Spectral Multi-Dimensional Scaling using Biharmonic Distance

Jun Yang

, Alexander Jesuorobo Obaseki

and Jim X Chen

School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China

Department of Computer Science, George Mason University, Fairfax, VA 22030-4444, U.S.A.

Keywords: Canonical Forms, Laplace-Beltrami Operator, Biharmonic Distance, Spectral Multidimensional Scaling

(S-MDS).

Abstract: The spectral property of the Laplace-Beltrami operator has become relevant in shape analysis. One of the

numerous methods that employ the strength of Laplace-Beltrami operator eigen-properties in shape analysis

is the spectral multidimensional scaling which maps the MDS problem into the eigenspace of its Laplace-

Beltrami operator. Using the biharmonic distance we show a further reduction in the complexities of the

canonical form of shapes making similarities and dissimilarities of isometric shapes more efficiently

computed. With the theoretical sound biharmonic distance we embed the intrinsic property of a given shape

into a Euclidean metric space. Utilizing the farthest-point sampling strategy to select a subset of sampled

points, we combine the potency of the spectral multidimensional scaling with global awareness of the

biharmonic distance operator to propose an approach which embeds canonical forms images that shows

further “resemblance” between isometric shapes. Experimental result shows an efficient and effective

approximation with both distinctive local features and yet a robust global property of both the model and

probe shapes. In comparison to a recent state-of-the-art work, the proposed approach can achieve

comparable or even better results and have practical computational efficiency as well.

1 INTRODUCTION

The problem of shape matching has become an

important research problem and a fundamental task

in a wide range of geometric applications including

but not limited to computer vision (Vankaick et al.,

2011), texture mapping (Sumner and Popovic, 2004),

mesh deformation (Kreavoy et al., 2003), morphing

(Alexa, 2002), and shape retrieval (Jain and Zhang,

2006).This problem can be defined as finding the

(dis)similarities between objects represented by

point clouds or triangle meshes, say two different

poses of the same object. Such a problem can be

reduced to establishing a correspondence between

two set of the mesh vertices, that is to say

establishing a meaningful mapping between them.

This mapping is either between coarse sets of feature

points selected on the meshes, or a dense continuous

one that involve all points on the two shapes.

The basic question is finding an effective, yet

accurate way to quantify the similarity between a

given reference surface; “the model” and some other

version (articulated) of the model; “the probe”. An

extrinsic property characterizes how a particular

surface is immersed into the ambient (Euclidean)

space and thus changes as the surface undergoes

transformations. Such a metric is not ideal to capture

distinction between shapes as they are significant

disparities between the extrinsic attributes of a shape

and its articulated version. A deformation that

preserves the intrinsic structure of the surface is

called an “isometry” (Bronstein et al., 2006).Thus

defining a computable deformation-invariant

measure of intrinsic similarity between the surfaces

becomes the task at hand.

2 RELATED WORK

Manifold learning refers to the process of non-linear

dimensionality reduction of data. When target space

of reduction (embedding) is Euclidean the procedure

is also known as flattening and the output is called

“canonical forms” (CF). Multi-Dimensional Scaling

(MDS) is a class of computationally efficient

methods used for embedding a canonical form.

One of the earliest methods finds a uniform

parameterization for convoluted surfaces that is

usually a priori in a more general surface matching

procedure (Schwartz et al. 1989). This result led to

Yang, J., Obaseki, A. and Chen, J.

Spectral Multi-Dimensional Scaling using Biharmonic Distance.

DOI: 10.5220/0007242901610168

In Proceedings of the 14th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2019), pages 161-168

ISBN: 978-989-758-354-4

 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reser ved

161

emergence of many efficient flattening algorithms

like in texture mapping (Zigelman et al., 2002), higher

dimensional Euclidean space embedding (Elad and

Kimmel, 2003) that captures the intrinsic geometric

structure of isometric surfaces and a more

generalized framework (Bronstein et al., 2006) that

uses Gromov-Hausdroff distance (Gromov, 1981) to

compute partial embedding distance for both full and

partial surface matching. The key to using MDS

algorithm for embedding is to obtain intrinsic

representation of the underlying surface which is

invariant to inelastic bending, and then interpolate

this representation to embed the surface in a new

ambient space such that the intrinsic geometry of the

surface is translated into its extrinsic geometry in the

new space. Conventionally, a set of inter-geodesic

distance between pairs of all surface points is used

as input. This approach though efficient, is

computationally expensive as the complexity

requirement in storing all pairwise distance is

quadratic in the number of data points which is

restrictive in shapes with substantial amount of

vertices.

In the last decade, a major breakthrough in the

eigenspace of the mesh Laplacian (Wolter et al., 2006)

has been exploited in variety of forms. Isospectral

properties of the eigenvectors know from linear

algebra provided theoretical foundations that have

been extended to correspondence between 3D

surfaces. Jain et al. (Jain et al., 2011) transformed 3D

meshes into the spectral domain, based on geodesic

affinities, and then matched the spectral embeddings

of the eigenvectors with respect to uniform scaling

and rigid-body transformation. Kim et al. (Kim et al.,

2011) approached the problem by blending a

collection of weighted low dimensional conformal

maps. The multi-scale geometry aware properties of

the Laplace Beltrami operator (LBO) are utilized to

infer and manipulate point-to-point maps between

shapes (Ovsjanikov et al., 2012). Rustamov

(Rustamov, 2007) introduced a deformation invariant

representation of surfaces similar to canonical forms

which is based on combining eigenvalues and

eigenvectors of LBO instead of geodesic distance. A

LBO decomposition method is exploited to construct

diffusion maps (Coifman and Lafon, 2006).

Descriptors like Heat kernel signature (Sun et al.,

2009), wave kernel signature (Aubry et al., 2011) are

all based on eigenfunctions of the LBO.

Similar to our approach, Aflalo et al. (Aflalo and

Kimmel, 2013) extracted the spectral data from the

LBO of pairwise geodesic distance of sampled

points, and then embedded the data into a low-

dimensional Euclidean space. Computed a small

fraction of the pairwise distances that was projected

onto the leading eigenfunctions of the LBO, thus,

efficiently reduced both the time and space

complexities of the flattening procedure. It seems

that they overcame a great amount of complexities,

however, the question of its reliance on geodesic

distance which has weak “global–awareness” and

significantly large topological sensitivity is a major

drawback to this approach.

In this paper, we argue that biharmonic distance

(Lipman et al., 2010) can serve as an efficient yet

accurate distance operator. The measure of

biharmonic distance on shapes are smooth functions,

thus are well suited for compact spectral

representation, and as such allow us to apply this

theoretically sound distance operator in a spectral

sense. Biharmonic distance is a metric structure that

is related to diffusion distance and commute-time

distance (Fouss et al., 2007; Yen et al., 2007) with a

slight modification in the eigenvalue normalization.

Our motivation to use this distance operator is based

on the fact that it finds a good trade-off between

local and global properties of the shape. Here, the

eigenvalue normalization decays slow enough to

get good local properties around source points and

fast enough to be globally aware of shape in far

areas.

3 COMPUTING BIHARMONIC

DISTANCE

Biharmonic distance is a distance operator endowed

with the fundamental properties required for shape

analysis, such as isometric invariant, practically

efficient, parameter-free, insensitive to noise and

topology, etc.

Let us consider a Riemannian manifold M

equipped with a metric G. The metric G induces a

Laplace-Beltrami operator (LBO) denoted by

The LBO is self-adjoint and defines a set of

functions called eigenfunctions, denoted by

, such

that

iii

Δ=

, where

is the eigenvalue

associated with

at vertex i.

Biharmonic distance operator is similar to

diffusion distance and the commute-time distance,

however there is slight modification on the

eigenvalue normalization. This normalization

is based on a kernel, which is Green’s function of

the biharmonic differential equation. In the

continuous setting, the squared distance is defined

GRAPP 2019 - 14th International Conference on Computer Graphics Theory and Applications

162

by using the eigenfunctions of the LBO (Lipman et

al., 2010):

(() ())

(, )

dxy

φφ

∞

−



(1)

The quadratic normalization as shown in the Eq.

(1) provides a good trade-off in the sense that it

decays slow enough to get good local properties

around the point and fast enough to be shape aware

in distance areas. The trade-off is intimately related

to the biharmonic equation. Expanding Eq. (1) we

obtain:

22 2

11 1

|()| |()| ()()

(, ) 2

(,) (,) 2 (,)

ii ii

ii i

BB B

yxy

dxy

gxx gyy gxy

φφ φφ

λλ λ

∞∞ ∞

== =

=+−

 

(2)

Using the Green’s function of the biharmonic

operator:

() ()

(, )

gxy

φφ

∞



(3)

The above equation satisfies the relation

()

(, ) () ()Δ=



xyf ydy f x (4)

for “smooth enough”  (Coifman and Lafon, 2006).

From Eq. (3), a discrete construction based on

the discrete Green’s function 



of the Bi-Laplacian

is derived from the well-known cotangent formula

discretization of the Laplace-Beltrami differential

operator on shapes (Grinspun et al., 2006; Meyer et al.,

2003).

()()

()

cot cot

∈

Δ= + −



ei i

iijijij

αβ

(5)

where

, denotes the discrete Laplacian evaluated

at vertex i (for

1, 2, , ,= iN

N is the number of

vertices),

is the Voronoi area at

shape vertex

(Grinspun et al., 2006) and angles

are the two

angles supporting the edge connecting vertices i and

j respectively.

Having discretized the Laplacian, the Green’s

function of the Bi-Laplacian,

∈ 

is defined

by discretizing the relation in Eq. (4) to obtain:

LgAf f

(6)

where

1×

∈ 

is an arbitrary vector in the image

. (See (Meyer et al., 2003) for prove).

Finally, having obtained

, the biharmonic

distance on the shape is defined from Eq. (3):

()

() ( ) ( )

,,,-2,=+

Bi j d d d

dvv gii g jj gij

(7)

4 SPECTRAL

MULTIDIMENSIONAL

SCALING

Let us consider the shape correspondence problem

that involves searching for the best point to point

matching of two given shapes, S and Q. The earliest

method of using multidimensional scaling (MDS) to

compute such an assignment was proposed by (Elad

and Kimmel, 2003). There, the pairwise geodesic

distances between all points on a 3D shape was

mapped to a simpler 3D Euclidean distance.

The spectral multidimensional scaling (Aflalo and

Kimmel, 2013) uses the fact that point to point

correspondence between two shapes induces a map

between the natural eigenspaces of the shapes, thus,

project the MDS problem into the data’s spectral

domain extracted from its Laplace-Beltrami

operator. In this framework, truncated

eigenfunctions were used to faithfully approximate

correspondence between the shapes. We will go

forward to briefly explain this approach.

Consider a manifold M, with n points

{

}

, P is

a subset of

{

}

such that

=≤

, and a

smooth function f is defined on

{

}

,=∈

VVpP.

Computing a smooth interpolation function requires

firstly constructing a continuous function h such that

() ()

,=∀∈



VfVpP. Then a smooth function

measure the smoothness of such a function, say up

to L

norm as

()

,=∇ =Δ



smooth

Ef fda ffda

The problem of smooth interpolation could be

rewritten as

()

() ()

min s.t. ,

→

==∀∈



smooth p p

Eh hVfVpP

. Then

we have

,∇=Δ



da h h da

. Thus the

interpolation problem could be written as:

() ()

min , s.t. ,

→

Δ=∀∈





hh da hV f V p P

(8)

In a discrete setting the problem in Eq. (8) above

can be rewritten as

min s.t. =

Wx Bx f

(9)

There the matrix B, represents a projection on the

basis vectors

, ∀∈

epP

, W is the conformal

discrete Laplacian without the area normalization

and f is the sampled vector

()

V . Their novel

Spectral Multi-Dimensional Scaling using Biharmonic Distance

163

technique was to introduce



, the spectral

projection of f (the eigenvectors of the LBO

{}

)

== =





xf f

φφ

Φα

, where

Δ=

Note that

represents the matrix of eigenfunctions

whose

column is

such that

. Thus

Eq. (9) is approximated as

min s.t.

WBf

αΦ Φα Φα

∈



(10)

Since

= W

ΛΦ Φ

, where

is the diagonal

matrix whose elements

are the corresponding

eigenvalues of linear transformation of the LBO L

Substituting

and adding the constraint check in

the target function the solution is rewritten as

()

TT TT

−

=+ + =

BBfMf

αμΛμΦ Φ Φ

(11)

The discretized smooth energy of the matrix D is

given by

()

()( )

smooth

EDtraceDWDAtraceDWDA

(12)

While the spectral projection of D

onto

, is

denoted in matrix form by

ΦαΦ

(13)

Substituting D into Eq. (12) we obtain the smooth

spectral interpolation as

() ()

()

min

∈

−





ij I

trace trace

DV V

αΛα αΛα

μΦαΦ

(14)

where



is the Frobenius norm, k is the number

of eigenfunctions.

A less accurate but efficient way to obtain an

approximation of the spectral interpolation matrix

is given as

(15)

This is obtained by interpolating the column

vectors f and

. Note in the above equation F is

simply a matrix of the sample points V

, V

and M

represents a matrix such that

from Eq. (11).

5 SPECTRAL MDS USING

BIHARMONIC DISTANCE

Following the method of spectral multidimensional

scaling, we apply biharmonic distance to compute

the canonical forms for non-rigid shapes. Given M, a

metric space endowed with a metric

:DM M×→

, and

{

}

,,,

VVV V= 

a finite

set of elements in M, the multidimensional scaling of

V in



involves finding a set of points

{

}

,,,

XXX X= 



whose pairwise

Euclidean distances

()

ij i j

dX X X X=−

are as

close as possible to

()

DV V for all (i, j).

For such an embedding, a family of MDS known

as classical scaling can be realized by the following

minimization program

min

XJDJ−

, where

represents a matrix defined by

()

ij i j

DDVV=

and

nnn

=−

, or

ij ij

=−

Using Classical scaling in Eq. (13), we find the

first k singular vectors and values of the matrix

DJ−

We utilize the farthest-point sampling strategy to

select a subset of p

sampled points, with indices P

of the data. Then we compute the biharmonic

distance between every two points of the sampled

data

()

Bi j

dvv MM∈×

()

,ij I P P∈= ×

. Since

we have solved the LBO to compute the biharmonic

distance as discussed in section 3, the LBO is not

required to re-compute, we thus, use the same data

to find k eigenfunctions in the eigenbasis

of the

LBO. Using the biharmonic distance we show a

further reduction in the complexity of the canonical

form of the shapes making comparison between

similar and dissimilar shapes more efficiently

computed. Using Eq. (14) and (15) we extract the

spectral interpolation matrix

from the computed

biharmonic distance and the eigenbasis

An outline of the steps to solve the canonical

form using biharmonic distance is shown in the

algorithm below:

Step1: Compute P; a subset of p

points sampled

from M.

Step2: Compute the matrix D of squared

biharmonic distances between every two points

()

,, ,

pp i Pj P∈∈.

Step3: Compute the matrices

containing

the k

eigenvectors and corresponding eigenvalues

of the Laplace-Beltrami operator of M.

Step4: Compute the matrix

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164

Step5: Compute the singular value decomposition

of the nxk matrix JQ=SUV

, where

nnn

=−

Step6: Compute the eigendecomposition of the

kxk matrix

UV VU

, such that

UV VU W W

αψ

Step7: Compute the matrix

QSW

= , such

that

QQ J J

ΦαΦ

Step8: Return the first d columns of the matrix

, where d is the embedding dimension.

6 EXPERIMENTAL RESULTS

AND DISCUSSION

In our evaluations and experiments, we utilized the

TOSCA (Bronstein et al., 2009) and SPACE

(Anguelov et al., 2004) shape databases for our shape

comparison experiments. TOSCA has 80 meshes

representing different classes of shapes, while

SCAPE has 72 meshes representing a human body

in different poses. All the meshes are fitted to

scanner data with a common template, and thus they

share the same mesh topology.

A qualitative evaluation of the canonical form is

represented in the Figure 1 showing our embedding

into canonical forms against that of spectral MDS

(Aflalo and Kimmel, 2013) for wolf and gorilla shapes

in various near-isometric positions. For both

approaches we selected 100 eigenfunctions for the

interpolation of the sampled points. The figure is

represented row-wise original shape model,

canonical forms of spectral MDS (Aflalo and Kimmel,

2013), canonical forms for our approach

respectively. Clearly our method shows a more

simplified canonical form and thus will produce

more efficient and accurate rigid alignment.

Next, we evaluate the distortion of the

embedding between two isometric shapes h: S→T

with respect to a “ground truth” we used a method

similar to (Kim et al., 2011). Here, we computed for

every point, p, on S in the ground truth

correspondence, the geodesic distance,

() ()

()

Strue

dhph p between the smoothness

function,

()

and its true correspondence,

()

true

. The difference between the geodesic

distance is added up in an error measure such that

()

() ()

()

true S true

Err h h d h p h p

∈



(16)

here

() ()

()

true

hp h p is normalized by the square

root of the area of the manifold S.

We generated a table to examine the distribution

of errors. Table 1 shows percentage correspondence

as a function of geodesic error. That is, the data of

varying geodesic error threshold,

, between the

model and probe

() ()

true

hp h p

against the average

percentage of points correspondence for which

() ()

()

Strue

dhph p

≤ . Taking an instance from

the “animal shapes” in Table 1, about 67% of

sample points had geodesic error below 0.1 for S-

MDS approach while for our method above 98% of

correspondences fell below the 0.1 geodesic error.

Another example from the “all shapes” table shows

100% of sample points had geodesic error below

0.15 for our approach when compared against 95%

for S-MDS. We also generated a graphical

representation of the data, where x-axis depicts

geodesic error threshold,

, and y-axis is the

average percentage of point correspondence that fall

below the threshold

. The top left, top right and

bottom left graphs in Figure 2 are graphical

representation of Table 1. Clearly, we can see that

the result of our method outperforms that of

spectral-MDS.

Overall our algorithm produces better results

when matching human shapes. Bottom right of

Figure 2 is a representation of the percentage of

correspondence measure of all six animal shapes of

the TOSCA database.

As the idea of multidimensional scaling is to find

a rigid alignment of the embedded image of the

shapes, in the next experiment, we used the Iterative

Closest Point (ICP) algorithm to compute such

alignment. Figure 3 is a picture of two near-

isometric wolf shapes before and after computing

their ICP alignment. We also performed a

comprehensive experiment on shapes from the

SPACE database. In this experiment, we selected the

first 50 eigenfunctions for our spectral interpolation.

We randomly selected a model and matched with

several probes. Having computed the canonical

forms, we computed the rigid alignment between the

canonical form images of the matching, and next we

used a relative straight forward scheme to find the

similarity measure between them. First, we

transformed the matrix of the output of the canonical

form S

and T into a vector s and t. And then given a

range of threshold, the dis(similarity) between them

by function

() () ()

t norm s norm t⋅⋅

is computed,

where “.” is the inner product between two vectors

and norm( )

is the Euclidean norm of the vector. The

Spectral Multi-Dimensional Scaling using Biharmonic Distance

165

range of threshold is between 0 and 1 such that the

closer the function is to 1 the more similar S and T

are, conversely, the closer the function is to 0 the

more dissimilar S and T are. The results from 10

similarity measures were sampled from our

experiment as shown in Table 2. Experimental

results from the table show an improvement in our

similarities measure when compared with that of the

spectral multidimensional scaling.

7 CONCLUSIONS

In this paper we argued that approaching the novel

method of spectral multidimensional scaling with a

theoretical sound distance operator in biharmonic

distance proves to further reduce the topological

complexities of the embeddings. Taking advantage

of the global shape awareness property of the

biharmonic distance operator, we were able to get a

minimal distorted canonical form thus making the

computing of a rigid assignment of canonical forms

more efficient and accurate. Experimental results

show a comparable and even better result to a state-

of-the-art method. Future prospect of this study

might include using other distance operators to

compute the dimension reduction problem that

might achieve a more favorable result.

Figure 1: Embedding of wolf and gorilla shapes into canonical forms. From top to bottom depicts original shape, followed

by canonical form obtained by Spectral MDS, Spectral MDS using biharmonic distance.

Table 1: Shows the data of varying geodesic error threshold D, between the model and probe against the average percentage

of point correspondence.

Animal Shapes Human Shapes All Shapes

Geodesic

error(D)

S-MDS

Pts.(%)

SMDS-Biharmonic

Pts. (%)

S-MDS

Pts. (%)

SMDS-Biharmonic

Pts. (%)

S-MDS

Pts. (%)

SMDS-Biharmonic

Pts. (%)

0 1.1183 1.3707 1.2298 1.2207 1.2413 1.3524

0.0125 4.4088 6.7875 4.452 7.3923 4.8266 7.1525

0.025 12.1579 19.1776 14.7096 20.9908 14.596 19.9216

0.0375 24.7667 40.4431 31.5826 46.061 30.4995 43.1803

0.05 39.3009 63.3398 51.1159 66.3549 48.8439 65.2796

0.0625 49.7646 81.8704 68.9267 81.5946 63.7864 82.0446

0.075 57.1189 91.3908 83.3491 92.5794 75.1602 92.2939

0.0875 62.7122 96.8784 90.7177 97.5841 81.9415 97.3656

0.1 67.3756 98.8709 94.6324 99.2309 86.438 99.0808

0.1125 71.5725 99.6609 98.0789 99.7937 90.1884 99.7827

0.125 76.0111 99.9399 99.2249 99.8841 93.1298 99.9328

0.1375 78.8878 99.9553 99.6353 99.9521 94.6593 99.9761

0.15 81.1456 99.9702 99.7597 100 95.6654 100

0.1625 83.5624 99.9702 99.881 100 96.7655 100

0.175 85.7585 100 99.9575 100 97.6749 100

0.1875 87.1764 100 99.9745 100 98.1592 100

0.2 89.0674 100 99.9915 100 98.8559 100

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166

Figure 2: Graphical representation evaluating the geodesic error of correspondence between different categories of shapes.

All graphs show percentage of correspondence between the thresholds of a normalized geodesics error(0-0.2). Top left, top

right and bottom left show comparison of the correspondence between regular S-MDS and S-MDS using biharmonic

distance. Bottom right shows the percentage of correspondence between six different shapes using S-MDS using

biharmonic distance.

Figure 3: An alignment of canonical forms using Iterative Closest Point (ICP) algorithm. From left to right are images of

pre-alignment and post-alignment of two wolf shapes.

Table 2: Comparison of similarity measure function. A

model is compared against ten probes with a threshold

between 0 and 1. The closer the value to 1 the more

similar the model and probe are.

Shape Similarity Measure

Spectral MDS Spectral MDS-Biharmonic

0.9034 0.9117

0.8761 0.8828

0.8720 0.8813

0.8398 0.8505

0.8840 0.8928

0.8565 0.8653

0.7255 0.7282

0.8746 0.8851

0.8987 0.9045

0.8741 0.8821

Spectral Multi-Dimensional Scaling using Biharmonic Distance

167

ACKNOWLEDGEMENTS

The authors would like to thank the reviewers for

their valuable comments. This work is supported by

National Natural Science Foundation of China under

grant No. 61862039, 61462059.

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