Wang Tile Modeling of Wall Patterns

Derouet-Jourdan, Alexandre; Mizoguchi, Yoshihiro; Salvati, Marc

doi:10.1007/978-981-10-1076-7_9

Cited by 6 publications

(24 citation statements)

References 8 publications

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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%

“…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%

“…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%

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Dappled Tiling

Kaji¹,

Derouet-Jourdan²,

Ochiai³

2018

Mathematical Insights Into Advanced Computer Graphics Techniques

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“…If we use a brick Wang tile set for tiling, we can tile any rectangular region larger than 2 squares by 2 squares with any boundary color conditions. An inductive proof has been presented in a previous paper [5].…”

Section: Introductionmentioning

confidence: 98%

“…Since a Wang tile set is a simple system that produces aperiodic patterns, there are several applications in computer graphics [3,8,9]. In 2015, A. Derouet-Jourdan et al introduced a class of Wang tiles in computer graphics to model wall pattern textures [5], which is referred to as a brick Wang tile set. Each tile represents a possible way to connect four bricks, where a brick is an element of the wall.…”

Section: Introductionmentioning

confidence: 99%

Verification of a brick Wang tiling algorithm

Matsushima¹,

Mizoguchi²,

Derouet-Jourdan³

EPiC Series in Computing

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We have implemented a certified Wang tiling program for tiling a rectangular region using a brick Wang tile set. A brick Wang tile set is a special Wang tile set introduced in 2015 by A. DerouetJourdan et al. in computer graphics to model the texture of wall patterns. We have implemented a tiling algorithm using the Coq proof assistant and have presented its proof, which assures the existence of a tiling of any brick Wang tile set for a rectangle of any size. The essential points of the proof are the existence of a tiling for a 2 × 2 rectangle and a simple induction process. Since the brick Wang tile set is a set of tiles of infinite types, the proof is not straightforward and there are many conditional branches in the proof of the algorithm. The certification with Coq assures that there are no lack of conditions.

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