A linear algorithm for Brick Wang tiling

Derouet-Jourdan, Alexandre; Kaji, Shizuo; Mizoguchi, Yoshihiro

doi:10.1007/s13160-019-00369-z

Cited by 3 publications

(17 citation statements)

References 13 publications

(18 reference statements)

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“…A method to create brick wall textures using the brick Wang tiles is introduced in A. Derouet-Jourdan et al [1] and studied further in [2,3]. In this method, each tile represents how the corners of four bricks meet.…”

Section: Example: Brick Wang Tilesmentioning

confidence: 99%

“…In [2], an algorithm to give a valid Wang tiling with W for any planar region which contains a cycle and with any boundary condition. In this paper, we give a different approach to give a brick pattern for a rectangular region in the plane using our dappled tiling algorithm.…”

Section: Example: Brick Wang Tilesmentioning

confidence: 99%

“…Figure 5. A brick Wang tile and a 3 × 3-tiling A problem of the previous algorithms in [1,2] is that it sometimes produces long traversal edges; horizontally consecutive occurrence of tiles with c 1 = c 3 or vertically consecutive occurrence of tiles with c 2 = c 4 . These are visually undesirable (see Fig.…”

Section: Example: Brick Wang Tilesmentioning

confidence: 99%

“…We overcome this problem by our L-dappled Figure 6. Brick Wall Patterns.Left: produced by the algorithms in [1,2], Middle: same as Left with emphasis on long traversal lines, Right: produced by our new algorithm tiling algorithm. First, we divide W into two classes W = W 0 W 1 , where W 0 consists of those with c 1 = c 3 and W 1 consists of those with c 2 = c 4 .…”

Section: Example: Brick Wang Tilesmentioning

confidence: 99%

“…Therefore, we consider an algorithm to produce an L-dappled tiling which takes a user specified (not necessarily L-dappled) tiling as input so that the user has some control over the output. We also discuss a concrete applications with the Brick Wang tiles ( [1,2,3]) in §4, and with flow generation in §5. Remark 1.5.…”

Section: Introductionmentioning

confidence: 99%

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Dappled Tiling

Kaji¹,

Derouet-Jourdan²,

Ochiai³

2018

Mathematical Insights Into Advanced Computer Graphics Techniques

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We consider a certain tiling problem of a planar region in which there are no long horizontal or vertical strips consisting of copies of the same tile. Intuitively speaking, we would like to create a dappled pattern with two or more kinds of tiles. We give an efficient algorithm to turn any tiling into one satisfying the condition, and discuss its applications in texturing. Proposition 1.3. Let N = m 2 n 2 . There exist at least |T | N tilings which are L-dappled. Proof. We will create draughtboard tilings. For each cell (2k, 2l), choose any tile t ∈ T and put the same tile at (2k + 1, 2l + 1) (if it exists). Pick any t , t ∈ T \ {t} and put them at (2k + 1, 2l) and (2k, 2l + 1) (if they exist). One can see that for any (i, j) ∈ G m,n the tile at (i − 1, j) or (i − 2, j) is different from the one 2010 Mathematics Subject Classification. 52C20, 68U05.

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Section: Example: Brick Wang Tilesmentioning

confidence: 99%

Section: Example: Brick Wang Tilesmentioning

confidence: 99%

Section: Example: Brick Wang Tilesmentioning

confidence: 99%

Section: Example: Brick Wang Tilesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dappled Tiling

Kaji¹,

Derouet-Jourdan²,

Ochiai³

2018

Mathematical Insights Into Advanced Computer Graphics Techniques

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Bounded Wang tilings with integer programming and graph-based heuristics

Tyburec

Zeman

2023

Sci Rep

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Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging applications in materials engineering, we consider the bounded version of the tiling problem and offer four integer programming formulations to construct valid or nearly-valid Wang tilings: a decision, maximum-rectangular tiling, maximum cover, and maximum adjacency constraint satisfaction formulations. To facilitate a finer control over the resulting tilings, we extend these programs with tile-based, color-based, packing, and variable-sized periodic constraints. Furthermore, we introduce an efficient heuristic algorithm for the maximum-cover variant based on the shortest path search in directed acyclic graphs and derive simple modifications to provide a 1/2 approximation guarantee for arbitrary tile sets, and a 2/3 guarantee for tile sets with cyclic transducers. Finally, we benchmark the performance of the integer programming formulations and of the heuristic algorithms showing that the heuristics provide very competitive outputs in a fraction of time. As a by-product, we reveal errors in two well-known aperiodic tile sets: the Knuth tile set contains a tile unusable in two-way infinite tilings, and the Lagae corner tile set is not aperiodic.

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On bounded Wang tilings

Tyburec¹,

Zeman²

2022

Preprint

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Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging applications in materials engineering, we consider the bounded version of the tiling problem and offer four integer programming formulations to construct valid or nearly-valid Wang tilings: a decision, maximum-rectangular tiling, maximum cover, and maximum adjacency constraint satisfaction formulations. To facilitate a finer control over the resulting tilings, we extend these programs with tile-based, color-based, packing, and variable-sized periodic constraints. Furthermore, we introduce an efficient heuristic algorithm for the maximum-cover variant based on the shortest path search in directed acyclic graphs and derive simple modifications to provide a 1/2 approximation guarantee for arbitrary tile sets, and a 2/3 guarantee for tile sets with cyclic transducers. Finally, we benchmark the performance of the integer programming formulations and of the heuristic algorithms showing that the heuristics provides very competitive outputs in a fraction of time. As a by-product, we reveal errors in two well-known aperiodic tile sets: the Knuth tile set contains a tile unusable in two-way infinite tilings, and the Lagae corner tile set is not aperiodic.

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A linear algorithm for Brick Wang tiling

Cited by 3 publications

References 13 publications

Dappled Tiling

Dappled Tiling

Bounded Wang tilings with integer programming and graph-based heuristics

On bounded Wang tilings

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