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Modeling and Simulation

The J1a triangulation: an adaptive triangulation in any dimension

Profile image of Marcelo SiqueiraMarcelo Siqueira

2006, Computers Graphics

Abstract

Spatial sampling methods have acquired great popularity due to the number of applications that need to triangulate portions of space in various dimensions. One limitation of the current techniques is the handling of the final models, which are large, complex and need to register neighborhood relationships explicitly. Additionally, most techniques are limited to Euclidean bidimensional or tridimensional spaces and many do not handle well adaptive refinement. This work presents a novel method for spatial decomposition based on simplicial meshes (the J a 1 triangulation) that is generally defined for Euclidean spaces of any dimension and is intrinsically adaptive. Additionally it offers algebraic mechanisms for the decomposition itself and for definition of neighbrs that allow to recover all the information on the resulting mesh via a set of rules. This way it is possible to balance the cost of storage and manipulation by calculating the needed information instead of storing it. Results additionally show good quality meshes with efficient calculation.

Figures (17)
Figure 1: Examples of k—dimensional faces in R?
Figure 1: Examples of k—dimensional faces in R?
If an 7-block is not adjacent to (i+ 1)-blocks then it is called basic block. Otherwise, it is called transition block. Transition blocks are responsible for the connection among basic blocks in different levels of refinement. Figure 2 shows a two-dimensional grid containing blocks in three levels of refinement. Figure 2: Basic (white) and trasition (gray) blocks in a Level 2 refinement of grid.
If an 7-block is not adjacent to (i+ 1)-blocks then it is called basic block. Otherwise, it is called transition block. Transition blocks are responsible for the connection among basic blocks in different levels of refinement. Figure 2 shows a two-dimensional grid containing blocks in three levels of refinement. Figure 2: Basic (white) and trasition (gray) blocks in a Level 2 refinement of grid.
Figure 3: Labeling the faces of a standard hypercube. (m — k)-faces, 1 < k < m, are derived from the labels of the (m — 1)-faces in accordance with their intersections. For example, in Figure 3, the highlighted vertex and edge are labeled {—2,1,3} and {1,2} respectively as they are generated by the intersection of the 2-faces {—2}, {1}, {3} and {1}, {2} respectively. Note that the label of a i-face is a set of numbers without ordering, i.e., the label {1,2} can also be represented by {2, 1}.
Figure 3: Labeling the faces of a standard hypercube. (m — k)-faces, 1 < k < m, are derived from the labels of the (m — 1)-faces in accordance with their intersections. For example, in Figure 3, the highlighted vertex and edge are labeled {—2,1,3} and {1,2} respectively as they are generated by the intersection of the 2-faces {—2}, {1}, {3} and {1}, {2} respectively. Note that the label of a i-face is a set of numbers without ordering, i.e., the label {1,2} can also be represented by {2, 1}.
Figure 4: Simplices generated from a) s; = (1,1) and m = (1,2); b) s; = (1,1) and mq = (2,1). In order to illustrate the construction above, suppose D? = [—1,1] x [—1,1], S$? = {(1,1), (-1, 1), (1, -1), (-1, -1)}, 71 = (1,2), and B? = {(1,0), (0,1)}. If s; = (1,1) then a 2-simplex in D? can be defined by the vertices vg = (0,0), v1 = vo + 1.(1,0) = (1,0), and v2 = vl + 1.(0,1) = (1,1), as shows Figure 4a). A permutation of the vectors in B? generates the 2-simplex given by the vertices v9 = (0,0), v1 = vo + 1.(0,1) = (0,1), and va = vy + 1.(1,0) = (1,1) as in Figure 4b). It is not difficult to see that each s; € S? jointly with all permutations of B? (given by II?) generates the simplices that triangulate a quadrant of D?.
Figure 4: Simplices generated from a) s; = (1,1) and m = (1,2); b) s; = (1,1) and mq = (2,1). In order to illustrate the construction above, suppose D? = [—1,1] x [—1,1], S$? = {(1,1), (-1, 1), (1, -1), (-1, -1)}, 71 = (1,2), and B? = {(1,0), (0,1)}. If s; = (1,1) then a 2-simplex in D? can be defined by the vertices vg = (0,0), v1 = vo + 1.(1,0) = (1,0), and v2 = vl + 1.(0,1) = (1,1), as shows Figure 4a). A permutation of the vectors in B? generates the 2-simplex given by the vertices v9 = (0,0), v1 = vo + 1.(0,1) = (0,1), and va = vy + 1.(1,0) = (1,1) as in Figure 4b). It is not difficult to see that each s; € S? jointly with all permutations of B? (given by II?) generates the simplices that triangulate a quadrant of D?.
It is important to note that the vertices [ug,...,Upn,..-,Um] of each simplex o generated from the scheme above are sorted in a such way that {vp,...,Um} generates the face of o that is contained in a refined face of D™. Let D™ be a transition block whose labels of its refined faces are in ®. For each permu- tation 7; € I’ and s; € S™ we can compute an h such that the vertices of the simplices in D™ can be specified as follows: Let D™ be a transition block whose labels of its refined faces are in ®. For each permu-
It is important to note that the vertices [ug,...,Upn,..-,Um] of each simplex o generated from the scheme above are sorted in a such way that {vp,...,Um} generates the face of o that is contained in a refined face of D™. Let D™ be a transition block whose labels of its refined faces are in ®. For each permu- tation 7; € I’ and s; € S™ we can compute an h such that the vertices of the simplices in D™ can be specified as follows: Let D™ be a transition block whose labels of its refined faces are in ®. For each permu-
Figure 5: Refinement of D™. a) Transition block with a two-dimensional and a one- dimensional refined face. b) Two tetrahedrons in a transition block. c) The boundary of the complete refinement of the transition block in a).
Figure 5: Refinement of D™. a) Transition block with a two-dimensional and a one- dimensional refined face. b) Two tetrahedrons in a transition block. c) The boundary of the complete refinement of the transition block in a).
Figure 6: Labeling of i-blocks. a)0-refinement in R?. b)1-refinement in R?
Figure 6: Labeling of i-blocks. a)0-refinement in R?. b)1-refinement in R?
Figure 7: Pivoting by a vertex. Let D™ be a basic block in G and o = [vo,..., Um] a simplex of the J? decomposition o D™. It is not difficult to see that the pivoted of o by vo is always in an adjacent block to D”™ and that the pivoted of o by the vertices v1,...,Um are all contained in D™. Based on that fact, we can generate a set of rules that, from the labels of a simplex, identify its pivoted. For example, consider the simplex o in Figure 7 generated from 7; = (2,1 and sz = (1,—1). The pivoted of o by the 2nd vertex (v1) is obtained by maintaining the label of the block (s;) and the level of refinement unchanged, only changing 7; to (1,2 (see Figure 7). If the pivoting rule is defined as a map piv : N4 > N*, that maps labels to labels then we can write:
Figure 7: Pivoting by a vertex. Let D™ be a basic block in G and o = [vo,..., Um] a simplex of the J? decomposition o D™. It is not difficult to see that the pivoted of o by vo is always in an adjacent block to D”™ and that the pivoted of o by the vertices v1,...,Um are all contained in D™. Based on that fact, we can generate a set of rules that, from the labels of a simplex, identify its pivoted. For example, consider the simplex o in Figure 7 generated from 7; = (2,1 and sz = (1,—1). The pivoted of o by the 2nd vertex (v1) is obtained by maintaining the label of the block (s;) and the level of refinement unchanged, only changing 7; to (1,2 (see Figure 7). If the pivoting rule is defined as a map piv : N4 > N*, that maps labels to labels then we can write:
Let o = [vo,..-,Um] be a simplex and N = H~1(0) be an implicit submanifold of M. In a similiar way as described in Section 2 we can define a linear aproximation for N in o as In this section we describe the adaptive mechanism implemented in code. As already mentioned, any adaptive strategy could be adopted, as for example curvature [22]. Here we choose to present a scheme for refinement based on submanifolds.
Let o = [vo,..-,Um] be a simplex and N = H~1(0) be an implicit submanifold of M. In a similiar way as described in Section 2 we can define a linear aproximation for N in o as In this section we describe the adaptive mechanism implemented in code. As already mentioned, any adaptive strategy could be adopted, as for example curvature [22]. Here we choose to present a scheme for refinement based on submanifolds.
Figure 9: Cylinder iso-surface extraction: a) initial grid 5 x 5 x 5 without refinement; b) two levels of a local refinement; c) marching cubes in a 40 x 40 x 40 grid.
Figure 9: Cylinder iso-surface extraction: a) initial grid 5 x 5 x 5 without refinement; b) two levels of a local refinement; c) marching cubes in a 40 x 40 x 40 grid.
Figure 10: Decomposition of a surface of a heart
Figure 10: Decomposition of a surface of a heart
Figure 11: Four levels of refinement around the sub-manifolds. a) Shading b) Mesh
Figure 11: Four levels of refinement around the sub-manifolds. a) Shading b) Mesh
Figure 12: a),b) and c) approximations with 1,2 and 3 levels of refinement; d),e) and f) meshes of a),b) and c) respectively.
Figure 12: a),b) and c) approximations with 1,2 and 3 levels of refinement; d),e) and f) meshes of a),b) and c) respectively.
Table 1: Computational times (seconds) to process the models in Figures 10, 12, and 13.
Table 1: Computational times (seconds) to process the models in Figures 10, 12, and 13.
Figure 13: A projection of S$! x S!
Figure 13: A projection of S$! x S!
Figure 14: Curvature based refinement of an implicit model, supporting correct detection of object topology. a) No refinement b)Two levels of refinement based on curvature radius
Figure 14: Curvature based refinement of an implicit model, supporting correct detection of object topology. a) No refinement b)Two levels of refinement based on curvature radius
References
References

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