LOCAL MULTIRESOLUTION OF A MESH

BASED ON

SUBDIVISION AND SURFACE

DISCONTINUITIES

Olivier Guillot and Jean-Paul Gourret

Laboratoire Informatique, Image, Interaction (L3i)

Université de La RochelleIUT, Department d’Informatique

15 Rue François de Vaux de Foletier 17026 La Rochelle Cedex, France

Keywords: Mesh, surface discontinuities, local multiresolution, analysis, synthesis, compression ratio.

Abstract: We build a local multiresolution of meshes when the connectivity is resulting from an enhanced

subdivision of a coarse mesh template. We use the concept of biorthogonality and lifting to develop a

set of filters for local analysis and local synthesis. The enhanced

subdivision, we developed, takes into

account natural surface discontinuities during the subdivision process. The multiresolution based on our

enhanced

subdivision permits to obtain a great compression ratio.

1 INTRODUCTION

A project of surface meshing for still and animated

images and associated software has been developed

at L3i for several years. Our goal is to build a

multiresolution analysis to compress mesh

information in order to permit fast transmission of

shapes on networks and to allow fast visualization

via levels of details. We use wavelet theory. The

wavelet functions are deduced from scaling

functions based on subdivisions.

There are two main categories of subdivision

schemes: the subdivision inserting vertices on edges

and the subdivision inserting vertices on faces. Each

one of these can use an approximation method or an

interpolation method. Among all the approximation

methods for subdivisions inserting vertices on edges,

we can cite (Doo et al. 1978), (Catmull and Clark

1978) and (Loop 1987). Among interpolation

methods for subdivisions inserting vertices on edges

the method of (Halstead et al. 1993) is a modified

Catmull-Clark subdivision. The most famous one is

the “butterfly” method (Dyn et al. 1990), which

gives a G1 continuity of the limit surface, with a

minimal number of neighbors and whatever the

connectivity of the vertices. A modified butterfly

method which ensures a better continuity has been

proposed by (Dyn et al. 1993) and (Zorin et al.

1996). Among approximation methods for

subdivisions inserting vertices on faces, we can cite

the

subdivision of Kobbelt (Kobbelt 2000).

Among interpolation methods, we can cite the

Subdivision of Labsik and Greiner (Labsik and

Greiner 2000). All these subdivision methods do not

take into account the natural discontinuities of the

surface.

Subdivision of meshes with natural surface

discontinuities has been studied by (Hoppe et al.

1994). The study is focused on subdivision inserting

vertices on edges and does not include a

multiresolution analysis.

The basic principles of multiresolution analysis and

wavelets were given by (Meyer 1986), (Meyer 1988)

for mathematical aspects, by (Mallat 1989) for

signals and images and by (Lounsbery et al. 1997)

and (Schröder and Sweldens 1995) for surfaces.

Schröder and Lounsbery worked on subdivision

inserting vertices on edges for meshes without

natural surface discontinuities.

A subdivision allows synthesizing a shape. It

increases the resolution of a coarse mesh, called

“control mesh”, and converges on a limit surface. A

multiresolution analysis permits to decrease the

resolution of a fine mesh without lost of information.

In the case of a subdivided mesh, the synthesis is

simply the subdivision of the coarse mesh and the

analysis consists in rebuilding the coarse mesh from

180

Guillot O. and Gourret J. (2007).

LOCAL MULTIRESOLUTION OF A MESH BASED ON â

LŽ3 SUBDIVISION AND SURFACE DISCONTINUITIES.

In Proceedings of the Second International Conference on Computer Graphics Theory and Applications - GM/R, pages 180-187

DOI: 10.5220/0002081701800187

 SciTePress

the subdivided mesh. In the case of a complex mesh

that is not the result of a subdivision but has the

connectivity of a subdivided mesh, the analysis

provides a low-resolution approximation mesh.

There exists remeshing methods to obtain this kind

of mesh (Eck et al. 1995), (Lee et al. 1998). In order

to rebuild the original mesh by the synthesis of the

low-resolution mesh, the analysis computes and

store errors, called details, at every level.

The multiresolution theory assures us that the

analyzed mesh is the best approximation of the

original mesh at this level for the chosen dot

product.

Because the subdivision tends to a smooth limit

surface, the analysis of a mesh representing a

smooth limit surface generates many null details.

That’s why the analysed version of a mesh and the

non null details can be stored more efficiently than

the original mesh.

In this paper we introduce a multiresolution analysis

of meshes when the connectivity is resulting from an

enhanced

subdivision taking into account

discontinuities. We choose a subdivision inserting

vertices on faces because the meshes are growing

slower than methods by insertion of vertices on

edges. We enhanced the original

subdivision of

Labsik and Greiner to take into account the natural

discontinuities of a surface, such as darts, creases

and boundaries (Guillot and Gourret 2006a) (Guillot

and Gourret 2006b). Having this subdivision

scheme, we build a multiresolution analysis, which

handles discontinuities. Moreover, due to the high

amount of data in recent meshes, we develop a local

analysis, i.e. the calculation should include only a

part of the data at a time.

The multiresolution analysis developed in this paper

works with every subdivision scheme. An analogous

approach is done by (Olsen et al. 2005). Their

calculation starts from a very small neighbourhood

which is recursively enhanced by a method similar

to the lifting scheme. So, the size of filter is growing

until an optimization of the magnitude of details is

reached. We build our wavelets with only one lifting

scheme. Then a recursive approach constrains the

number of null details. Moreover our method uses

natural discontinuities to minimize magnitude of

details.

In section 2, we present the principles of a global

multiresolution and some definitions. In section 3,

we explain how to process the local calculus of

scaling and wavelet functions using biorthogonality

and lifting scheme. In Section 4 we describe how to

perform locally a synthesis and in section 5 we

describe how to perform locally an analysis. Section

6 is dedicated to results, and section 7 is dedicated to

conclusion and future works.

2 MULTIRESOLUTION

ANALYSIS BASED ON

SUBDIVISION

2.1 Our Enhanced

Subdivision

The enhanced

subdivision is based on the

insertion of a new vertex in each triangular face. We

start from a control mesh at subdivision level j=0. A

vertex is inserted in each face. Then new faces are

created joining the new vertices to the initial vertices

and to the new vertices in immediate

neighbourhood.

Doing this, a new vertex always shares 6 faces.

The mesh resulting from one subdivision of level j is

called the level j+1.

A mesh M

of level j has n

vertices and f

faces. We

call Y

the set of vertices in the mesh of level j and

its connectivity, so M

= (K

, Y

) represent the

level j.

A direct property is that

j+1

– n

= f

= 3.f

j-1

= …= 3

Far of natural surface discontinuities and far of

extraordinary vertices (non 6 connected vertices),

our enhanced

subdivision uses the method of

Labsik-Greiner or of Kobbelt (for accuracy we

introduce the name Labsik-Greiner formula and

Kobbelt formula in this paper). Otherwise we

developed a formulation explained in (Guillot and

Gourret 2006b).

2.2 Multiresolution Analysis

Generally a mesh M is obtained from a cloud of

points acquired by a 3D scanner. So the number of

vertices and the connectivity are not the result of

recursive subdivisions (K ≠ K

). Our multiresolution

analysis method needs that the connectivity of the

starting high level mesh is the result of a recursive

subdivision (K = K

). We suppose that K = K

what follows. It means for example that most of the

vertices are 6-connected and that extraordinary

vertices, of connectivity different of 6, are

sufficiently spaced. Note that as said in section 2.1, a

new vertex always shares 6 faces: it is always of

LOCAL MULTIRESOLUTION OF A MESH BASED ON v3 SUBDIVISION AND SURFACE DISCONTINUITIES

181

connectivity 6, so the extraordinary vertices are not

introduced by the subdvisivion process. They are

defined in the control mesh M

and they remain

through the recursive subdivisions.

Figure 1: Analysis from level 2 to level 0.

The analysis of a level j+2, builds a level j+1 and a

set of details called Z

j+1

. Recursively the level j+1

can be analysed as a level j and a set of details Z

Let’s call A

the matrix that transform Y

j+1

into Y

and B

the matrix that transform Y

j+1

into Z

The detail Z

j+1

has exactly three times more vertices

than Z

. We show in Figure 1 the analysis from level

2 to level 0.

Let’s call Y

(n) the n

vertex of Y

. The details Z

are

considered as a list of virtual points (points not

connected via K

In what follows V

is the i

vertex of Y

, V’

is the i

vertex of Y

j+1

and W

is the (i - n

)

vertex of Z

2.3 Multiresolution Synthesis

The synthesis is the reconstruction of the level j+1

from the level j and the details Z

. A second

synthesis rebuilds the level j+2 from the level j+1

rebuilt and from the details Z

j+1

. Thus the level j+2

can be obtained from the level j and the details Z

and Z

j+1

Figure 2: Synthesis from level 0 to level 2.

Let’s call P

the matrix that subdivide Y

and call Q

the matrix that uses the details Z

. The action of Q

on Z

gives the difference between Y

j+1

and the

subdivision of Y

. We show in Figure 2 the synthesis

from level 0 to level 2.

3 LOCAL COMPUTATION OF

SCALING AND WAVELET

FUNCTIONS

In order to build a multiresolution, we need a scaling

function φ and a wavelet function ψ. They define the

global matrices P

and Q

as shown in Figure 3. We

never compute the whole P

or Q

matrices, only

rows or columns of these matrices.

Figure 3: Definition of the global matrices P

and Q

3.1 Local Computation of φ

Note that “local computation of φ

” means that the

computation itself is local, not the function φ

. Let’s

V’

be a vertex in Y

j+1

and D’

be the d-disk in Y

j+1

centred on V’

. The d-disk centered on V’

is the set

of vertices connected to V’

by less than d edges.

In what follows, we will suppose that every

calculation is local around a vertex, which means

that to synthesize a vertex V’

in Y

j+1

, we need to

consider some vertices in Y

and some points in Z

The vertices of Y

and the points of Z

are in

= {V

; V’

є D’

} U { W

; W’

є D’

If i is in [1,n

], φ

represents the influence of the

vertex V

in the computation of the new vertices in a

one level subdivision, it is the scaling function

associated with V

. The influence of the vertex V

local, so we can compute it locally, in fact V

only

influences the vertices of D’

The scaling function φ

depends on the connectivity

of the vertices around D’

For example, when we use the Labsik-Greiner

interpolation formula with twelve neighbours

(Figure 4) the disk D’

is shown in Figure 5. Note

that because the Labsik-Greiner formula is an

interpolation method only black vertices are

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications

182

influenced by V

, white vertices are not influenced

by V

Figure 4: Stencil of Labsik-Greiner formula (the twelve

circled neighbours are weighted a, b or c to calculate the

black vertex V’

Figure 5: 3-Disk D’

for Labsik-Greiner formula (V

influences black vertices, with a, b, c weights).

For example, when we use Kobbelt formula (Figure

6), D’

is the 2-disk shown in Figure 7. Note that

because Kobbelt is an approximation method, black

and white vertices are influenced by V

Figure 6: Stencils for kobbelt formula.

Figure 7: D’

for the Kobbelt formula.

D’

depends on the method of subdivision and on the

proximity or not of natural discontinuities or

extraordinary vertices. See (Guillot and Gourret

2006b) for more explanations on the enhanced

subdivision. Calculations are now implemented

in our software MEFP3C (Khamlichi and Gourret

2004). Note that taking discontinuities into account

enlarges the size of D’

For every V’

in Y

j+1

, we note φ

(k) the influence of

on the computation of V’

. If V’

is not in D’

, V

has no influence, so φ

(k) = 0. If V’

is in D’

, φ

(k)

can be obtained by simulating the calculation of V’

in the subdivision algorithm, we obtain the weight of

in the stencil around V’

3.2 Local Computation of ψ

Note that “local computation of ψ

” means that the

computation itself is local, not the function ψ

. For k

in [1 , n

j+1

- n

] ψ

is the wavelet function associated

with W

= Z

(k - n

)).

Knowing the φ

function, we should build the ψ

function using the global orthogonal condition

between φ

and ψ

For all i in [1,n

], for all k in [1 , n

j+1

- n

]

< φ

, ψ

> = 0.

But a global computation is too expensive. So we

release this constraint to something local.

We use the concept of biorthogonality with lazy

wavelet (Sweldens 1996). For ψ

lazy k

we choose the

dirac δk. Let’s V

, V

and V

be the vertices of the

face of Y

in which we insert W’

. Our lifting

operation to enhance the orthogonality of the lazy

wavelets consists in writing :

= ψ

lazy k

+ α. φ

+ β. φ

+ γ. φ

, where α, β, γ are

real numbers that we compute for every k and every

j writing the system :

< φ

, ψ

> = 0

< φ

, ψ

> = 0

< φ

, ψ

> = 0

It is a 3x3 system, easily solved, done for every k.

is known by its values on every vertex of Y

j+1

it can be seen as a vector of R

nj+1

. Let’s consider ψ

as a vector of R

nj+1

. To calculate < φ

, ψ

> we use

the Euclidean inner product of φ

and ψ

as vectors

of R

nj+1

. We do not use the usual inner product first

defined by Lounsbery.

Our wavelet function is the sum of three scaling

functions. Thus we can compute locally ψ

. An

example for the regular case of the Labsik-Greiner

formula is shown in Figure 8.

LOCAL MULTIRESOLUTION OF A MESH BASED ON v3 SUBDIVISION AND SURFACE DISCONTINUITIES

183

Figure 8: 4-Disk D’

of the wavelet function ψ

associated

with the Labsik-Greiner formula.

4 LOCAL SYNTHESIS

The synthesis is the sum of the result of the P

and Q

matrices applied to Y

. The synthesis is computed

with the matrices P

and Q

as show in Figure 9.

Figure 9: Synthesis from level 0 to level 1.

The action of P

on Y

is exactly the result of the

subdivision which is a local computation, i.e. the

computation of Y

j+1

is local around Y

To compute the action of Q

on Z

, we need the i

row of Q

. We have already assumed that this row

has only non null factors in the columns representing

the vertices of D’

. Let’s W

be a point of Z

. We

know how to compute ψ

where only the i

coefficient interests us.

The calculation of the i

row of Q

needs just the

calculation of ψ

for every W

in Z

Figure 10: Computation scheme of Y

= Q

5 LOCAL ANALYSIS

The analysis is globally computed with the matrices

and B

as shown in Figure 11.

Figure 11: Analysis from level 1 to level 0.

and B

are usually computed as the inverse of the

global matrix [PQ]

whose properties are shown in

Figure 12.

Figure 12: Global matrices.

5.1 Local Analysis: the Built of Y

Let’s V

be in Y

. The i

row of A

applied on Y

j+1

gives us V

We assume that the action of A

around V

can be

computed locally on D’

. If A

is the term of A

the i

row and the k

column, we can assume that

= 0 if Y

j+1

(k) is not in D’

. To compute the result

of the i

row of A

, we need the (A

)

for k verifying

j+1

(k) in D’

. In section 3 we saw that we can

compute φ

and ψ

for every k.

Let’s A

be the i

row of A

with only the

coefficients corresponding to a column h that verify

j+1

(h) in D’

Let’s call (P

)

kєD’i

the matrix containing the P

where k is in D’

. It is also the matrix containing the

when Y

j+1

(k) in D’

, with just the rows of number

h verifying Y

j+1

(h) in D’

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications

184

From the equation A

= Id we have just kept some

rows and some columns so we can write the

following equation:

.(P

)

kєD’i

= (0, 0, 0, …, 1, 0, 0, …, 0)

The one is placed as i in D’

In the same way we build (Q

)

kєD’i

as the matrix

containing the ψ

when Y

j+1

(k) is in D’

with just the

rows of number h when Y

j+1

(h) is in D’

. The

equation A

= 0 gives

.(Q

)

kєD’i

= (0, …, 0) .

The matrix [(P

)

kєD’i

, (Q

)

kєD’i

] is a square matrix (of

size given by the number of vertices in D’

). We

compute A

with the inverse of [(P

)

kєD’i

, (Q

)

kєD’i

Eventually, V

= A

. Y

j+1

5.2 Local Analysis: the Built of Z

The i

row of B

applied on Y

j+1

gives us W

i+nj

Let’s use again the matrices B

, (P

)

kєD’i

and

)

kєD’i

, from the equations:

= 0 and B

= Id

we can write:

. (P

)

kєD’i

= 0 and B

. (Q

)

kєD’i

= (0, 0, …, 1, 0,

…,0) with the one placed as i in D’

By just inverting [(P

)

kєD’i

, (Q

)

kєD’i

] we obtain B

and so W

6 RESULTS

Examples presented are not realistic ones. There are

only given to prove working of our method. We

show in Figure 13, two examples. In order to get

meshes with the connectivity of a subdivided mesh

that are not just the result of a previous subdivision,

we created meshes with discontinuities and we

subdivided them 4 times with our approximation

enhanced

subdivision scheme. Then we have

implemented our multiresolution analysis from our

interpolating enhanced

subdivision scheme.

The meshes to analyse have the connectivity of

meshes subdivided four times, so we can analyse

them 4 times.

We show in Figure 14 the result of four analyses.

Note that because our enhanced

subdivision is

a subdivision inserting vertices on faces, the

discontinuities do not belong to edges of the meshes

of odd level 1 and 3 (Guillot and Gourret 2006b),

only the even level produced by the analysis (level 0

and 2) should be visualized as shown in Figure 15.

Figure 13: The construction of the meshes to analyse

(construction with our enhanced

approximation

method).

Figure 14: The result of four analyses with our

enhanced

interpolation method.

6.1 Compression

With the mesh of level 0 and the four details Z

, Z

and Z

, we can rebuild exactly the original mesh.

The size of (Y

) is the same as the size of (Y

, Z

). Because the subdivision tends to a

smooth limit surface, the analysis of a mesh

representing a smooth limit surface generates many

approximately null details. Considering some of this

details as null do not generate a great difference

between the rebuilt mesh and the original mesh. That

is why the analysed version of a mesh and the non

null details can be stored more efficiently than the

original mesh. We developed a constraint on the

details (not explained in this paper) in order to

ensure that the rebuilt mesh will not differ from the

original mesh by more than a given tolerance

epsilon. It means that for every vertex i, |Y

original

rebuilt

| ≤ epsilon * diameter of the bounding sphere.

We show in Table 1, the percentage of details kept

because they cannot be considered as null factors for

LOCAL MULTIRESOLUTION OF A MESH BASED ON v3 SUBDIVISION AND SURFACE DISCONTINUITIES

185

the 2 examples. The third column when the analysis

takes into account natural surface discontinuities.

Our multiresolution analysis method based on the

standard Labsik-Greiner formula, modified around

extraordinary vertices, and without natural surface

discontinuities processing permits to obtain a good

compression ratio as shown in the second column of

table 1.

Our multiresolution analysis method based on our

enhanced

subdivision, with natural surface

discontinuities, permits us to obtain an even better

compression ratio.

We show in Figure 16 a third example, which is the

second example at level 4, deformed by MEFP3C.

Table 1: Compression ratio in order to archive a precision

of ε.

є without

discontinuities

with

discontinuities

Example 1 1% 81% 87%

Example 2 1% 81% 88%

Example 2 1‰ 59% 63%

Example 3 1% 65% 65%

7 CONCLUSION AND FUTURE

WORK

The local multiresolution analysis of meshes

presented in this paper uses our enhanced

subdivision. Because the calculations are local, the

algorithm could be parallelized on multiprocessor

computers.

Without discontinuities, the disk D’

to synthesize a

vertex V’

is a 3-disk for the scaling function and a

4-disk for the wavelet function. Because our

software deduces ψ from φ, it is only necessary to

impose the disk size for φ. Without discontinuities,

the disk D’

to analyze a vertex V’

is a 4-disk. This

result is an experimental result.

The multiresolution analysis developed in this paper

works with every subdivision scheme.

Our multiresolution analysis method based on our

enhanced

subdivision which takes into account

natural surface discontinuities permits us to obtain a

great compression ratio.

We are presently working on boundaries that are

handled by our enhanced subdivision but not yet

implemented in our multiresolution analysis and we

are also working on remeshing algorithms. Then we

will be able to process realistic shapes such as faces

and bodies.

ACKNOWLEDGEMENTS

This work was supported by a French region Poitou-

Charentes grant.

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