Creating Curvature Adapted

Subdivision Control Meshes from Scan Data

Simon Kloiber

and Ursula H. Augsd

orfer

Institute of Computer Graphics and Knowledge Visualisation,

Graz University of Technology, Inffeldgasse 16c, Graz, Austria

Keywords:

Quad-dominant Remeshing, Surface Reconstruction.

Abstract:

Often, designers have real-life models which need to be converted to a mathematical representation for fur-

ther processing. For the designer to be able to manipulate the data sensibly and in a controlled manner the

number of data points have to be reduced. However, if the new reduced representation of the shape is sparse

everywhere, high frequency detail in the model will be lost. In this work we modify an existing quad meshing

algorithm to convert a dense triangle mesh capturing the shape of the real-life model to a quad-dominant mesh

of varying density. Our distribution of vertices allows to represent high frequency features in the surface,

without increasing the density of the mesh elsewhere unnecessarily. Our quad mesh approximates the scan

data up to a predeﬁned error margin. This quad mesh is then transformed into a subdivision control mesh,

which corresponds to a limit subdivision surface which closely resembles the scan data.

1 INTRODUCTION

Often, designers create real-life models of their ideas.

The real-life model is then scanned to obtain a dig-

ital representation in form of a point cloud, which

can be converted to a triangle mesh. Because these

meshes closely capture the shape of the real-life mod-

els, we will refer to these dense triangular meshes as

reference meshes. To be able to further reﬁne the de-

sign within a design pipeline, a mathematical repre-

sentation of the shape has to be found. Subdivision

surfaces are able to model complex shapes with a

single surface. They are deﬁned by a coarse set of

control points and a subdivision algorithm. To con-

tinue the design with subdivision surfaces, the ref-

erence mesh has to be converted into a subdivision

control mesh. The extraction of an adaptive subdivi-

sion control mesh from dense triangle meshes is the

focus in this paper. There are many different sub-

division schemes resulting in limit surfaces with dif-

ferent mathematical properties. While the algorithm

presented is applicable to any subdivision scheme, we

will base all examples on the Catmull-Clark scheme

since it is the de facto standard subdivision scheme in

the CAD and entertainment industry. Catmull-Clark

subdivision operates on quadrilateral meshes.

https://orcid.org/0000-0003-1186-7630

The paper is structured as follows. In Section 2 we

discuss related work on the topic of quad mesh gener-

ation. In Section 3 we describe how we extend an ex-

isting algorithm for isotropic quad meshing to create

non-uniform isotropic quad meshes that are sensitive

to the curvature of the surface and adapt the size of

the quadrilaterals such that the error of the extracted

quad-dominant mesh to the reference mesh does not

exceed a given threshold. We also describe a simple

way to convert a given quadrilateral polyhedron to a

Catmull-Clark control mesh so that the limit surface

closely describes the shape captured in the reference

mesh. We present our results in Section 4. We close

with the conclusion in Section 5.

2 QUADRILATERAL MESHING

Because Catmull-Clark subdivision is based on a

quadrilateral mesh, we ﬁrst need to compute a quad

mesh (also called quadrangulation or quad meshing)

from a given input mesh. Quad meshing is a topic of

interest in many applications, like texture mapping,

parameterisation and ﬁnite element analysis and there

are numerous different approaches and methods. A

comprehensive survey is e.g. provided by (Bommes

et al., 2013b).

Quad meshing methods either deﬁne parametrisa-

Kloiber, S. and Augsdörfer, U.

Creating Curvature Adapted Subdivision Control Meshes from Scan Data.

DOI: 10.5220/0008989601490159

In Proceedings of the 15th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2020) - Volume 1: GRAPP, pages

149-159

ISBN: 978-989-758-402-2; ISSN: 2184-4321

149

tions or curves, or employ triangle decimation to cre-

ate quads. Parametrisations can be created through

global optimisations (Ray et al., 2006; Hormann and

Greiner, 2000; K

alberer et al., 2007; Bommes et al.,

2009; Bommes et al., 2013a; Liu et al., 2011; Myles

et al., 2010) or by deﬁning a coarse layout of charts

which are then parametrised (Tong et al., 2006; Dong

et al., 2006; Zhang et al., 2010; Zhang et al., 2013).

Another approach is deﬁning curves on the reference

mesh in a cross-based layout. Creating pure quad

meshes is difﬁcult with straightforward curve based

approaches because additional structures are needed

(Campen et al., 2012). Quad-dominant approaches

integrating curves are more prevalent (Alliez et al.,

2003; Marinov and Kobbelt, 2004; Dong et al., 2005).

A third approach employs triangle mesh decimation

(Lai et al., 2010), where quadrilateral faces are cre-

ated by combining triangles, and makes use of the fact

that creating a triangulation is easier than creating a

quadrangulation.

Some of these approaches allow anisotropy and

adapt the mesh density according to curvature:

(Zhang et al., 2010) create their pure quad mesh

via parametrising the quasi-dual Morse-Smale Com-

plex of a standing wave function with manual size

control. (Alliez et al., 2003) create quad-dominant

meshes from genus-0, non-closed input meshes. They

use a smoothed curvature tensor ﬁeld to guide curves

in anisotropic regions and point sampling in spher-

ical regions. The layout of the curves (i.e. den-

sity, anisotropy, geometric accuracy) can be inﬂu-

enced manually. (Marinov and Kobbelt, 2004) cre-

ate a parametrization on an arbitrary input mesh on

which curvature lines are traced. The traced lines sat-

isfy a manually set sampling density. (Dong et al.,

2005) create a smooth harmonic scalar ﬁeld over the

mesh. They compute a gradient vector ﬁeld and use a

second vector ﬁeld which is orthogonal to the gradient

ﬁeld to trace integral lines. The spacing of these lines

is curvature sensitive and may be controlled manu-

ally. (Lai et al., 2010) remesh the given triangle mesh

such that the density of the newly created triangle

mesh adapts to curvature. They then perform a re-

laxation on the triangle mesh which aligns the edges

of the triangles such that non-aligned edges can be

dropped to form curvature aligned quads. (Kovacs

et al., 2011) introduce an anisotropic metric to fea-

ture aligned and harmonic parametrisations. This im-

proves the approximation quality of these isotropic

approaches. The quad-dominant remeshing algorithm

by (Jakob et al., 2015) creates isotropic uniform quad-

dominant meshes through quick, local optimisation of

a parametrisation from triangle or point cloud refer-

ence meshes.

The results of uniform quad meshing methods

cannot create sparse subdivision control meshes that

have a low approximation error. We present a non-

uniform quad meshing method, which adapts the den-

sity of control points to the degree of detail present in

the original surface. To ensure an accurate representa-

tion of surface features, we adjust the size of mesh el-

ements to reﬂect curvature variations, but also aim to

align the mesh with shape features. This is achieved

by modifying the algorithm presented in (Jakob et al.,

2015) to respect the curvature detail of the original

scan data.

3 ADAPTIVE QUAD MESHING

This section provides a comprehensive guide through

our changes to Instant Meshes (Jakob et al., 2015).

We created a work-chain that is able to transform ref-

erence meshes into subdivision control meshes with

densities based on the curvature information of the

referenced surface. The local approach of optimising

the parametrization used by Instant Meshes is ideal

for using curvature information to shape the grid size

locally around each vertex. By adapting the local grid

size to the curvature of the shape we create a mesh

with large similar sized regular quadrilateral faces in

ﬂat regions with low curvature, and small quadrilat-

eral faces in high curvature regions in order to cap-

ture high frequency details in the model. We con-

strain the orientation ﬁeld estimation and feed the

edge lengths into the position ﬁeld estimation. The

resulting mesh is post-processed to guarantee mani-

foldedness, to align quads with ridges, and to improve

the topology of ﬂat areas. Finally, the extracted mesh

is transformed into a subdivision control mesh such

that its limit surface corresponds to the reference sur-

face given by the reference mesh.

The following steps are performed to extract

a quad-dominant subdivision control mesh with

curvature-based densities:

1. Build a multi-resolution hierarchy of the mesh.

For each hierarchy level, estimate the curvature,

compute the scales, and detect ridge vertices.

2. Optimise the orientation ﬁeld.

3. Optimise the position ﬁeld.

4. Extract the mesh.

5. Perform mesh clean-up.

6. Perform mesh smoothing.

7. Create the subdivision control mesh.

Changes in the density of Catmull-Clark subdivi-

sion control meshes are difﬁcult to achieve because of

GRAPP 2020 - 15th International Conference on Computer Graphics Theory and Applications

150

their quad-based nature. Transitions from low to high

densities have to be done with care because they will

introduce additional extraordinary vertices (EVs). An

extraordinary vertex has a valency other than regular,

that is either more or less than four. Typically, the EVs

are avoided, since they are associated with curvature

problems in the limit surface. In this work, however,

we accept the appearance of EVs in order to capture

shape information encapsulated in high frequency re-

gions of the surface.

3.1 Curvature Estimation

There are two ways to estimate curvature on a piece-

wise linear mesh: Fitting parametric surfaces locally

around each vertex or estimating curvature in a dis-

crete sense based on the neighbourhood of a vertex

(Petitjean, 2002; Gatzke and Grimm, 2006). Paramet-

ric surface ﬁtting generally produces better results at

the expense of computation time compared to other

methods. Most discrete methods are linear in time

because they compute sums over neighbours, but are

more susceptible to noise (Gatzke and Grimm, 2006).

We employ principal curvatures to ﬁt a parametric

surface, because they are more precise and uniform

compared to other methods. The improved preci-

sion warranted the increased time complexity (a fac-

tor of three for an input mesh with 50000 vertices and

100000 faces).

The parametric surface ﬁtting method of cubic or-

der developed in (Goldfeather and Interrante, 2004)

estimates principal curvature directions well and com-

putes satisfying results. To estimate the principal di-

rections and curvature, a cubic order B-spline patch

is ﬁtted onto the neighbourhood of a vertex. For this,

neighbouring vertices are brought into a local coordi-

nate frame to create a system of equations where the

neighbours describe the shape of the parametric sur-

face. We found a 2-ring neighbourhood works well.

Every neighbouring point introduces three equations

with one equation for the position and two equations

for the normals. The least-squares solution of this sys-

tem of linear equations is the Weingarten curvature

matrix of the cubic surface. The Eigenvalues of this

matrix are the principal curvatures and its Eigenvec-

tors are the corresponding principal directions.

From the estimated curvature, we take the largest

absolute principal curvature as our guide for edge

lengths. This ensures that the highest frequency de-

tail is always respected.

3.2 Curvature Averaging

The curvature information might not be as smooth as

needed for good results. This may be due to noise in

the reference mesh or multiple different features com-

ing together in the mesh. To address these issues, we

tried different averaging methods to achieve a smooth

curvature distribution. We also tried different weigh-

ing schemes, either in order to increase the inﬂuence

of high curvature regions or based on similarity (be

it curvature or normal similarity). We found that uni-

form weights for all neighbours in a 2-ring neighbour-

hood produced the most consistent results.

3.3 Error Estimation and Edge Length

A scale parameter employed in the work of (Jakob

et al., 2015) deﬁnes a global desired edge length of

the extracted mesh. This scale inﬂuences the position

ﬁeld calculation and the extraction itself. In the po-

sition ﬁeld, the scale deﬁnes the positional invariance

points of the ﬁeld while in the extraction it is used

to classify edges and to remove unnecessary edges.

To adjust scale based on curvature, the ﬁrst step is

unravelling the global scale by giving each point its

own. The transformation of curvature to scale is a

one-dimensional map and is very similar to a simple

transfer function in volume rendering (Ljung et al.,

2016).

All information necessary to deﬁne a desired edge

length for a given vertex that ﬁts the complexity of the

underlying surface is provided by the curvature esti-

mation. The relation of a tangent to the osculating

circle provides an error metric. We can use it to esti-

mate the error when moving along a curve in direction

of the tangent direction away from the vertex. As il-

lustrated in Figure 1, for a vertex v with curvature κ

along a tangential curve on the surface, the osculating

circle has radius

|κ|

. When we move away from the

point along this curve by x, we denote the estimated

distance to the curve as ε. Users can select the desired

error, which drives the edge length computation.

(Alliez et al., 2003) have used a similar estima-

tion that places the constructed edge such that its end-

points are on opposing sides of the circle (i.e. the

vertex is its centre). This way, the same error ε will

allow an edge length of 2x(ε). Their derivation creates

edges that are larger and become smaller too slowly.

Hence, the factor of two is not incorporated in our

edge length computation.

Because the derived edge lengths tend to inﬁn-

ity as the curvature approaches zero, we cap edge

lengths to remain smaller than ﬁve times the global

edge length. In areas where the edge length is too

Creating Curvature Adapted Subdivision Control Meshes from Scan Data

151

r =

|κ|

= (r − ε)

+ x

= r

− (r − ε)

|κ|

− (

|κ|

− ε)

|κ|

ε − ε

x(ε) =

|κ|

ε − ε

0 ≤ ε ≤

|κ|

Figure 1: The derivation of the maximum edge length x(ε)

with an error of ε, for tangential edges going out of vertex v

with curvature κ.

small for the density of the reference mesh, the posi-

tion ﬁeld can either not form clusters or forms clus-

ters that are too small to extract vertices. If not spec-

iﬁed differently by a user, (Jakob et al., 2015) set the

global edge length such that the expected number of

extracted vertices is 1/16th of the number of refer-

ence vertices. This expectation has become less pre-

cise due to varying quad sizes but is still a good in-

dicator to ensure that edge lengths do not become too

small. Setting the minimal edge length to be between

70% and ﬁve times the global edge length has proven

to be a good estimate.

3.4 Hierarchy

The method in (Jakob et al., 2015) creates a multi-

resolution hierarchy of the reference mesh to improve

stability and convergence of the orientation and posi-

tion ﬁeld estimation. The lowest level of the hierarchy

contains all vertices of the reference mesh. New lev-

els of the hierarchy are created by merging pairs of

vertices with the highest weights. Weights are calcu-

lated via the dot product of vertex normals, scaled by

the relation of maximum over minimum area of their

barycentric cells to ensure a balanced hierarchy tree.

The curvature on each level of the hierarchy

is estimated based on the vertices and their adja-

cency. Since vertices are merged through averages,

the higher level mesh might have areas with different

curvatures than the original mesh. Whether the cur-

vature information of higher levels resembles the cur-

vature of the original mesh depends on the weighting

that decides which points are supposed to be merged

on each level.

3.5 Orientation Field Optimisation

While the orientation ﬁeld computed in (Jakob et al.,

2015) produces satisfying orientation ﬁelds, it is not

without ﬂaws. Especially for rotation symmetric ob-

jects, the algorithm cannot always direct the orien-

tation ﬁeld so that the ﬂow lines align with feature

lines (e.g. ridges, creases). To address this, we in-

troduce constraints for vertices lying on feature lines

based on their principal directions. It is important to

note that these constraints have to be used with care

because they may greatly disturb/constrain the ori-

entation ﬁeld optimisation. We use constraints from

anisotropic regions of the mesh because the principal

directions are well deﬁned in these regions.

To classify highly anisotropic regions, a threshold

for the difference between minimal and maximal cur-

vature needs to be deﬁned and the maximal curvature

must exceed some extent to make sure the anisotropic

region is on a feature line with high curvature in one

direction. To classify anisotropy, we use a scale-

invariant quotient (Rugis and Klette, 2006), which is

computed from the principal curvatures, κ

and κ

and is computed as κ

min(|κ

|,|κ

max(|κ

|,|κ

. We deﬁne the

threshold for high curvature by the median curvature

of all principal curvatures for all vertices. The median

of the curvature distribution is less affected by out-

liers but depends on the shape of the reference mesh

and its distribution of curvature. It proved to be sta-

ble enough as a measure for high curvature. We can

now deﬁne candidate vertices for feature lines to have

< 0.1 and max(|κ

|,|κ

|) > median. Only one path

of vertices is needed to constrain the orientation along

a feature line but the candidate vertices are not lim-

ited to a single path yet. To remedy this, only vertices

v, which have max(|κ

|,|κ

|) > max(|κ

|,|κ

|) for

each of its neighbours v

and have at least one neigh-

bour which is a feature candidate are true feature ver-

tices. True feature vertices require a feature candidate

neighbour to avoid stray feature candidates coming

from noise in the curvature estimation. This classi-

ﬁcation is simple to compute and sufﬁces to give a

minimal set of constraints for the orientation ﬁeld op-

timisation. The merits of classifying feature vertices

and adding constraints to the orientation ﬁeld optimi-

sation can be seen in Figure 2.

3.6 Position Field Optimisation

The position ﬁeld optimisation of the algorithm of

(Jakob et al., 2015) calculates a local parametrization

for every vertex. A positional vote for each vertex is

calculated via a sum of positions of its neighbours.

For each neighbour, an optimal position of the vertex

GRAPP 2020 - 15th International Conference on Computer Graphics Theory and Applications

152

Figure 2: By adding a sparse set of constraints through the

principal directions of vertices that lie on ridges, the orien-

tation ﬁeld optimisation is better aligned to surface features.

and the neighbour is calculated based on their associ-

ated edge length.

3.7 Extraction

The extraction of the ﬁnal mesh is done in two stages:

extracting a graph of vertices and their neighbourhood

and then extracting a set of quad-dominant faces from

this neighbourhood. The following three changes

were made to the mesh extraction mechanism.

Minimal Vertex Cluster Size. In (Jakob et al., 2015)

extracted vertices are discarded when the size of their

cluster in the parametrization is smaller than 1/10 of

the average cluster of all extracted vertices to avoid

stray extracted vertices. For most meshes, 1/10 is

appropriate. However, when the extracted mesh has

large faces, the clusters of their vertices may distort

the average such that vertex clusters are discarded that

would still be needed. To give control over this be-

haviour, we added an option to change the fraction of

the cluster size below which vertices are discarded.

Snapping Flat Triangles. When extracting vertices,

the original method (Jakob et al., 2015) may produce

skinny triangles (Botsch and Kobbelt, 2002). These

triangles may cause problems during subsequent pro-

cessing stages of the mesh. As in the original work,

we change their neighbourhood and positioning so

they form a line instead of a triangle.

In (Jakob et al., 2015) three neighbouring vertices

are considered a skinny triangle if the length of any

edge of the triangle is below 30% of the global edge

length scale. However, since the scale is not global

anymore, this threshold does not work and each ex-

tracted vertex needs its own scale. The new scales for

the extracted vertices are based on the scales of the

vertices which contribute to their position in the po-

sition ﬁeld parametrization. They are computed as a

weighted sum over all scales of contributing vertices,

using the same weights as the computation of the ex-

tracted vertex positions. We deﬁne the new threshold

for skinny triangles to be the sum of 10% of each ex-

tracted vertex’ scale. It is equivalent to the original

threshold when all vertices have the same scale.

If a skinny triangle is encountered, the vertex for

which the scale was calculated is moved. If this vertex

is close to one of the other two vertices of the triangle

then these two are merged and their position is aver-

aged. When the vertex to be moved is not close to any

of the other vertices forming the triangle it is moved

so that all three vertices are collinear. Unfortunately,

this simple threshold is sometimes moving vertices

away from feature lines of the reference mesh, which

decreases the quality of the mesh.

To prevent bad snapping, we set the position of

merged vertices to be the position of the vertex with

greater principal curvature instead of computing an

averaged position. Collinear snapping is only per-

formed if the extracted vertex to be moved does not

lie on a feature line (i.e., it has less than two contribut-

ing input feature vertices) or all three vertices of the

extracted triangle lie on a feature line (i.e., more than

one feature vertex).

Creating Quads. When extracting faces, the algo-

rithm of Jakob et al. (Jakob et al., 2015) searches for

polygons in the adjacency graph of the extracted ver-

tices. A face is created if the polygon is a triangle,

a quad or a pentagon. Selecting quads for ﬁlling n-

gons follows a greedy scoring system. For an n-gon

,...,v

n−1

, the score is calculated as a sum of dif-

ferences between the four next angles’ and a right an-

gle, starting from i:

score

∑

j=0

|90

◦

− ](v

where v

= v

i+ j (mod n)

, v

= v

i+ j+1 (mod n)

, and v

i+ j+2 (mod n)

In (Jakob et al., 2015), no reasoning for this scoring is

given and it does not try to ﬁnd perfect quads as only

the ﬁrst two angles (i.e. j = 0, j = 1) are part of the

quad.

To favour optimal quads, we use a new scor-

ing system which does not use four consecutive

angles but the four angles of the quad in ques-

tion. We therefore deﬁne the score as above but

use the following expressions for the angles: v

i+ j (mod n)

, v

= v

i+( j+1 (mod 4)) (mod n)

, and v

i+( j+2 (mod 4)) (mod n)

This scoring now ﬁlls the space with the most rectan-

gular quad ﬁrst. Observations on the behaviour sug-

gest that the best quads are also well aligned with the

principal directions and that no additional considera-

tion for this alignment is necessary.

Creating Curvature Adapted Subdivision Control Meshes from Scan Data

153

3.8 Ensuring Manifoldness

Because of the local nature of the position ﬁeld op-

timisation and the greedy face creation, extracting a

manifold quad-dominant mesh is not always possible.

To remove the non-manifold parts of the mesh, Jakob

et al. perform two passes where faces introducing

non-manifold edges and then all faces surrounding

non-manifold vertices are deleted. However, deleting

affected faces results in holes. We changed the algo-

rithm presented in (Jakob et al., 2015) to repair non-

manifoldness instead of deleting it. Non-manifold

normals (i.e. neighbouring faces with opposite nor-

mals) are not created during the extraction and do not

need to be handled.

Non-manifold Edges. Non-manifold edges can

be detected in the directed edge data-structure (Cam-

pagna et al., 1998) used to represent the extracted

mesh, when the same directed edge is introduced by

two faces. Removal of these edges is done by chang-

ing the affected faces or by removing one face. The

method to repair a non-manifold edge is as follows:

delete all vertices in both faces which are not part of

the non-manifold edge and are adjacent to only one or

two faces in total. If no vertex is deleted, delete one

face as a last resort (if the non-manifold edge is a di-

agonal on one face, delete this face - otherwise delete

a random face). When the deleted vertices leave faces

with 2 or fewer vertices, they are deleted as well.

Non-manifold Vertices. Non-manifold vertices can

be detected by creating a list of all faces adjacent to

the vertex, starting at a random neighbouring face,

traversing the neighbourhood of the vertex and re-

moving all encountered faces off the list. If the vertex

is non-manifold, it has more than one neighbouring

face loop. So, if the list is not empty after all faces

were visited, the vertex is non-manifold. To ﬁnd all

loops, this loop-search is repeated until all neighbour-

ing faces have been assigned to a face loop. For each

additional face loop, a new vertex is created and all

instances of the non-manifold vertex are positioned to

be the average of all adjacent vertices in their respec-

tive face loop.

While this method removes all non-manifold el-

ements, it cannot guarantee that no holes are cre-

ated, because there are cases where faces cannot be

changed and must be deleted to ensure manifoldness.

Since our goal is a closed mesh, the algorithm de-

scribed in Section 3.7 is used to ﬁnd and patch poten-

tial holes. To ensure that the hole-ﬁlling does not in-

troduce non-manifoldness, one more iteration of non-

manifold clean-up is performed.

Figure 3: The cases considered in the elimination of T-

junctions in the mesh as described in Section 3.9. Not all

cases improve the quality of the mesh: Combining two

quads (top right) is not feasible because this would intro-

duce more non-quadrilateral faces to the mesh. Combining

a quad and a 5-gon (second right) would create a triangle

and a 5-gon, which is not favourable as well. While two

5-gons are possible in theory, it has not occurred during yet.

3.9 Mesh Post-processing

After the non-manifold clean-up, the extracted mesh

still has potential for improvement. The follow-

ing three additional passes for optimisation are per-

formed.

Eliminating T-junctions. T-junctions in the ex-

tracted mesh induce extraordinary vertices in the sub-

division control mesh. We can avoid T-junctions by

altering the topology of the mesh as shown in Figure

3. To prevent losing detail, the two faces on either side

of the edge ending in a T-junction must lie on similar

planes and their edges adjacent to the T-vertex must

lie, approximately, on a line. For the purposes of this

clean-up, two faces lie on similar planes when their

normals differ by less than ﬁve degrees. Two outgo-

ing edges from the T-vertex lie on similar lines when

the angle α between them is 175

◦

< α < 185

◦

. These

angles have shown to be a ﬁtting threshold. Figure 3

shows the different combinations of faces and their

resolved topology. This elimination of T-junctions

tends to reduce the number of non-quads and reduces

the number of extraordinary vertices in the subdivi-

sion control mesh by one.

Feature Aligned Quad Edges. To ensure align-

ment of edges with surface features, two different

approaches are used: The mis-aligned quad face is

split into two triangles along the face diagonal closest

to the feature or the quad face is removed by merg-

ing both endpoints of the diagonal on the feature line

along the middle of the diagonal. Although removal

does not introduce non-quads, it must be done with

caution, because the quad may lie on a high curvature

part of the feature line and removing it could increase

GRAPP 2020 - 15th International Conference on Computer Graphics Theory and Applications

154

the error to the reference mesh. Hence, only quads

where the diagonal has a pair of adjacent edges which

lie on a common line are removed, and all others are

split into two triangles. We consider a pair of out-

going edges from both ends of the diagonal to be on

the same line if, as before, the angle α between the

edges is 175

◦

< α < 185

◦

. This angular threshold was

determined empirically through observations over all

compared meshes.

Finding these quads is a delicate task because

false-positives must be avoided. We used the follow-

ing criteria as a classiﬁcation of mis-aligned quads:

• The average angle between the orientation ﬁeld of

both end-vertices and the face diagonal (with 90

◦

rotation symmetry) must be smaller than 15

◦

• The angle of the two triangles on both sides of the

diagonal on the feature must be greater than 50

◦

• The average largest principal curvature of both

end-vertices of the diagonal must be greater than

the curvature of the other face diagonal.

• The closest face of the reference mesh must be

closer to the middle of the face diagonal than the

middle of the other diagonal.

The ﬁrst three criteria make sure that only quads

on sharp feature lines are considered. The fourth cri-

teria excludes false-positives where the removal or

split would be performed on the wrong diagonal and

thus worsening the result.

This post-processing improves the feature align-

ment of the extracted mesh. Edges follow the feature

lines again and when a quad is removed, the endpoints

of the diagonal are combined. The outcome of both

approaches of ensuring feature alignment are shown

in Figure 4.

Flat Neighbouring Triangles. We further reduce

the number of non-quad faces in the mesh by merging

neighbouring triangles which lie on a similar plane.

This post-processing is applied before and after elim-

inating T-junctions and aligning edges with features.

Two triangles are considered lying on a similar plane

based on the same threshold as in Section 3.9.

3.10 Smoothing

The position ﬁeld optimisation does not always pro-

duce perfect quads. Especially in areas where large

edge lengths meet short edge lengths, some quads

might be very distorted. To improve the quality of dis-

torted quads without losing detail, tangential smooth-

ing is performed. New vertex points are computed

by averaging over all neighbours in a 1-ring neigh-

bourhood and then projecting them onto the tangen-

(a) Quad to be removed (b) After quad removal

Figure 4: The meshes and their corresponding subdivision

meshes are shown before and after post-processing to avoid

mis-aligned edges. Figures (a) and (c) show meshes with

mis-aligned edges together with their corresponding subdi-

vision mesh. Figures (b) and (d) show the meshes together

with their subdivision meshes after post-processing. The

ridge lines are kept intact by removing or splitting the quads

as described in Section 3.9.

tial plane of the original vertex. To preserve the fea-

tures of the mesh, vertices are not smoothed when

the largest angle between the normals of their neigh-

bouring faces is greater than 50

◦

. Three iterations of

smoothing are able to restore distorted quads while

largely preserving the difference in edge length.

3.11 Subdivision Control Mesh

The next step is creating a Catmull-Clark subdivision

control mesh from the extracted quad-dominant mesh.

A subdivision control mesh is created via iteratively

moving the vertices of the quad-dominant mesh along

their normal until the limit point closely approximates

the input mesh as described previously in (Thaller

et al., 2016). The algorithm to derive a subdivision

control mesh from a polygon mesh is as follows:

mesh ← output of instant meshes

while Less than 20 iterations do

for all vertices ∈ mesh do

LimitPoint ← limit point of vertex

I ← intersection of reference mesh and

Ray[LimitPoint, normal]

NewVertexPos ← vertex + (I − LimitPoint)

end for

if max

v∈mesh

|NewVertexPos − OldVertexPos| < ε

then

break

end if

mesh ← all NewVertexPos

end while

The algorithm stops either after 20 iterations or when

Creating Curvature Adapted Subdivision Control Meshes from Scan Data

155

the maximum distance between old and new posi-

tions for vertices are below a threshold ε. Our ε is

−4

× the greatest dimension of the reference meshes

bounding box.

To apply this algorithm, the limit points of the ex-

tracted mesh need to be calculated. Catmull-Clark

subdivision surfaces correspond to bi-cubic B-splines

in the regular region. In non-regular regions, for ver-

tices with valences not equal to four, (Halstead et al.,

1993) have derived a linear combination to compute

the limit points from the control points.

4 RESULTS

The meshes resulting from our work are able to rep-

resent reference meshes up to a user-deﬁned desired

error, with an emphasis on distributing vertices so that

high frequency details are preserved.

A change in density in quad meshes leads to ver-

tices with a valence different to four or to the intro-

duction of non-quads. Therefore, our resulting quad-

dominant meshes have more extraordinary vertices

compared with other approaches. They have uniform

quads in regions of similar curvature of the reference

mesh and adapt to changes in curvature with non-

quads or EVs. The number of EVs and non-quads of

our method depends on the number of transitions be-

tween low and high curvature regions because quad-

dominant meshes need non-quads or EVs to make the

corresponding transition of edge lengths.

Most isotropic approaches aim for pure quad or

quad-dominant meshes while trying to minimise the

number of EVs. This limits their abilities to adapt

to curvature. Compared to our method, the isotropic

meshes of Jakob et al. (Jakob et al., 2015) are more

regular with fewer EVs and non-quads, and have

quads with better quality. However, they are less

dense in high curvature regions and thus cannot rep-

resent high frequency detail as well (c.f. Figure 5).

Additionally, due to the dependence on curvature es-

timation, our approach is slower, but still interactive

(timings with 16GB RAM, Intel i7-4930K CPU):

#Input Vertices Jakob et al. (ms) Our work(ms)

14000 500 1300

50000 850 2700

100000 1300 4600

To compare the original work of Jakob et al. and

our contribution, we extracted meshes with a similar

number of vertices through both approaches. We used

the default setting for the original work but changed

the desired number of extracted vertices to match our

resulting meshes. Since the algorithm can only be

Our work:

Jakob et al.:

2108 vertices

96.2% reduction

2076 vertices

96.3% reduction

Mesh Name: Bimba

Figure 5: Comparison of subdivision surfaces for the origi-

nal work (top) and our method (bottom) with a high density

and the rendered original scan data (left).

set to approximate a number of vertices, we cannot

compute the exact same number of vertices for both

approaches. Both types of meshes were converted

to subdivision control meshes. A visual compari-

son of the resulting subdivision surfaces demonstrates

clearly how the algorithm presented in this paper is

able to capture high frequency details present in the

original data with a similar number of vertices.

To quantify the improvement, we employ a

number of error metrics to compare subdivision

surface derived using the approach presented in this

paper to those derived from quad meshes using Jakob

et al.’s approach. Averaged over all meshes compared

in this paper, the following error can be observed:

Error Metric EV’s

Hausdorff

Distance

Avg. Quad

Quality

Avg. Face

Centroid Dist.

Jakob et al. 0.28% 0.468 0.489 0.122

Our algorithm 0.89% 0.463 1.291 0.115

% Difference +246.01% -2.90% +180.79% -6.06%

Here, the average quad quality is an adapted quad

quality measure taken from (Marinov and Kobbelt,

2006) and the average face centroid distance to

the reference mesh is used to avoid large quads in

detailed regions. The adaptation of the quad quality

measure comes from omitting the deviation from a

square, since our goal is not a perfect isotropic mesh.

The introduced curvature-dependent vertex densi-

ties improve the Hausdorff distances between limit

surface and reference mesh. It also shows how the

focus on adaptive edge length increases the number

GRAPP 2020 - 15th International Conference on Computer Graphics Theory and Applications

156

Our work:

Jakob et al.:

723 vertices

98.5% reduction

706 vertices

98.5% reduction

Mesh Name: Igea

High Error

Low Error

Figure 6: On the left is the rendered input mesh resulting from scan capture. 2nd from left: The subdivision control mesh

generated from the scan data. The limit surface derived from control mesh is shown 3rd from left. On the right we used colour

to highlight the error between the scan data and the subdivision limit surface. Our algorithm creates a control mesh which is

dense in regions where it is required to capture ﬁne detail, but coarse in regions of low curvature (bottom row). The resulting

limit surface captures high frequency features at a considerably higher precision opposed to the original approach by Jakob

et al., who are more concerned with regularity of the underlying mesh. Despite a similar number of vertices, our algorithm

captures the scar around the mouth quite well, while using a coarse grid in regions with less detail.

of extraordinary vertices and deteriorates quad qual-

ity. Additionally, the proposed algorithm struggles

when two very ﬂat areas meet at a sharp crease. The

problem arises from the very concentrated high cur-

vature region of the sharp crease being surrounded by

nearly zero curvature. When the disparity between

large edge lengths and small edge lengths is too sud-

den, the position ﬁeld optimisation may not produce

optimal results. To tackle this, however, curvature can

either be smoothed stronger for more gradual curva-

ture changes, or the desired error can be set lower to

create smaller quads overall.

Figure 6 and Figure 7 compare the different

meshes and errors between the original and our ap-

proach. They show the improved accuracy in detailed

regions and more irregular meshes that our curvature

adapted edge lengths produce, when compared to the

isotropic approach in (Jakob et al., 2015). Figure 8

compares the results of both approaches for a variety

of meshes. The most notable differences are in re-

gions with ﬁne detail.

All our additional methods and many options for

the behaviour of the algorithms are added to the soft-

ware created in (Jakob et al., 2015). With the embed-

ded batch mode, a user may extract quad-dominant or

subdivision control meshes quickly based on our set

of default options. Because of the variety of options,

it is also possible to ﬁnd a set of parameter setting to

obtain improved results for speciﬁc meshes. Because

the extraction is fast and initialisation is random, try-

ing different options to ﬁnd good solutions is easy.

5 CONCLUSION

We have created an adaptive quad-dominant meshing

method with a focus on curvature-dependent density

to accurately capture high frequency detail in the orig-

inal data. A subsequent conversion of the extracted

quad meshes to subdivision control meshes yields

subdivision surfaces that faithfully describe even de-

tailed features of the original shape. Possible defects

in the parametrisation have been addressed through

post-processing steps. Our algorithm presented in this

paper enables the creation of coarse subdivision con-

trol meshes with corresponding subdivision limit sur-

faces which precisely model arbitrary shapes with a

low number of vertices, while still accurately repre-

senting the reference shape, by offering higher den-

sity of the control mesh in regions with detailed fea-

tures. The trade-off, however, is higher irregularity of

the derived subdivision control mesh.

The design is very ﬂexible and may be adapted

to more specialised applications. Our focus was on

an automated process that is able to produce satisfy-

ing results for a variety of inputs. Further work may

Creating Curvature Adapted Subdivision Control Meshes from Scan Data

157

Our work:

Jakob et al.:

2283 vertices

96.1% reduction

2358 vertices

95.9% reduction

Mesh Name: Rampant

High Error

Low Error

Figure 7: Same setup as Figure 6. We can see that the high frequency areas are better preserved with out approach (bottom),

however, low frequency areas around the neck and behind the saddle lose some detail as their curvature is too low. This shows

how our method distributes the expected error more evenly over the mesh, while still maintaining a low number of vertices.

Jakob et al.

Our work

1406 vertices

1483 vertices

Omotondo

Jakob et al.

Our work

2062 vertices

1991 vertices

Stanford Bunny

Jakob et al.

Our work

4349 vertices

4249 vertices

David

Jakob et al.

Our work

4202 vertices

4176 vertices

Carter

Jakob et al.

Our work

1736 vertices 1666 vertices

Rocker Arm

Jakob et al.

Our work

2893 vertices 2995 vertices

Vase Lion

Jakob et al.

Our work

2567 vertices 2400 vertices

Botijo

Jakob et al.

Our work

2320 vertices 2404 vertices

Gargoyle

Figure 8: An overview over a variety of resulting subdivision meshed for both approaches and the mesh names. Vertex

numbers represent the extracted control meshes that were used to generate the subdivision meshes. All input meshes were

taken from the supplementary material of (Jakob et al., 2015).

include more post-processing to ensure that the ex-

tracted meshes fulﬁl the demands of more speciﬁc

applications, more specialised settings for a subset

of meshes or different constraints on the parametri-

sation.

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