Placement Rule for Texels

Repetitions of various texels create periodicity, structure and symmetry in a texture. A special placement rule is derived to model spatial relationship between texels, i.e. how the image locations of multiple occurrences of a single texel or different texels are related in forming a texture pattern.

In a stochastic texture, texels appear with no explicit placement rules and have mostly weak spatial interrelations. As shown later, a synthetic texture of this class can be formed by sampling, repeating and randomly placing of the distinct training texels.

In contrast, a regular texture involves strict placement rules reflecting the underlying strong periodicity. Due to the assumption of translation-invariant pixel interaction, only translation symmetry is taken into account in estimating the placement rule for texels for textures in this class.

In a regular texture, there exists a underlying placement grid guiding the repetitions of texels. Each grid cell is a compact bounding parallelogram around a texel mask, specified by parameters $\Psi=(\theta_x,\theta_y, m, n)$, where $\theta_x$ and $\theta_y$ are guiding angles that give orientations of cell sides with respect to the image coordinate axes, and $m$ and $n$ are the side lengths, i.e. the mask's spans along the orientational directions. The bounding parallelogram could be decided using an invariant fitting algorithm, which is related to a canonical frame of the texel mask [103]. Figure 6.9 shows a hexagonal texel mask of texture D34, its bounding parallelogram and related parameters $\Psi$. In general, the grid with parallelogram cells represents any five possible types of translation lattice known in the theory of wallpaper groups for repeated patterns [89].

The placement grid is related to a coordinate system with non-orthogonal basis vectors, which leads to a hexagonal tessellation of the image plane [94]. Figure 6.10 shows a tessellation of texture D34 in line with a placement grid defined by the cell in Fig 6.9.

Figure 6.10 also shows that, with the tessellation, each texel is associated with a relative shift, $(\delta_{x},\delta_{y})$, of its centre $(x,y)$, with respect to the origin made coincident with the centre of the closest grid cell:

$$\begin{array}{ccc} \delta_{x}& = &(y \cdot \sin \theta_y + x ... ...\cdot \cos \theta_y - x \cdot \sin \theta_x)\bmod n \end{array}$$ (6.4.1)

The placement rule is to repeat each training texel at arbitrary locations having the same relative shift with respect to the placement grid. Due to an infinite number of absolute image locations related to a same relative shift, the rule reflects translation symmetry of regular textures. It also suggests that an arbitrarily large texture can be synthesised by expanding the image lattice in line with the infinite placement grid.