The printed diagram is a propagation subdivision mask: it shows the influence of one old vertex on several new vertices. Its entries do not all belong to a single averaging stencil. The old vertex is retained, and a new vertex is inserted at each incident face centre. Reading the contributions arriving at a target givesThus each new vertex-vertex averages its old value and its four edge neighbors, while each new face-vertex is the centroid of its four old corners. Both stencils are convex combinations, reproduce constants and commute with affine maps. The neighboring contributions go to old vertices; the diagonal contributions go to face centres. The propagation coefficients sum to two because one refinement doubles the number of lattice vertices, not because either stencil fails to sum to one.
For calculations, describe this quincunx subdivision usingThe refined square lattice is rotated by and its spacing is reduced by . In its integer coordinates, the one-step mask isIndeed, a unit step in these new coordinates is an old face-centre displacement, whereas a diagonal step reaches an old edge neighbor. The even parity class contains the central and four diagonal coefficients and sums to one; the odd class contains the four axial coefficients and also sums to one. This checks the two target stencils independently.
Write the Laurent generating function of the one-step subdivision mask asFor quincunx subdivision, . A second step gives . Setting and , its fine-grid offset is , since . Hence the binary mask is the convolution of with its -transformed copy:Expanding gives the centered binary propagation subdivision mask, with row and column offsets running from to :The numerator entries sum to , so , appropriate for a binary surface refinement. More specifically, each of the four parity classes sums to one after division by , checking constant reproduction at every type of new lattice point.
In a regular region, moving one control point adds its displacement multiplied by a translate of the twice-smoothed four-direction box spline. Measure parameter distances in original grid-edge units, with the affected control point at the origin. The convex hull of the binary mask offsets isAfter successive binary refinements the physical displacements are scaled by . Their possible accumulated offsets therefore form . The box spline description confirms that this zonotope is the actual support, not merely a loose enclosure. Equivalently,It is an octagon with vertices . Its enclosing square is six original grid intervals wide in each direction. Cutting four right triangles of area two from this square gives area parameter-square units. Outside this octagon the control-point displacement has no effect; its closure is the support, although the basis function can vanish on its boundary.
The regular influence footprint is this width-six, area-28 parameter octagon. For a nonplanar control mesh, the affected part of the spatial surface is the image of the octagon, not necessarily a planar octagon of the same metric size. Near an extraordinary subdivision vertex, use the actual refinement neighborhood instead of assuming a globally regular lattice footprint.
The factorization of the binary subdivision mask identifies eight box spline directions. Apart from the monomial that centers the support, it isThe four normalized factors correspond respectively to averaging in the directions , each with multiplicity two. The factor four is the determinant of the binary dilation in two dimensions. The centering monomial translates the resulting box spline and changes neither its polynomial degree nor its smoothness. Thus this is the twice-smoothed four-direction box spline, rather than a tensor-product bicubic B-spline.
A two-dimensional box spline built from spanning directions is piecewise polynomial of total degree at most . Here there are eight directions, counting multiplicities, so the regular pieces have total degree six. This is total degree, not separate degree six in each parameter.
For the box spline smoothness criterion, let be the smallest number of directions whose removal leaves a set that does not span the plane. In this direction multiset, a line can contain at most two of the eight vectors, since each of the four distinct directions appears twice. At least six vectors must therefore be removed, and removing the six outside one chosen direction does achieve loss of rank. Hence , givingIn particular all fourth partial derivatives are continuous across the piece boundaries; generic fifth derivatives need not be. The box spline criterion and this degree-six four-direction example are documented in the primary paper 4–8 Subdivision. The factor count also explains the large support: the extra smoothing comes from repeating all four directional averaging factors. This regular-grid calculation does not establish at an extraordinary subdivision vertex.
Choose coefficients using the local subdivision matrix, not just by copying a pleasant-looking regular stencil. Because the grids alternate, form the matrix for a complete two-step cycle around the valence- extraordinary subdivision vertex, retaining a large enough neighborhood that it maps into itself. The three local coefficients enter this matrix through the specified vertex and face rules.
First require constant reproduction and commutation with affine maps: every target stencil must sum to one, and the valence-four rules must reduce to the regular ones already found. A propagation mask need not itself sum to one. For example, with an old-vertex target rule , normalization requires ; a face-centroid target rule separately has corner weights summing to one. If face weights depend on incident valences, impose their sum on each actual face, rather than incorrectly using a propagation-column normalization. Rotational and reflection symmetry around the vertex should preserve an isotropic treatment of its sectors. Nonnegative coefficients are desirable for convex hull bounds and stability, although they are not by themselves a smoothness proof.
For convergence, seek a simple eigenvalue one corresponding to the constant eigenvector, with all remaining eigenvalues of modulus below one. For a well-defined tangent plane, seek two independent, semisimple tangent modes with a common positive subdominant eigenvalue:The angular modes should be the first sine and cosine modes around the ring. On the regular lattice the full two-step tangent scale is , which provides a useful consistency check. These requirements suppress persistent unwanted modes and ensure that generic first-order geometry is dominated by a two-dimensional tangent space.
Then check that the characteristic map of a subdivision surface built from those tangent eigenvectors is regular and injective on the punctured neighborhood. A dominant tangent pair together with a regular, nonfolding characteristic map is the crucial target; eigenvalues alone are insufficient. This check rules out folds or vanishing tangent area that a spectrum cannot detect. For reasonable curvature behavior, also try to keep second-order modes at or below in modulus, with equality and appropriate mode structure when nonzero finite curvature is desired. Avoid negative or complex modes that produce visible alternating ripples, excessive shrinkage and strongly anisotropic shape response.
These are selection criteria, not a uniquely determined triple of coefficients. The disappearance of extraordinary faces after one step simplifies the subsequent local matrix, but the extraordinary vertex remains and its smoothness still needs these tests. In particular, the regular result cannot simply be assigned to arbitrary valence.
Articles by others on the same topic
There are currently no matching articles.