SIMPLE FLEXIBLE SKINNING

BASED ON MANIFOLD MODELING

Franck H

etroy

1,2

, C

edric G

erot

, Lin Lu

and Boris Thibert

Universit

e de Grenoble & CNRS, Laboratoire Jean Kuntzmann, Grenoble, France

INRIA Grenoble - Rh

one-Alpes, Grenoble, France

Universit

e de Grenoble & CNRS, GIPSA-Lab, Grenoble, France

Department of Computer Science, The University of Hong Kong, Hong Kong, China

Keywords:

Skinning, Manifold atlas, Covering.

Abstract:

In this paper we propose a simple framework to compute ﬂexible skinning weights, which allows the creation

from quasi-rigid to soft deformations. We decompose the input mesh into a set of overlapping regions, in a

way similar to the constructive manifold approach. Regions are associated to skeleton bones, and overlaps

contain vertices inﬂuenced by several bones. A smooth transition function is then deﬁned on overlaps, and is

used to compute skinning weights. The size of overlaps can be tuned by the user, enabling an easy control of

the desired type of deformations.

1 INTRODUCTION

Skeletal animation is a widespread technique to

deform articulated shapes. It uses a joint hierarchy

called skeleton; during the animation, joints are

translated and/or rotated then each vertex of the

shape (usually represented by a mesh) is deformed

with respect to the closest joints. The process that

describes the skin deformation is called skinning.

Many skinning techniques attach joint (or bone)

weights to each vertex of the mesh; a weight speciﬁes

the amount of inﬂuence of the corresponding joint on

the vertex. Deﬁning proper values for joint weights

is often time-consuming for the animator. Usually,

weights are deﬁned using the Euclidean distance

between the vertices and the joints. A basic painting

tool (or equivalent) can be applied manually to

quantify which vertices are inﬂuenced by a given

joint. Careful manual tuning is then required to set

up weights that give the desired deformation.

In this paper, we propose a simple framework to

automatically compute skinning weights, with a

user control on the type of deformation. We get

inspiration from the concept of constructive manifold

atlas (Grimm and Zorin, 2005). Contrary to piece-

wise modeling, an atlas allows to construct a surface

from pieces of surface which overlap substantially

instead of abutting only along their boundaries. As a

consequence, when one piece is stretched or moved,

the overlapping pieces follow this deformation or

motion. We use this idea to compute skinning weights

for any shape, proceeding in two steps. Firstly, a

covering of the mesh, with regions associated to

skeleton bones, is deﬁned (Section 3). This covering

can be controlled on the overlapping areas. Secondly,

a partition of the unity is deﬁned on this covering for

each vertex of the mesh, providing the weights for

the skinning (Section 4).

Our weight computation scheme is both simple and

fast. Control is easy since only one parameter has to

be tuned in order to move from a quasi-rigid defor-

mation to a soft one, and no manually tuned example

nor additional tool is required as input. We demon-

strate the effectiveness of our framework on a set of

examples (Section 5).

2 RELATED WORK

2.1 Flexible Skinning

Most skinning weight computation methods try

to generate ideal weights for realistic character

animation. They can rely on geometric features,

such as the medial axis of the object (Bloomenthal,

259

Hétroy F., Gérot C., Lu L. and Thibert B. (2009).

SIMPLE FLEXIBLE SKINNING BASED ON MANIFOLD MODELING.

In Proceedings of the Fourth International Conference on Computer Graphics Theory and Applications, pages 259-265

DOI: 10.5220/0001767602590265

 SciTePress

(a) (b)

Figure 1: Rest pose, medium deformation and large deformation around an elbow. K = 0.1, 0.5,1.0 and 2.0 for (a), (b), (c)

and (d) respectively. Overlap areas are shown in black. Overlaps and weights were computed using a geodesic distance, and

deformations were created using the technique of (Kavan et al., 2007).

2002) or a mesh segmentation (Katz and Tal, 2003;

Attene et al., 2006), or on example poses (e.g. (Merry

et al., 2006; Wang et al., 2007; Weber et al., 2007)).

An increasingly popular solution is to solve a heat

equation for each joint in order to automatically

set the weights associated to this joint (Baran and

Popovi

c, 2007; Weber et al., 2007). However, these

solutions usually do not allow for ﬂexible skinning.

To the best of our knowledge, only a few skinning

methods allow different kinds of deformations. One

of them is to use spline-aligned deformations instead

of the traditional Linear Blend Skinning (LBS),

which can be mixed with user-designed deformation

styles (Forstmann et al., 2007). Another solution is to

compute the set of possible new locations for a vertex

deformed with LBS and let the user choose the one

he wants (Mohr et al., 2003). Recently, Rohmer et

al. proposed a local volume preservation technique

which enables the creation of both rubber-like and

realistic deformations for organic shapes, depending

on the correction map applied to skinning weights

(Rohmer et al., 2008). The solution we suggest is

more ﬂexible in the sense that any deformation, from

quasi-rigid to soft, can be created, and any skinning

method can be used: for instance LBS, (Merry et al.,

2006; Kavan et al., 2007). It also lies in the general

(rigid) skeleton-based animation framework, and do

not need the creation of new tools such as spline

curves.

Our method can be related to the “mesh forging”

approach of Bendels and Klein (Bendels and Klein,

2003), except that we propose a Hermite function as a

transition function between two bones, while they let

the user draw the function.

2.2 Modeling with an Atlas

Surface modeling with an atlas has properties which

lends itself to the skinning problem. Indeed construc-

tive manifold deﬁnitions (Grimm and Zorin, 2005)

represent a surface as a set of blended embedded

planar disks. The blending is performed as a convex

combination whose weights are deﬁned as a partition

of the unity overall the planar disks. Hence the

surface is made up with 3D regions which overlap

substantially and are glued together. As a conse-

quence, when an embedded planar disk is stretched

or moved, the overlapping regions are stretched or

moved accordingly. Deﬁning such a set of regions

per joint of the skeleton provides a skinning.

However, this construction makes sense only if

the planar disks are linked together with transition

functions. These functions indicate which embedded

points have to be combined together in the blending

process. To do so, either a proto-manifold associated

with a mesh with a large number of pieces (at least

one per vertex) is deﬁned (Grimm and Hughes, 1995;

Navau and Garcia, 2000; Ying and Zorin, 2004),

or a pre-deﬁned manifold with a small number of

pieces, but in general not adapted to the particular

geometry to be represented is used (Grimm, 2004).

GRAPP 2009 - International Conference on Computer Graphics Theory and Applications

260

These constructions target a global highly-continuous

parameterization of the surface. This implies major

contraints on the deﬁnition of the transition functions.

Reversely, an atlas can be constructed from the ﬁnal

surface to be represented. The global parameteriza-

tion of the surface is used for high-quality sampling,

texture mapping or reparameterization (Praun et al.,

2000). In this case again, the components of the

atlas have to be deﬁned explicitly and with continuity

constraints.

Real-time constraints impose to deal with small struc-

tures and to consider meshes as C

-surfaces. Hence,

we propose to adapt this parameterization-oriented

framework onto a lighter one, sufﬁcient for skinning

and providing a better control on the overlapping in-

ﬂuences of different skeleton bones than other skin-

ning algorithms.

3 C

ATLAS DEFINITION

Our work takes as input a closed mesh and an em-

bedded animation skeleton. As stated in Section 2.2,

we adapt the manifold modeling with an atlas onto

a lighter framework sufﬁcient for skinning. Follow-

ing constructive manifold approach, we decompose

the mesh into overlapping regions. Despite the fact

that these regions are not necessarily homeomorphic

to discs, they will be interpreted as charts with transi-

tion functions implicitely deﬁned by the shared faces.

In order to control these overlapping areas, we ﬁrst

segment the mesh into a partition of regions associ-

ated to skeleton bones, and then stretch these regions

onto a covering of the mesh. Note that regions are

not restricted to cylindrical shapes with at most two

boundaries.

3.1 Initial Mesh Segmentation

To decompose the mesh into overlapping regions, we

need as a preprocess its segmentation into regions

associated to skeleton bones. Any skeleton-based

segmentation method can be used, such as for

instance (Katz and Tal, 2003; de Goes et al., 2008)

which also use segmentation to create animations. In

our implementation, we use a simple yet robust au-

tomatic mesh segmentation algorithm. Our approach

is to ﬁrst ﬁnd the boundaries of the regions, which

should be associated to skeleton joints since regions

are associated to skeleton bones. The boundary B

associated to joint J is deﬁned as the intersection

between the input mesh and a plane P going through

J and orthogonal to a plane Q (see Figure 2). Q is

deﬁned by the two bones incident to J. In case more

than two bones are incident to J (this is for instance

the case of the pelvis joint for a human model), we

can use the skeleton’s hierarchy to select two of them.

There is an inﬁnite number of possible planes P, but

each one can be deﬁned by its normal n, which lies in

the plane Q. In practice we compute a discrete set of

planes P

,...,P

k−1

, by selecting a random n

normal

vector and then rotating it around J with an angle

2πi/k, 1 ≤ i < k. Then we keep the plane such that

the length of the corresponding boundary curve B is

minimum.

Figure 2: Each boundary is deﬁned with respect to a plane

P going through a joint J.

Although this method is quite simple, it provides seg-

mentations which are robust to noise on the input

mesh, to the initial pose of the character and to the

location of joints, as can be seen on Figure 3. Once

again, we emphasize that any other skeleton-based

segmentation method can be applied instead of this

one, as a pre-processing step for overlap generation.

3.2 Overlap Generation

We now describe how we generate a mesh decompo-

sition into overlapping pieces from this segmentation.

Suppose that the mesh M is decomposed into r

regions {R

}

j=1

; we note {B

}

i=1

the b boundaries

between these regions. Besides, each B

has 2 adja-

cent regions denoted by R

and R

; each R

has m

boundaries, denoted by {B

j,k

}, with ∀ j,k,B

j,k

= B

k, j

Generation of overlaps consists in growing each

region R

into a new region R

with a distance

criterion: R

is connected and R

⊂ R

. This is done

by integrating vertices of neighbouring regions to

. Thus, each boundary B

j,k

of R

is modiﬁed into

a new boundary B

j,k

of R

, with B

j,k

6= B

k, j

(the

new boundary of R

). Vertices between B

j,k

and B

k, j

are in the overlap area of R

and R

. Note that a

whole region R

may belong to the overlap area of a

SIMPLE FLEXIBLE SKINNING BASED ON MANIFOLD MODELING

261

(a) (b)

Figure 3: Segmentation results for (a) a human model, (b)

the same model with a different pose (and no hand nor arm

joint in the skeleton), (c) the same model with different right

shoulder and left hip joint locations, and (d) the same model

with noised vertex locations.

neighbouring region R

(see Figure 4).

To compute the overlap areas, we compute for each

vertex v of the mesh its distance to all B

j,k

, and we

let the user choose a size parameter K. Then, we

use the length L

j,k

of B

j,k

as the criterion to generate

the overlap area between R

and R

: we mark each

vertex with distance to B

j,k

lower than K ∗ L

j,k

/π

as in this overlap area. (Baran and Popovi

c, 2007)

claims that the range of a transition between two

bones (that is to say, the area of the region where

vertices are inﬂuenced by both bones) must be

roughly proportional to the distance from the joint to

the surface. This corresponds to K = 0.5.

In our implementation, the same parameter is used

for all areas, but other solutions can be applied: for

instance, K can be chosen according to the type

of skeleton joint, in case semantic information is

attached to joints (Aujay et al., 2007).

Different kinds of distances can be used: Euclidean

distance, approximated geodesic distance or distance

based on a harmonic function, for instance. We tested

several of them and discuss results in Section 5.2.

4 COMPUTATION OF SKINNING

WEIGHTS

In the manifold constructive approach, a partition

of the unity deﬁned on a proto-manifold is used to

blend embedded pieces. In the same way, we deﬁne

skinning weights as a partition of the unity on the

covering deﬁned in Section 3.

We deﬁne weights that depend on the mesh covering

}

j=1

deﬁned in Section 3.2. In each extended

region R

a distance map d

(v) is speciﬁed. It gives

to every vertex v of the region R

its distance to

the boundary of the region (computed as the lowest

distance from v to all B

j,k

). As in Section 3.2, this

can be a Euclidean or geodesic distance, or anything

else. We tested Euclidean, approximated geodesic

and harmonic distances; see Section 5.2 for results

and a discussion.

Let δ

be the maximal distance to the boundary in R

= max

v∈R

(v). Let s(l) be the cubic function

which satisﬁes the Hermite conditions s(0) = 0,

s(1) = 1, s

(0) = s

(1) = 0: s(l) = −2l

+ 3l

. This

cubic function lets us deﬁne weights which decrease

smoothly towards 0 as the vertex v is closer to the

region boundary, providing visually better results

(see Section 5.3). However, weights can be deﬁned

with any function such that s(0) = 0 and s(1) = 1.

We deﬁne unnormalized weights σ

(v) as

(v) = s(

(v)

). Let I (v) be the set of

indices of regions the vertex v belongs to

I (v) = { j ∈ {1, . ..,n} : v ∈ R

}. Normalized

weights ω

(v) are then deﬁned as ω

(v) =

(v)

∑

i∈I (v)

(v)

Because the regions R

deﬁne a covering of the

surface and s in monotonic from [0, 1] onto [0, 1], the

denominator is never equal to zero and ω

(v) ∈ [0,1].

Moreover these well-deﬁned weights deﬁne a parti-

tion of the unity associated to this covering: for every

vertex v of the mesh,

∑

j∈I (v)

(v) = 1.

Note that for non-overlapped vertices, I (v) is reduced

to a singleton { j} and ω

(v) = 1. For a vertex v

belonging to the boundary of a region R

, we have

(v) = 0, thus σ

(v) = 0 and ω

(v) = 0.

GRAPP 2009 - International Conference on Computer Graphics Theory and Applications

262

5 RESULTS AND DISCUSSION

Some deformation results are shown on Figures 1, 4,

6 and 7. In all cases the Dual Quaternion technique

(Kavan et al., 2007) was used to deform the meshes.

The segmentation pre-processing step is done in

real-time, and so is done the weight computation.

Time to compute the overlap areas highly depends

on the chosen distance function: it is almost real

time using a Euclidean distance, but lasts a few

seconds using an approximated geodesic distance, on

a low-end PC.

Figure 4 shows the mesh covering deﬁned for two

standard models, and examples of deformations that

can be generated in a few minutes using our frame-

work. K was set to 0.5 (resp. 0.2) for all joints of

the human (resp. Homer) model. Overlaps as well

as skinning weights were computed with an approx-

imated geodesic distance, using Dijkstra’s algorithm

on the mesh’s vertices. As input we only used the

two mesh models and their corresponding animation

skeletons.

(a) (b)

Figure 4: Computed covering (a,c) and deformation (b,d)

for two models. Overlap areas are shown in black. Note

that some vertices may belong to three or more overlapping

areas, especially around the spine and the pelvis.

5.1 Inﬂuence of the Overlap Size

As can be seen on Figure 1, the overlap size K ∗L

j,k

/π

inﬂuences the behavior of the deformation around a

joint. For a small value of K, only a few number of

vertices around the joint are smoothly bended out: the

deformation is quasi-rigid. As K becomes larger, the

deformation becomes elastic. Thus tuning K allows

for various kinds of deformations.

5.2 Choice of the Distance Function

As stated in Section 3, several distance functions

can be used to compute both overlap areas and

skinning weights. Using the Euclidean distance

is the simplest and fastest solution. However, in

some cases it generates artefacts (see Figure 5). For

instance, if some part of the input mesh is close to

a joint related to other regions, vertices in this part

can be wrongly set to be in an overlap area of the

joint. This drawback can sometimes be corrected

using the skeleton’s hierarchy, by preventing vertices

from belonging to overlap areas of joints that are

far from their bone in the hierarchy, but this is not

always possible. Euclidean distance can also generate

artefacts for weight computation, in case of curved

regions: see for instance Figure 5 (b).

(a) (b)

Figure 5: Artefacts using the Euclidean distance (overlap

areas are shown in black). (a) For the overlap generation:

an overlap area around a joint can be disconnected. (b) For

the weight computation: the point represented by a square

is closest to the boundary of the region than the point repre-

sented by a triangle.

Figure 6 shows the deformation around a pelvis joint

using Euclidean (ﬁrst row), approximate geodesic

(second row) or harmonic (third row) distance. Ap-

proximated geodesic distance has been computed

with Dijkstra’s algorithm. Following an idea from

(Aujay et al., 2007), we set two boundary conditions

for the computation of the harmonic distance: the

points on boundary curves have zero distance and the

farthest points to these curves have a distance set to

their approximated geodesic distance to these curves.

SIMPLE FLEXIBLE SKINNING BASED ON MANIFOLD MODELING

263

Although the overlap areas between the three regions

(waist and both thighs) are quite similar, a small arte-

fact can be noticed for the Euclidean distance, due to

the high inﬂuence the right thigh has on vertices close

to the left thigh/pelvis boundary.

(a) (b) (c)

(d) (e) (f)

Figure 6: Overlap areas (a,b,c) and deformation (d,e,f)

around a pelvis joint using Euclidean (a,d), approximated

geodesic (b,e) and harmonic (c,f) distance. K was set to 0.5

in the ﬁrst two cases, and to 0.4 in the harmonic case.

5.3 Choice of the Weight Function

Results of deformations using a linear function in-

stead of s to compute the skinning weights are shown

on Figure 7 (a,b). They look much less natural (com-

pare with Figure 1 (b,d)), because of the sharp de-

crease or increase of inﬂuence of bones near the over-

lap boundaries. On the contrary, our cubic function s

increases very slowly around l = 0 and l = 1, leading

to visually better results.

(a) (b) (c) (d)

Figure 7: (a,b) Deformations using a linear function in-

stead of a cubic one with K = 0.5 (a) or K = 2 (b). We

used approximate geodesic distance to compute overlap ar-

eas and weights. (c) Deformation using Blender’s paint

tool. (d) Deformation using harmonic weights (Baran and

Popovi

c, 2007).

5.4 Comparison with Standard

Methods

Deformations obtained using two standard weight

computation methods and Dual Quaternion technique

are shown on Figure 7 (c,d). Using the paint tool

(available in common software such as Autodesk’s

Maya or Blender), it took approximately half an hour

to get a relatively decent result. The painted area cor-

responds to the overlap area shown on Figure 1 (b).

The use of a harmonic function (Baran and Popovi

2007) is as fast as our technique, but do not allow for

accurate control over the size of the deformed region.

6 CONCLUSIONS

We have presented a simple way to compute ﬂexible

skinning weights for skeleton-based animation, based

on the concept of manifold modeling. Starting from

a segmentation of the input mesh into regions corre-

sponding to skeleton bones, we generate overlaps by

extending each region around joints. Size of these

overlaps is controlled by a simple parameter, that can

be user-chosen or automatically computed. Then,

vertices belonging to an overlap area are inﬂuenced

by bones related to all regions that overlap. Skinning

weights are deﬁned using a simple smooth function

based on the distance to the overlap boundary.

Results show that this framework allows to create

from quasi-rigid to soft deformations, depending on

the overlap size. Using a geodesic distance instead

of a Euclidean one to create overlaps and compute

skinning weights is more time-consuming, but avoids

some artefacts. We believe our method can be

especially useful for non-expert animators, since it

is simple (only one parameter is to set) and fast to use.

Further work includes anatomic information into the

overlapping width deﬁnition. Such information can

be derived from semantic information associated with

skeleton (Aujay et al., 2007). Besides, providing

a skinning framework for multiresolution animated

meshes, founded on our pseudo-parameterization on

the initial mesh, would be a further development in

the similarity with manifold parameterization.

ACKNOWLEDGEMENTS

This work was partially supported by the IMAG,

ELESA and INRIA through the MEGA project and

GRAPP 2009 - International Conference on Computer Graphics Theory and Applications

264

the ANR through the MADRAS project (ANR-07-

MDCO-015). Part of this work was done while Lin

Lu was visiting INRIA with an INRIA Internship

grant.

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