## 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

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.

### 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.

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.

### iii.2 Generation of a single Tetrahedron

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

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.

### 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.

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.

### 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.

### iii.7 Ambiguous Connectivity

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.

## 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

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.

## 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

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 |

## References

- (1) B. R. Schlei, “Verfahren zur Hyperflächenkonstruktion in n Dimensionen,” german patent applications No. 10 2011 050 721.3 and No. 10 2011 051 203.9, submitted on May 30, 2011, and June 20, 2011, and PCT application No. PCT/EP2012/058873 submitted on May 14, 2012, by GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstr. 1, 64291 Darmstadt, Germany, respectively.
- (2) X. A. Li, Editor, Adaptive Radiation Therapy (Imaging in Medical Diagnosis and Therapy, CRC Press Inc. 2011.
- (3) A. Mundt, J. Roeske, Image-Guided Radiation Therapy (IGRT): A Clinical Perspective, Mcgraw-Hill Professional 2011.
- (4) S. S. Lo, B. S. Teh, J. J. Lu, T. E. Schefter, Editors, Stereotactic Body Radiation Therapy (Medical Radiology / Radiation Oncology), Berlin: Springer 2012.
- (5) B. L. Friman, C. Höhne, J. Knoll, S. Leupold, J. Randrup, R. Rapp, P. Senger, editors, The CBM Physics Book: Compressed Baryonic Matter in Laboratory Experiments (Lecture Notes in Physics), Berlin: Springer, 2011.
- (6) R. B. Clare, D. Strottman, “Relativstic Hydrodynamics and Heavy Ion Collisions,” Phys. Rep. 141 (1986) 177 – 280.
- (7) L. P. Csernai, Introduction to Relativistic Heavy Ion Collisions, John Wiley & Sons, 1994.
- (8) F. Cooper, G. Frye, E. Schonberg, “Landau’s Hydrodynamic Model of Particle Production and Electron-Positron Annihilation into Hadrons,” Phys. Rev. D11 (1975) 192 – 213.
- (9) Yun Cheng, L. P. Csernai, V. K. Magas, B. R. Schlei, and D. Strottman, “Matching Stages of Heavy-Ion Collision Models,” Phys. Rev. C81, 064910 (2010), doi: 10.1103/PhysRevC.81.064910.
- (10) B. R. Schlei, “Extracting the Equation of State of Nuclear Matter through Hydrodynamical Analysis,” Heavy Ion Phys. 5, 403 – 415 (1997).
- (11) B. R. Schlei, D. Strottman, “Predictions for Au+Au Collisions from Relativistic Hydrodynamics,” Phys. Rev. C59, R9 – R12 (1999).
- (12) B. R. Schlei, “Volume-Enclosing Surface Extraction,” Computers & Graphics 36 (2012) 111 130, doi: 10.1016/j.cag.2011.12.008.
- (13) W. E. Lorenzen and H. E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” Comput. Graph. 21 (1987), pp. 163 – 169.
- (14) P. Bourke, “Polygonising a scalar field,” May 1994; for more detail, cf., http://paulbourke.net/geometry/polygonise/
- (15) J.-P. Thirion, A. Gourdon, “The 3D Marching Lines Algorithm,” Graphical Models and Image Processing, 58, pp. 503 – 509, 1996.
- (16) J. Bloomenthal, “An Implicit Surface Polygonizer,” in Graphics Gems IV, ed. P. S. Heckbert, AP Professional, Boston, 1994, pp. 324 – 349.
- (17) H. Petersen, J. Steinheimer, G. Burau, M. Bleicher, H. Stöcker, “A Fully Integrated Transport Approach to Heavy Ion Reactions with an Intermediate Hydrodynamic Stage,” Phys. Rev. C78, 044901 (2008).
- (18) P. Bhaniramka, R. Wenger, R. Crawfis, “Isosurface Construction in Any Dimension Using Convex Hulls,” IEEE Trans. Visualization and Computer Graphics 10 (2004), doi: 10.1109/TVCG.2004.1260765.
- (19) Márta Fidrich, “Iso-Surface Extraction in 4D with Applications related to Scale Space,” Unité de recherche INRIA Sophia-Antipolis, Rapport de Recherche No. 2833, Programme 4, March 1996.
- (20) B. R. Schlei, “Hyper-Surface Extraction in Four Dimensions,” Theoretical Division - Self Assessment, Special Feature, a portion of LA-UR-04-2143, Los Alamos (2004) 168.
- (21) B. R. Schlei, “A New Computational Framework for 2D Shape-Enclosing Contours,” Image and Vision Computing 27 (2009) 637 647, doi: 10.1016/j.imavis.2008.06.014.
- (22) L. Dorst, D. Fontijne, S. Mann, Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry, Morgan Kaufmann, 2007.
- (23) C. Perwass, Geometric Algebra with Applications in Engineering, Geometry and Computing, Springer, 2009.
- (24) B. R. Schlei, “STEVE - Space-Time-Enclosing Volume Extraction Algorithm, Version 1.0,” Los Alamos Computer Code LA-CC-04-056, Los Alamos National Laboratory.

Comments

There are no comments yet.