Riemannian Filters for Multi-variate Mesh Signals

Teodor Cioaca

, Bogdan Dumitrescu

and Mihai-Sorin Stupariu

University Politehnica of Bucharest, Bucharest, Romania

University of Bucharest, Bucharest, Romania

Keywords:

Graph Wavelets, Multi-variate Mesh Signals, Riemannian Mean Shift.

Abstract:

Designing ﬁlters over irregular non-Euclidean domains requires algorithms that take into account the intrinsic

curvature of these domains. We propose a new ﬁltering method based on Riemannian weighted averages. The

resulting ﬁlters are non-Euclidean adaptations of the mean shift and blurring mean shift algorithms. We also

introduce a hybrid, efﬁcient computing strategy by combining these iterative ﬁltering methods with wavelet

multi-resolution editing. The applications of our ﬁlters include multi-variate mesh data smoothing, denoising,

attribute enhancement and curvature ﬁltering.

1 INTRODUCTION

Filtering operations are essential tools of signal pro-

cessing. For signals sampled over 1-D and 2-D reg-

ular, ﬂat domains the existing literature offers an ex-

tensive collection of ﬁltering methods. In the case of

graphs and meshes, where the domain can be highly

irregular and intrinsically curved, it is not immedi-

ately possible to adapt and apply even the simpler

mean or Gaussian blurring ﬁlters. The main difﬁcul-

ties stem from the fact that non-Euclidean domains

are no longer closed under linear combinations of

points.

In this work, we develop a solution for the numer-

ical evaluation of a mean shift ﬁlter on discretely sam-

pled 2-dimensional smooth Riemannian manifolds.

The principle we use to overcome the difﬁculties in

working on non-Euclidean spaces is to perform the

ﬁlter iterations in the local tangent space and then

project the result onto the underlying manifold. This

same strategy is used in the continuous case, in the

Weiszfeld algorithm (Aftab et al., 2015). We extend

the technique to meshes with per-vertex attributes.

This way, we design a family of mean shift algorithms

that function on curved domains, even under condi-

tions of heavy noise corruption.

1.1 Related Work

Since its introduction by (Fukunaga and Hostetler,

1975), there have been many attempts to adapt the

mean shift algorithm to geometrically complex do-

mains. (Yamauchi et al., 2005) use an Euclidean ver-

sion of the mean shift algorithm to segment meshes

by clustering faces. Although their method handles

geometry and normal orientations differently using

composite kernels, the ﬁltered output is not guaran-

teed to lie on the same support mesh. (Zhang et al.,

2008) approach the segmentation problem via mean

shift by enhancing and ﬁltering the per-vertex cur-

vature estimates. This solution does not evaluate

geodesic distances between samples, instead measur-

ing only topological distances. (Cerveri et al., 2012)

perform curvature mean shift ﬁltering by also treat-

ing the feature spaces as entirely Euclidean where

the vertex coordinates and curvature values make up

the coordinates of a 4-dimensional space. (Shamir

et al., 2006) perform curvature mean shift using lo-

cal geodesic parameterizations. Still, their solution

requires rendering an entire mesh neighborhood to

a texture and performing image mean shift instead.

(Subbarao and Meer, 2009) correctly generalize the

mean shift ﬁlter to certain families of Riemannian

manifolds such as matrix Lie groups, Grassmann

manifolds, essential matrices and symmetric positive

deﬁnite matrices. (Cetingul and Vidal, 2009) propose

an intrinsic mean shift algorithm for Grassmann and

Stiefel manifolds. (Caseiro et al., 2012) introduce a

mapping of arbitrary Riemannian manifolds into a re-

producing kernel Hilbert space via a heat kernel ob-

taining a semi-intrinsic mean shift formulation.

(Solomon et al., 2014) introduce a generalized bi-

lateral and mean shift ﬁltering framework for multi-

variate data on domains that admit a Laplacian oper-

228

Cioaca T., Dumitrescu B. and Stupariu M.

Riemannian Filters for Multi-variate Mesh Signals.

DOI: 10.5220/0006128602280235

In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2017), pages 228-235

ISBN: 978-989-758-224-0

ator. As such, their method can be applied to graphs,

meshes, images and point clouds. The main drawback

of this method is that the resulting mean shift algo-

rithm is not a sliding window mode-ﬁnding algorithm.

In essence, the framework computes a weighted aver-

age of the attributes with respect to the neighborhood

of a ﬁxed sample on a curved domain. Opting for a

sliding window formulation leads to a better adaption

to the local structure of the data, as emphasized in

(Comaniciu and Meer, 2002).

1.2 Our Contributions

We generalize the works of (Pennec, 2006), (Fletcher

et al., 2008) and (Wang and Carreira-Perpin, 2010) to

generic mesh domains by eliminating the need of a

priori knowing a closed-form expressions for the ex-

ponential and logarithm maps on the underlying man-

ifold. In this work, we describe:

• a rigorous formulation for the mean shift imple-

mentation on general triangular meshes with both

geometry and attribute vertex data;

• a computation acceleration algorithm using a

graph-wavelet decomposition to ﬁlter a coarser

model and propagate changes through wavelet

synthesis, thus effectively reducing the size of the

problem.

The remainder of this paper is organized as fol-

lows. In section 2 we describe the mean shift ﬁlter and

how it can be extended to mesh domains. A hybrid

technique for accelerating the ﬁlter computations via

wavelet multiresolution is presented in section 3. We

further present mesh normal denoising and attribute

ﬁltering applications in section 4 and conclude this

research in section 5.

2 FILTERING ON

NON-EUCLIDEAN DOMAINS

In this section we discuss the key ingredients required

to compute Riemannian weighted averages. Although

the main technique resembles the Euclidean case, the

computation on Riemannian manifolds is based on the

Weiszfeld algorithm (Aftab et al., 2015). For gen-

eral triangular meshes, the exponential and logarithm

maps are the only algorithm operations that cannot be

directly computed (since there is no closed-form ex-

pression available). Instead, we propose evaluating

the logarithm map using the geodesic polar coordi-

nates algorithm of (Melvær and Reimers, 2012). For

the exponential map, the straightest geodesics algo-

rithm described by (Polthier and Schmies, 1998) is

the most efﬁcient choice. Both algorithms work on

triangular meshes and do not depend upon the genus

of the underlying mesh or upon the regularity of the

network.

2.1 Euclidean Mean Shift

Let x

∈ R

be a set of n data samples. It can be

assumed that the x

points are sampled from a ran-

dom variable whose density can be approximated us-

ing kernel estimators. The expression for this density

estimate at a point x ∈ R

is given by

h,K

(x) =

K,d

∑

i=1



x − x



, (1)

where h is the bandwidth parameter, k(·) is a kernel

proﬁle function corresponding to a multi-variate ker-

nel K(x) := k (kxk

) and c

K,d

is a normalization

constant.

The principle behind mean shift ﬁltering is to re-

place each sample x with the closest mode of the prob-

ability density estimate. If

x is such a mode, then

∇

h,K

(

x) = 0. The problem of ﬁnding

x is usually

solved through an iterative gradient ascent process,

i.e.

(k+1)

← x

(k)

+ m

h,G

(k)

), (2)

where

h,G

(x) =

∑

i=1





x−x





− x)

∑

i=1





x−x





, (3)

with g(·) the kernel proﬁle function of a multivariate

kernel G such that g(·) = −k

(·). In many practical

situations, the individual coordinates of the samples

have different interpretations. The most common ex-

ample is that of an image where spatial and range co-

ordinates are concatenated. Formally, x =





with x

∈ R

and p + q = d. Given the dif-

ferent semantics of these coordinates, a concatenated

kernel (as proposed by (Comaniciu and Meer, 2002))

can be used to achieve better bandwidth selection con-

trol. For spatial and range coordinates, equation (1)

becomes

s,r

(x) =

∑

i=1



− x



− x



(4)

where

C =

K,d

and h

are the spatial and

range bandwidth parameters.

2.2 Riemannian Mean Shift

We now consider the case when the sampled space

is inherently curve, i.e. {x

} ⊂ M for i ∈ 1,n and M

Riemannian Filters for Multi-variate Mesh Signals

229

a 2-dimensional Riemannian manifold embedded in

. Further, let us consider x

to be the vertices of a

mesh network. The iterative process in equation (2)

no longer moves the samples along the surface of the

domain manifold. Furthermore, the deﬁnition of the

kernel function must take into account the intrinsic

geodesic distance measured across the manifold. For

Riemannian manifolds, the gradient of the geodesic

distance function d(p,q) for p ﬁxed is given by

∇

(d(p, q)) =

log

(q)



log

(q)



, (5)

where log

(q) is the logarithm map. As in the Eu-

clidean case, the samples may correspond to differ-

ent semantic spaces. In this situation, M is a sub-

manifold of M

× M

, where M

and M

are spatial

and range manifolds. An important result is that the

tangent space of the product can be decomposed as

a direct sum of the individual tangent planes, estab-

lishing a mapping T

)⊕T

) → T

×M

If π

: M

×M

→ M

is the canonical projection, then

dπ

(x) is also a projection from T

×M

) to T

Since the gradient of the density function is a vec-

tor in T

×M

), then dπ

(x)



∇

s,r

(x)



∈ T

Further details on how to evaluate these quantities are

provided in the Appendix. Using this projection prin-

ciple, we now formulate the mean shift vector equiv-

alent of (3) in T

), i.e.

(x) =

∑

i=1



)





)



log

)

∑

i=1



)





)



(6)

Since the Riemannian mean shift vector lies in the

tangent space T

, we must project the result

back onto the manifold. This projection is achieved

through the exponential map, exp

). We are now

able to assemble the mean shift algorithm for Rieman-

nian manifolds, which is similar to how the weighted

median is computed using the Weiszfeld algorithm

(see (Fletcher et al., 2009)). The iterative update

equation, illustrated in ﬁgure 1, is the following

(k+1)

← exp

(k)



(k)

)



. (7)

In the above equation, the exponential maps canoni-

cally a vector from T

to a point on the M subman-

ifold of M

× M

The Riemannian mean shift presented in algo-

rithm 1 is formulated as a blurring ﬁlter. By setting

the total number of blurring passes N

= 1, we obtain

the classical mean shift algorithm. Performing more

than one ﬁltering pass and using the ﬁltered output as

the input of a next pass, the blurring effects become

Algorithm 1: Blurring Riemannian mean shift.

Require: {x

} ⊂ M, i ∈ 1,n .

Ensure: {y

} ⊂ M, ı ∈ 1,n, the modes of

K,h

cor-

responding to {x

for pass ← 1 ...N

for i ← 1 . . . n do

(0)

← x

,k ← 0

repeat

(x) =

∑

j=1



)





)



log

)

∑

j=1



)





)



(k+1)

← exp

(k)



∗

(k)

)



k ← k +1

until ky

(k+1)

− y

(k)

k ≤ ε or k > k

max

← y

(k)

end for

for i ← 1 . . . n do

← y

end for

log

exp

Figure 1: Computing the tangent-space mean shift vector

approximation via the log map, as described in equation (6).

The resulting mean shift vector is projected via the exp map

as detailed in equation (7).

more pronounced. The k

max

parameter is internally

used to control the maximum number of gradient as-

cent iterations, while ε is the error threshold used to

stop this reﬁnement.

3 COMBINED WAVELET

FILTERING

Wavelet multiresolution analysis allows building a

hiearchical approximation of the initial mesh in a ﬁne-

to-coarse fashion. Let M := M

 M

L−1

 ...  M

be the resulting chain of L successive approximations,

GRAPP 2017 - International Conference on Computer Graphics Theory and Applications

230

where M := M

is the initial mesh that is subjected

to the multiresolution analysis procedure. In general,

this process implies repeating the following decompo-

sition M

k+1

= M

∪

⊕

, where D

is the set of detail

vectors that are removed from M

k+1

in order to pro-

duce the lower resolution approximation, M

. Using

the lifting scheme, this analysis operation becomes

trivially invertible. The lifting scheme design de-

Algorithm 2: Hybrid wavelet-based ﬁltering strategy.

Require: M = (V,E,F) high resolution mesh, L

number of intermediate approximations

Ensure:

M :=



L−1

 .. . 

a ﬁltered hi-

erarchical approximation

for k ← n,1 do

= M

k−1

∪

⊕

k−1

end for

CALL algorithm 1 M

→

for k ← 1,n do

k−1

∪

⊕

k−1

 D

k−1

end for

veloped by (Cioaca et al., 2016) is a light-weight so-

lution for performing analysis on multi-variate mesh

data. Carrying out the intensive mean shift computa-

tions on the coarsest approximation, M

, we obtain a

new mesh,

. We then recover the high resolution

model through wavelet synthesis. This is achieved

by adding back missing details with a certain cho-

sen amplitude, α

∈ R

. The synthesis thus con-

sists of update sequences

k−1

∪

⊕

k−1

D

k−1

where α

k−1

D

k−1

is the set of difference vectors uni-

formly scaled by the α

k−1

factor. This whole process

is sketched in algorithm 2.

4 RESULTS AND DISCUSSION

4.1 Mesh Smoothing Via Normal

Filtering

We test the non-linear ﬁlter through a series of

normal-based mesh smoothing experiments. We ap-

ply the mean shift algorithm, treating the per-vertex

normals as additional attribute data. Reﬁning these

normals allows for smoothing and repairing the mesh.

The algorithm of (Lee and Wang, 2005) serves this

purpose by iteratively altering the vertex positions to

align the one-ring face normals with the ﬁltered sam-

ples. While (Solomon et al., 2014) chose to treat face

normals separately, we have found that it is possible to

ﬁlter the per-vertex normals instead and approximate

the face normal by averaging the three normals of its

vertices. This choice eliminates the need of remesh-

ing the input model and allows treating normal infor-

mation as any kind of attribute data.

To ensure the exponential and logarithm map

computations are not inﬂuenced by the artiﬁcial geo-

metric noise, the base model was subjected to a Lapla-

cian smoothing pass using the method of (Taubin,

1995). We however note that the mean shift ﬁlter

was not found to lead to a signiﬁcant degradation of

the normal ﬁeld, but, when extending the geodesic

neighborhood radius, the exponential map samples

became unreliable. The Laplacian smoothed output

is solely used for computing the exponential and log-

arithm maps and does not constitute the input of the

algorithm described by (Lee and Wang, 2005).

The models used in our tests (also used in

(Solomon et al., 2014) and depicted in ﬁgure 2) were

rescaled to ﬁt within the unit cube. The denoising ex-

periments were performed by artiﬁcially adding ver-

tex displacements having a uniform distribution of

magnitudes half the size of the average edge length.

This choice of the artiﬁcial noise distribution is iden-

tical to that proposed in (Solomon et al., 2014) and al-

lows for a qualitative assessment between the results

described in this work and our own experiments.

The methods selected for comparison are the com-

bined vertex and tangent plane projection feature

preserving smoothing of (Jones et al., 2003), the

prescribed mean curvature ﬂow of (Hildebrandt and

Polthier, 2004), the bilateral normal ﬁlter of (Zheng

et al., 2011) and the bilateral and mean shift frame-

work of (Solomon et al., 2014).

To achieve normal-based mesh ﬁltering we per-

formed three alternating passes of algorithm 1 and the

vertex relaxation routine of (Lee and Wang, 2005).

To penalize both normal dissimilarity and sample dis-

tance, we used the following kernel expression in

equation (6)

exp



−

)



exp



−1

(0.5 + 0.5n

)



(8)

where n

is the unit normal vector at a point x.

The mean shift is performed inside a geodesic ra-

dius r = 4AEL, using the bandwidth parameter values

= 3AEL, h

= 0.45, where AEL denotes the mesh

average edge length. The iteration count is controlled

by setting k

max

= 50 and ε = AEL/5.

The denoised models are depicted in ﬁgure 3.

Table 1 compares our method to several state-

of-the-art mesh denoising algorithms using the root

mean square distance, computed using MeshLab

(Cignoni et al., 2008)). Observing the smoothed

meshes (ﬁgure 3), we conclude that the ﬁltering al-

gorithm of (Zheng et al., 2011) is more efﬁcient

Riemannian Filters for Multi-variate Mesh Signals

231

Figure 2: Mesh models with added uniform noise (vertex

displacements half the mean edge length).

Table 1: Root mean square distance values for the mesh de-

noising experiments (scaled by 10

). Models: (R)amesses,

(Fr)og, (B)ust, (F)andisk, (S)tar and (D)ouble torus. Meth-

ods, top to bottom: (Jones et al., 2003), (Hildebrandt and

Polthier, 2004), (Zheng et al., 2011), bilateral and mean

shift ﬁlters of(Solomon et al., 2014), our Riemannian mean

shift ﬁlter.

(R) (Fr) (B) (F) (S) (D)

0.36 0.36 0.47 0.58 0.44 1.2

0.34 - 0.52 0.47 0.34 0.45

0.23 0.30 0.39 0.33 0.55 0.47

0.24 0.27 0.40 0.38 0.41 0.5

0.24 0.31 0.35 0.31 0.41 0.42

0.26 0.24 0.41 0.35 0.35 0.44

than the methods of (Jones et al., 2003), (Hilde-

brandt and Polthier, 2004) and than the bilateral ﬁl-

ter of (Solomon et al., 2014). The mean shift ﬁlter

of (Solomon et al., 2014) tends to produce smoother

meshes, with fewer isolated artifacts. Overall, our ﬁl-

ter produces smooth output surfaces with a few per-

ceivable, low frequency noise artifacts. Compared to

other methods, the high frequency noise components

are suppressed. The visual quality and numerical re-

sults recommend our ﬁltering technique for situations

where a uniﬁed ﬁltering framework that can handle

both geometry and attribute smoothing tasks.

4.2 Combined Wavelet Attribute

Filtering

Given the iterative nature of the mean shift algorithm,

it is important to understand the computational com-

plexity of the underlying operations. The exponential

map can be found in linear time and is straightfor-

ward to compute. The logarithm map, on the other

hand, falls in the O(nlog(n)) family. Keeping a small

geodesic window radius helps reducing the impact on

scalability. If algorithm 2 is used, inputs with several

millions of samples can be efﬁciently handled.

4.2.1 Application to Terrain Modeling

In practice, large input models are commonplace.

For an applied proof of concept, we propose ﬁlter-

ing terrain models digitized from LiDAR scans. De-

noising or smoothing this type of data is a com-

mon GIS task, but the computations tend to be slow

given the size of the input. For our numerical ex-

periments we chose two terrain samples: one rep-

resenting a fragment of the Great Smoky Moun-

tains (having 270,000 points), obtained through the

http://www.opentopography.org/ portal, and an-

other one representing a fragment of the Romanian

Carpathian Mountains (having 11.8 million points),

obtained through a custom aerial scan. These models

include both geometric coordinates, as well as vegeta-

tion information in the form of a height offset (above

ground) between 0 and 30 meters and a class index

that ranges between the scalar values of 2 and 15.

To incorporate a spatial and range dissimilarity pe-

nalizing effect in the ﬁlter iteration, we opt for a prod-

uct of Gaussians kernel, i.e.

∗

(x) := exp



−



exp



−



. (9)

We ran the single-pass mean shift algorithm (MS) set-

ting the parameters h

= 8AEL, h

= 4, ε = AEL/20

and k

max

= 50. To achieve a blurring effect (BMS),

we set h

= 5AEL, h

= 5, ε = AEL/5 and k

max

= 5

and execute a sequence of 10 blurring passes inside

algorithm 1. We summarize the execution times of

both algorithm variations for the terrain models in ta-

ble 2 using different values for the number of levels

parameter, L, of algorithm 2.

Figure 4 illustrates the terrain models before and

after subjecting them to the combined wavelet and

GRAPP 2017 - International Conference on Computer Graphics Theory and Applications

232

Figure 3: Denoising experiments. A: from left to right (Jones et al., 2003), (Hildebrandt and Polthier, 2004), (Zheng et al.,

2011), bilateral and mean shift of (Solomon et al., 2014) and our mean shift ﬁlter. B: from left to right and from top to bottom

(Jones et al., 2003), (Hildebrandt and Polthier, 2004), (Zheng et al., 2011), mean shift of (Solomon et al., 2014) and our mean

shift ﬁlter. C: from left to right and from top to bottom (Jones et al., 2003), (Hildebrandt and Polthier, 2004), (Zheng et al.,

2011), bilateral and mean shift of (Solomon et al., 2014) and our mean shift ﬁlter.

Riemannian mean shift ﬁlters (i.e. algorithms 1 and

2). Setting the α

= 1 is equivalent to adding back the

missing geometry and attribute details, while setting

= 0 suppresses the higher frequency components,

including noise. In our experiments, we ﬁlter and de-

noise these models by completely suppressing the de-

tails. The recovered, high resolution models exhibit

smoother vegetation boundary contours.

Table 2: Algorithm benchmarks for the LiDAR point sets.

Model L Vertex count MS BMS

Frog 2 5242 5s 16s

Smoky 6 40438 40s 120s

Carpathians 12 229095 260s 780s

The execution times of our algorithms, detailed

in table 2, are sensibly lower than those reported by

(Shamir et al., 2006). In total, the authors report

an execution time of 10 minutes for a mesh with

20,000 vertices on a 2GHz machine. In our set-up,

we have benchmarked single-threaded implementa-

tions of both algorithms. The machine used to evalu-

ate the performance on the datasets from table 2 was

running at a frequency of 3GHz. All algorithms were

implemented in standard C++.

4.2.2 Curvature Filtering

Curvature ﬁltering and curvature-based segmentation

are other common mesh processing applications. In

our last experiment, we compute the absolute discrete

curvature and map it to a vertex color attribute us-

ing MeshLab (Cignoni et al., 2008). We then run the

ﬁltering algorithm on the bust and frog meshes, set-

ting the α

factors from algorithm 2 to 0, thus com-

pletely suppressing the higher frequency information.

Both models are subjected to a sequence of only 2

wavelet analysis steps, reducing the vertex count to

approximately 56% of the initial model. More specif-

Riemannian Filters for Multi-variate Mesh Signals

233

Figure 4: Rows: the Smoky Mountains fragment and the

Carpathian Mountains fragment. Columns, left to right:

original data, low resolution ﬁltered with mean shift, low

resolution ﬁltered with blurring mean shift.

Figure 5: Curvature ﬁltering using algorithm 2 with α

0. Left to right: initial, mean shift and blurring mean shift

ﬁltered color-coded absolute curvature values.

ically, the frog model is reduced from 9815 points

to 5242 points. The algorithm parameters are h

3AEL, h

= 0.25 for the single pass mean shift and

= 5AEL , h

= 0.1 for the blurring mean shift with

10 passes. The curvature ﬁltering results for the frog

and bust models are shown in ﬁgure 5.

5 CONCLUSIONS

The ﬁlter described in our work is a rigorous discrete

formulation of the mean shift method on Riemannian

manifolds. We are not aware of any to-date adap-

tations that are set up in general context of meshes

viewed as sampled Riemannian manifolds. Our ap-

proach does not require prior knowledge about the

characteristics of the manifold subjected to these ﬁl-

ters (e.g. the exponential and logarithm maps need not

have a known closed-form expression). Furthermore,

the iterative process considers both the geometry and

connectivity of the underlying mesh, guaranteeing the

ﬁlters output samples that lie on the surface, along

geodesic paths that originate from a seed point. This

is an improvement over existing literature where there

is either no such a guarantee or where the iterations

are performed in a Euclidean fashion.

Reducing the size of the input through wavelet

analysis is an effective compromise that allows per-

forming the time-consuming ﬁltering operations on a

coarse approximation. In our experiments, we have

successfully recovered sets subjected to up to 12 con-

secutive analysis passes, combining the effects from

both the low resolution mean ﬁlter and from suppress-

ing the high frequency details (which often are af-

fected by noise).

The main limitations are the need to use pre-

smoothed geometry for robust exponential map eval-

uations and the subtle data dependency imposed by

the selection of the bandwidth parameters h

and h

Since our mean shift implementation also requires

projecting the gradient of the density function onto

the tangent space of the spatial coordinate manifold,

the question whether performing the ﬁlter iterations

on the entire tangent space product can improve the

results represents a research direction. We thus con-

sider these points to constitute directions of improve-

ment and future research.

ACKNOWLEDGEMENTS

The authors would like to thank Dr. Justin Solomon

for providing numerical results for the mesh smooth-

ing methods presented in (Jones et al., 2003), (Hilde-

brandt and Polthier, 2004), (Zheng et al., 2011) and

(Solomon et al., 2014).

This work was supported by the Swiss Enlarge-

ment Contribution in the framework of the Romanian-

Swiss Research Program, project WindLand, project

code: IZERZO 142168/1 and 22 RO-CH/RSRP.

REFERENCES

Aftab, K., Hartley, R., and Trumpf, J. (2015). General-

ized Weiszfeld algorithms for Lq optimization. IEEE

Transactions on Pattern Analysis and Machine Intel-

ligence, 37:728–745.

Caseiro, R., Henriques, J. a. F., Martins, P., and Batista, J.

(2012). Semi-intrinsic mean shift on riemannian man-

ifolds. In Proceedings of the 12th European Confer-

ence on Computer Vision - Volume Part I , ECCV’12,

pages 342–355, Berlin, Heidelberg. Springer-Verlag.

GRAPP 2017 - International Conference on Computer Graphics Theory and Applications

234

Cerveri, P., Manzotti, A., Marchente, M., Confalonieri, N.,

and Baroni, G. (2012). Mean-shifted surface curva-

ture algorithm for automatic bone shape segmentation

in orthopedic surgery planning: a sensitivity analy-

sis. Computer Aided Surgery, 17(3):128–141. PMID:

22462564.

Cetingul, H. E. and Vidal, R. (2009). Intrinsic mean shift

for clustering on stiefel and grassmann manifolds. In

2009 IEEE Conference on Computer Vision and Pat-

tern Recognition, pages 1896–1902.

Cignoni, P., Corsini, M., and Ranzuglia, G. (2008). Mesh-

Lab: an Open-Source 3D Mesh Processing System.

Ercim News, 2008.

Cioaca, T., Dumitrescu, B., and Stupariu, M.-S. (2016).

Graph-based wavelet representation of multi-variate

terrain data. Computer Graphics Forum, 35(1):44–58.

Comaniciu, D. and Meer, P. (2002). Mean shift: A robust

approach toward feature space analysis. IEEE Trans.

Pattern Anal. Mach. Intell., 24(5):603–619.

Fletcher, P. T., Venkatasubramanian, S., and Joshi, S.

(2008). Robust statistics on riemannian manifolds via

the geometric median. In Computer Vision and Pat-

tern Recognition, 2008. CVPR 2008. IEEE Confer-

ence on, pages 1–8.

Fletcher, P. T., Venkatasubramanian, S., and Joshi, S.

(2009). The geometric median on riemannian mani-

folds with application to robust atlas estimation. Neu-

roImage, 45(1, Supplement 1):S143 – S152. Mathe-

matics in Brain Imaging.

Fukunaga, K. and Hostetler, L. (1975). The estimation of

the gradient of a density function, with applications in

pattern recognition. IEEE Transactions on Informa-

tion Theory, 21(1):32–40.

Hildebrandt, K. and Polthier, K. (2004). Anisotropic Fil-

tering of Non-Linear Surface Features. Computer

Graphics Forum, 23:391–400.

Jones, T. R., Durand, F., and Desbrun, M. (2003). Non-

iterative, feature-preserving mesh smoothing. ACM

Trans. Graph., 22(3):943–949.

Lee, K. and Wang, W. (2005). Feature-preserving mesh

denoising via bilateral normal ﬁltering. In 9th In-

ternational Conference on Computer-Aided Design

and Computer Graphics, CAD/Graphics 2005, Hong

Kong, China, 7-10 December, 2005, page 6.

Melvær, E. L. and Reimers, M. (2012). Geodesic polar co-

ordinates on polygonal meshes. Computer Graphics

Forum, 31(8):2423–2435.

Pennec, X. (2006). Intrinsic statistics on riemannian man-

ifolds: Basic tools for geometric measurements. J.

Math. Imaging Vis., 25(1):127–154.

Polthier, K. and Schmies, M. (1998). Straightest Geodesics

on Polyhedral Surfaces, pages 135–150. Springer

Berlin Heidelberg.

Shamir, A., Shapira, L., and Cohen-Or, D. (2006). Mesh

analysis using geodesic mean-shift. Vis. Comput.,

22(2):99–108.

Solomon, J., Crane, K., Butscher, A., and Wojtan, C.

(2014). A general framework for bilateral and mean

shift ﬁltering. ArXiv e-print 1405.4734, 32.

Subbarao, R. and Meer, P. (2009). Nonlinear mean shift

over riemannian manifolds. International Journal of

Computer Vision, 84(1):1–20.

Taubin, G. (1995). Curve and surface smoothing without

shrinkage. In Proceedings of the Fifth International

Conference on Computer Vision, ICCV ’95, pages

852–852, Washington, DC, USA. IEEE Computer So-

ciety.

Wang, W. and Carreira-Perpin, M. . (2010). Manifold blur-

ring mean shift algorithms for manifold denoising.

In Computer Vision and Pattern Recognition (CVPR),

2010 IEEE Conference on, pages 1759–1766.

Yamauchi, H., Lee, S., Lee, Y., Ohtake, Y., Belyaev, A., and

Seidel, H.-P. (2005). Feature sensitive mesh segmen-

tation with mean shift. In Proceedings of the Inter-

national Conference on Shape Modeling and Applica-

tions 2005, SMI ’05, pages 238—245, Washington,

DC, USA. IEEE Computer Society.

Zhang, X., Li, G., Xiong, Y., and He, F. (2008). 3D Mesh

Segmentation Using Mean-Shifted Curvature, pages

465–474. Springer Berlin Heidelberg, Berlin, Heidel-

berg.

Zheng, Y., Fu, H., Au, O. K.-C., and Tai, C.-L. (2011).

Bilateral normal ﬁltering for mesh denoising. IEEE

Transactions on Visualization and Computer Graph-

ics, 17(10):1521–1530.

APPENDIX

We evaluate the gradient of (4) on M

× M

∇

s,r

(x) =

∑

i=1



)





)





log

),0



∑

i=1



)





)





,log

)



(10)

Through projection onto T

we obtain

dπ

(x)



∇

s,r

(x)



∑

i=1



)





)



log

). (11)

Riemannian Filters for Multi-variate Mesh Signals

235