senting empty space, have been developed in [11][13][14]. These
methods are practical and commonly used nowadays. Surface
shapes, however, are infinitely thin in space. To sample this thin
shape, one needs infinitely high sampling rates. To avoid such a
difficulty, Kaufman’s algorithm [13][14] increases the thickness of
a surface, and therefore, to some extent, band limits the frequency
spectrum before the sampling, or scan-conversion, process. Huang
et. al [11] discovered and proved the sufficient and necessary
thickness of the surface shape that guarantees a correct discrete
topology in the resulting volume representation. Unfortunately, all
these methods are based on binary volume representations, which
are highly susceptible to aliasing artifacts. To address this issue,
Sramek and Kaufman initiated data representations in non-binary
formats [21]. In their paper, they show one has to use higher order
smoothing functions to pre-filter and band limit the spectrum of
the volume. Later, an incremental voxelization method for non-
binary volume is reported in [5].
Over the years, in addition to the search for optimal voxeliza-
tion, the community has also been exploring other representations
of surfaces. Distance fields are scalar fields, with each element in
the 3D volume representing the minimal distance to a certain
shape. It is common practice to use signed values to distinguish
between interior and exterior of the shape. Compared to the sur-
face shapes that correspond to impulses in 3D space, distance
fields are much smoother. For shapes without sharp corners and
edges, both the surface position and gradient can be reconstructed
relatively accurately using a distance field [1][9]. When corners
and sharp edges are introduced, high frequency components are
also brought into the spectrum. To preserve such details, super-
sampling with exceptionally high volume resolution, as well as
low-pass filtering, is necessary to achieve an alias-free representa-
tion.
In this paper, we present a novel scheme for representing dis-
tance fields. We build our volumetric representation of distance
fields based on a complete distance definition. Our approach is dis-
parate from the theory of linear sampling. We name our distance
field representation a “complete distance field representation
(CDFR)”, because once the distance volume is constructed, we can
extract any distance contour to any requirement of accuracy. As a
comparison, conventional approaches based on a single valued dis-
tance field can only achieve higher accuracy by re-building the
whole distance volume at an increased resolution. However, in
most cases building a high resolution distance volume is non-triv-
ial both in computational time and storage space. A recent work
[15] stores the estimated local edge positions in x, y and z direc-
tions with each voxel and can extract triangle meshes from volume
data at a much improved quality compared to conventional meth-
ods [2][10]. However, their distance field representation is not
complete and the surface extraction is still based on estimation.
Our method only relies on exact computations, and, different from
the approach in [2][10][15], uses a point-based approach to repre-
sent the extracted contour surfaces.
A Complete Distance Field Representation
Jian Huang
1
, Yan Li
2
, Roger Crawfis
1
, Shao-Chiung Lu
3
and Shuh-Yuan Liou
4
1
Computer and Information Science, The Ohio State University, Columbus, OH
2
Electrical Engineering, The Ohio State University, Columbus, OH
3
Visteon Inc., Dearborn, MI
4
Ford Motor Company, Dearborn, MI
Abstract
Distance fields are an important volume representation. A high
quality distance field facilitates accurate surface characterization
and gradient estimation. However, due to Nyquist’s Law, no
existing volumetric methods based on the linear sampling theory
can fully capture surface details, such as corners and edges, in
3D space. We propose a novel complete distance field
representation (CDFR) that does not rely on Nyquist’s sampling
theory. To accomplish this, we construct a volume where each
voxel has a complete description of all portions of surface that
affect the local distance field. For any desired distance, we are
able to extract a surface contour in true Euclidean distance, at
any level of accuracy, from the same CDFR representation. Such
point-based iso-distance contours have faithful per-point
gradients and can be interactively visualized using splatting,
providing per-point shaded image quality. We also demonstrate
applying CDFR to a cutting edge design for manufacturing
application involving high-complexity parts at un-precedented
accuracy using only commonly available computational
resources.
CR Categories: I.3.6 [Computer graphics]: Methodology and
techniques - Graphics data structures; I.3.5 Computational
Geometry and Object Modeling - Object modeling
Keywords: distance fields, volume modeling, polygonal
surfaces, point-based models, graphics.
1. Introduction
A voxel-based volume, as a 3D raster, holds discrete sample
points representing a certain multi-dimensional entity. In an alias-
free volume discretization, only frequency components below half
the Nyquist sampling rate would be stored. As a natural description
of solid physical entities, volume representations have found appli-
cations in a variety of areas, including medicine, mechanical engi-
neering, scientific computing and simulations. In order to utilize
volume technologies, it has been common to convert surface mod-
els, such as a polygonal mesh exported by a CAD package, to a
volume representation. In this process, first, one needs to voxelize
the surface model into a hollow volume representing the surface
shape [11][13][14]. Second, a distance transform is computed to
construct a solid volume that encompasses a distance or thickness
field recording distances to the surface. Euclidean distance has not
been commonly used due to both efficiency concerns and the fact
that accuracy is already compromised in the binary surface volume
model. Instead, most applications use less accurate distance heuris-
tics such as Manhattan or chessboard distance, or a Chamfer dis-
tance [3].
Voxelization techniques that convert surface shapes into
binary volumes, with 1’s representing occupancy and 0’s repre-
247
Presented at IEEE Visualization 2001
October 21 - October 26, 2001
San Diego Paradise Point Resort, San Diego
Hierarchical data structures applied to represent distance
fields efficiently have been reported in [7], where adaptively sam-
pled distance fields (ADF) are introduced. ADFs help in reducing
volume storage size when fewer details are locally present. The
specific ADF implementation described in [7] relies on a single
valued distance representation, therefore that implementation still
depends on a band-limited spectrum that discards all details
beyond the cut-off bandwidth supported by the leaf level in the tree
structure. Our CDFR algorithm can be embedded into the ADF
structure for an exact distance field representation that is efficient
both in terms of storage and processing time.
Conventionally, distance fields are most often rendered with
ray-casting approaches[1][5][7][9]. High interactive rates are often
not attainable. CDFR can also be rendered with a slightly modified
ray-casting scheme. However, for efficient rendering of distance
contours, we also discuss a method to reconstruct a point-based
contour of any distance value from CDFR. Such sparse point-based
models can be efficiently rendered with splatting [12].
The paper is organized in the following way. We give a brief
introduction to single valued distance fields and the limitations of
such representations in Section2. In Section3, we present a com-
plete distance definition (CDD) and complete distance field repre-
sentation (CDFR) based on CDD. We prove the correctness of our
schemes in constructing CDFRs and extracting point-based con-
tours from CDFRs in Section4. Results on a variety of data sets
are shown in Section5. Finally, we conclude and discuss future
work in Section6.
2. Distance Fields
Traditionally, distance fields are defined as spatial fields of
scalar distances to a surface geometry or shape. Each element in a
distance field specifies its minimum distance to the shape. As long
as the shape is represented by an oriented manifold, positive and
negative distances can be used to distinguish outside and inside of
the shape, for instance, using negative values on the outside and
positive on the inside. Distance fields have a number of applica-
tions in constructive solid geometry [1][7], surface reconstruction
and normal estimation [9] and morphing [1][3]. Distance fields are
also applied to concurrent engineering [16] where simulations and
analysis involving the interior of geometries, such as die-casting
simulation or thickness analysis of parts [22], are routine.
For an alias-free sampling of a signal, Nyquist’s Law dictates
that the sampling rate must be at least two times the highest fre-
quency component in the signal. In spatial domain, geometry is
infinitestismally thin, and has an infinitely wide frequency spec-
trum. The sharp details on the surface, such as corners and edges,
also reside on the high ends in the spectrum. Even with an over-
whelmingly large volume resolution, one still needs extensive low-
pass filtering to limit the bandwidth of the geometric shape. These
low-pass filtering operations, with either simple box filters
[11][13][14] or specifically designed higher order filters [21], inev-
itably cause a loss of the exact surface details. Converting the sur-
face shape to a distance field, which is smoother, provides a way to
exactly locate the surface [9] during reconstruction. But the under-
lying assumption of having a completely smooth surface that is
free from sharp corners and edges is unrealistic for most scenarios.
Frisken et. al [7] developed a well analyzed framework for
adaptively sampled distance fields (ADF), by which one can build
hierarchies of distance fields at different levels of detail and be
able to cross over different levels of detail as needed. They also
vary sampling rates according to the amount of details that are
available locally. They used tri-linear interpolation to reconstruct
distances, and were able to demonstrate a suite of applications with
impressive rendering quality. However, ADF [7] does not funda-
mentally solve the problem of losing surface details in discrete rep-
resentations. After the leaf level of ADF is constructed, the loss in
surface details is final and irreversible. When the primary goal of
an application shifts from visual quality to accuracy, ADFs with
trilinear interpolation may not satisfy the accuracy needs with a
guarantee, simply because the true distance fields are not linear
where corners or edges are present. What the hierarchies provide is
an ability to save computational and storage resources when less
details are encountered. For models with fine details everywhere,
the ADF eventually resorts to an extremely large voxelization. For
the applications where accuracy is highly sought after, current
ADFs based on single valued distances incur overwhelming costs,
because most practical geometrical models are rich in details at a
wide range of scale.
A high quality distance field should be accurate, efficiently
stored and can be efficiently processed. There are two fundamental
issues involved in building a high quality distance field. First, we
need an accurate way to represent the distance from an arbitrary
point in 3D space to an arbitrary shape. Second, how should we
optimally organize the distance representations in space?
This first question is our focus in this paper. We intend to
show CDFR as a fundamental fix that preserves all geometic
details in the true distance field. Only exact computations are used.
ADF, as a systematic framework, addresses the second fundamen-
tal issue, by providing an adaptive and smooth transition between
resolution levels, depending on the amount of surface details avail-
able.
3. Complete Distance Definition (CDD) & Com-
plete Distance Field Representation (CDFR)
In this paper, we propose a complete distance definition
(CDD). Corresponding to different surface representations, such as
parametric surfaces, implicit surfaces or subdivision/polygonal
mesh surfaces, there could be different instantiations of CDDs. For
this paper, we focus on a simple case where the surface is repre-
sented by polygonal meshes. When accuracy is paramount, Euclid-
ean distances are preferred over other distance metrics, such as
Chamfer distance or Manhattan distance. The CDD distances are
true Euclidean distances. Before discussing CDD, we will discuss
a few observations that motivated our research on CDFR.
3.1 Some Observations
Distance fields are very smooth in some simple scenarios. For
instance, suppose in a 1-dimensional space, we have an impulse.
It’s frequency components extend to infinity. There is no way to
use a finite sampling frequency to sample the impulse without
aliasing. But on the other hand, as illustrated in Fig. 1, the signed
distance field of the impulse is a linear function which extends
from negative infinity to positive infinity. Sampling this linear
function can be accurate with a relatively low sampling rate.
Unfortunately, this feature does not hold in higher dimensions
where corners are present. As presented in [11], when extended
into 2D or 3D, the discrepancies and discontinuity on corners
makes the distance field non-smooth. For instance, in the triangle
in Fig. 2, we have a rather faithful sampling in the light grey grids
on the edges, because the geometry is locally linear
*
. But the sam-
*
Locally linear: reconstruction is accurate with linear interpolations.
248
pling is not sufficient in the dark grey grids that have corners. The
non-linear distance fields within the dark grey grids make it impos-
sible to accurately recover the correct distance distribution from
the grid samples.
According to Nyquist’s Law, to sample such complicated dis-
tance fields, one must low-pass filter the corners and smooth out
the sharpness. Using ADF [7], for the grids on the corners, a higher
resolution would be used, whereas in the light grey grids, a much
lower resolution might suffice.
To capture the exact location of the impulse in Fig. 1, we do
not have to use sampling. Alternatively, all one needs is to place an
anchor point at some location, and record the signed distance from
the anchor to the impulse. In this method, preserving the exact
position of the impulse is made straight-forward. This observation
motivated our research towards a new distance representation for
distance fields.
3.2 Complete Distance Definition (CDD)
CDD is a set of parameters describing both the distance from
a 3D point to a surface geometry primitive and the geometry prim-
itive itself. Specifically, when the shape is represented as a mesh of
triangles, CDD reduces to a tuple that consists of a scalar canonical
distance value, and a description of the triangle with a vertex list
and an edge list:
(1)
The value distance is the true Euclidean distance from the
voxel center to a finite triangle. This distance is defined in the
pseudo code in Fig. 3.
While the return value is the CDD distance, the input parame-
ters include a triangle, tri, and a 3D point, pnt. If pnt orthogonally
projects into tri (case C
1
), the return value is the orthogonal dis-
tance from pnt to the plane where tri lies. Otherwise, we check
whether pnt orthogonally projects onto any of the three edges. If
yes (case C
2
), then the returned distance value is the shortest dis-
tance from pnt to an edge that pnt projects orthogonally onto. In
case neither C
1
or C
2
applies (case C
3
), the distance is the minimal
distance from pnt to the three vertices. This definition of distance
to a finite triangle is further illustrated in Fig. 4.
We can still use positive and negative distance to distinguish
inside and outside. We term the triangle that is the closest to pnt as
the base triangle of pnt. If pnt is closest to a triangle and the dis-
tance is of case C
1
, then this triangle is pnt’s base triangle. If pnt’s
distance is not case C
1
, rather, it’s case C
2
or C
3
, looking for pnt’s
base triangle is more complicated. For C
2
cases, let’s label the pro-
jection point of pnt on that corresponding edge as, p’, and we
record the vector pointing from p’ to pnt as V. Between the two tri-
angles sharing that edge, the triangle with a normal direction closer
to V’s direction, i.e. larger absolute dot product value,
, is pnt’s base triangle. Very similarly, in C
3
cases,
among the triangles sharing that closest vertex, we can easily find
out the base triangle of pnt by comparing dot product values. We
are interested in finding out pnt’s base triangle, because by using
the outward normal direction of the base triangle and the relative
position of pnt, we can determine the sign of the distance at pnt
without ambiguity.
To better illustrate the process in determining the distance
sign, in Fig. 5a, we show several 3D examples, shown in 2D. p
1
through p
6
are 2D points. t
1
through t
6
are triangles that form the
surface mesh. p
2, 3, 4 and 5
are all case C
1
. From the normal direc-
tion of t
2
, we can tell p
2
is outside, p
3
is inside. Similarly, using the
normal of t
3
and t
5
, one can tell that p
4
is inside, and p
5
is outside,
respectively. p
1
and p
6
are both C
2
. We show an enlarged view of
these two cases in Fig. 5b. By comparing the dot products, we can
tell p
1
’s sign is determined by t
2
, and for p
6
, it is decided by t
1
.
Finally, to save space, we store the description of all triangles
in a separate array and only keep a triangle index in a CDD tuple.
f(x)
Fw
()
spatial impulse
distance field of the impulse
Figure 1: Without low-pass filtering, it’s impossible to sample
the impulse (left), but we can sample its distance field (right).
0
0
0
0
∞
∞
∞
∞
–
∞
–
∞
–
∞
–
2
w
2
------
∞
Figure 2: Corners in the triangle cause complexity in the
distance field, resulting in an aliased spectrum after sampled.
distance
v
1
v
2
v
3
,,
〈〉
e
1
e
2
e
3
,,
〈〉
,,
〈〉
float CDD (triangle tri, vec3 pnt)
{
float mindist = MAXIMUM;
if (pnt projects orthogonally into tri’s interior) // C
1
mindist = distance from pnt to tri’s plane;
else
{
for each edge of tri, e
i
,
if (pnt projects orthogonally onto e
i
) // C
2
mindist = min(mindist, distance from pnt to e
i
);
for each vertex of tri, v
i
, // C3
mindist = min(mindist, distance from pnt to v
i
);
}
return mindist;
}
Figure 3: Definition of distance from a point to a finite triangle
C1
C2
C2
C2
C3
C3
C3
Figure 4: If pnt projects into tri, it’s case C
1
. Otherwise, pnt is
either C
2
or C
3
, depending on whether it’s closer to an edge or a
vertex. This diagram is drawn in 2D for ease of illustration.
pnt
pnt
pnt
VNormal
•
249
3.3 A Complete Distance Field Representation
(CDFR)
In this section, we show the process that uses CDD to build a
complete distance field representation (CDFR), allowing exact
capture of all geometric details, e.g. sharp corners and edges, to
any level of accuracy.
Given a surface mesh, in the voxelization step, we store CDD
tuples with each surface voxel, rather than single valued distances.
For each triangle touching a surface voxel, a CDD tuple is
stored with that voxel. The end result of the voxelization step
leaves all surface voxels with a list of CDD tuples, sorted in
ascending order by distance values. Fig. 6 provides an example of
voxelizing a single surface voxel, Vox. There are three triangles
touching Vox. T
2
is case C
1
, with T
1
and T
3
being case C
2
or C
3
.
The minimal distance of Vox, measured from the center of Vox is
d
2
. As a result, Vox has a sorted list of 3 CDD tuples.
At the end of voxelization, we have a volume where each
voxel which the surface intersects contains a list of polygons cut-
ting through it.
3.4 Distance Transform
For a distance transform, initially, we use an outside flooding
algorithm to eliminate all outside voxels from our computation.
For the remaining voxels, a contour by contour CDD propagation
is performed from the surface voxels to the interior. During this
process, a voxel looks for CDD tuples that have been newly propa-
gated to anyone of its 26-neighbors [11]. It inherits all new CDD
tuples from its neighbors, and for each triangle, it computes the
true Euclidean distance from its own position. An updated list of
CDD tuples are then sorted into ascending order and the first CDD
tuple in the list contains the current distance, cur_distance, of this
voxel. All the CDD tuples that contain a distance value within the
range:
[cur_distance, cur_distance + *voxel size]
are stored with that voxel. This is a sufficient range to guarantee
correctness in the distance transform, as we will prove in Section4.
The CDD tuples out of this range are discarded. This process of
distance transform iterates until no voxels find new CDD tuples
from its 26-neighbors affecting its current CDD tuples list.
3.5 Reconstructing A Distance Contour
The most frequent way in which a distance field is used is by
reconstructing or extracting an iso-distance contour. For instance, a
user asks the following request, “show me the zero distance con-
tour with an error tolerance of 0.5mm”. The conventional way of
reconstructing sub-voxel distance is to trilinearly interpolate in-
between voxels [7]. Often times this reconstruction step is embed-
ded in ray-casting procedures at rendering time. While this works
for some applications, there is no guarantee on the level of accu-
racy. From CDFR, we extract a distance contour with a fulfillment
of an arbitrarily high accuracy requirement. We store the extracted
distance contours as point-based models [6][20], so that we can
render the contours at high interactive rates with splatting
[4][12][18].
The extracting procedure works as following. Given a
requested interior thickness, t (t>0), we traverse these voxels with
a distance value in the range:
(2)
The requested iso-contour will pass through the span of these
voxels. Unlike marching cubes [10], We do not use conditions like
and , because the
underlying assumption of having a linear function is not true in our
case. There could be cases where the 8 corner voxels are just sur-
rounding the maximal thickness point in the model, and none of the
8 voxels exactly captures that maximum.
After identifying the relevant voxels, we then subdivide the
voxels into sub-voxels, or points [2]. We only extract the sub-vox-
els that are close to the desired surface into a point-based iso-dis-
tance contour represented the surface. In order to support the error
tolerance, E, picked by users, the size of each sub-voxel must be:
(3)
For each sub-voxel, or point, we compute the signed true dis-
tance for all the CDD tuples resident on each of the 8 cornering
voxels. The points that have the minimal positive distance value
within the range [t - E/2, t + E/2] are extracted into the point-based
iso-distance contour.
3.6 High Quality Gradients
Besides using the distance contour for analysis, visualizations
of the distance contours are also highly desired in applications. For
point-based models, having high quality normal information on
each point is essential for high image quality.
Our CDFR offers an additional advantage in this perspective.
When extracting the distance contour from the base triangle of
p1
p2
p3
p4
p6
p5
p1
t1
t2
t5
t3
t4
t6
t1
t1
t2
t6
p6
Figure 5: (a) 2D illustrations of the process to determine the sign
of the distance of a point. The solid black arrows depict the
outward normal direction of each triangle. Points p
2
through p
5
project into the triangles, i.e. case C
1
. The signs of the distances
of p
2
through p
5
are determined by evaluating the normal
direction of each point’s base triangle. p
1
and p
6
are examples of
C
2
cases. (b) Enlarged view of p
1
and p
6
. Both p
1
and p
6
are
outside.
V
V
N
1
N
2
N
1
N
6
<
VN
1
•
VN
2
•
T
2
is pnt’s base trian-
gle, p1 is outside
>
VN
1
•
VN
6
•
T
1
is pnt’s base trian-
gle, p6 is outside
(a)
(b)
T
1
T
2
d
3
T
3
d
2
d
1
Vox
CDD Tuple List of Vox:
{d
2
, T
2
}
{d
1
, T
1
}
{d
3
, T
3
}
// in ascending order
Figure 6: A 2D illustration of building a CDD tuple list for a
surface voxel, Vox. There are 3 triangles intersecting Vox. The
CDD tuple list is organized in ascending distance order, with the
minimal distance of Vox being d
2
.
3
t
3
2
-------voxelsize– t
3
2
-------voxelsize+,
minimal thickness
t
≤
maximal thickness
t
≥
3
2
-------subvoxelsizeE<
250
each sub-voxel, the normal of this point is computed. If this point
is of case C
1
to its base triangle, then the normal of the base trian-
gle is this point’s true gradient. If the point is one of the cases C
2
or
C
3
, the gradient is the vector V in Fig. 5. For instance, in a C
2
case,
the 3D point, P, first gets projected onto the closest edge. The gra-
dient is the vector connecting P and its projection. In C
3
cases, the
gradient direction is obtained by connecting P and the closest ver-
tex. Therefore, the normal vectors computed for the whole point-
based model is continuous and accurate. High quality per point
shading is thus supported.
4. Proof of Sufficiency
To prove the correctness of CDFR, we need a proof of suffi-
ciency. That is, when we need to reconstruct the local distance
field in the span of any voxel, all the surface primitives affecting
this local area are present on that voxel.
A surface primitive, such as a triangle, affects a local field in
3D space by being the closest surface triangle to at least one posi-
tion in this local area. Based upon this observation, we devise our
proof of sufficiency with a proof by contradiction:
Suppose in the CDFR, R, there exists a local voxel, V, in
whose span there exists at least one point, P(x,y,z), whose base tri-
angle, T, is not resident on the voxel, V.
Without loss of generality, we write the distance from P to T
as D. All distance fields are continuous functions, although they
may not have continuous derivatives. For a point,
P’(x+dx,y+dy,z+dz), that is closely neighboring P, the minimal
distance from P’ to T is bounded by:
(4)
Due to deduction, when P’ incrementally moves from P
towards V’s center point, it logically follows that the distance from
V’s center point to T is bounded by:
(5)
Equation (5) can be rewritten as:
(6)
However, the minimum distance to P, which is D, must also
be smaller than , with minD denoting the mini-
mum distance of the surface to V. Therefore, the distance of T to V,
must be within the following range:
(7)
Since P is in the span of V, the maximum possible distance is
voxel size, the range in (7) is actually a subset of:
(8)
Contradiction. Since during our distance propagation process,
Equation (8) is exactly the range that we maintain on each voxel.
Hence, triangle T must be resident on voxel V. The assumed case
can not exist. Proof completed.
We do not claim our storage is minimal. We might have kept
more CDD tuples on each voxel than necessary. However, enforc-
ing that minimality would introduce more complexity. As long as
we use a triangle index in CDD tuples instead of complete descrip-
tion of each triangle, the extra storage cost that we spend is low.
We have traded for simplicity in implementation.
5. Results and Analysis
The resolution of CDFR volumes do not affect the accuracy
of the distance field. Also, the CDFR construction step is indepen-
dent from the step that reconstructs iso-distance contours. Before
we analyze the performance of our approach, we show images of
distance contours on a few sample parts to demonstrate the accu-
rate Euclidean distance fields obtained. All point-based models are
rendered with splatting [12]. All results are collected on a SGI
Octane with a 300MHz processor and 512MB memory. Table 1
provides a full description of the models used for test and analysis
of our algorithm. We have also applied our algorithm to a very
complicated part, Engine Cylinder Head, for heavy section detec-
tion. The details of the Engine Cylinder Head model is described in
Section5.5.
5.1 Experimental Data Sets
We first look at some simple cases. Cubes and tetrahedral
cells are the simplest. They are convex and symmetric. With a true
Euclidean distance field, the thickness contours of different values
are in the exact shape of the outer surfaces, including the sharp
edges and corners (Fig. 7).
Concavities cause additional complexity in distance fields.
Two concave examples, a one-ended tooth and a six-pointed star
are shown in Fig. 8. For the one-ended tooth, we choose a small
thickness value to extract a contour close to the surface, while for
the six-pointed star, a larger thickness is chosen. The evolving
effects in Euclidean distance fields are interesting, with corners
being smoothed out on the interior distance contours in both Fig.
8a and Fig. 8b. Small thickness contours closer to the surfaces
retain more detail of the surface shape. The shape of the deeper
contours manifest more global features of the shape (Fig. 8).
All four models have a CDFR at a low resolution of
. To reconstruct the thickness contours in Fig. 7 and
8, we set the accuracy to 1/500 of the longest dimension of the
model. For all 4 models, there are less than 400K points in the
point-based contour. We obtained 3 frames/second rendering rates
of the accurate distance contours with per point shading.
Ddx
2
dy
2
dz
2
++– Ddx
2
dy
2
dz
2
+++[,]
Dsd
p
v
∫
– Dsd
p
v
∫
+[,]
DdistVP
,
()
–
DdistVP
,
()
+
[,]
minDdistVP
,
()
+
minDminD
2
distVP
,
()
+
[,]
3() 2⁄
minDminD 3voxelsize+[,]
Model No.
Triangles
Bounding Box
Size (x,y,z) (inch)
Maximal
Thickness(inch)
Cube 12 (5, 5, 5) 2.5
Tetrahedron 4 (1, 1, 1) 0.2
1-Tooth 16 (1, 2, 2) 0.48
6-Star 48 (1, 3, 3.46) 0.49
Connector 242 (6.9, 2.0, 2.9) 0.50
Brevi 1812 (38.1, 34.9, 96.0) 13.00
TABLE 1. Physical Information of Test Models.
Figure 7: A cube and tetrahedron, with the surface mesh shown
in semi-transparency. The distance contours (shown in red, per-
point shaded) are of thickness (a) 0.6 inch and (b) 0.1 inch.
(a)
(b)
3232
×
32
×
251