branch	90–98% coverage, linear runtime in #candidates (Xu et al., 2020)
Hierarchical TAS	Avoiding consistent overlap	Guarantees finite, unique assembly (Chen et al., 2014)
IFS/monotiles	Overlap-resolved top tilings	Aperiodic monotile tilings, infinite shape diversity (Barnsley et al., 16 Apr 2025)

Selected empirical results:

• HOTA (Hierarchical Overlap-Tiling Aggregation) in flood mapping achieves IoU improvement of 6.59 pp over U-Net baseline and 3.84 pp over SegFormer baseline (Jia et al., 10 Jul 2025).
• ACPT in MobilePicasso recovers nearly all top quality (PSNR, SSIM) of naive 50% overlap while delivering a >50× runtime reduction and minimal memory overhead (Kwon et al., 7 Oct 2025).
• PolyMage-GPU’s hybrid and warp-overlap tiling reduces synchronization stalling and global memory traffic, yielding absolute speedups over best-in-class DSL kernels (Jangda et al., 2019).
• In UAV-based dense object detection, overlap-aware tiling together with clustering gates boosts recall substantially with manageable precision trade-off; semantic gating dominates post-processing cost (Xiao, 13 Sep 2025).
• In TilinGNN, explicit overlap penalties and two-branch GCN structure are essential: ablation studies show coverage loss or increased overlaps if these components are disabled (Xu et al., 2020).

5. Symbolic Dynamics, Substitution Tilings, and Overlap Coincidence

Overlap-aware tiling is central to modern developments in symbolic dynamics and substitution tiling theory:

• Overlap Coincidence: For self-affine substitutions, overlap coincidence is equivalent to pure discrete dynamical spectrum. It requires that for every geometric overlap, iteration under substitution eventually produces an exact tile-level coincidence (Akiyama et al., 2014).
• Strong and Simultaneous Coincidence: Under trivial height group conditions, overlap coincidence implies strong coincidence (for every pair of prototiles, iterates share at least one tile) and simultaneous coincidence (all tile types share a common tile under substitution iterates). These concepts control eigenstructure and quasicrystal diffraction properties (Akiyama et al., 2014).
• Weighted and Overlapping Substitutions: Overlapping substitutions generalize classical tiling rules to allow controlled tile intersections. The substitution matrix may have non-integer entries, reflecting partial overlap weights, and PF theory yields patch frequencies and scale invariants (Akiyama et al., 2024).
• IFS and Monotile Aperiodicity: Overlap-resolved inverse compositions from attractor "top" partitions yield aperiodic, non-periodic tilings by monotiles (e.g. the hat tile), with the limit tiles often having fractal boundaries (Barnsley et al., 16 Apr 2025).

6. Design Principles, Pitfalls, and Theoretical Implications

Correctly managing overlap is critical for:

• Avoidance of Uncontrolled Assembly Growth: Consistent overlap between a tile and its translation in hierarchical systems can trigger pathologically large or non-unique assemblies. Systems producing unique finite assemblies must forbid such overlaps (Chen et al., 2014).
• Artifact-Free Image Partitioning: In image segmentation and super-resolution, tile overlaps must be balanced to avoid seam artifacts, and blending schemes must ensure coverage and feature continuity (Kwon et al., 7 Oct 2025, Jia et al., 10 Jul 2025).
• Termination Proofs: Overlap closures in string rewriting systems are crucial for rigorous proofs of local and relative termination. Languaging and automata completion encode all possible overlaps to ensure soundness (Geser et al., 2020).
• Consistency in Substitution Tilings: Consistency criteria (open-set, linear GIFS, finite-type, OSC) are required in overlaps of substitution rules to avoid contradictory or locally consistent but globally inconsistent patches (Akiyama et al., 2024).

A plausible implication is that overlap-aware design in both discrete and continuous tiling spaces is foundational for controlling complexity, aperiodicity, and correctness across combinatorial, symbolic, and computational classes.

7. Future Directions and Open Problems

Research in overlap-aware tiling continues to drive advances in theoretical mathematics, algorithmics, and applications:

• Extension to overlapping substitutions beyond the Pisot family and towards heavy overlaps, e.g., Bernoulli convolutions or coverings (like Gummelt’s decagon) (Akiyama et al., 2024).
• Complexity characterization and synthesis of overlap-free tile sets for arbitrary shapes, including potential PSPACE-completeness results for detecting repetitious assemblies (Chen et al., 2014).
• Generalization into higher dimensions, robustness to spurious attachments, and error-tolerant hierarchical systems.
• Integration of temporal cues and lightweight embedding for real-time semantic gating in object detection (Xiao, 13 Sep 2025).
• Unification of symbolic and geometric frameworks, e.g., linking Delone multi-set inflation symmetry to overlapping substitution tilings (Akiyama et al., 2024).
• Algorithmic optimizations for overlap-aware fusion, parallel tile evaluation, and feature caching in neural segmentation and GPU pipelines (Jia et al., 10 Jul 2025, Jangda et al., 2019).
• Elucidation of equivalences and converse results in substitution tiling dynamics, such as proving strong $\Rightarrow$ overlap coincidence in higher dimensions (Akiyama et al., 2014).

The field remains active at the intersection of computational mathematics, symbolic dynamics, geometric analysis, computer vision, and high-performance computing, with overlap-aware tiling as a central unifying theme.