Geometric Approach to Estimation of Volumetric Distortions

Alexander Naitsat

1

, Emil Saucan

2

and Yehoshua Y. Zeevi

1

1

Electrical Engineering Department, Technion, Haifa, Israel

2

Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Keywords:

Conformal Geometry, Isometric Distortion, Volume Parameterization, Volumetric Meshes, Geometric

Computing, Computer Graphics.

Abstract:

The problem of measuring geometrical distortions is not trivial for volumetric domains. There exist intrinsic

restrictions and constrains on higher dimensional mappings. Moreover, according to Liouville theorem, most

existing techniques for 2D data can not be directly applied to volumetric objects. In this work we approximate

continuous deformations by piecewise afﬁne functions deﬁned on tetrahedral meshes. Our aim is to study

a few types of geometrical distortions that can be expressed as functions of singular values of a Jacobian.

We employ the proposed methods of estimating conformal and isometric distortions to analyze volumetric

data. In particular, we examine parametrization of tetrahedral models to a ball. Distortions produced by the

resulting spatial mappings depict intrinsic structure of domains, and therefore can be employed in such tasks

as detection of abnormalities and comparison (i.e. similarity assessment) between 3D objects. This geometric

approach and results are highly relevant to various applications in Computer Vision, Computer Graphics, 3D

Printing and Medical Imaging.

1 INTRODUCTION

In mapping between volumetric domains, we are in-

terested in quantifying distortions of the intrinsic geo-

metrical properties, i.e. angles and Euclidean lengths,

referred to as conformal and isometric distortions, re-

spectively. Obeying the condition of angle preserving

transformations, conformal maps are desirable in dig-

ital geometry processing and in computer graphics,

since they do not exhibit shear and, therefore, pre-

serve different vertex properties as well as the topol-

ogy of the mesh itself. They are also instrumental in

image processing and in computer vision.

A conformal mapping f of a domain D ⊂ R

n

is

deﬁned as a smooth bijective function, which at any

point x ∈ D scales the space uniformly in every direc-

tion. This can be stated formally as

kd f

x

· h

1

k = kd f

x

· h

2

k, (1)

where d f

x

denotes the Jacobian matrix at a point x of

a function f : D → R

n

and h

1

,h

2

∈ R

n

are arbitrary

unit vectors. Following this annotation, a conformal

map f (x) is isometric in D if

∀x ∈ D : |detd f

x

| = 1 . (2)

Most of the methods used to construct confor-

mal and almost isometric parametrization of a sur-

face can’t be generalized for dealing with volumes,

due to the classical theorem of Liouville (K

¨

uhnel and

Rademacher, 2007), which states that every confor-

mal mapping of a domain in R

n

,n ≥ 3, is a restric-

tion of M

¨

obius transformation, i.e. a composition of

conformal afﬁne transformations and inversions in a

sphere. Therefore, the problem of computing perfect

conformal functions in higher dimensions is reduced

to minimizing the amount of distortion produced by a

spatial mapping.

Conformal and isometric properties of smooth

mappings in R

n

can be studied by at least two alter-

native approaches. The ﬁrst approach considers topo-

logical properties of a function along the integrable

curves. The other approach, which we shall employ,

examines the properties of a map in inﬁnitely small

neighborhoods. To this end, we examine the behavior

of a discrete map f in nearest neighborhood repre-

sented by a ring of tetrahedral cells. Our measure-

ments of distortions are based on the singular value

analysis of the Jacobian d f .

The advantage of this method over other tech-

niques that are used in volumetric modeling (e.g. dis-

crete harmonic energy; (Wang et al., 2003)) is that

we can employ the same framework to deal with a

wide range of spatial distortions and energies. For in-

stance, the aspect-ratio distortion (Aigerman and Lip-

Naitsat, A., Saucan, E. and Zeevi, Y.

Geometric Approach to Estimation of Volumetric Distortions.

DOI: 10.5220/0005778201030110

In Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2016) - Volume 1: GRAPP, pages 105-112

ISBN: 978-989-758-175-5

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

105

man, 2013), the certain types of rigid and afﬁne en-

ergies (Kovalsky et al., 2014), the n-D comformality

distortion (Lee et al., 2015), the angle and the volume

energies (Paill

´

e and Poulin, 2012) all of which are ex-

pressed as functions of singular values.

While there is an abundance of studies and ap-

plications of conformal mappings in 2D, general

volumetric domains can be mapped only quasi-

conformally. Therefore one of the challenges in 3D

is to measure the minimal conformal distortion asso-

ciated with mapping of one domain into another.

We examine the problem of measuring volumetric

distortions, produced by discrete transformations be-

tween tetrahedral meshes, and addresses issues con-

cerning relations between volumetric distortions and

assessment of geometric similarity between 3D ob-

jects.

2 VOLUMETRIC DISTORTIONS

Let f (x) = Ax + b be a full rank n-dimensional afﬁne

function. A number of geometry processing problems

associated with f are naturally described in terms of

singular values of its linear part. Singular values of a

matrix are non-negative numbers usually denoted by

σ

1

(A) ≥ σ

2

(A) ≥ ... ≥ σ

n

(A); they are unique up to

the order, and satisfy

kAk = σ

1

, `(A) = σ

n

, | det A| =

n

∏

i=1

σ

i

, (3)

where

kAk := max

khk=1

kA · hk, (4)

is also known as the spectral norm of A, and

l(A) := min

khk=1

kA · hk. (5)

The geometrical interpretation is that A transforms

vectors of some unitary basis {v

i

}

i=1,..,n

to the vectors

of other unitary basis {u

i

}

i=1,..,n

, multiplied by the

corresponding singular value σ

i

. Putting these vectors

in rows and columns, respectively, yields a singular

value decomposition (SVD) of A

A = UDiag(σ

1

,...,σ

n

)V

∗

. (6)

The latter plays an important role in image processing

and numerical analysis.

When dealing with volumetric mappings, our pri-

mary interest is in studying distortions of ﬁrst order

differential properties, such as angles and Euclidean

lengths. This lends itself to the deﬁnition of qc (quasi-

conformal) and qi (quasi-isometric) dilatations of a

linear map A, which basically are measures of how

far A is from being conformal and isometric mapping,

respectively. The basic method to assess the degree

of conformality of a linear map A, is to consider the

ratio of singular values

H(A) =

kAk

`(A)

, (7)

also known as the condition number.

A more accurate approach, which follows from

the theory of n-dimensional mappings, is to consider

the following quantity, called qc-dilatation,

K(A) = max

|det(A)|

`(A)

n

,

kAk

n

|det(A)|

. (8)

Quasi-isometric dilatation of a function f is de-

ﬁned as a minimal number C ∈ {1,∞}, such that

1

C

kp

1

− p

2

k ≤ k f (p

1

) − f (p

2

))k ≤ Ckp

1

− p

2

k,

for any p

1

, p

2

∈ D (see (Saucan et al., 2008)). In par-

ticular, for a linear function A, Eq. (4) and Eq. (5)

imply

C(A) = max{`(A)

−1

, kAk}. (9)

Dilatations deﬁned above are related to each other

via the following inequality

K(A) ≤ H

n−1

(A) ≤ C

2(n−1)

(A). (10)

Now, suppose that f is a local diffeomorphism of

an open domain D ⊂ R

n

, that is for each point in D

there is a neighborhood where f is a smooth bijec-

tive mapping with a smooth inverse. We refer to such

mapping simply as a deformation function. Clearly,

d f

x

is a full rank linear transformation, and thus we

deﬁne the maximal qc-dilatation of f as

K( f ) = sup

x∈D

K(d f

x

), (11)

and the maximal qi-dilatation as

C( f ) = sup

x∈D

C(d f

x

). (12)

We call a deformation function f qc-mapping if

K( f ) < ∞, and qi-mapping if C( f ) < ∞.

In the sequel we shall assume that n = 3 and that

all the above conditions for f are satisﬁed. The ba-

sic common method to represent volumetric data in

computer science and engineering is to triangulate a

continuous volume into tetrahedrons. This is a pow-

erful method applicable for various 3D topologies.

Employing this approach to approximate a continuous

deformation f : D → D

0

yields the so-called simplicial

map f

s

.

Let (V,E,F,T ) be a tetrahedral representation of

D, where V , E, F and T are the vertex, edge, face

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

106

(a) Domain (b) Image (c) qc-dilatation K (d) qi-dilatation C

(e) σ

1

(f) σ

3

(in cross section) (g) Image (h) K

Figure 1: Radial stretching of the hexahedral volume of the cube [−1,1]

3

onto the unit ball and onto the round cylinder B ×

[−1,1]. Figures 1(c)-1(f) and 1(h) depict distribution diagrams of the scalar values associated with volumetric distortion: qc

and qi dilatations and singular values σ

i

of a Jacobian. Equivalent tetrahedral meshes are obtained by hexahedral triangulation.

and tetrahedra sets of the mesh, respectively. Then,

for a vertex v ∈ V located at the position x we set

f

s

(v) = f (x). Next, we extent f

s

to the entire do-

main by a piecewise linear extension of the vertex val-

ues (see the exact formula, Eq. (13)). The following

section uses simplicial maps to estimate differential

properties of continuous deformations.

2.1 Computing Singular Values

We can estimate dilatations of a simplicial map f at

vertex v, placed at a position x ∈ R

3

, based on estima-

tion of the Jacobian matrix d f

x

. First, we estimate the

average Jacobian matrix inside a given tetrahedron τ,

denoted by d f

τ

. Then we deﬁne the vertex Jacobian

matrix d f

v

as an average over d f

τ

in the neighbouring

tetrahedra.

Let f

(1)

, f

(2)

, f

(3)

be the coordinate components

of the map f , where each one is a function from V to

R. We can express d f

v

as the matrix whose rows are

the average gradient vectors of f

(i)

.

Suppose r is a point inside tetrahedral cell τ,

which consists of vertices v

1

,v

2

,v

3

,v

4

. Let λ

j

be a

face against v

j

. Let s

j

be the vector along the normal

of λ

j

, such that ks

j

k = Area(λ

j

). If v

4

is set to be the

origin of τ, then applying 3D barycentric coordinates

for f yields

f

(i)

(r) =

1

3m(τ)

4

∑

j=1

(v

4

− r) · s

j

f

(i)

(v

j

). (13)

This implies that the gradient is constant inside τ,

hence our estimates yield

∇ f

(i)

(τ) = −

1

3m(τ)

4

∑

j=1

s

j

f

(i)

(v

j

); (14)

and

∇ f

(i)

(v) =

∑

c∈Ring(v)

∇ f

(i)

(c)w(c), (15)

where Ring(v) are the neighbouring tetrahedrons of v,

and w(c) are the chosen normalized weights.

The singular values σ

1

= kd f

v

k and σ

3

= l(d f

v

)

are then approximated as the maximum and minimum

of kd f

v

· h

j

k, sampled at chosen directions h

1

,...,h

m

.

The intermediate singular value is then

σ

2

=

|det(d f

v

)|

σ

1

σ

3

, (16)

where the determinant can be computed directly from

the estimates of the vertex Jacobian. Alternatively, it

may be computed as a weighted average over deter-

minants det(d f

τ

), where the last quantity may be ap-

proximated as a volume ratio between the target and

the source tetrahedra :

|det(d f

τ

)| =

volume( f (τ))

volume(τ)

. (17)

Geometric Approach to Estimation of Volumetric Distortions

107

(a) Domain (b) Image, a = 1 (c) Cross section of the domain (d) a = 1

(e) a = 2 (f) a = 0.5 (g) Mean curvature (h) Dihedral angle (i) qc-dilatation

Figure 2: Parametrization of tetrahedral model mapped into a ball, computed for three different values of parameter a. Figures

2(a), 2(b) and 2(i) depict entire tetrahedral models of the domain and of the image. Figures 2(c)-2(f) show cross sections

delineated by yz plane. Colors depict and correspond to diagrams of cylindrical coordinates of the domain for Figures 2(a)-

2(f), and distribution diagrams of mean curvature and dihedral angles of the boundary and qc-dilations for Figures 2(g)-2(i),

respectively.

(a) (0.2,10) (b) (0.1,20) (c) (0.5,10)

Figure 3: Images of the unit sphere under Ω

A,ω

functions.

The shown pairs of numbers corresponds to the values of A

and ω , respectively.

[See (Naitsat et al., 2015; Naitsat et al., 2014) for de-

tails about estimation of singular values and the Jaco-

bian on tetrahedral and hexahedral meshes.]

Combining the above leads to numerical estimates

of local volumetric distortions at vertex positions. We

denote the corresponding qc and qi dilatations for a

chosen vertex v by K( f , v) and C( f ,v), respectively.

2.2 Bounded Distortion Mappings

A variety of techniques for morphing of polygonal

meshes are used in computer graphics to measure

similarities between objects. Let us consider the fol-

lowing related problem for volumetric domains.

Assume D

1

,D

2

⊂ R

3

are smooth domains that

have the same topology and similar geometrical prop-

erties. Then, according to the basic geometric intu-

ition, there exists a deformation map f : D

1

7→ D

2

with bounded dilatations. The proof and exact condi-

tions required for the existence can be found at (Cara-

man, 1974; V

¨

ais

¨

al

¨

a, 1971). We consider the minimal

conformal and isomtertic distortions that are required

to map one domain into another. The correspond-

ing quantities are called quasi-conformal and quasi-

isometric coefﬁcients of D

1

and D

2

, respectively, and

they are deﬁned as

K(D

1

,D

2

) = inf K( f ), (18)

C(D

1

,D

2

) = infC( f ), (19)

where the inﬁmum is taken over all smooth bijective

mappings f : D

1

→ D

2

.

We expect that the amount of distortion produced

by morphing domain D

1

into D

2

will provide a mea-

sure of resemblance between their geometries. For

instance, if D

2

is image of D

1

under a similarity trans-

formation, then K(D

1

,D

2

) = 1. In a general case

K(D

1

,D

2

) ≥ 1 and it measures how close are these

domains to being conformally-equivalent.

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

108

(a) Domain, K (b) image, K (c) domain, C (d) image, C

(e) domain, K (f) image, K (g) domain, C (h) image, C (i) mean curvature on the do-

main’s surface

(j) Domain, K (k) domain, C (l) mean curvature (m) Gaussian curvature

Figure 4: Mapping into a ball of volumes contained inside the wave-like surfaces Ω

A,ω

(S

3

) deﬁned by Eq. (25) for the

following values of the amplitude A and the frequency ω : A = 0.5,ω = 10 for Figures 4(a)-4(i) and A = 0.1, ω = 20 for

Figures 4(j)-4(m). The ﬁrst two lines alternatively show domain and image of the deformation. Figures 4(a)-4(d) and 4(j)-

4(k) show entire tetrahedral models, while 4(e)-4(h) depict cross sections. The rest of the Figures 4(i), 4(l) and 4(m) are

diagrams of curvature distribution on boundary surfaces.

To access ﬁne geometrical features of the object

from qi-coefﬁcients, we must take into account rela-

tive sizes of the given domains. Let us measure the

size of a domain D

i

by the radios r

i

of the bounding

sphere, i.e, the smallest sphere containing D

i

. Given

a deformation mapping f : D

1

→ D

2

, it is often useful

to consider the so called k-bounded qi-dilation of f

C

0

( f ) =

C( f )

k

, (20)

where k is deﬁned by

k(D

1

,D

2

) = max

r

1

r

2

,

r

2

r

1

. (21)

We deﬁne the k-bounded qi-coefﬁcient C

0

(D

1

,D

2

) ac-

cordingly. This approach is important for a general

case. However to make things simpler when dealing

with tetrahedral models, we assume that k(D

1

,D

2

) =

1. This assumption is easily achieved in modeling

tools by scaling the data during the preprocessing.

Dilatation coefﬁcients can be often derived from

comparison between domain’s boundaries. For exam-

ple, consider deformation of a cube onto the round

cylinder shown in Figure 1. This transformation is

a radial stretching of the square in planes R

2

× {z},

with edge 2l into the circle of radius l, which can be

written explicitly in cylindrical coordinates as

(r,ϕ,z) 7→

lr

R(ϕ)

,ϕ,z

, (22)

Geometric Approach to Estimation of Volumetric Distortions

109

where R(ϕ) is a distance from the origin to the bound-

ary of the cube, measured at angle ϕ. Without loss of

generality, we focus on compact domains in R

3

with

non-empty interior that contains the origin.

The mapping (22) for l = 1 can be generalized to

construct parametrization of compact volumes onto

the unit cylinder by

(r,ϕ,z) 7→

r

R(ϕ)

a

,ϕ,z

, (23)

where a > 0 is a chosen parameter.

The same technique of radial stretching can be ap-

plied to produce volumetric parametrization into the

unit ball. Consider a domain D that satisﬁes the fol-

lowing geometrical condition: for any ζ ∈ ∂D, the an-

gle between the line segment ζ0 and tangent plane at

ζ is larger than or equal to some constant α > 0. We

construct a smooth parametrization of D into the unit

ball written in the spherical coordinates (r,ϕ,θ). First,

set ζ(ϕ, θ) to be the furthest intersection point of ∂D

and the ray along the ϕ,θ direction, then we deﬁne the

family of parametrization functions f

a

: D → B

3

by

(r,ϕ,θ) 7→

r

kζ(ϕ,θ)k

a

,ϕ,θ

, (24)

where a > 0 is the same parameter as in Eq. (23).

If the domain is star-shaped, namely if for any x ∈

D: 0x ⊂ D, then the resulting parametrization is onto.

Moreover, Caraman (Caraman, 1974, p. 408) has

shown in this case that f

a

is a qc-mapping and gave

an accurate approximation of K( f

a

) as a function of a

and angle α from the geometrical condition. In partic-

ular, if a → 1 and α → π/2, then K( f

a

) approaches 1,

since the limit function appears to be a uniform scal-

ing of a ball to the unit ball B

3

.

Figures 1 and 2 show cylindrical and spherical

parametrization of models. Both parametrizations ap-

plicable to analysis of volumetric data. However,

parametrization to a cylinder requires an accurate

choice of the rotation axis, where deformation to a

ball is invariant to the initial orientation of objects.

Therefore, this work is focused on radial stretching

into a ball. Figures 2(a)-2(f) depict the geometrical

meaning of the parameter a, which affects the amount

of distortion along the radial axis. In particular, when

a decreases, the interior volume is stretched toward

the surface, while raising values of a squeezes the in-

terior toward the origin. This behavior can be em-

ployed to adjust the amount of local distortions. In

the sequel we assume a = 1, unless stated otherwise.

2.3 Results and Explanations

We present results of numerical computations of volu-

metric distortions for various domains along with the

qualitative interpretation of the outcomes.

Figure 2 summarizes results for the parametriza-

tion of the cat model to the unit ball, represented by

a tetrahedral mesh. Note the correlation between ar-

eas on the surface of higher values of dilatations and

areas of higher values of mean curvature and dihe-

dral angles shown in Figures 2(g) and 2(h). This

phenomenon is further emphasized in Figure 4. The

relation between curvature and measured volumetric

distortions follows from the existence of conformal

and isometric invariants. Similarly, dihedral angle im-

poses tight bounds on qc-coefﬁcients of polyhedral

domains. These facts has been demonstrated for basic

deformations in (Naitsat et al., 2015).

Next, we choose series of symmetrical volumetric

domains D

A,ω

whose boundaries ∂D

A,ω

are set to be

an image of the following wave functions:

Ω

A,ω

(r,ϕ,θ) = (r + A cos(ωϕ)cos(ωθ),ϕ,θ) , (25)

where r is a constant. In other words, we deﬁne a si-

nusoid height function on a sphere S

3

r

with amplitude

A and frequency ω (see Figure 3). Our aim is to es-

timate an impact that distinct geometrical features of

a wavelike surface have on the dilatation coefﬁcients.

Although, Eq. (11) and Eq. (12) deﬁne K( f ) and

C( f ) as a maximum of the corresponding local dila-

tions, the average value of the volumetric distortion is

more relevant for such practical applications as shape

classiﬁcation and comparison. Let us denote the aver-

age value of a scalar function F : D ⊂ R

3

→ R by

¯

F =

1

volume(D)

Z

D

F(x)dx (26)

In order to compute

¯

K and

¯

C for a tetrahedral mesh

(V,E,F, T ) of a domain D, we approximate continu-

ous integration as follows :

Z

D

F(x)dx ≈

∑

v∈V

F(v)·volume(Cent(v)), (27)

where Cent(v) is a barycentric cell of a vertex v,

obtained by connecting middle points of edges shar-

ing v. Namely, for each τ ∈ Ring(v) we construct a

sub-tetrahedron by connecting v and middle points of

edges sharing v. Cent(v) is deﬁned as the union of

these sub-tetrahedrons over Ring(v).

Denote by D

A,ω

the volume enclosed by the im-

ages of the unit sphere under Ω

A,ω

. Figure 4 con-

siders deformations f

1

: D

A,ω

→ B

3

deﬁned by Eq.

(24). Observing these ﬁgures, we see concentration

of conformal distortion on the surface of the model,

while isometric distortion is spread through the entire

volume. The phenomenon is related to the choice of

a deformation function f

1

from Eq. (24) which is a

uniform scaling when restricted to a radial segment

{(r,ϕ,θ)|r ∈ [0,t]}.

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

110

Figure 5: Average qi and qc dilatations

¯

K and

¯

C as a function of the amplitude A and the frequency ω of Ω

A,ω

deﬁned

according to Eq. (25). Dilatations are computed for parametrization (24) of domains D

A,ω

, which are interior volumes of

Ω

A,ω

(S

3

). The results are shown for the constant frequency ω = 10 and the constant amplitude A = 0.1, respectively.

Figure 6: Average volumetric distortions

¯

K and

¯

C produced by parametrization of 3D models into a ball. The corresponding

distances between models in (

¯

K,

¯

C)-plane reveal similarities of global geometrical features. The mesh data used in this

simulation was adopted from (Sumner and Popovi

´

c, 2004).

The corresponding charts in Figure 5 show de-

pendence between the wave parameters and the aver-

age level of volumetric distortions. As expected, for

ω = const, we see that

¯

C has a linear dependence on

the amplitude, while

¯

K appears to be a quadratic func-

tion of A. In contrast, when the amplitude is constant,

both

¯

C and

¯

K are nearly linear functions of ω. This

is because conformal distortion is intensiﬁed near the

surface, which for A = const is restrained to the same

area 1 − A ≤ r ≤ 1 + A. 3D shapes can be analyzed

in the spectral domain as a composition of compo-

nents that represent features of high and low frequen-

cies. These features can be estimated by the amount

of distortion required to round the shape. This con-

cept is illustrated in Figure 6 for three pairs of polyg-

onal models. All the pairs, except the last one, are

mutual images under a nearly isometric bending. Av-

erage distortions are measured and displayed on the

(

¯

K,

¯

C)-plane for a deformation map (24) of the en-

closed volumes into a ball. The vertical displacement

of the resulting points in the distortion plane reveals

how far are shapes from being isometrically equiva-

lent, while the Euclidean distance between the points

accesses the resemblance of global features and align-

ment of objects’ medial axes.

3 CONCLUSIONS

This paper presents quantitative method of computing

the major volumetric distortion: qc and qi dilatations.

Our technique is based on estimation of the Jacobian

and the corresponding singular values for a simplicial

map.

Among other applications, our approach can be

used to produce desirable deformations and param-

eterizations of 3D domains, by minimizing the distor-

tion functional

Σ

v∈V

K( f , v) + λC( f , v) , (28)

Geometric Approach to Estimation of Volumetric Distortions

111

under the given conditions. The latest research in

the area (Kovalsky et al., 2014), deals with the min-

imization of the condition number (7), which is not

always sufﬁciently accurate to properly measure de-

formations in 3D (see (Naitsat, 2015, p. 13) ). We

believe that using our metrics it is possible to achieve

more desirable results.

We have demonstrated in Section 2.2 basic

parametrization techniques for volumetric objects.

Dilatations produced by the corresponding mappings

are tightly related to the distortion of several geomet-

rical properties of a boundary surface, such as curva-

ture and dihedral angles. In particular, we have illus-

trated in Section 2.3 correlations between an average

distortion, the amplitude and the frequency of wave-

like surfaces. These consequences may be targeted

toward spectral analysis of the closed shapes that can

be represented as an image of a sphere under a com-

position

Ω

A

1

,ω

1

◦ Ω

A

2

,ω

2

◦ ... , (29)

which can be considered, in turn, as an analogue of

Fourier series in the spherical coordinates. Likewise,

future analysis may consider distortions produced by

parametrization into non-symmetrical domains and its

relation to global properties of shapes. For example, a

volumetric parametrization into cylinder and shape’s

symmetry relative to the rotation axis.

As has been illustrated in Figure 6, these facts can

be instrumental in quantitative assessment of similar-

ity of objects undergoing elastic deformations. Un-

like most existing algorithms for classiﬁcation and

comparison of polygonal meshes, our approach can

be applied both for closed simply-connected surfaces

and for volumetric domains with more sophisticated

boundaries.

ACKNOWLEDGEMENTS

This research has been supported by the Israeli Min-

istry of Economics; OMEK consortium and by the

Ollendorff Minerva Center for Vision and Image Sci-

ences.

REFERENCES

Aigerman, N. and Lipman, Y. (2013). Injective and

bounded distortion mappings in 3D. ACM Trans.

Graph, 32(4):106:1–106:14.

Caraman, P. (1974). n-dimensional quasiconformal (QCF)

mappings. Revised, enlarged and translated from

the Roumanian by the author. Editura Academiei

Rom

ˆ

ane, Bucharest, Abacus Press, Tunbridge Wells

Haessner Publishing, Inc., Newfoundland, NJ, 1974.

Kovalsky, S. Z., Aigerman, N., Basri, R., and Lipman, Y.

(2014). Controlling singular values with semideﬁ-

nite programming. ACM Transactions on Graphics

(TOG), 33(4):68.

K

¨

uhnel, W. and Rademacher, H.-B. (2007). Liou-

ville’s theorem in conformal geometry. Journal de

math

´

ematiques pures et appliqu

´

ees, 88(3):251–260.

Lee, Y. T., Lam, K. C., and Lui, L. M. (2015). Landmark-

matching transformation with large deformation via n-

dimensional quasi-conformal maps. J. Sci. Comput.

(Accepted for publication), pages 1–29.

Naitsat, A. (2015). Quasi-conformal mappings for volu-

metric deformations in geometric modeling. Master’s

thesis.

Naitsat, A., Saucan, E., and Y, Y. Z. (2014). Computing

quasi-conformal maps in 3d with applications to geo-

metric modeling and imaging. In 2014 IEEE Conven-

tion, Israel, pages 1–5.

Naitsat, A., Saucan, E., and Zeevi, Y. Y. (2015). Volu-

metric quasi-conformal mappings - quasi-conformal

mappings for volume deformation with applications to

geometric modeling. In Proceedings of VISIGRAPP

2015, pages 46–57.

Paill

´

e, G.-P. and Poulin, P. (2012). As-conformal-as-

possible discrete volumetric mapping. Computers &

Graphics, 36(5):427–433.

Saucan, E., Appleboim, E., Barak-Shimron, E., Lev, R.,

and Zeevi, Y. Y. (2008). Local versus global in quasi-

conformal mapping for medical imaging. Journal of

Mathematical Imaging and Vision, 32(3):293–311.

Sumner, R. W. and Popovi

´

c, J. (2004). Deformation transfer

for triangle meshes. ACM Transactions on Graphics

(TOG), 23(3):399–405.

V

¨

ais

¨

al

¨

a, J. (1971). Lectures on n-Dimensional Quasicon-

formal Mappings. Springer-Verlag Berlin Heidelberg

New York.

Wang, Y., Gu, X., Yau, S.-T., et al. (2003). Volumetric har-

monic map. In Communications in Information & Sys-

tems, volume 3, pages 191–202. International Press of

Boston.

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

112