STEVE - Space-Time-Enclosing Volume Extraction

02/22/2013 ∙ by B. R. Schlei, et al. ∙ Apple Inc 0

The novel STEVE (i.e., Space-Time-Enclosing Volume Extraction) algorithm is described here for the very first time. It generates iso-valued hypersurfaces that may be implicitly contained in four-dimensional (4D) data sets, such as temporal sequences of three-dimensional images from time-varying computed tomography. Any final hypersurface that will be generated by STEVE is guaranteed to be free from accidental rifts, i.e., it always fully encloses a region in the 4D space under consideration. Furthermore, the information of the interior/exterior of the enclosed regions is propagated to each one of the tetrahedrons, which are embedded into 4D and which in their union represent the final, iso-valued hypersurface(s). STEVE is usually executed in a purely data-driven mode, and it uses lesser computational resources than other techniques that also generate simplex-based manifolds of codimension 1.

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I Introduction

The following presentation is subject to a pending patent application (cf., Ref. PAT11 ).
Recent advances in the field of computed tomography (cf., e.g., Ref.s XLI11 SSL12 ) have made available high-quality D (i.e., three spatial dimensions (D) and one temporal dimension (D)) reconstructed sets of measured time-varying voxel data. Similarly structured data may be generated in other scientific fields also, e.g., in the field of theoretical heavy-ion physics. There, so-called fireballs (i.e., extremely hot zones of strongly compressed nuclear matter that are formed by nuclear collisions) expand and cool down while emitting subatomic particles FRIM10 . For the proper fireball expansion modeling, relativistic fluid simulation codes are employed (cf., e.g., Ref.s STRO86 ; CSER94 ), which generate various – real-valued – field quantities (e.g., temperature, density, energy density, etc.) on a cartesian spatial (D) grid for the discretized (D) time, i.e., at fixed time steps.
The D history data of the fireball simulation can be used by a theoretician to calculate various observables (e.g., subatomic particle production rates). In doing so, it is very often necessary to determine an isotherme (cf., e.g., Ref.s COOP75 ; CHEN10 ), i.e., a manifold of codimension – or iso-hypersurface – at fixed temperature, which is implicitly contained in the discretized relativistic fluid history. If only so-called central heavy-ion collisions (i.e., the impact parameter of the collision equals to zero) are simulated, it is sufficient to model D (i.e., D spatial cartesian coordinates: radius, , and the beam axis position, , of the incident nuclei; plus D time) relativistic fluids, because of the rotational symmetry of the system under consideration (cf., e.g., Ref.s BRS97 ; BRS99 ). Then the task of iso-hypersurface extraction reduces to the task of iso-surface extraction in D for voxelized data BRS12 .
Direct D methods (i.e., those which do not extrude the data to higher dimensions for the purpose of iso-surface construction) can be subdivided basically into three classes (cf., Ref. BRS12 and Ref.s therein for more detail): those algorithms, which (i) use templates (cf., e.g., the Marching Cubes algorithm LORE87 ; BOUR94 ), (ii) introduce polarities at the voxel sites, and which implicitly solve spatial ambiguities (cf., e.g., the Marching Lines algorithm THIR96 ; BLOO94 ), and (iii) are protomesh-based, and which solve spatial ambiguities explicitly (cf., e.g., VESTA BRS12 ). However, the majority of real heavy-ion collisions has an impact parameter that usually deviates strongly from zero. Hence, the rotational symmetry is absent (cf., e.g., Ref.s CHEN10 ; HPET08 ), which may force a theoretician to perform a full-fledged D simulation of a heavy-ion collision with – perhaps – a subsequent numerical iso-hypersurface construction.
Similarly, one cannot always assume that D computed tomography data have internal symmetrical features (except, e.g., for the trivial case of temporally static data). The ability to extract iso-hypersurfaces from such (discretized) data may allow for an explicit, continuous D shape-representation (e.g., a continuous chronological evolution of D shapes). This subject of numerical iso-hypersurface extraction in D is not new (cf., below). As in the D case, in D there also seem to be three classes of direct methods to construct iso-hypersurfaces from toxel (i.e., time-varying voxel) data sets. For a template-based D algorithm, cf., e.g., Ref. BHAN04 . An algorithm that generalizes the ideas of the D Marching Lines algorithm into D is provided by the much earlier work of Fidrich FIDR96 . Here, however, we shall present in great detail – for the very first time – the protomesh-based “Space-Time-Enclosing Volume Extraction” algorithm (in the following shortly referred to as STEVE; cf., Ref. BRS04 for its first announcement).
This paper is organized as follows. First, we shall discuss the initial data mesh features. In particular, we shall compare proper D, D, and D meshes and stress certain analogies, because we would like to compare a variety of features of STEVE eventually with its D and D counterparts, DICONEX BRS09 and VESTA BRS12 , respectively. Next, the volume extraction framework will be presented. In doing so, we are going to introduce for each D -neighborhood of toxels a proper indexing scheme and a complete, corresponding vector path table (cf., Table I, in the Appendix) that will provide all possible links for contributing volume segments. Particular emphasis will be put onto the treatment of topological ambiguities. The latter will make clear that more than one solutions are generally possible if one is faced with the task of iso-hypersurface construction in discretized D spaces. In an application related section, we shall explore projections of constructed hypersurfaces. Finally, this paper will conclude with a short summary.

Ii Initial Mesh Features

Figure 1: Coordinate systems in (a) D with a single pixel; (b) D with a single voxel; and (c) D with a single toxel.

In the following, we shall take a closer look at the geometrical structure of the underlying meshes (or grids) in D, D, and D, which support the data that should be processed. In particular, we shall consider here only homogeneous, -cubical (cartesian) grids, with referring to the integral dimension of the space under consideration. In doing so, we shall use two different approaches. Either, we shall look at the data globally, or we shall look at a -neighborhood of the data. In the following, we shall use in the first case the term “global view” (GV), and in the latter case we shall use the term “neighborhood view” (NV), respectively.

Figure 2: Pairs of picture elements that are in direct contact in (a) 2D; (b) 3D; and (c) 4D (see text).

ii.1 Pixels, Voxels, and Toxels

In Fig. 1, the various coordinate systems that we are going to use ( in D, in D, and in D, respectively) are shown together with samples of single, corresponding picture elements (in GV). The centers of the elements are each marked with a sphere. In Fig. 1.a, we show a pixel (i.e., picture element) that is represented by a square, which is surrounded by its four edges. In Fig. 1.b, we show a voxel (i.e., volume pixel) that represented by (the projection of) a D cube, which is surrounded in D by its six squares. In Fig. 1.c, we show a toxel (i.e., – without loss of generality – time-varying voxel) that is represented by (the projection of) a tesseract (or -cube, or D hypercube), and is surrounded in D by its eight cubes. Note, that each of these picture elements are already perfectly enclosed by their surrounding shape elements. However, for an iso-valued contour, surface, or hypersurface, respectively, a different supporting point set is generally favored, than just the end points of the lines and/or the corner points of the surrounding shape elements themselves.

ii.2 Protomesh Building Blocks

In the following, we shall prepare ourselves for the construction of iso-valued hypersurfaces while using a protomesh-based technique that is derived from the initially identified surrounding boundary structures. For the anticipated type of construction of contours in D and surfaces in D, we would like to refer the reader to the detailed descriptions of DICONEX BRS09 and VESTA BRS12 , respectively. However, here we shall shortly review the basics that have led to both, DICONEX and VESTA, because we shall proceed analogously in D.
In Fig. 2, we show in GV pairs of picture elements in D, D and D, respectively, which are in direct contact. In each dimension, one element is active, i.e., it is marked for enclosure (indicated by the spheres), and the other element is considered inactive, i.e., it should not be enclosed (no spheres are placed in their centers). Each center of an active element is the origin of a range vector, which ends in the center of the inactive neighboring picture element. The range vectors (dark gray) define the bounds within which a support point (black dots) of the final extracted, iso-valued manifolds may be positioned. The element pairs are separated in D (cf., Fig 2.a) by a single vector (black), which has the active pixel at its left, and in D (cf., Fig 2.b) by a single square with four dashed lines (light gray) connecting the center of the square (black dot) with the middle points (light gray dots) of its four edges. In D (cf., Fig 2.c) the element pairs are separated by a single cube with six dashed lines (light gray) connecting the center of the cube (black dot) with its face centers (light gray dots), i.e., its six surrounding squares.

Figure 3: Initial building blocks for contour, surface and hypersurface construction in (a) for D, in (b) & (c) for D, and in (d) – (f) for D, respectively (see text).

These transitions from an active to an inactive picture element are of particular importance, because the final manifolds of codimension should be located in their vicinity. The transitions are represented by the geometrical structures that are described above (cf., Fig 2). These structures form the initial building blocks (cf., Fig 3) for the algorithms under consideration. Regardless of the dimension of the space, the centers (black dots) of the building blocks are going to be the support points of the final shape-enclosing manifolds. The light gray dots indicate the points of contact, where a building block will connect to another one when the manifolds are constructed. The range vectors (dark gray) define the bounds for the final support points (black dots), and they also indicate the interior of the shapes, which ought to be enclosed, through their orientation.
The building blocks are shown (each in GV) in Fig 3.a, for D as implemented in the DICONEX algorithm; in Fig 3.b, for D as implemented in VESTA (without internal vector paths); in Fig 3.c, for D as implemented in VESTA with vector paths; in Fig 3.d, for D as implemented in the STEVE algorithm (without internal vector paths); in Fig 3.e, for D as exploded view for positive orientation (, i.e., the range vector points into positive -, -, -, or -direction, respectively) with the vector paths as implemented in Table I (cf., the Appendix); in Fig 3.f, for D as exploded view for negative orientation (, i.e., the range vector points into negative -, -, -, or -direction, respectively) with the vector paths as implemented in Table I (cf., the Appendix). Note that in Fig.s 3.e and 3.f, the tiny black arrows mark vector pairs, which belong together.

Iii The Volume Extraction Framework

Eventually, the properly connected sets of building blocks, which have been introduced in the previous subsection, form the protomeshes that will be processed further. For the D and D processing, we would like to refer the reader to Ref.s BRS09 and BRS12 , (i.e., the detailed descriptions of DICONEX and VESTA, respectively). In this section, however, we shall describe – among many other things – how STEVE processes the identified initial boundary volumes that are embedded in D spaces.

iii.1 Indexing Scheme and Vector Paths

It is sufficient (particularly here, in D) to discuss all transitions from active to inactive toxels within a -neighborhood (or D-cell), because here we deal only with homogeneous, tesseract-like D grids. Such an approach has the clear advantage that we can label each relevant D point of the D-cell with an unique index.
In Fig. 4, we show the notations, i.e., identities (IDs), which we have chosen for a given D-cell, in NV. In Fig. 4.a, all sixteen toxel site IDs are shown. No.s through represent the “past,” whereas no.s through represent the “future,” while using – without loss of generality – time as the fourth dimension. In Fig. 4.b, all thirty-two potential boundary cube centers (i.e., final manifold support points for a single D-cell) are indicated. And finally, in Fig. 4.c, the all twenty-four connectivity points (i.e., potential points of ambiguity) for neighboring boundary cubes are shown. As an example, Fig. 4.d, shows for toxel no.  four range vectors (with positive (i.e., ) orientations each in the -, -, -, and -directions), superimposed with their corresponding boundary cube centers. Note that the central cubes in Fig.s 4.b–d represent the “present.”
As Fig. 4.d already suggests, each of the sixteen toxels may contribute within the -cell with exactly four boundary volume octants (i.e., an eights of a full boundary volume). Furthermore, each boundary volume octant is represented within STEVE by a triplet of vector pairs, which each connects a connectivity point through the corresponding boundary cube center to another connectivity point (cf., Fig.s 3.e and 3.f). In total, one has different possible vector paths within a given toxel neighborhood. The indexing scheme as shown in Fig. 4 and the table of the vector paths (cf., Table I, in the Appendix) form two of the three main ingredients that STEVE uses for the purpose of iso-hypersurface construction.
The third main ingredient for STEVE is a connectivity diagram (cf., Fig. 13.c) that helps to resolve topological ambiguities, when connectivity points will become points of ambiguity. The latter will be discussed in subsection III.G.

Figure 4: Indexing scheme for a -neighborhood of toxels; (a) toxel site IDs; (b) boundary cube centers; (c) connectivity points; (d) range vectors for toxel no. , superimposed with corresponding boundary cube centers.

iii.2 Generation of a single Tetrahedron

Figure 5: (a) A sixteenth of a toxel within a -cell at site no. ; detailed views of the corresponding boundary volumes (b) no. ; (c) no. ; (d) no. ; (e) no. (cf., Table I).
Figure 6: Continuation of Fig. 5: (a) the four initial cyclic vector paths within a -cell. The final tetrahedron together with the final cyclic vector path (b) (embedded into the D-cell); (c) ; (d) ; (e) . (f) The final tetrahedron, where each of its edges represents a pair of anti-parallel vectors.
Figure 7: Resulting manifolds of codimension for a single (a) pixel in D, (b) voxel in D, and (c) toxel in D, respectively.

In this subsection, we shall demonstrate the extraction of an iso-hypersurface for a single toxel (cf., Fig. 1.c).
In D, a single pixel is represented within a -neighborhood by a quarter of its (quadratic) area. In D, a single voxel is represented within a -neighborhood by an eights of its (cubic) volume. Analogously in D, a single toxel is represented within a -neighborhood by a sixteenth of its (-cubic) space-time (if we choose three spatial and one temporal dimensions). Hence, we need sixteen different -cells for the proper construction of the complete hypersurface of a single toxel. In order to demonstrate how STEVE uses the previously introduced indexing scheme (cf., Fig. 4) in combination with the vector path table (cf., Table I, in the Appendix), we shall process here a single sixteenth of a toxel.
As an example, we shall consider within a -cell the four boundary cubes for the single toxel at site ID no.  (cf., Fig. 5.a). These are the volumes no. (in the positive -direction; cf., Fig. 5.b), no. (in the negative -direction; cf., Fig. 5.c), no. (in the negative -direction; cf., Fig. 5.d), and no. (in the positive -direction; cf., Fig. 5.e). In total we obtain initial vectors (cf., Fig. 6.a), which – in a first step – can be combined into the four initial cyclic vector paths: , , , and , respectively.
Note that at the connectivity points, vectors are only connected such that the end-point of a predecessor connects to the starting-point of a linked successor (which never must be anti-parallel to the preceeding vector). The vector connectivities at the boundary cube centers are predefined through the vector path table (cf., Table I, in the Appendix).
In a second step, all connectivity points (i.e., those with point IDs above ; cf., Fig. 4.c) will be discarded. As a result, one obtains a single tetrahedron that is represented by its four final, reduced cyclic vector paths (cf., Fig.s 6.b–f). Each of these cyclic vector paths represents a triangle, which is embedded into D. Note that the initial orientations of the boundary volumes have been passed on, such that a consistent evaluation of -normal vectors is possible (cf., e.g., Ref.s DORS07 ; PERW09 ).
If we repeat the above processing for the remaining fifteen toxel sites within the -cell, we end up in total with sixteen oriented tetrahedrons that represent in their union the final STEVE-hypersurface for a single, isolated active toxel (cf., Fig. 7.c). Note that in D the orientations of the surface elements (i.e., triangles) of the tetrahedrons are not drawn, because each triangle is oriented both ways in the figure. For comparison we also show in Fig. 7.a the DICONEX-contour (consisting of four oriented edges, i.e., vectors) for a single, isolated pixel BRS09 . And in Fig. 7.b, we show the VESTA-surface (consisting of eight oriented triangles) for a single, isolated voxel BRS12 . Note that in the D case, each edge of the octahedron in the figure represents two anti-parallel vectors.

iii.3 Isochronous Hypersurface Sections

Figure 8: Generation of an isochronous hypersurface section within a -cell: (a) initial boundary cubes; (b) initial cyclic vector paths; (c) final cube-shaped hypersurface element, with one final cyclic (clockwise oriented) vector path indicating the orientation of the hypersurface section.

While considering time as the fourth dimension, a so-called isochronous hypersurface section (cf., Ref. HPET08 , for an application) can be generated, which also has a very simple geometry within a given -cell (i.e., similar to the generated single tetrahedrons in the previous subsection). If all eight toxel sites of the “past” are active, and simultaneously all eight toxel sites of the “future” are inactive (cf., Fig. 8), or vice versa (cf., Fig. 9), then one obtains as a result a cube-shaped element (as shown in the figures at the intermediate time, here, the “present”), which simply fills the whole D space at the fixed time, .
Note that in each case, the initial eight boundary cubes yield after consideration of both, the indexing scheme (cf., Fig. 4) and the vector path table (cf., Table I), six initial cyclic vector paths. The orientations of the final vector paths are inherited from the initial ones (i.e., either or ; cf., Fig.s 3.e and 3.f, and Table I) after the removal of the connectivity points. In Fig.s 8.b and 9.b, the tiny black arrows mark the initial vectors, which lead to the final vector -cycles, which are pronounced in Fig.s 8.c and 9.c, respectively.

Figure 9: As in Fig. 8, but with an inverted toxel site occupancy: (a) initial boundary cubes; (b) initial cyclic vector paths; (c) final cubic shaped hypersurface element, with one final cyclic (counterclockwise oriented) vector path indicating the orientation of the hypersurface section.

iii.4 Subspaces and Bounding Shapes

In the previous two subsections, we have encountered two rather simple D shapes as resulting hypersurface sections, i.e., a tetrahedron and a cube, respectively. The -cells within which STEVE determines iso-hypersurface sections, each decompose into eight -cubes (or cubes), i.e., D subspaces. Both previously determined hypersurface sections, i.e., the tetrahedron and the cube, are bounded by either four triangles (i.e., -cycles; cf., Fig. 10.c) or six squares (i.e., -cycles; cf., Fig. 10.e), respectively, which themselves are embedded into the corresponding D subspaces.

Figure 10: Bounding shapes (tiling sets) for DICONEX, VESTA, and STEVE: (a) – (b) two different 2D contour sections for 2D (sub)spaces, which can be combined into the following 3D-cells; (c) – (p) fourteen different 3D surface cycles for 3D (sub)spaces.

In fact, all possible hypersurface sections that can be generated with the STEVE algorithm will be bounded by the -cycles (), which VESTA BRS12 would create, if it were processing the active voxels within properly arranged -cells (i.e., -neighborhoods of voxels). This observation agrees with the fact that the number of vector paths (i.e., ) for the STEVE algorithm (cf., Table I, in the Appendix) equals to eight times of the number of vector paths (i.e., ) for the Marching VESTA (cf., Ref. BRS12 , Table 1).
In a lower-dimensional analogy, the DICONEX BRS09 algorithm determines all (properly oriented) line segments within the cube-bounding D subspaces, i.e., its six squares. In Fig. 10, the complete tiling sets of sections of the manifolds of codimension for 2D and 3D (sub)spaces as determined by the DICONEX, VESTA, and STEVE algorithms are depicted. Note that multiple sections could be generated within in a given (sub)space (for more detail, cf., Ref. BRS12 ).

iii.5 Decomposition of Hypersurface Sections

Due to the various D bounding shapes as shown in Fig.s 10.c–10.p, the hypersurface sections which STEVE computes could be very complex-shaped polyhedrons (that are embedded into D). For visualization purposes, or, e.g., for the purpose of -normal vector determination (cf., e.g., Ref.s DORS07 ; PERW09 ) it may be desirable to decompose the polyhedrons into a set of tetrahedrons, because these simplices are better suited for the considered tasks.
As an example, we show in Fig. 11, how to decompose a single cube into twenty-four tetrahedrons. In Fig. 11.a, a cube is shown, which consists of six D tilings (-cycles) as shown in Fig. 10.e. In Fig. 11.b, for each -cycle, its center of mass point (face center) is determined. In Fig. 11.c, the surface cycles are decomposed into triangles, while connecting the face centers with the corresponding -cycle support points. Note that triangles will not be decomposed any further. Furthermore, each newly drawn line actually represents a pair of anti-parallel vectors.
Next, in Fig. 11.d, all triangles, which enclose a particular single volume are collected into an object along with the information, to which -cycles the triangles belong; for all face centers of each of the found objects, the absolute center of the enclosed volume is determined. In Fig. 11.e, lines are introduced that connect the face centers with the absolute volume center. Finally, in Fig. 11.f, the additional connections of all -cycle support points with the absolute volume center finally yields the twenty-four properly oriented tetrahedrons.
Within STEVE, this approach has been extended to the more or less complex shaped polyhedrons. Note that polyhedrons, which are embedded into D, could be decomposed while using fewer tetrahedrons also. E.g., one could decompose a cube into just five tetrahedrons. However, one should be aware, that such a choice may introduce directional ambiguities into the space-time considered here. For the remainder of this paper, we shall apply the variant as depicted in Fig. 11. As a result, we shall always obtain a directionally most unbiased decomposition. Note that single tetrahedrons will not be decomposed any further.

Figure 11: Decomposition of a cube into twenty-four tetrahedrons (see text).

iii.6 Generation of a triangular Strut

In another example, we briefly demonstrate the generation of a hypersurface section that has the shape of a triangular strut. In Fig. 12, two toxels (in NV) at site IDs no.  and no.  are in contact through a single volume. This volume of contact is visible within the cube that represents here the “present” (cf., Fig. 12.a). Once again, the application of both, the indexing scheme (cf., Fig. 4) and the vector path table (cf., Table I), yields five initial cyclic vector paths (cf., Fig. 12.b). Fig. 12.c shows the final trianglular strut-shaped hypersurface element, with two final cyclic vector paths indicating its orientation. Note that this hypersurface element is bounded by two -cycles, as shown in Fig. 10.c, and by three -cycles, as shown in Fig. 10.d, respectively. Finally, in Fig. 12.d, a decomposition of the hypersurface element into fourteen oriented tetrahedrons is shown as described in the previous subsection (cf., Fig. 11).
Until now, we did not encounter any particular topological ambiguities while constructing the heretofore discussed hypersurface sections. However, this is going to change in the next subsection.

Figure 12: Generation of a triangular strut within a -cell: (a) initial boundary cubes; (b) initial cyclic vector paths; (c) final triangular strut-shaped hypersurface element, with two final cyclic vector paths indicating the orientation of the hypersurface section; (d) as in (c), but with a hypersurface section that is decomposed into fourteen tetrahedrons.

iii.7 Ambiguous Connectivity

Figure 13: Encounter of a topological ambiguity within a -cell: (a) initial boundary cubes; (b) initial cyclic vector paths; (c) connectivity diagram.
Figure 14: Continuation of Fig. 13: (a) two resulting tetrahedrons from the “disconnect” mode; (b) a final, more complex-shaped hypersurface element, resulting from the “connect” mode; (c) as in (d), but with a hypersurface section that is decomposed into sixteen tetrahedrons.
Figure 15: Hypersurface section generation for two toxels, which are in contact through a single edge: (a) initial boundary cubes; (b) initial cyclic vector paths; (c) two resulting tetrahedrons.

The here considered discretized spaces could lead to topological ambiguities. In D this is the case, if two active pixels share only one common point (i.e., a vertex; cf., Ref. BRS09 for more detail). In D this is the case, if two active voxels share only one common edge (cf., Ref. BRS12 for more detail). Analogously in D this is the case, if two active toxels share only one common face (square).
In Fig. 13.a, we show a -cell with two active toxels at site IDs no.  and no. . The toxels are in contact through a single face. Within the cube that here represents the “present” this surface of contact is visible. In Fig. 13.b, we observe an ambiguous configuration (white dot) for the initial cyclic vector paths. Apparently, here this particular connectivity point has turned into a so-called “point of ambiguity” (POA; cf., Ref. BRS12 ). Each incoming (white, straight) vector can connect to either one of the two (black, straight) vectors, depending on the chosen connectivity mode (cf., Fig. 13.c). Using the field values for the toxels at positions I, II, III, and IV in the figure, i.e., the connectivity diagram, one may assign their average field value to the POA. E.g., for an average field value below (above) the desired isovalue of the hypersurface, one generates the local “disconnect” (“connect”) mode, while pursuing the white (black) bent directed path for each incoming vector consistently. This treatment (also known as “mixed” mode BRS12 treatment) will allow for an automated, robust resolution of all encountered ambiguities.
Note that one may enforce the connectivity modes also globally onto the complete considered D data set, by either always selecting the “disconnect”, or the “connect” mode. In Fig. 14.a, we show two final hypersurface elements (tetrahedrons), which result from the application of the “disconnect” mode. Alternatively, we show in Fig. 14.b one final hypersurface element, which results from the application of the “connect” mode. This latter hypersurface element is bounded by four -cycles as shown in Fig. 10.c, and by two -cycles as shown in Fig. 10.h, respectively. In Fig. 14.c, it has been decomposed into sixteen oriented tetrahedrons as described above (cf., Subsection III.E). Note that each drawing in Fig. 14 shows two final cyclic vector paths indicating the orientation of the hypersurface sections.

iii.8 Disjunct Hypersurface Sections

In D, VESTA never connects two voxels, which are in contact only through a single vertex (cf., Ref. BRS12 for more detail). In D, we have a similar situation when considering toxels that are in direct contact.
In Fig.s 16 and 17, the process of hypersurface section generation is shown for two toxels that are in contact only through a single edge and through a single vertex, respectively. In both cases, we simply obtain two tetrahedrons as final hypersurface sections, since the initial cyclic vector paths form two disjunct sets with four cyclic paths each. Apparently, toxel pairs with such a weak connectivity will always result in two separate hypersurface segments. This concludes the technical section of this paper.

Figure 16: Hypersurface section generation for two toxels, which are in contact through a single vertex: (a) initial boundary cubes; (b) initial cyclic vector paths; (c) two resulting tetrahedrons.

Iv Continuous Evolution of Surfaces

This section is more related to applications. Here, we shall discuss a few use cases for hypersurfaces (or hypersurface sections) that have been generated while applying STEVE to D data.

iv.1 A single -Neighborhood

Figure 17: Continuous transformation of a single quadriliteral into two separate triangles (see text).

In a first example, we shall extract continuously evolving surfaces from a hypersurface segment, which has been generated with STEVE for a single -cell. In Fig. 17, we show a sequence of one and the same -cell (i.e., in NV) with six activated toxel sites each. In the “past” (left side) the four toxels with IDs no.s  and  are active, whereas in the “future” (right side) only the two toxels with IDs no.  and no.  are active (cf., Fig. 4.a). STEVE has been applied to this configuration in its global “disconnect” mode, in order to determine the corresponding hypersurface section (i.e., the networks of black lines in the figure). For each tesseract (except for the first and the last one), an additional intersecting cube is drawn for various fixed times . In fact, the parameter, , indicates the relative time (that ranges between the extremes of and ), where the hypersurface section has been intersected.
Fig. 17 shows a temporal evolution of slices (i.e., D surface segments) of the generated hypersurface segment, which has been properly decomposed into a set of connected tetrahedrons (cf., subsection III.E). Each tetrahedron – when intersected – could contribute to a time slice either with a single point, a single edge, a single triangle, a single quadrilateral, or the whole volume of the tetrahedron. Here it is demonstrated, how a single square may transform continuously and smoothly into two separate triangles. Hence, the depicted D hypersurface section establishes a correspondence between the surfaces in the “past” and in the “future” (and between those anywhere in between).

iv.2 Two consecutive 3D Data Sets

In this subsection, we provide lastly an example for freezeout hypersurface (FOHS) extraction from D hydrodynamic simulation data (cf., e.g., Ref. CHEN10 and Ref.s therein). On a cartesian D grid, a relativistic fluid has been propagated numerically. The spheres in Fig.s 18.a and 18.h represent grid points above a certain threshold temperature at the two subsequent time steps (in GV) where the considered fireball decays into two pieces. In particular, STEVE has been used (in its global “disconnect” mode) to determine the (continuous) iso-thermal hypersurface that lies between the two shown D data sets, i.e., a FOHS (of fixed temperature).
In doing so, STEVE has made use of the range vectors (cf.

, e.g., Fig.s 2.c and 4.c) in order to determine a (with respect to linear interpolation) most correctly located (and thereby smoother) iso-hypersurface (which is

not shown here). Note that STEVE provides interpolations for all other field values, which have been associated with each toxel, as well. For the purpose of FOHS, STEVE will not just be applied to two subsequent time steps as shown in the figure, but to the full set of D simulation data. The resulting total FOHS will allow for the further calculation of various observables FRIM10 ; STRO86 ; CSER94 ; COOP75 ; CHEN10 ; BRS97 ; BRS99 .
In Fig.s 18.b–18.g, various time projections of the chosen FOHS section are shown for visualization purposes. Note that (as it has been the case in the previous subsection) any finite number of chronologically evolving D surfaces can be extracted.

Figure 18: (Taken from Ref. BRS04 .) Generation of chronologically evolving freezeout hypersurfaces: (a) & (h) selected grid points of the discretized D temperature fields at two subsequent time steps; (b) – (g) various time projections of the generated iso-hypersurface.

V Summary

In summary, the D protomesh-based iso-hypersurface construction algorithm STEVE has been described here for the very first time in great detail. Where D template-based approaches (cf., e.g., Ref. BHAN04 ) may require the storage of (due to possible topological ambiguities at least) volume tile templates, the STEVE algorithm basically requires the storage of vector paths only.
We would like to stress that – similar to the cases of iso-contour and iso-surface extraction from D pixel and D voxel data sets, respectively (cf., DICONEX BRS09 and VESTA BRS12 ), – more than one iso-hypersurface solutions are possible. This is simply a consequence of the discretization of the D spaces under consideration, where topological ambiguities will allow for two different global treatments (i.e., global “disconnect” or global “connect”connectivity modes), and/or an additional local treatment (i.e., local “mixed” connectivity mode). The proper treatment of possible ambiguities ensures that accidental rifts in the final hypersurfaces will be avoided by all means.
STEVE constructs iso-hypersurfaces from the grounds up. I.e., it continuously transforms the set of initially identified octants (i.e., eights of a cube) into the final set of tetrahedrons. This set represents the final hypersurface or a finite set of hypersurfaces. Note that the initially given information about the interior/exterior of the enclosed D regions will be propagated to the final result.
Finally, we would like to point out that this work represents – particularly because of the explicit definition of all possible vector paths (cf., Table I, in the Appendix) – also a self-contained and complete instruction manual for iso-hypersurface extraction from homogeneous, tesseract-like structured D spaces, e.g., like time-varying computed tomographic D images.

Vi Acknowledgement

Some initial work has been supported by the Department of Energy under contract W-7405-ENG-36. The particular funding has resulted in a first version of ANSI C-based software STEVE04 , where ambiguities are resolved globally only.

Vii Appendix

Figure 19: Orientations of bounding elements: (a) in D, for a single rectangular surface tile; (b) in D, for a triangular strut-shaped hypersurface segment.

In D, a shape could be enclosed either with a clockwise or a counter-clockwise contour. The DICONEX BRS09 algorithm uses the latter, i.e., shapes will always be enclosed with counter-clockwise contours. Similarly, in D the normal vectors of a shape-enclosing surface could either point to the interior or to the exterior of the enclosed shape. VESTA BRS12 uses the latter, i.e., normal vectors of shape-enclosing surfaces always point to the shapes’ exterior. Using a right-hand rule, this has direct consequences for the orientations of the three vectors, which circumscribe triangles (i.e., the simplices that in their union represent a surface or a set of surfaces) and, hence, for the orientations of the initial voxel face vectors (for more detail, cf., Ref. BRS12 ).
Analogously, one has in D also two choices for the orientation of the vector paths as shown in Table I. I.e., the first three path columns that here refer to , instead could have referred to ; and the last three path columns that here refer to , instead could have referred to (cf., Fig.s 3.e and 3.f). In Fig. 19, we visualize how our particular choice for the orientations of hypersurface segments came about.
In Fig. 19.a we show a surface tile in D, similar to the one that is shown in Fig. 10.d. Its orientation is such that the bounding line element (cf., Fig. 10.a) in the -plane at has a normal, which points towards the enclosed voxel, whereas the bounding line element in the -plane at has a normal, which points away from the enclosed voxel. In analogy to the D case (as implemented by VESTA BRS12 ), we show in Fig. 19.b a hypersurface tile that is embedded into D (cf., Fig. 12.c). Its orientation is chosen such that the bounding surface element (cf., Fig. 10.c) in the -subspace at has a normal, which points towards the enclosed toxel, whereas the bounding surface element in the -subspace at has a normal, which points away from the enclosed toxel.

Orientation
Center ID Path 1 Path 2 Path 3 Path 1 Path 2 Path 3
Table 1: Triplets of directed paths (cf., Fig.s 3.e and 3.f) for the octants of the oriented boundary volumes, which have their centers at the predefined locations as depicted in Fig. 4.b. The start and end points of the paths are the potential points of ambiguity, as shown in Fig. 4.c.

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