3D mesh denoising using normal based myriad filter

Mohammed El Hassouni DESTEC, FLSHR

University of Mohammed V-Agdal- Rabat, Morocco

Mohamed.Elhassouni@gmail.com

Hocine Cherifi LE2I UMR 5158

University of Burgundy Dijon, France

Hocine.Cherifi@u-bourgogne.fr

Bouchra El Maroufi LRIT, FSR

University of Mohammed V-Agdal- Rabat, Morocco

el.maroufi.bouchra@gmail.com

Abstract—In this paper, we propose a new filtering scheme for denoising of 3D objects which are represented by a triangular mesh. This scheme consists on applying myriad filter to face normals and then updating the vertices positions in order to preserve the original shape of the object. The choice of the Myriad is justified by the assumption of Cauchy distributed angles between surface normals. This filter improves the performance of a normal-based method which is adapted to the underlying mesh structure. To evaluate these methods of filtering, we use three error metrics. The first is based on the vertices, the second is based on the normals and the third is based on Hausdorff distance. Experimental results demonstrate the effectiveness of our proposed method in comparison with the existing methods.

Keywords- Mesh denoising, myriad filter, median filter, normal based filter.

I. INTRODUCTION

The current graphic data processing tools allow the design and the visualization of realistic and precise 3D models. These 3D models are digital representations of either the real world or an imaginary world. The techniques of acquisition or design of the 3D models (modelers, scanners, sensors) generally produce sets of very dense data containing both geometrical and appearance attributes. The geometrical attributes describe the shape and dimensions of the object and include the data relating to a unit of points on the surface of the modeled object. The attributes of appearance contain information which describes the appearance of the object such as colors and textures.

These 3D models can be applied in various fields such as

the medical imaging, video games and cultural heritage... etc [1]. These 3D data are generally represented by polygonal meshes defined by a unit of vertex and faces. The most frequently used meshes for the representation of objects in 3D space are the triangular surface meshes.

The presence of noise in surfaces of 3D objects is a

problem that should not be ignored. The noise affecting these surfaces can be topological or geometrical. In the first case it is due to algorithms used to extract the meshes starting from groups of vertices while in the second case it depend on measurements errors and sampling of the data in the various treatments [2].

To eliminate this noise Taubin [3] applied signal processing methods to surfaces of 3D objects. This study has

encouraged many researchers to develop extensions of image processing methods in order to deal with 3D objects. Among these methods, there are those based on Wiener filter [4], Laplacian flow [5] which adjusts simultaneously the place of each vertex of mesh on the geometrical centre of its neighboring vertex, median filter [5], and Alpha-Trimming filter [6] which is similar to the nonlinear diffusion of the normals with an automatic choice of threshold. The only difference is that instead of using the nonlinear average, it uses the linear average and the non-iterative method based on robust statistics and local predictive factors of first order of the surface to preserve the geometric structure of the data [7]. There are other approaches for denoising 3D objects such as adaptive filtering MMSE [8]. The filter presented in [9] can be considered as weighted combination of an average filter and a min filter. Other approaches are based on bilateral filtering by identification of the characteristics [10], the non local average [11] and adaptive filtering by a transform in volumetric distance for the conservation of the characteristics [12].

Most of these methods use angles between normals

without any information about their statistical behavior. In [21] authors have studied the distribution of angles in order to determine sharpness value. They used Gaussian and Laplacian Distributions which despite being a rough approximation of the angle PDF leads to significant improvement. We believe that heavy tailed distributions can be a better fit of the angle based PDF. In this context, we propose a new normal based filter which uses the Cauchy function as the statistical distribution of angles. The filtered normal of each triangle is computed by using the Myriad value. It is determined by Cauchy parameter location estimation which is based on ML method. The myriad filter has been extensively used for signal [18][19], image [20] [21] and video [23] denoising in the presence of impulsive noise and especially the alpha-stable[16][17]. Once, the normals are filtered and the vertices positions are updated, we use L2 based metrics [13] and the distance of Hausdorff [22] in order to quantify the perceived quality of the filtered objects.

This article is organized as follows: Section 2 presents the problem formulation. In Section 3, we review some 3D mesh denoising techniques; Section 4 presents the proposed approaches; Section 5 presents the used error metrics. In Section 6, we provide experimental results to demonstrate

2011 Seventh International Conference on Signal Image Technology & Internet-Based Systems

978-0-7695-4635-3/11 $26.00 © 2011 IEEE

DOI 10.1109/SITIS.2011.26

431

the performance of the proposed methods in 3D mesh filtering. Section 7 deals with some concluding remarks.

II. PROBLEM FORMULATION

3D objects are usually represented as polygonal or triangle meshes. A triangle mesh is a triple M= (P, �, T) where:

• P = {P1, ……,Pn} denotes the set of vertices; • � = {eij} represents the set of edges; • T = {T1, ……,Tn} describes the set of triangles.

Each edge connects a pair of vertices (Pi, Pj). The neighbouring of a vertex is the set P*= {Pj∈ P: Pi ~ Pj}. The degree di of a vertex Pi is the number of the neighbours Pj. N(Pi) is the set of the neighbouring vertices of Pi. N(Ti) is the set of the neighbouring triangles of Ti . We denote by A(Ti) and n(Ti) the area and the unit normal of Ti, respectively. The normal n at a vertex Pi is obtained by averaging the normals of its neighbouring triangles and is given by

( )( )*

1

j i

i ji T T P

n n Td

∈

= � (1.1)

The mean edge length l of the mesh is given by

�∈

=ε

εije

ijel1

(1.2)

During acquisition of a 3D model, the measurements are perturbed by an additive noise:

η+= 'PP (1.3)

Where the vertex P includes the original vertex P’ and the random noise η . This noise is generally considered as a

Gaussian additive noise. For that, several methods of filtering of the meshes were proposed to filter and decrease the noise affecting the 3D models.

III. RELATED WORK

In this section, we present the methods based on the normals such as the mean, the median, the min and the adaptive MMSE filters

Consider an oriented triangle mesh. Let T and Ui be a mesh triangles, n(T) and n(Ui) be the unit normal of T and Ui

respectively, A(T) be the area of T, and C(T) be the centroid of T. Denote by N(T) the set of all mesh triangles that have a common edge or vertex with T (see Fig. 1)T

Figure 1. Left: Triangular mesh. Right: updating mesh vertex position.

1) Mean Filter: The mesh mean filtering scheme

includes three steps [5]: Step 1: For each mesh triangle T, compute the averaged normal m(T) :

( )( )

( ) ( )( )�� ∈

=TNU

iii i

UnUAUA

Tm1

(1.4)

Step 2: Normalize the averaged normal m (T):

( ) ( )( )Tm

TmTm ← (1.5)

Step 3: Update each vertex in the mesh:

( )( ) ( )

1new oldP P A T v T

A T← + ��

(1.6)

With

( ) ( ) ( )v T PC m T m T=����

(1.7)

( )Tv denotes the projection of the vector PC����

onto the

direction of m(T), as shown by the right image of Fig. 1.

2) Min filter : The process of min filtering differs from the average filtering only at step1. Instead of computing the average of the normals, we determine the narrowest normal for each face, by using the following steps [9]:

- Compute of angle Φ between n(T) and n(Ui). - Research of the minimal angle: If Φ is the

minimal angle in N (T) then n(T) is replaced by n(Ui).

432

3) Angle Median Filter: This method is similar to min

filtering; the only difference is that instead of using the narrowest normal we compute the median normal trought the angle median filter [5]:

( ) ( )( )ii UnTn ,∠=θ (1.8)

If �i is the median angle in N(T) then n(T) is replaced by n(Ui).

4) Adaptive MMSE Filter: This filter differs from the average filter only at step 1. The new normal m(T) for each triangle T is calculated by [8]:

( )

( )

( ) ( )

2 2 2

2 22 2 2

2 2

ou 0

1 et 0

lj n lj lj

j n nj lj n lj lj

lj lj

M T

m Tn T M T

σ σ σ

σ σσ σ σ

σ σ

� > =�� �=�

− + ≤ ≠� �� ��

(1.9)

( )( ) ( )

( )�

�−

=

−

==1

0

1

0N

ii

N

iiji

lj

UA

UnUA

TM (1.10)

__ _

�n2 represents the variance of additive noise and �lj

2 denotes the variance of neighbouring mesh normals which is changed according to elements of normal vector. Thus,

�lj2

is computed as follows :

( ) ( )

( )( )TM

UA

UnUA

ljN

ii

N

iiji

lj2

1

0

1

0

2

2 −=

�

�−

=

−

=σ (1.11)

5) Sharpness based filter: The sharpness filter is a five

steps algorithm described below: Step 1 : compute the average normal im for each triangle T using the following relation:

( )( )

( )

( )( )

i N T j

i N Tj

ij iU

i

ij iU

W n Um T

W n U

∈

∈

=�

� (1.12)

Where 1

( )iji

WN T

=

And ( )iN T denotes the set of neighbouring triangles of a

triangle T and ijW is the associate weight to each normal of

the triangle T. Step 2: Determine the normals of neighbouring triangles

iμ for each triangle T as follows:

(1) Compute the angle ( ) ( )( )ii UnTn ,∠=θ

(2) Find the minimal value( )

min min ii

i iU N Tθ

θ� �

= � ∈ �

Step 3: Compute the local sharpness value

( )

( )( )

2min1

i i

i i iU N Ti

SN T

θ θ∈

= −� (1.13)

Step 4: Compute the new normal for each triangle T by:

( )

( ) (1 ( ))i i i in W S m W S

nn T

n

μ� = + −��� ′ =���

�

���

� (1.14)

W(S) is a weighting function. Step 5: To update the vertices positions, use the following relation:

( ) ( )

( )( ) ( )1

i N Pi N P

i i ij i iUijU

P P W PC n U n UW

∈∈

′ ′ ′= + ∗′��

���� (1.15)

Go to next iteration if a stable final state was not reached. The procedure stops if all the face normals satisfy the condition

1 ( ) ( )n T n T ε′− ∗ < , where � is a predefined tolerance for

the steady state. Sharpness value is defined as a measure of the distribution of angles between the normals of triangles polygons.

IV. NORMAL BASED MYRIAD FILTER

In this section, we first study the statistical behavior of the angles between normals. For modeling these angles, we use the Cauchy distribution. Based on this assumption, we compute the normal based myriad filter.

A. Angle statistics distribution

A random variable X has a Cauchy if it admits a density with respect to Lebesque measure, depending on two parameters xo and a (a>0) defined by:

( )0 2

0

1, ,

1

f x x ax x

aa

π

= �−� �

+� �� �� �� �

(1.16)

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This distribution is symmetric with respect to x0 (location parameter). The parameter a controls the spread of the function (scale parameter).

B. Myriad value estimation

The mean and median are the maximum likelihood (MLE) estimates of the location parameter of the Gaussian and Laplacian distributions respectively. For the Cauchy distribution it is given the Myriad value of samples.

If we consider a sequence of variable X1 ... XN which follows a Cauchy distribution with location parameter � and a scale factor k:

( )( )

22

1,

kf x

k xβ

π β

� �= ∗� � + −

(1.17)

Myriad value of the sample is the value that maximizes the function of the maximum likelihood as follows:

( )1

ˆ arg max ,N

k ii

f xβ β=

= ∏ (1.18)

This is equivalent to minimize:

( ) ( )( )22

1

N

k ii

G k xβ β=

= + −∏ (1.19)

Thus the estimate of the value becomes a myriad:

{ }

( )

( )( )

1 2 3

22

1

22

1

ˆ myriad ; , , ,.......,

arg min log

arg min

k N

N

ii

N

ii

k x x x x

k x

k x

β

β

β

=

=

=

�= + −� �

= + −

�

�

(1.20)

With k denotes the linearity parameter of the samples Myriad filter. This parameter will play a fundamental role in this filter and can conduct to a lot of special cases.

C. Myriad filter

Step 1: Compute of angle � between n(T) and n(Ui). Step 2: Angle Myriad Filter: This method is similar to min and median filtering; the only difference is that instead of seeking the narrowest and the median normal we determine the Myriad normal by applying the angle Myriad filter:

( ) ( )( ),i in T n Uθ = ∠ (1.21)

If �i is the Myriad angle in N(T) then n(T) is replaced by n(Ui). Step 3: Update each vertex in the mesh as follows:

( )

( ) ( )1

new oldP P A T v TA T

← + �� (1.22)

With

( ) ( ) ( )v T PC m T m T=����

(1.23)

V. ERROR METRICS

To quantify the performance of the proposed approach in comparison with the method based on the vertices we compute the vertex-position error metric, the normal error metric [13] and the Hausdorff distance.

A. Vertex-position error metric

Consider an original model M’ and M the model after adding noise or applying several smoothing iterations. P is a vertex of M. Let set ( )dist ,P M ′ equal to the distance

between P and a triangle of the ideal mesh M’ closest to P. Our L2 vertex-position error metric is given by

( )( ) ( )

2'1dist ,

3vP M

A P P MA M

ε∈

= � (1.24)

Where A (P) is the summation of areas of all triangles incident on P and A (M) is the total area of M.

B. Normal error metric

The face-normal error metric is defined by

( )( ) ( ) ( )�

∈

−=MT

f TnTnTAMA

2'1ε (1.25)

Here T and T’ are triangles of the meshes M and M’ respectively; n(T) and n(T’) are the unit normals of T and T’

respectively and A(T) is the total area of T.

C. Hausdorff distance

The Hausdorff distance is an effective method to measure the distance between the 3D surfaces represented by triangular meshes. To measure� the Hausdorff distance� between two� meshes S and S ′ , we must first measure the distance of a vertex of

the mesh { },S V E= to { },S V E′ ′ ′= with the following

equation:

( ) ( ), min , e P S d P P P V′ ′ ′ ′= ∈ (1.26)

With ( ),d P P′ is the Euclidean distance between two

vertices P and P′ .

434

( ) ( ) ( ) ( )2 2 2

,d P P x x y y z z′ ′ ′ ′= − + − + − (1.27)

In practice to calculate the distance between a peak P and the surface S 'we must first make a dense sampling of S' for all P 'to ensure the accuracy of calculation and then measuring the distance:

( ) ( ), max , E S S e P S P V′ ′= ∈ (1.28)��

VI. EXPERIMENTAL RESULTS

This section presents simulation results where the proposed method is applied to noisy 3D models obtained by adding Gaussian noise. The standard deviation of Gaussian noise is given by

noise lσ = × (23)

Where l is the mean edge length of the mesh. The used objects on our experimentations are:

• Cow with 46 433 vertices and 92 864 faces, • and Shell with 915 vertices and 1680 faces

We first conducted a study of the statistical properties of the angles between surface normals. From the results illustrated in Figure 2, we see that the Cauchy distribution approximates the data better than the normal distribution regardless of the object and the standard deviation of Gaussian noise used. These results have been justified by computing the distance between the data and the two distributions (the distance to measure whether two histograms are "close" to each other). Indeed, Table 1 clearly shows that the distance between the data and the Cauchy distribution is smaller than the distance between the data and the normal distribution and this hold for the different levels of noise. These results demonstrate the better fit of the Cauchy distribution as compared to the the normal distribution.

0 20 40 60 80 100 120 140 160 1800

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045approximation of the distribution using the Cauchy function

My data

Cauchy pdfNormale pdf

0 20 40 60 80 100 120 140 160 1800

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04approximation of the distribution using the Cauchy function

My data

Cauchy pdfNormale pdf

(a) (b)

0 20 40 60 80 100 120 140 160 1800

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045approximation of the distribution using the Cauchy function

My data

Cauchy pdfNormale pdf

0 20 40 60 80 100 120 140 160 1800

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04approximation of the distribution using the Cauchy function

My data

Cauchy pdfNormale pdf

(c) (d)

Figure 2. Goodness of fit for angles distribution for Shell object. Data histogram in blue normal distribution estimate in green and Cauchy

distribution estimate in red for different values of noise standard deviations. a) �=0.2, b)�=0.3, c) �=0.5, d) �=0.6

TABLE I. KULLBACK LEIBLER DISTANCE VALUE BETWEEN EMPIRICAL DATA AND STUDIED DENSITIES FOR SHELL OBJECT.

Noise standard deviation

Distance between empirical data and

Cauchy density

Distance between empirical data and

Normal density

0.1 0.1470 0.3543

0.2 0.1523 0.3278

0.3 0.1792 0.2837

0.4 0.2172 0.2537

0.5 0.2360 0.2375

0.6 0.2360 0.2365

0 2 4 6 8 10 12 14 16 18 205.1

5.12

5.14

5.16

5.18

5.2

5.22x 10

4

filter iterations

vert

ex-b

ased

L2 e

rror

vertex-based L2 error

0 2 4 6 8 10 12 14 16 18 204.39

4.4

4.41

4.42

4.43

4.44

4.45

filter iterations

Hau

sdor

ff d

ista

nce

Hausdorff distance between the original object and filtered object

(a) (b)

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8x 10

-4

filter iterations

Nor

mal

bas

ed L

2 err

or

Normal based L2 error

(c)

Figure 3. Evolution of metric errors for Cow object filtered by Myriad filter. (a vertex based metric, (b Hausdorff distance and c) normal based

metric.

Before giving comparative results, we studied the evolution of the error metrics as a function of the iteration number. Results show that little iteration is needed in order to reach a low error value for the three metrics. Figures 3 and 4 illustrate this behavior for respectively the myriad and median filter. After the third iteration for both filters the curves converge to an asymptotical value. Using more iteration does not increase the performance significantly. It only produces an over-smoothing of the object. In the following experiments we therefore fixed the number of iterations to 3. We also compared the normal based filters described in the related work section. The notation of these filters is resumed in Table 2. Typical results are reported in table 3 and 4 for the object cow.

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In Table 3 and 4, we notice that the proposed method (Myriad filter) gives the smallest error metrics comparing to alternative filters.

0 2 4 6 8 10 12 14 16 18 205.32

5.34

5.36

5.38

5.4

5.42

5.44x 10

4

filter iterations

vert

ex-b

ased

L2 e

rror

vertex-based L2 error

0 2 4 6 8 10 12 14 16 18 204.49

4.5

4.51

4.52

4.53

4.54

4.55

filter iterations

Hau

sdor

ff d

ista

nce

Hausdorff distance between the original object and filtered object

(a) (b)

0 2 4 6 8 10 12 14 16 18 20

1.4

1.5

1.6

1.7

1.8

1.9

2x 10

-3

filter iterations

Nor

mal

bas

ed L

2 err

or

Normal based L2 error

(c)

Figure 4. Evolution of metric errors for Cow object filtered by angle median filter. (a vertex based metric, (b Hausdorff distance and c) normal

based metric.

This behaviour is always observed independently of the noise level and the object nature. Local sharpness filter (E) and angle median filter (C) are the closest to the myriad filter in terms of performances.

TABLE II. NOTATIONS

A Mean filter

B Angle min filter

C Angle median filter

D Adaptive MMSE filter

E Local Sharpness filter

F Myriad filter

To facilitate the comparison their metric values are mentioned in bold. Note that Myriad filter seems more performing as the noise level increase this may be due to its robustness in highly impulsive environments. Figure 5 and 6 are given to assess the visual impact of the filtering. We choose to report results only for angle median filer and myriad filter as they give comparable results in terms of error metric. We can see that for in both case the Myriad filter processed objects exhibit a more appealing visual appearance.

VII. CONCLUSION

In this paper, we introduced a normal-based Myriad filter for 3D mesh denoising. This filter is efficient for 3D mesh denoising strategy to fully preserve the geometric structure of the 3D mesh data. The experimental results clearly show an improvement of the performance of the proposed

approach in comparison with the median and other normal-based filters. This improvement is obtained with the same level of complexity by using a more appropriate characterization of the angle probability distribution function

TABLE III. ERROR METRICS OF THE FILTERED COW OBJECT � = 0.6

Methods �v �f Hausdorff

A 0.0087 0.0734 0.0589

B 0.0086 0.0738 0.0572

C 0.0074 0.0575 0.0468

D 0.0084 0.0701 0.0506

E 0.0072 0.0557 0.0443

F 0.0068 0.0500 0.0402

TABLE IV. ERROR METRICS OF THE FILTERED NOISY COW OBJECT � = 0.8

Methods �v �f Hausdorff

A 0.0098 0.0845 0.0792

B 0.0092 0.0833 0.0748

C 0.0078 0.0625 0.0578

D 0.0089 0.0732 0.0713

E 0.0078 0.0630 0.0569

F 0.0072 0.0603 0.0551

.

(a) (b)

(c) (d)

Figure 5. (a) Original Shell object, (b) Noisy model (0.6), (c) Filtered

model by Angle median filter, (d) Filtered model by myriad filter

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Figure 6. (a) Original Cow object, (b) Noisy model (0.6), (C) Filtered

model by Angle median filter, (d) Filtered model by myriad filter

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