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plot of the maximum of these functions and the associated partition of the pixels into
three distinct materials via three bi-linear contours that meet at a common point.
2.1.2 Characterization of the Contours
The multi-material contours produced by this method have several important prop-
erties. First, the contours are continuous across cells sharing a common face. This
fact follows from the observation that two cells sharing a common face have the same
scalars and material indices on that face. Since the restriction of the tri-linear func-
tions used in defining the multi-material contour on this face depend only on the
scalar and material indices on that face, the multi-material contours must agree.
Piecewise Tri-Iinear Surfaces
Inside a single cell, the resulting contours are simply piecewise contours of various
tri-linear functions. To understand why, note that the contours bounding the region
associated with a material with index к are simply surfaces where the tri-linear func-
tion tk(x) and another tri-linear function tj(x) both reach the maximum. Therefore,
this contour is an iso-surface of the form
tfe(τ) = tj(x')
We have used a GPU-based, volume rendering approach to find the contour in
our implementation. However, it is possible to solve this classification problem using
polygonal methods such as Dual Contouring. Under Dual Contouring, we find a
point within the cell that best describes the intersection of all the pairwise tri-linear
surfaces. More formally, let M be the set of materials within a cell and let x be a
point inside the cell. Consider the function
EW = ∑ (it(τ)-f>W)2 (2.3)
j,kEM,j≠k