Screened Poisson Surface Reconstruction
MICHAEL KAZHDAN
Johns Hopkins University
and
HUGUES HOPPE
Microsoft Research
Poisson surface reconstruction creates watertight surfaces from oriented
point sets. In this work we extend the technique to explicitly incorporate
the points as interpolation constraints. The extension can be interpreted as
a generalization of the underlying mathematical framework to a screened
Poisson equation. In contrast to other image and geometry processing
techniques, the screening term is defined over a sparse set of points rather
than over the full domain. We show that these sparse constraints can
nonetheless be integrated efficiently. Because the modified linear system
retains the same finite-element discretization, the sparsity structure is
unchanged, and the system can still be solved using a multigrid approach.
Moreover we present several algorithmic improvements that together
reduce the time complexity of the solver to linear in the number of points,
thereby enabling faster, higher-quality surface reconstructions.
Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Compu-
tational Geometry and Object Modeling
Additional Key Words and Phrases: screened Poisson equation, adaptive
octree, finite elements, surface fitting
ACM Reference Format:
Kazhdan, M., and Hoppe, H. Screened Poisson surface reconstruction.
ACM Trans. Graph. NN, N, Article NN (Month YYYY), PP pages.
DOI = 10.1145/XXXXXXX.YYYYYYY
http://doi.acm.org/10.1145/XXXXXXX.YYYYYYY
1. INTRODUCTION
Poisson surface reconstruction [Kazhdan et al. 2006] is a well
known technique for creating watertight surfaces from oriented
point samples acquired with 3D range scanners. The technique
is resilient to noisy data and misregistration artifacts. However,
as noted by several researchers, it suffers from a tendency to
over-smooth the data [Alliez et al. 2007; Manson et al. 2008;
Calakli and Taubin 2011; Berger et al. 2011; Digne et al. 2011].
In this work, we explore modifying the Poisson reconstruc-
tion algorithm to incorporate positional constraints. This mod-
ification is inspired by the recent reconstruction technique of
Calakli and Taubin [2011]. It also relates to recent work in im-
age and geometry processing [Nehab et al. 2005; Bhat et al. 2008;
Chuang and Kazhdan 2011], in which a data fidelity term is used
to “screen” the associated Poisson equation. In our surface recon-
struction context, this screening term corresponds to a soft con-
straint that encourages the reconstructed isosurface to pass through
the input points.
The approach we propose differs from the traditional screened
Poisson formulation in that the position and gradient constraints
are defined over different domain types. Whereas gradients are
constrained over the full 3D space, positional constraints are
introduced only over the input points, which lie near a 2D manifold.
We show how these two types of constraints can be efficiently
integrated, so that we can leverage the original multigrid structure
to solve the linear system without incurring a significant overhead
in space or time.
To demonstrate the benefits of screening, Figure 1 compares results
of the traditional Poisson surface reconstruction and the screened
Poisson formulation on a subset of 11.4M points from the scan of
Michelangelo’s David [Levoy et al. 2000]. Both reconstructions are
computed over a spatial octree of depth 10, corresponding to an
effective voxel resolution of 1024
3
. Screening generates a model
that better captures the input data (as visualized by the surface
cross-sections overlaid with the projection of nearby samples),
even though both reconstructions have similar complexity (6.8M
and 6.9M triangles respectively) and required similar processing
time (230 and 272 seconds respectively, without parallelization).
1
Another contribution of our work is to modify both the octree
structure and the multigrid implementation to reduce the time
complexity of solving the Poisson system from log-linear to linear
in the number of input points. Moreover we show that hierarchical
point clustering enables screened Poisson reconstruction to attain
this same linear complexity.
2. RELATED WORK
Reconstructing surfaces from scanned points is an important and
extensively studied problem in computer graphics. The numerous
approaches can be broadly categorized as follows.
Combinatorial Algorithms. Many schemes form a triangula-
tion using a subset of the input points [Cazals and Giesen 2006].
Space is often discretized using a tetrahedralization or a voxel
grid, and the resulting elements are partitioned into inside and
outside regions using an analysis of cells [Amenta et al. 2001;
Boissonnat and Oudot 2005; Podolak and Rusinkiewicz 2005],
eigenvector computation [Kolluri et al. 2004], or graph cut
[Labatut et al. 2009; Hornung and Kobbelt 2006].
Implicit Functions. In the presence of sampling noise, a common
approach is to fit the points using the zero set of an implicit func-
tion, such as a sum of radial bases [Carr et al. 2001] or piecewise
polynomial functions [Ohtake et al. 2005; Nagai et al. 2009]. Many
techniques estimate a signed-distance function [Hoppe et al. 1992;
1
The performance of the unscreened solver is measured using our imple-
mentation with screening weight set to zero. The implementation of the
original Poisson reconstruction runs in 412 seconds.
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
2
•
M. Kazhdan and H. Hoppe
Fig. 1: Reconstruction of the David head
‡
, comparing traditional Poisson surface reconstruction (left) and screened Poisson surface reconstruction which
incorporates point constraints (center). The rightmost diagram plots pixel depth (z) values along the colored segments together with the positions of nearby
samples. The introduction of point constraints significantly improves fit accuracy, sharpening the reconstruction without amplifying noise.
Bajaj et al. 1995; Curless and Levoy 1996]. If the input points are
unoriented, an important step is to correctly infer the sign of the
resulting distance field [Mullen et al. 2010].
Our work extends Poisson surface reconstruction [Kazhdan et al.
2006], in which the implicit function corresponds to the model’s
indicator function χ. The function χ is often defined to have value 1
inside and value 0 outside the model. To simplify the derivations, in
this paper we define χ to be
1
2
inside and
−1
2
outside, so that its zero
isosurface passes near the points. The function χ is solved using a
Laplacian system discretized over a multiresolution B-spline basis,
as reviewed in Section 3.
Alliez et al. [2007] form a Laplacian system over a tetrahedral-
ization, and constrain the solution’s biharmonic energy; the de-
sired function is obtained as the solution to an eigenvector prob-
lem. Manson et al. [2008] represent the indicator function χ using
a wavelet basis, and efficiently compute the basis coefficients using
simple local sums over an adapted octree.
Calakli and Taubin [2011] optimize a signed-distance function
to have value zero at the points, have derivatives that agree
with the point normals, and minimize a Hessian smoothness
norm. The resulting optimization involves a bilaplacian operator,
which requires estimating derivatives of higher order than in the
Laplacian. The reconstructed surfaces are shown to have good
accuracy, strongly suggesting the importance of explicitly fitting
the points within the optimization. This motivated us to explore
whether a Laplacian system could be extended in this respect, and
also be compatible with a multigrid solver.
Screened Poisson Surface Fitting. The method of Nehab et al.
[2005], which simultaneously fits position and normal constraints,
may also be viewed as the solution of a screened Poisson equation.
The fitting algorithm assumes that a 2D parametric domain (i.e.,
a plane or triangle mesh) is already established. The position and
derivative constraints are both defined over this 2D domain.
In contrast, in Poisson surface reconstruction the 2D domain
manifold is initially unknown, and therefore the goal is to infer an
indicator function χ rather than a parametric function. This leads
to a hybrid problem with derivative (Laplacian) constraints defined
densely over 3D and position constraints defined sparsely on the set
of points sampled near the unknown 2D manifold.
3. REVIEW OF POISSON SURFACE
RECONSTRUCTION
The approach of Poisson surface reconstruction is based on the
observation that the (inward pointing) normal field of the boundary
of a solid can be interpreted as the gradient of the solid’s
indicator function. Thus, given a set of oriented points sampling the
boundary, a watertight mesh can be obtained by (1) transforming
the oriented point samples into a continuous vector field in 3D,
(2) finding a scalar function whose gradients best match the vector
field, and (3) extracting the appropriate isosurface.
Because our work focuses primarily on the second step, we review
it here in more detail.
Scalar Function Fitting. Given a vector field
~
V : R
3
→ R
3
, the
goal is to solve for the scalar function χ : R
3
→ R minimizing:
E(χ) =
Z
k∇χ(p) −
~
V (p)k
2
d p. (1)
Using the Euler-Lagrange formulation, the minimum is obtained
by solving the Poisson equation:
∆χ = ∇ ·
~
V .
System Discretization. The Galerkin formulation is used to
transform this into a finite-dimensional system [Fletcher 1984].
First, a basis {B
1
,...,B
N
} : R
3
→ R is chosen, namely a collection
of trivariate (usually triquadratic) B-spline functions. With respect
to this basis, the discretization becomes:
h∆χ,B
i
i
[0,1]
3
= h∇ ·
~
V , B
i
i
[0,1]
3
1 ≤ i ≤ N
where h·, ·i
[0,1]
3
is the standard inner-product on the space of
(scalar- and vector-valued) functions defined on the unit cube:
hF,Gi
[0,1]
3
=
Z
[0,1]
3
F(p) · G(p) d p,
h
~
U,
~
V i
[0,1]
3
=
Z
[0,1]
3
h
~
U(p),
~
V (p)i d p.
Since the solution is itself expressed in terms of the basis functions:
χ(p) =
N
∑
i=1
x
i
B
i
(p),
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
Screened Poisson Surface Reconstruction
•
3
finding the coefficients {x
i
} of the solution reduces to solving the
linear system Ax = b where:
A
i j
= h∇B
i
,∇B
j
i
[0,1]
3
and b
i
= h
~
V , ∇B
i
i
[0,1]
3
. (2)
The basis functions {B
1
,...,B
N
} are chosen to be compactly
supported, so most pairs of functions do not have overlapping
support, and thus the matrix A is sparse.
Because the solution is expected to be smooth away from the input
samples, the linear system is discretized by first adapting an octree
to the input samples and then associating an (appropriately scaled
and translated) trivariate B-spline function to each octree node.
This provides high-resolution detail in the vicinity of the surface
while reducing the overall dimensionality of the system.
System Solution. Given the hierarchy defined by an octree of
depth D, a multigrid approach is used to solve the linear system.
The basis functions are partitioned according to the depths of their
associated nodes and, for each depth d, a linear system A
d
x
d
= b
d
is defined using the corresponding B-splines {B
d
1
,...,B
d
N
d
}, such
that χ(p) =
∑
D
d=0
∑
i
x
d
i
B
d
i
(p).
Because the octree-selected B-spline functions do not form a
complete grid at each depth, it is generally not possible to prolong
the solution x
d
at depth d into the solution x
d+1
at depth d + 1.
(The B-spline associated with a given node is a sum of B-spline
functions associated not only with its own child nodes, but also
with child nodes of its neighbors.) Instead, the constraints at depth
d + 1 are adjusted to account for the part of the solution already
realized at coarser depths.
Pseudocode for a cascadic solver, where the solution is only relaxed
on the up-stroke of the V-cycle, is given in Algorithm 1.
Algorithm 1: Cascadic Poisson Solver
1 For d ∈ {0, ... ,D} Iterate from coarse to fine
2 For d
0
∈ {0,. ..,d − 1} Remove the constraints
3 b
d
= b
d
− A
dd
0
x
d
0
met at coarser depths
4 Relax A
d
x
d
= b
d
Adjust the system at depth d
Here, A
dd
0
is the N
d
× N
d
0
matrix used to transform solution
coefficients at depth d
0
into constraints at depth d:
A
dd
0
i j
= h∇B
d
i
,∇B
d
0
j
i
[0,1]
3
.
Note that, by definition, A
d
= A
dd
.
Isosurface Extraction. Solving the Poisson equation, one obtains
a function χ that approximates the indicator function. Ideally, the
function’s zero level-set should therefore correspond to the desired
surface. In practice however, the function χ can differ from the true
indicator function due to several sources of error:
—The point sampling may be noisy, possibly containing outliers.
—The Galerkin discretization is only an approximation of the
continuous problem.
—The point sampling density is approximated during octree
construction.
To mitigate these errors, in [Kazhdan et al. 2006] the implicit
function is adjusted by globally subtracting the average value of
the function at the input samples.
4. INCORPORATING POINT CONSTRAINTS
The original Poisson surface reconstruction algorithm adjusts the
implicit function using a single global offset such that its average
value at all points is zero. However, the presence of errors can cause
the implicit function to drift so that no global offset is satisfactory.
Instead, we seek to explicitly interpolate the points.
Given the set of input points P with weights w : P → R
≥0
, we
add to the energy of Equation 1 a term that penalizes the function’s
deviation from zero at the samples:
E(χ)=
Z
k
~
V (p)−∇χ(p)k
2
d p +
α · Area(P )
∑
p∈P
w(p)
∑
p∈P
w(p)χ
2
(p) (3)
where α is a weight that trades off the importance of fitting the
gradients and fitting the values, and Area(P ) is the area of the
reconstructed surface, estimated by computing the local sampling
density as in [Kazhdan et al. 2006]. In our implementation, we
set the per-sample weights w(p) = 1, although one can also use
confidence values if these are available.
The energy can be expressed concisely as
E(χ) = h
~
V − ∇χ,
~
V − ∇χi
[0,1]
3
+ αhχ, χi
(w,P)
(4)
where h·,·i
(w,P)
is the bilinear, symmetric, positive, semi-definite
form on the space of functions in the unit-cube, obtained by taking
the weighted sum of function values:
hF,Gi
(w,P)
=
Area(P)
∑
p∈P
w(p)
∑
p∈P
w(p) · F(p) · G(p).
4.1 Interpretation as a Screened Poisson Equation
The energy in Equation 4 combines a gradient constraint integrated
over the spatial domain with a value constraint summed at
discrete points. As shown in the appendix, its minimization can be
interpreted as a screened Poisson equation (∆ − α
˜
I)χ = ∇ ·
~
V with
an appropriately defined operator
˜
I.
4.2 Discretization
We apply a discretization similar to that in Section 3 to the
minimization of the energy in Equation 4. The coefficients of the
solution χ with respect to the basis {B
1
,...,B
N
} are again obtained
by solving a linear system of the form Ax = b. The right-hand-side b
is unchanged because the constrained value at the sample points is
zero. Matrix A now includes the point constraints:
A
i j
= h∇B
i
,∇B
j
i
[0,1]
3
+ αhB
i
,B
j
i
(w,P)
. (5)
Note that incorporating the point constraints does not change the
sparsity of matrix A because B
i
(p) · B
j
(p) is nonzero only if the
supports of the two functions overlap, in which case the Poisson
equation has already introduced a nonzero entry in the matrix.
As in Section 3, we solve this linear system using a cascadic
multigrid algorithm – iterating over the octree depths from coarsest
to finest, adjusting the constraints, and relaxing the system. Similar
to Equation 5, the matrix used to transform a solution at depth d
0
to
a constraint at depth d is expressed as:
A
dd
0
i j
= h∇B
d
i
,∇B
d
0
j
i
[0,1]
3
+ αhB
d
i
,B
d
0
j
i
(w,P)
.
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
4
•
M. Kazhdan and H. Hoppe
Fig. 2: Visualizations of the reconstructed implicit function along a planar
slice through the cow
‡
(shown in blue on the left), for the original
Poisson solver, and for the screened Poisson solver without and with scale-
independent screening.
This operator adjusts the constraint b
d
(line 3 of Algorithm 1)
not only by removing the Poisson constraints met at coarser
resolutions, but also by modifying the constrained values at points
where the coarser solution does not evaluate to zero.
4.3 Scale-Independent Screening
To balance the two energy terms in Equation 3, it is desirable to
adjust the screening parameter α such that (1) the reconstructed
surface shape is invariant under scaling of the input points with
respect to the solver domain, and (2) the prolongation of a solution
at a coarse depth is an accurate estimate of the solution at a
finer depth in the cascadic multigrid approach. We achieve both
these goals by adjusting the relative weighting of position and
gradient constraints across the different octree depths. Noting that
the magnitude of the gradient constraint scales with resolution, we
double the weight of the interpolation constraint with each depth:
A
dd
0
i j
= h∇B
d
i
,∇B
d
0
j
i
[0,1]
3
+ 2
d
αhB
d
i
,B
d
0
j
i
(w,P)
.
The adaptive weight of 2
d
is chosen to keep the Laplacian and
screening constraints around the surface in balance. To see this,
assume that the points are locally planar, and consider the row of
the system matrix corresponding to an octree node overlapping
the points. The coefficients of the system in that row are the
sum of Laplacian and screening terms. If we consider the rows
corresponding to the child nodes that overlap the surface, we find
that the contribution from the Laplacian constraints scales by a
factor of 1/2 while the contribution from the screening term scales
by a factor of 1/4.
2
Thus, scaling the screening weights by a factor
of two with each resolution keeps the two terms in balance.
Figure 2 shows the benefit of scale-independent screening in
reconstructing a cow model. The leftmost image shows a plane
passing through the bounding cube of the cow, and the images
to the right show the values of the computed indicator function
along that plane, for different implementations of the solver.
As the figure shows, the unscreened Poisson solver provides a
good approximation of the indicator functions, with values inside
(resp. outside) the surface approximately 1/2 (resp. -1/2). However,
applying the same solver to the screened Poisson equation (second
from right) provides a solution that is only correct near the input
samples and returns to zero near the faces of the bounding cube,
2
For the Laplacian term, the Laplacian scales by a factor of 4 with
refinement, and volumetric integrals scale by a factor of 1/8. For the
screening term, area integrals scale by a factor of 1/4.
potentially resulting in spurious surface sheets away from the
surface. It is only with scale-independent screening (right) that we
obtain a high-quality solution to the screened Poisson equation.
Scale-Independence. Using this resolution adaptive weighting,
our system has the property that the reconstruction obtained by
solving at depth D is identical to the reconstruction that would be
obtained by scaling the point set by 1/2 and solving at depth D+1.
To see this, we consider the two energies that guide the reconstruc-
tion, E
~
V
(χ) measuring the extent to which the gradients of the so-
lution match the prescribed vector field, and E
(w,P)
(χ) measuring
the extent to which the solution meets the screening constraint:
E
~
V
(χ) =
Z
~
V (p) − ∇χ(p)
2
d p
E
(w,P)
(χ) =
Area(P)
∑
p∈P
w(p)
∑
p∈P
w(p)χ
2
(p).
Scaling by 1/2, we obtain a new point set ( ˜w,
˜
P) with positions
scaled by 1/2, unchanged weights, ˜w(p) = w(2p), and scaled area,
Area(
˜
P) = Area(P)/4; a new scalar field,
˜
χ(p) = χ(2p); and
a new vector field,
˜
~
V (p) = 2
~
V (2p). Computing the corresponding
energies, we get:
E
˜
~
V
(
˜
χ) =
1
2
E
~
V
(χ) and E
( ˜w,
˜
P)
(
˜
χ) =
1
4
E
(w,P)
(χ).
Thus, scaling the screening weight by a factor of two with each
successive depth ensures that the sum of energies is unchanged (up
to multiplication by a constant) so the minimizer remains the same.
4.4 Boundary Conditions
In order to define the linear system, it is necessary to define the
behavior of the function space along the boundary of the integration
domain. In the original Poisson reconstruction the authors imposed
Dirichlet boundary conditions, forcing the implicit function to have
a value of
−1
2
along the boundary. In the present work we extend the
implementation to support Neumann boundary conditions as well,
forcing the normal derivative to be zero along the boundary.
In principle these two boundary conditions are equivalent for
watertight surfaces, since the indicator function has a constant
negative value outside the model. However, in the presence of
missing data we find Neumann constraints to be less restrictive
because they only require that the implicit function have zero
derivative across the boundary of the integration domain, a property
that is compatible with the gradient constraint since the guiding
vector field
~
V is set to zero away from the samples. (Note that when
the surface does cross the boundary of the domain, the Neumann
boundary constraints create a bias to crossing the domain boundary
orthogonally.)
Figure 3 shows the practical implications of this choice when
reconstructing the Angel model, which was only scanned from
the front. The left image shows the original point set and the
reconstructions using Dirichlet and Neumann boundary conditions
are shown to the right. As the figure shows, imposing Dirichlet
constraints creates a water-tight surface that closes off before
reaching the boundary while using Neumann constraints allows the
surface to extend out to the boundary of the domain.
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
Screened Poisson Surface Reconstruction
•
5
Fig. 3: Reconstructions of the Angel point set
‡
(left) using Dirichlet (center)
and Neumann (right) boundary conditions.
Similar results can be seen at the bases of the models in Figures 1
and 4a, with the original Poisson reconstructions obtained using
Dirichlet constraints and the screened reconstructions obtained
using Neumann constraints.
5. IMPROVED ALGORITHMIC COMPLEXITY
In this section we discuss the efficiency of our reconstruction al-
gorithm. We begin by analyzing the complexity of the algorithm
described above. Then, we present two algorithmic improvements.
The first describes how hierarchical clustering can be used to re-
duce the screening overhead at coarser resolutions. The second ap-
plies to both the unscreened and screened solver implementations,
showing that the asymptotic time complexity in both cases can be
reduced to be linear in the number of input points.
5.1 Efficiency of basic solver
Let us begin by analyzing the computational complexity of the
unscreened and screened solvers. We assume that the points P are
evenly distributed over a surface, so that the depth of the adapted
octree is D = O(log |P|) and the number of octree nodes at depth d
is O(4
d
).
We also note that the number of nonzero entries in matrix A
dd
0
is O(4
d
), since the matrix has O(4
d
) rows and each row has at
most 5
3
nonzero entries. (Since we use second-order B-splines,
basis functions are supported within their one-ring neighborhoods
and the support of two functions will overlap only if one is within
the two-ring neighborhood of the other.)
Assuming that the matrices A
dd
0
have already been computed, the
computational complexity for the different steps in Algorithm 1 is:
Step 3: O(4
d
) – since A
dd
0
has O(4
d
) nonzero entries.
Step 4: O(4
d
) – since A
d
has O(4
d
) nonzero entries and the
number of relaxation steps performed is constant.
Steps 2-3:
∑
d−1
d
0
=0
O(4
d
) = O(4
d
· d).
Steps 2-4: O(4
d
· d + 4
d
) = O(4
d
· d).
Steps 1-4:
∑
D
d=0
O(4
d
· d) = O(4
D
· D) = O(|P| · log |P|).
There still remains the computation of matrices A
dd
0
.
For the unscreened solver, the complexity of computing A
dd
0
is O(4
d
), since each entry can be computed in constant time. Thus,
the overall time complexity remains O(|P| · log |P |).
For the screened solver, the complexity of computing A
dd
0
is O(|P|) since defining the coefficients requires accumulating
the screening contribution from each of the points, and each point
contributes to a constant number of rows. Thus, the overall time
complexity is dominated by the cost of evaluating the coefficients
of A
dd
0
which is:
D
∑
d=0
d−1
∑
d
0
=0
O(|P|) = O(|P| · D
2
) = O(|P| · log
2
|P|).
5.2 Hierarchical Clustering of Point Constraints
Our first modification is based on the observation that since the
basis functions at coarser resolutions are smooth, it is unnecessary
to constrain them at the precise sample locations. Instead, we
cluster the weighted points as in [Rusinkiewicz and Levoy 2000].
Specifically, for each depth d, we define (w
d
,P
d
) where p
i
∈ P
d
is the weighted average position of the points falling into octree
node i at depth d, and w
d
(p
i
) is the sum of the associated weights.
3
If all input points have weight w(p) = 1, then w
d
(p
i
) is simply the
number of points falling into node i.
This alters the computation of the system matrix coefficients:
A
dd
0
i j
= h∇B
d
i
,∇B
d
0
j
i
[0,1]
3
+ 2
d
αhB
d
i
,B
d
0
j
i
(w
d
,P
d
)
.
Note that since d > d
0
, the value hB
d
i
,B
d
0
j
i
(w
d
,P
d
)
is obtained by
summing over points stored with the finer resolution.
In particular, the complexity of computing A
dd
0
for the screened
solver becomes O(|P
d
|) = O(4
d
), which is the same as that of the
unscreened solver, and both implementations now have an overall
time complexity of O(|P| · log |P |).
On typical examples, hierarchical clustering reduces execution time
by a factor of almost two, and the reconstructed surface is visually
indistinguishable.
5.3 Conforming Octrees
To account for the adaptivity of the octree, Algorithm 1 subtracts
off the constraints met at all coarser resolutions before relaxing
at a given depth (steps 2-3), resulting in an algorithm with log-
linear time complexity. We obtain an implementation with linear
complexity by forcing the octree to be conforming. Specifically,
we define two octree cells to be mutually visible if the supports of
their associated B-splines overlap, and we require that if a cell at
depth d is in the octree, then all visible cells at depth d −1 must also
be in the tree. Making the tree conforming requires the addition of
new nodes at coarser depths, but this still results in O(4
d
) nodes at
depth d.
While the conforming octree does not satisfy the condition that
a coarser solution can be prolonged into a finer one, it has the
property that the solution obtained at depths {0, . . .,d − 1} that
is visible to a node at depth d can be expressed entirely in terms
of the coefficients at depth d − 1. Using an accumulation vector
to store the visible part of the solution, we obtain the linear-time
implementation in Algorithm 2.
3
Note that the weight w
d
(p) is unrelated to the screening weight 2
d
introduced in Section 4.3 for scale-independent screening.
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
![Fig. 8: Reconstruction of the noisy Eagle dataset‡ using the original scheme [Kazhdan et al. 2006] and our new algorithm, with various settings of the screening parameter (α) and samples per node (SPN). (The highlighted results are those with the default parameters used for evaluation in Section 6.1.)](/figures/fig-8-reconstruction-of-the-noisy-eagle-dataset-using-the-38dxaiet.webp)
![Fig. 9: Reconstruction of the noisy Eagle dataset‡ using the Smoothed Signed Distance reconstruction [Calakli and Taubin 2011] using various settings of the Hessian weighting parameter (h). (The highlighted results are those with the default parameters used for evaluation in Section 6.1.)](/figures/fig-9-reconstruction-of-the-noisy-eagle-dataset-using-the-2t8w2kd9.webp)
