SURFACE APPROXIMATION TO POINT CLOUD
DATA USING VOLUME MODELING
Adam Huang & Gregory M. Nielson
Arizona State University, Tempe, AZ, USA
Abstract:
Given a collection of unorganised points in space, we present a new method of
constructing a surface which approximates this point cloud. The surface is
defined implicitly as the isosurface of a trivariate volume model. The volume
model is piecewise linear and obtained as a least squares fit to data derived
from the point cloud. The original point cloud input is assigned a zero value.
Additional points are derived for the interior and exterior and assigned positive
and negative values respectively.
Key words:
surface approximation, unorganised points, point clouds
1.
INTRODUCTION
We describe a new method for fitting an approximating surface to a point
cloud of data. A point cloud simply consists of a collection of unorganized
points ( x s , y s , z s ) s = 1, , S . We assume that these points are on or near a
surface. We find an approximation to this surface as the isosurface of a
trivariate, volume model, F ( x, y, z ) = 0 . The function F is a piecewise
linear trivariate function and is obtained by using least squares fitting applied
to the original data which are assigned zero values and certain additional
data points which are interior and exterior to the surface and assigned
positive and negative values respectively. Results of this approach are
shown in Figures 1 and 2. The input for Figure 2 is the Stanford bunny data
set which is obtained with a laser scanner. The data of Figure 1 is obtained
by sampling a mathematical function of two linked tori. Once the sample
points are obtained no additional information about the source or how the
data was collected is used. This, linked tori example, points out that our
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Adam Huang & Gregory M. Nielson
modelling technique can produce surface with disjoint, linked segments with
non-trivial genus.
Figure 1. Point cloud input on the left and the resulting smooth shaded triangular mesh
surface on the right.
Figure 2. Point cloud input on the left and the resulting triangular mesh surface
approximation on the right.
Fitting surfaces that interpolate or approximate point clouds had been
studied for over a dozen years. See [5], [3], [2] and [1] for example. A good
survey on the subject is [6]. The current paper is based upon [4] and is
organized as follows. In Section 2, we describe the details of the piece-wise
linear volume model and discuss methods for computing optimal
approximations. Section 3 is devoted to a description of the various methods
SURFACE APPROXIMATION TO POINT CLOUD DATA USING
VOLUME MODELING
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we use for obtaining the additional interior and exterior data points. More
information on results is covered in Section 4.
2.
THE VOLUME MODEL FUNCTION AND THE
FITTING PROCESS
We view the input data ( x s , y s , z s ) s = 1, , S as a collection of
unorganized sample points on a surface of interest. We assume no other
input information. All properties that are used in the creation of a surface
which approximates this point cloud are derived from the original data. Our
general approach is to compute a trivariate volume model, F ( x, y , z ) , so that
the isosurface S = {( x, y, z ) such that F ( x, y, z ) = 0} approximates the point
cloud input. The function F ( x, y , z ) is determined in a least squares fitting
sense to be approximately 0 for surface points, -1 for interior points and +1
for exterior points. The three distinct function values correspond to points
( x r , y r , z r ) “interior to”, ( x s , y s , z s ) “on”, or ( xt , y t , z t ) “exterior to” the
surfaces respectively. The “on” points are the given unorganized data while
“exterior” and “interior” are derived from them. How we derive these
additional points is covered in the next section. For the remainder of this
section, we assume that these three types of data points are already available.
The form of F is a piece-wise linear function. We write F in the form
F ( x, y , z ) =
ai , j ,k bi , j ,k ( x, y, z )
i, j ,k
where bi , j ,k is +1 at a grid point with index (i, j, k ) and zero at all other grid
points. The grid will consist of a regular, Cartesian grid, of size N × N × N
and additionally the centers of these cells. See in Figure 3.
Figure 3. The grid points which serve to define the volume model.
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Adam Huang & Gregory M. Nielson
The entire domain is decomposed into tetrahedra with edges joining each of
the N × N × N grid (gray points in Figure 3) with the center points (black
points in Figure 3). The function, F, is linear over each of these tetrahedra.
The coefficients of the volume model are determined by mininmizing the
quantity
2
Φ (ai , j , k ) =
R
ai , j , k bi , j , k ( xr , yr , zr ) + 1
r =1 i , j , k
2
S
+
ai , j , k bi , j , k ( xs , ys , zs )
s =1 i , j , k
2
T
+
ai , j , k bi , j , k ( xt , yt , zt ) − 1
t =1
i, j,k
This is a quadratic form in the variables a i , j ,k . The values of the coeficients
which minimize this quantity must be zeros of the gradient of Φ and so we
have the linear system of normal equations,
∂Φ
(ai , j , k ) = 0
∂a , m , n
This linear system is symmetric and sparse. With the grid system and basis
functions we use, each row can have at most 9 non-zero elements. We use a
data structure that is related to the grid over which the basis functions are
defined and only keeps track of the non-zero entries of the coefficient matrix
of the normal equations. The Gauss-Seidel iteration scheme is used to solve
this sparse linear system.
After solving for the a i , j ,k ’s, an approximation surface can be
constructed by computing the isosurface F ( x, y , z ) = 0 . We use a simple
marching tetrahedra approach where the tetrahedra are the same as used in
the description of the volume model.
3.
SAMPLING
In this section we will discuss some schemes based on ray casting
concepts to generate a good sample set of points from unorganized data. A
good sample should be correct-valued and well-distributed. By the term of
SURFACE APPROXIMATION TO POINT CLOUD DATA USING
VOLUME MODELING
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correct-valued, we assure that the function values of “interior”, “on”, and
“exterior” points err at most within the range of a grid unit. By the term of
well-distributed, we require that there are a moderate number of sample
points in every grid unit. An ill-distributed sample might result in some zero
coefficients in the linear system and will result in an erroneous solution.
In order to generate correct-valued and well-distribution sample points
from unorganized surface data, we introduce a new instrument called oneeighth cubes. As the name suggests, one-eighth cubes are made by dividing
basic cubes into eight equal sized cubes which have two grid points as their
vertices as shown in Figure 4. These small cubes are named as one-eighth
cubes or 1/8 cubes.
Figure 4. One-eighth cubes.
By dividing the grid system into 8 smaller cubes, we can generate sample
points that satisfy the well-distribution property. We assign an InOut tag to
each 1/8 cube to record the status within each 1/8 cube in the system. Before
performing any sampling algorithm, a procedure is performed to initialize all
InOut tags to be “inside”. The procedure then browses through 1/8 cubes to
update the tags to be “on” for those 1/8 cubes with data points inside. The
initialization procedure is:
for each 1/8 cube
cube->InOut = “inside”; //initialization
for each data point
{
cube = find_cube(point); // find the 1/8 cube with point inside
if(cube->InOut = = “inside”)
cube->InOut = “on”; // set InOut tag to “on” for 1/8 cubes with
// data points inside
}
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Adam Huang & Gregory M. Nielson
The result is shown in Figure 5 where dark gray areas are “inside”
and light gray areas are “on”.
In the rest of this section we will discuss three sampling methods based
on casting rays on or away from surface data points.
On
Inside
Outside
Figure 5. The beginning state where all non-surface points are initialised to “inside”.
3.1 Parallel Ray Casting
The parallel ray casting algorithm casts a set of parallel rays toward
objects to detect the possible surface existence. Although the direction of
parallel rays can be randomly chosen, we will only use positive and negative
x, y, and z direction rays for simplicity and efficiency.
At this stage we assume that the initialization procedure has been
performed and has resulted in a closed space by the “on” cubes without
holes. We assert that "without holes" means a point inside the space
enclosed by the “on” cubes cannot go outside without passing through any
“on” cube's vertices, edges, or faces. (See Figure 5)
The positive x direction rays are performed first as shown in Figure 6
from the left side. Before a ray hits any “on” cube, the cubes it travels
through must be “outside.” The InOut tag is assigned by “outside” which is
denoted by the white areas in Figure 6. At the same time, equally spaced
“exterior” points are created with function values equal to 1. The ray
advances from one cube to the next and repeats the same procedure until it
hits an “on” cube or reaches the last cube. Repeat other directions in the
same way except when it hits an “outside” cube it passes to the next cube
without creating new sample points. Figure 6 shows the result after applying
the six basic ray directions.
SURFACE APPROXIMATION TO POINT CLOUD DATA USING
VOLUME MODELING
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The order of performing these six basic directions will not change the
result. For special cases such as the bunny data set that has holes in the
bottom, the initialized space is no longer a close space. To prevent the ray
casting and marking system from mistakenly eroding the “inside” cubes, the
positive z direction rays are omitted. Similarly for any data set that misses
part of data from a particular view angle, a carefully chosen set of ray
directions is necessary.
On
Inside
Outside
On
Inside
Figure 6. The results of the Parallel Ray Casting to find exterior points.
3.2 An Optional Mobile Ray Source
On
Inside
Outside
Figure 7. Illustrating the results of the Mobile Ray Source.
Outside
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Adam Huang & Gregory M. Nielson
For objects with concave shapes, parallel rays in x, y, and z directions
may not detect their hidden contours. To improve the process, an optional
mobile ray source can travel each “outside” cube and from its current cube
location it again can cast the six basic direction rays (positive and negative x,
y, and z) to find the hidden cubes. The same process is repeated until no
more hidden cubes can be found. Figure 7 shows how the mobile ray source
finds the hidden cubes.
The reason that this process is suggested as an option is because some
data sets such as the bunny data may miss some data from certain angles.
Without modifying the marking system we use so far, the mobile ray source
will erode the “inside” body by going inside the object and marking every
“inside” cube as an “outside” cube. There will be only “on” and “outside”
cubes left at the end. A possible remedy to prevent this problem is to add
one type of cubes, call “reserved” cubes, which are assigned by the user to
seal the holes where data points are missing. Therefore, the new mobile ray
source cannot pass through both “on” and “reserved” cubes. This method
will require human manipulation and the result will depend on how the
“reserved” cubes are chosen.
Although we treat this mobile ray source algorithm as a secondary
method, it actually can find all correct “outside” cubes alone without
performing any parallel ray casting. We can choose a seed as the beginning
point, say the cube at (0,0,0), and apply the algorithm continually until no
more new “outside” cubes are found. The reason we did not choose this
method is that the parallel ray casting provides more flexibility in dealing
with missing data. Combining both parallel ray casting and mobile ray
source provides a more versatile solution than mobile ray source itself.
The “inside” cubes which remain untouched by parallel ray casting and
mobile ray source can be filled with equally spaced “interior” points with
function value equal to −1. The procedure is similar to the one for creating
exterior points except for the opposite function value.
So far, we have succeeded in identifying the correct types of all 1/8 cubes
by applying parallel ray casting and mobile ray source algorithms combined
with the tag checking system. A simple function that generates equalspaced, single-typed sample points is available for both “inside” and
“outside” cubes.
The only task left is how to generate correct
interior/exterior sample points within “on” cubes. In the next section we will
introduce another algorithm, local scattered spherical ray casting, and show
how to generate multiple-typed sample points within “on” cubes correctly by
casting rays from data points to neighboring cubes.
SURFACE APPROXIMATION TO POINT CLOUD DATA USING
VOLUME MODELING
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3.3 Local Scattered Spherical Ray Casting
The idea of casting rays from data points is borrowed from wave theory.
Local scattered spherical ray casting is a simplified example of the point
light source model which sends out light in sphere-shaped wave fronts
shown in Figure 8. Every “on” point will be a spherical light source. These
spherical wave fronts will propagate away from their origin. As we
gradually increase the radius of a sphere, the wave front will eventually hit
all its neighbor cubes and return their types to help us generate sample points
with correct function values. Although the idea of a wave front will work
for any direction and to any distance until stopped by other “on” cubes, we
do not want to over-sample within a 1/8 cube. Therefore, as the term “local”
suggests, the wave front will travel for only a short distance within the light
source's own “on” cube. The other term “scattered” suggests that the
spherical wave front will be replaced by moderately few rays in certain
directions only.
There are 26 possible neighbors a cube can have. They include 12 cubes
which share an edge, 8 which share a vertex, and 6 which share a face with
the cube. Therefore, there are 26 possible directions we can cast. If we
consider the positive and negative directions as the same, the number
direction can be reduced to 13.
After performing parallel ray casting and mobile ray source algorithms,
we have marked all 1/8 cubes into three types, “inside”, “outside”, and “on”.
We name “on” cubes as ambiguous neighbors because we cannot tell
whether the trace along this particular direction is inside or outside of the
object. In addition, the cube in which the light source is located is also
ambiguous itself. If the details of the surface have fine shapes within the
cube, they are beyond the current grid size ability to capture them.
As we have 13 directions to cast rays, each direction has a pair of
neighbors on the opposite sides. As we mark 1/8 cubes into three types,
there are 6 possible combinations of a pair of neighbors. The 6 cases can be
divided into two groups. The first group includes + 0, − 0, and + −, where
+, −, and 0 are short notes for “inside”, “outside”, and “on”. Sample points
can be generated in these three cases from data points along the particular
direction to the cubes' boundary. From each “on” point to its positive (
negative) neighbor we create equally spaced exterior (interior) points. The
second group includes + +, − −, and 0 0. The cases of + + and − − are
the result of fine shapes or possible hole problems. The case of 0 0 is the
result of fine shapes or simply indicates that the ray in this particular
direction does not have enough information to generate correct sample
points.
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Adam Huang & Gregory M. Nielson
If at least 1 of these 13 pairs belongs to first group, the cube can be
correctly sampled. For the cubes that have a pair of neighbors belonging to
the second group, we only take the given data points as “on” sample points
On
Inside
Outside
Figure 8. Illustrating Spherical Ray Casting
4.
EXAMPLES
The bunny data set of Figure 2 is gathered from 10 range images
which we register into a set of 362,272 unorganized points. Additional
interior and exterior points were added by using a combination of parallel
ray casting and local spherical ray casting. The volume model is based upon
a 36 × 36 × 36 grid and has 97,309 unknown parameters. We used an iterative
method to solve the 97,309 × 97,309 system of equations in 8.9 minutes on
200MHz PC. For the example of Figure 1, we randomly sampled two
parametric tori to produce 200,000 points. A combination of parallel ray
casting, mobile ray source and local spherical ray casting was used to obtain
the additional interior and exterior points. The grid size for the volume
model is 24 × 24 × 24 and the linear system of equations with 29,449
unknowns was solved in 4.1 minutes.
ACKNOWLEDGEMENTS
The bunny data set of Figure 2 is from the Computer Graphics Group,
Stanford University. The tori data sets were suggested by Gerald Farin.
SURFACE APPROXIMATION TO POINT CLOUD DATA USING
VOLUME MODELING
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This research was supported by NSF IIS-9980166 & ACI-0083609, ONR
N00014-00-1-0281 and DARPA MDA972-00-1-0027.
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