3. Rules

• Setup

  Make Boxels (2d Voxels) by drawing a line from each data reading to its neighboor such that you form a system of boxes with the readings at the verticies of the boxes.

Later we will demonstrate how to do this on non-rectalinear or irregular grids, such as a delaunay triangularization. x

• Boxelize

• Two hit rule

• Four hit rule

  • Find Critical Value

    Disambiguation Rule:

    For each 4-hit face we linearly interpolate $\delta$ across each edge. Now interpolate the position of the point p, called the critical point, such that both the vertical and horizontal lines (with respect to the two coordinate directions on the face) that pass through this point intersect identical values on the opposite edges. Interpolate the value $c = \delta(p)$ of this point, called the disambiguation value, and we extend the domain of $\delta$ to include p (we call this the extended $\delta$). For specific $\tau$, if $\tau > c$ regard the Lows as adjacent (knit High), else regard the Highs as adjacent (knit Low).

    See figure 4 for clarification.

    Assume without loss of generality that the face is in the x, y plane and that the origin of the face is x=i y=j. Let $\delta (i, j) = A$, $\delta (i+1, j) = B$, $\delta (i+1, j+1) = C$, $\delta (i, j+1) = D$.

    he disambiguation value $\delta(i+x_0, j+y_0)$ is A+ (D-A)x₀.

    For example the disambiguation point of figure 20 is x = 1/3 , y= 1/2 and the disambiguation value is 60.

  • Find Critical Point

    The location of the critical point is given by i + x₀, j + y₀, where x₀= (A-B) / ((A+C) - (B+D)) and y₀= (A-D) / ((A+C) - (B+D)). T

  • Knit High Above, Knit Low Below

    Another way to see this is to draw four new boxels inside the 4 hit boxel, and note that the internal edges have NO gradient (difference along their edges). Therefor since each boxel must have two hits, the hit must be in the adjacent edge.