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Figure from:Multiresolution isosurface extraction with adaptive skeleton climbing

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Figure 4: (a): The lign is subdivided into length-maximal dikes. (b): The dikes visited when walking through the lign. Fig. 4(a) illustrates that a lign is subdivided into length- maximal dikes (shown in black in Fig. 4(b)). In this 9-voxel lign example, the lign is subdivided into 4 dikes. The first two dikes are unit dikes, since the isosurface crosses both of them. Although the isosurface crosses the rest of the seg- ment only once, it is still subdivided into two dikes due to the binary edge constraint imposed by the binary tree orga- nization. Hence the subdivision may not be always minimal. But this restriction simplifies the merging process in the 2D adaptive skeleton climbing discussed in next section.

Figure 4 (a): The lign is subdivided into length-maximal dikes. (b): The dikes visited when walking through the lign. Fig. 4(a) illustrates that a lign is subdivided into length- maximal dikes (shown in black in Fig. 4(b)). In this 9-voxel lign example, the lign is subdivided into 4 dikes. The first two dikes are unit dikes, since the isosurface crosses both of them. Although the isosurface crosses the rest of the seg- ment only once, it is still subdivided into two dikes due to the binary edge constraint imposed by the binary tree orga- nization. Hence the subdivision may not be always minimal. But this restriction simplifies the merging process in the 2D adaptive skeleton climbing discussed in next section.

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Figure 1: Overview of adaptive skeleton climbing.
Figure 1: Overview of adaptive skeleton climbing.
Figure 2: Basic 1D data structures.
Figure 2: Basic 1D data structures.
Figure 3: Binary tree organization of 1D voxel data. lign of 2” + 1 samples. For simplicity, let N = 2” for short hand. The occupancy array of a lign describes the presence of iso-points (1D analogy of 3D isosurface) on its dikes. With this occupancy array, we can accurately locate the position of iso-point and how the isosurface crosses the lign. Now, let us denote the voxel sample with value above or equal to the threshold (7) as e, and sample with value below 7 as o. Then the binary value of the :*" entry in the occupancy array means:
Figure 3: Binary tree organization of 1D voxel data. lign of 2” + 1 samples. For simplicity, let N = 2” for short hand. The occupancy array of a lign describes the presence of iso-points (1D analogy of 3D isosurface) on its dikes. With this occupancy array, we can accurately locate the position of iso-point and how the isosurface crosses the lign. Now, let us denote the voxel sample with value above or equal to the threshold (7) as e, and sample with value below 7 as o. Then the binary value of the :*" entry in the occupancy array means:
Figure 5: The 2D data structures. T g 2 he 1D data structures allow us to group voxels into length- maximal simple segments (dikes). Similarly, in the 2D, we want to group voxels to form size-maximal simple rectan- es. Consider an (N-+1)x(N-+1) farm of voxel samples, with N+1 horizontal and N-+1 vertical ligns, each with its own occupancy and simple dike arrays. First, let’s define the D terminologies and data structures. A strip (Fig. 5(a)) con- sists of two consecutive ligns. A plot (Fig. 5(b)) is analogous to the dike which consists of two consecutive dikes. Plots are also organized by a binary tree. Similarly a plot is simple if and only if its two dikes are also simple. Hence, we can define a simple plot array which is similar to the simple dike array. Since the shorter dike has a larger dike ID, the length-maximal simple plots can be easily found by
Figure 5: The 2D data structures. T g 2 he 1D data structures allow us to group voxels into length- maximal simple segments (dikes). Similarly, in the 2D, we want to group voxels to form size-maximal simple rectan- es. Consider an (N-+1)x(N-+1) farm of voxel samples, with N+1 horizontal and N-+1 vertical ligns, each with its own occupancy and simple dike arrays. First, let’s define the D terminologies and data structures. A strip (Fig. 5(a)) con- sists of two consecutive ligns. A plot (Fig. 5(b)) is analogous to the dike which consists of two consecutive dikes. Plots are also organized by a binary tree. Similarly a plot is simple if and only if its two dikes are also simple. Hence, we can define a simple plot array which is similar to the simple dike array. Since the shorter dike has a larger dike ID, the length-maximal simple plots can be easily found by
Figure 6: Length-maximal plots from consecutive ligns. Fig. 6 shows one such operation graphically. The calcu- lated plots are overlaid with the voxel samples in Fig. 6(b). Note that each plotis crossed at most twice by the isosurface.
Figure 6: Length-maximal plots from consecutive ligns. Fig. 6 shows one such operation graphically. The calcu- lated plots are overlaid with the voxel samples in Fig. 6(b). Note that each plotis crossed at most twice by the isosurface.
Figure 9: P lots are first merged to form rectangle. The rect- angle is then subdivided along y direction to satisfy the bi- nary edge constraint applied to the y direction. length and align to each other. A candidate rectangle is then formed. Since the binary edge constraint is also applied to the vertical direction, this rectangle is subdivided to form size-maximal padis (Fig. 9).
Figure 9: P lots are first merged to form rectangle. The rect- angle is then subdivided along y direction to satisfy the bi- nary edge constraint applied to the y direction. length and align to each other. A candidate rectangle is then formed. Since the binary edge constraint is also applied to the vertical direction, this rectangle is subdivided to form size-maximal padis (Fig. 9).
Figure 11: Storing the padi layout in layout arrays. Figure 10: Example result of running algorithm ASC2D.
Figure 11: Storing the padi layout in layout arrays. Figure 10: Example result of running algorithm ASC2D.
Figure 12: Generate iso-lines by a 16-entry table. Two am- biguous cases lead to subsampling.
Figure 12: Generate iso-lines by a 16-entry table. Two am- biguous cases lead to subsampling.
Figure 13: Bad ambiguity resolution by subsampling. The ambiguity with two diagonally opposite « comers can sometimes be resolved by subsampling at the center of the padi. However, wrong iso-lines will still be generated in some cases (Fig. 13). Where connectivity is crucial, software should warm the user of ambiguous cases and offer finer, more CPU-costly tools for local investigation. In many cases the warning is as useful to the surgeon, geologist or other user as any silently-attempted best guess by the software.
Figure 13: Bad ambiguity resolution by subsampling. The ambiguity with two diagonally opposite « comers can sometimes be resolved by subsampling at the center of the padi. However, wrong iso-lines will still be generated in some cases (Fig. 13). Where connectivity is crucial, software should warm the user of ambiguous cases and offer finer, more CPU-costly tools for local investigation. In many cases the warning is as useful to the surgeon, geologist or other user as any silently-attempted best guess by the software.
Figure 14: Data structures for the 3D algorithm. Considera (N+1) x (N+1) x (N+1) voxel sample grid. For gies the sake of discussions, we now define the 3D terminolo- and data structures. All terminologies are graphically illustrated by a3 x 3 x 3 volume in the Fig. 14(a). A farm contains (N+1) x (N-+1) voxels ona 2D grid. A slab, anal- ogous to a strip containing two consecutive ligns, consists of two consecutive farms. A brick in the slab has two match- ing padis in two consecutive farms as faces. A highrice is a rectangular box composed of stacked bricks.
Figure 14: Data structures for the 3D algorithm. Considera (N+1) x (N+1) x (N+1) voxel sample grid. For gies the sake of discussions, we now define the 3D terminolo- and data structures. All terminologies are graphically illustrated by a3 x 3 x 3 volume in the Fig. 14(a). A farm contains (N+1) x (N-+1) voxels ona 2D grid. A slab, anal- ogous to a strip containing two consecutive ligns, consists of two consecutive farms. A brick in the slab has two match- ing padis in two consecutive farms as faces. A highrice is a rectangular box composed of stacked bricks.
Figure 17: Merging bricks to form highrices Next, neighbor bricks merge to form highrices in 3D (Fig. 17), analogous to merging plots to form padis in the 2D case (Fig. 7). Again there is no unique merging mule (Fig. 17), and again we prefer a fast heuristic to a search for e suboptimal subdivision. For each size-maximal xy-brick on each slab, we stack the xy-bricks upward along z direction until no more simple bricks available. Then this temporary box is subdivided along z direction to fulfill the binary edge constraint (Fig. 18). During the highrice formation, the gen- erated highrices may overlap with each other or one may be enclosed by another. A ny enclosed highrice will be removed. Overlapping highrices are clipped.
Figure 17: Merging bricks to form highrices Next, neighbor bricks merge to form highrices in 3D (Fig. 17), analogous to merging plots to form padis in the 2D case (Fig. 7). Again there is no unique merging mule (Fig. 17), and again we prefer a fast heuristic to a search for e suboptimal subdivision. For each size-maximal xy-brick on each slab, we stack the xy-bricks upward along z direction until no more simple bricks available. Then this temporary box is subdivided along z direction to fulfill the binary edge constraint (Fig. 18). During the highrice formation, the gen- erated highrices may overlap with each other or one may be enclosed by another. A ny enclosed highrice will be removed. Overlapping highrices are clipped.
Figure 19: Sharing information between neighbor high- rices. Fig. 19(a) shows a larg e highrice next to a small one. The isosurface crosses the plane separating the two high- rices (Fig. 19(b)). If triang es are emitted for each highrice without the knowledge of their neighbors, gaps will appear in the generated triangular iso-lines generated on the mesh. This is because the linear highrice surfaces may not match each other geometrically (Fig. 19(c)), even though they are topologically correct. To prevent this mismatch, information must be shared between adjacent highrices.
Figure 19: Sharing information between neighbor high- rices. Fig. 19(a) shows a larg e highrice next to a small one. The isosurface crosses the plane separating the two high- rices (Fig. 19(b)). If triang es are emitted for each highrice without the knowledge of their neighbors, gaps will appear in the generated triangular iso-lines generated on the mesh. This is because the linear highrice surfaces may not match each other geometrically (Fig. 19(c)), even though they are topologically correct. To prevent this mismatch, information must be shared between adjacent highrices.
Figure 20: Once we reinitialize the layout arrays to store the 3D highrices’ layout, ASC2D can be run to generate padis that fit into the surface boundaries of both highrices.
Figure 20: Once we reinitialize the layout arrays to store the 3D highrices’ layout, ASC2D can be run to generate padis that fit into the surface boundaries of both highrices.
Figure 22: In each iteration, one triangle is emitted and one vertex is removed. Given an edge loop consisting of several vertices v;, we emit triangles as follows. In each iteration, three consecutive vertices vj, vig1 and vi42 are selected and one triangle is generated (Fig. 22). The vertex vi41 is then removed from the edge loop. The algorithm continues until only two ver tices are left.
Figure 22: In each iteration, one triangle is emitted and one vertex is removed. Given an edge loop consisting of several vertices v;, we emit triangles as follows. In each iteration, three consecutive vertices vj, vig1 and vi42 are selected and one triangle is generated (Fig. 22). The vertex vi41 is then removed from the edge loop. The algorithm continues until only two ver tices are left.
Figure 21: The six faces of a highrice are tiled with padis after information sharing.
Figure 21: The six faces of a highrice are tiled with padis after information sharing.
counts, but with different geometry (Fig. 23(b) and (c)). To generate a mesh that closely approximates the true isosur face, we make use of the gradient. We reject any triangle with planar normal vector 7, that largely deviates from the gradients g at three vertices. The deviation is measured by the dot product of 7, and g;. A threshold is used as a criteria. The threshold constraint will be relaxed if no triangle can be generated under current constraint. Figure 23: Triangulate the edge loop to emit triangles.
counts, but with different geometry (Fig. 23(b) and (c)). To generate a mesh that closely approximates the true isosur face, we make use of the gradient. We reject any triangle with planar normal vector 7, that largely deviates from the gradients g at three vertices. The deviation is measured by the dot product of 7, and g;. A threshold is used as a criteria. The threshold constraint will be relaxed if no triangle can be generated under current constraint. Figure 23: Triangulate the edge loop to emit triangles.
Figure 24: Graphical presentation of the results shown in Table 1. (a) Graph of triangle count. (b) Graph of CPU time. Table 1: Comparison of marching cubes (MC) and adaptive skeleton climbing (ASC), with block sizes N = 1,2,4,8, and marching cubes, in term of triangle count (A) and CPU time.
Figure 24: Graphical presentation of the results shown in Table 1. (a) Graph of triangle count. (b) Graph of CPU time. Table 1: Comparison of marching cubes (MC) and adaptive skeleton climbing (ASC), with block sizes N = 1,2,4,8, and marching cubes, in term of triangle count (A) and CPU time.
[Poston, Wong & Heng] Figure 25. Visual comparison of the effects of block size, for an algebraic surface, a landscape, and CT data for bones and blood vessels in the hea (a)-(e): Mathematical data set ”knoto4”. (f)-(j): Landscape data "Mt. Alps”.
[Poston, Wong & Heng] Figure 25. Visual comparison of the effects of block size, for an algebraic surface, a landscape, and CT data for bones and blood vessels in the hea (a)-(e): Mathematical data set ”knoto4”. (f)-(j): Landscape data "Mt. Alps”.