PARAMETRIZATION, ALIGNMENT AND SHAPE OF SPHERICAL

SURFACES

Xiuwen Liu, John Bowers

Department of Computer Science, Florida State University, Tallahassee, FL, 32306, USA

Washington Mio

Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

Keywords:

Surface parametrization, surface alignment, shape geodesics, shape of surfaces, shape metric, texture maps.

Abstract:

We develop parametrization and alignment techniques for shapes of spherical surfaces in 3D space with the

goals of quantifying shape similarities and dissimilarities and modeling shape variations observed within a

class of objects. The parametrization techniques are reﬁnements of methods due to Praun and Hoppe and

yield parametric mesh representations of spherical surfaces. The main new element is an automated technique

to align parametric meshes for shape interpolation and comparison. We sample aligned surfaces at the vertices

of a dense common mesh structure to obtain a representation of the shapes as organized point-clouds. We

apply Kendall’s shape theory to these dense point clouds to deﬁne geodesic shape distance, to obtain geodesic

interpolations, and to study statistical properties of shapes that are relevant to problems in computer vision.

Applications to the construction of compatible texture maps for a family of surfaces are also discussed.

1 INTRODUCTION

The representation and analysis of 3D geometric

structures are fundamental problems in areas such as

computer vision and medical imaging. The practical

relevance of these problems is increasing vertically in

tandem with the fast evolution of scanning technolo-

gies that allow us to acquire geometric structural data

both at the macroscopic and microscopic levels. The

geometry of the outer contour of an object is of par-

ticular interest as it carries a rich amount of 3D visual

information that can enable us to classify and discern

objects, analyze patterns of variation, and model the

temporal evolution of their shapes in the presence of

dynamics. In this paper, we develop techniques for

constructing parametrizations and for aligning closed

surfaces of genus zero, which are presented as meshes

in 3D Euclidean space R

3

. An example of a closed

surface of genus zero – that is, a surface that can

be obtained as a deformation of a round sphere – is

shown in Figure 1. We apply the parametrization and

alignment methods to the following problems: (i) the

construction of compatible low-distortion meshes for

a family of surfaces; (ii) the interpolation of spheri-

cal shapes; (iii) the development of metrics to quan-

tify shape similarity and dissimilarity; (iv) the con-

struction of compatible texture maps for a family of

shapes; (v) the calculation of mean shapes. These

problems are all relevant to computer vision as they

arise in the modeling of variations in shape and ap-

pearance, for example, in 3D object recognition.

Unlike a curve that has a natural parametriza-

tion by the arc-length parameter, no such special

parametrization exists for a surface. As a conse-

quence, discrete models of surfaces used in prac-

tice often adopt representations involving highly non-

uniform samplings, especially for surfaces whose ge-

ometry exhibit thin and elongated parts, high curva-

ture areas, or other sharp features. Meshes represent-

ing such surfaces tend to exhibit many regions that

are either undersampled or oversampled and contain

many triangles with undesirable aspect ratios. To ad-

dress these issues, parametrizations of spherical sur-

faces by mappings that minimize the average geomet-

ric distortion have been investigated in (Praun and

Hoppe, 2003). Let S

2

denote the unit sphere cen-

tered at the origin in R

3

and let φ: S

2

→ M be a

parametrization of a spherical surface M embedded

in R

3

. The inﬁnitesimal distortion produced by the

mapping φ at x ∈ S

2

was quantiﬁed by the sum of

199

Liu X., Bowers J. and Mio W. (2007).

PARAMETRIZATION, ALIGNMENT AND SHAPE OF SPHERICAL SURFACES.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 199-206

Copyright

c

SciTePress

Figure 1: The contour of a bunny as a deformation of a round sphere.

the squares of the singular values of the derivative

dφ(x), the linearization of φ at x. However, this mea-

sure of distortion is heavily biased towards stretch-

ing and only mildly penalizes compression. This has

the practical effect of controlling undersampling of

the parametrizations – but not oversampling – and

allowing the parametrizations to distort aspect ratios

in a signiﬁcant way. A solution to this problem was

proposed in (Praun and Hoppe, 2003), but the cost

function utilized is highly asymmetric with respect to

stretching and compression. In this paper, our con-

tribution to the construction of spherical parametriza-

tions is a reﬁnement of the model of Praun and Hoppe

based on a fully symmetric measurement of distortion

that leads to more uniform meshes. The process of

remeshing a surface with this technique to improve

regularity is illustrated in Figure 3. From a more

technical perspective, we also introduce some alterna-

tive computational strategies for the implementation

of the parametrization algorithm.

The main new element of this paper is an algo-

rithm to align parametric surfaces of genus zero in a

fully automated manner. Alignment is of basic impor-

tance in shape interpolation and in the development

of shape metrics. Without alignment, interpolations

tend to be intuitively incorrect (Alexa, 2000). In sev-

eral previous works on shape interpolation, such as

(Alexa, 2000) and (Asirvatham et al., 2005), align-

ment is done with the aid of manually chosen land-

mark points. Here, alignment is based on an optimal

matching of the outward unit normal ﬁelds to the sur-

faces, which capture important geometric properties

to ﬁrst order. Applied to a family of spherical sur-

faces, the parametrizaton and alignment methods al-

low us to obtain compatible parametrizations for the

family that, on average, optimally match correspond-

ing features and minimize metric distortions. Once

such family of parametrizations has been constructed,

one can remesh the given surfaces using a common

reference spherical triangulation of S

2

and use these

meshes for multiple purposes such as interpolating

shapes, deﬁning shape metrics, studying the statis-

tics of shapes, and applying texture maps to the entire

family in a compatible fashion.

The paper is organized as follows. In Section

2, we describe an algorithmic procedure to construct

“regular” parametrizations of surfaces of genus zero.

Sections 3 and 4 are devoted to the problem of align-

ing surfaces. A brief review of Kendall’s shape model

is presented in Section 5.1, which is followed by ap-

plications to shapes of surfaces. Mean shapes are dis-

cussed in Section 6, as an introduction to the statisti-

cal study of shapes of surfaces.

2 PARAMETRIZATIONS

Let a closed surface of genus zero in R

3

be presented

as a mesh M. A spherical parametrization of M con-

sists of a spherical triangulation K of the sphere S

2

to-

gether with an embedding φ: S

2

→ M ⊆ R

3

such that

φ respects the mesh structures. Thus, vertices, edges

(that is, great circles connecting adjacent vertices) and

spherical triangles of K are mapped to the correspond-

ing structures in M. To construct parametrizations, we

normalize M by scaling it to have a ﬁxed total area,

say, 4π.

For each x ∈M, let J

φ

−1

(x) denote the Jacobian of

the inverse map φ

−1

: M →S

2

at x, and let 0 < γ(x) ≤

Γ(x) be the singular values of J

φ

−1

(x). In (Praun and

Hoppe, 2003), the metric distortion of φ is quantiﬁed

by the average value of

1

γ

2

(x)

+

1

Γ

2

(x)

(1)

over M. If p = φ

−1

(x) ∈ S

2

, then the singular values

of J

φ

(p) are 1/γ(x) and 1/Γ(x). Thus, (1) is heav-

ily biased toward stretching due to φ, since sizable

stretching corresponds to small values of γ, while high

compression relates to large values of Γ. To address

this issue, a term proportional to the sixth power of Γ

is added to (1), but the proposed solution is rather ad

hoc and highly asymmetric with respect to stretching

and compression. We modify the cost function to the

average value of

log

2

γ(x) + log

2

Γ(x), (2)

which is perfectly symmetric in that stretching and

compression by the same factor are penalized iden-

tically. Note that, on a logarithmic scale, the mag-

nitude of the singular values of J

φ

−1

(x) and J

φ

(p) are

the same. Moreover, (2) vanishes precisely in the case

where both singular values are 1; that is, if there is no

inﬁnitesimal distortion at x.

To estimate the cost function, Praun and Hoppe

subdivide the mesh K until each triangle is “sufﬁ-

ciently” planar and approximate each spherical trian-

gle τ in the subdivided mesh by the planar triangle

τ

1

⊂ R

3

spanned by its vertices. Let T be a triangle

in the associated subdivision of M that corresponds

to τ via φ. Then, φ

−1

is approximated by the lin-

ear map A

T

: T → τ

1

determined by the vertex cor-

respondence. The singular values of J

φ

−1

(p) are ap-

proximated by the singular values of A

T

over the tri-

angle T, for any x ∈ T. However, if the triangles have

bad aspect ratio, large errors may occur in the esti-

mation of the singular values even after taking ﬁne

subdivisions of K. To cope with this problem, they

precede the subdivision of K with another subdivi-

sion step that attempts to improve aspect ratios. We

adopt a different approach to a reliable estimation of

the singular values. We replace the linear approxima-

tion to each spherical triangle with the planar triangle

obtained via the exponential map, as described next.

Let v be the vertex of τ determined by its two longest

sides of length ℓ

1

and ℓ

2

, respectively, and let θ be the

internal angle at v. We approximate τ with the plane

triangle τ

2

determined by ℓ

1

,ℓ

2

and θ, as indicated

in Figure 2. Note that τ

2

is well deﬁned up to rigid

motions, which will not affect singular values. The

advantage of τ

2

is that, for small triangles, it gives

a better approximation of the geometry of τ even for

nearly degenerate triangles. As before, the singular

values are estimated using the linear map T → τ

2

de-

termined by the vertex correspondence.

θ

v

Figure 2: A spherical triangle ﬂattened via the exponential

map.

Praun and Hoppe employ a coarse-to-ﬁne strategy

– analogous to that adopted in (Hormann et al., 1999)

and (Sander et al., 2002) – to construct a spherical em-

bedding that parameterizes M. The mesh M is deci-

mated sequentially until it is reduced to a tetrahedron.

This gives a multiresolution representation of M as

a sequence of meshes. A spherical parametrization

that minimizes the cost function is ﬁrst constructed

at the coarse tetrahedral level and then progressively

reﬁned through the various intermediate levels to a

parametrization of M with minimal distortion. We re-

fer the reader to (Praun and Hoppe, 2003) for further

details. Figures 3 (a) and (b) show a mesh on a bunny

and the result of remeshing the shape by transferring

a regular mesh on the sphere via a minimal distor-

tion parametrization, respectively. A close-up view is

shown on panels (c) and (d). Note that the triangles

are much more regular in the remeshed surface.

(a) (b)

(c) (d)

Figure 3: Remeshing the surface of a bunny: (a) the original

mesh; (b) transferring a regular spherical mesh from S

2

with

a minimal-distortion parametrization; (c) and (d) are close-

up views.

Similar to (Gu et al., 2002), minimal-distortion

parametrizations can be used to transfer texture pat-

terns from the sphere S

2

to a shape as illustrated in

Figure 4.

3 SHAPE ALIGNMENT

Let M

1

and M

2

be meshes in 3D space representing

spherical surfaces. Using the procedure described in

Section 2, construct parametrizations φ

i

: S

2

→ M

i

,

for i = 1,2. Before aligning the shapes, we remesh

them with respect to a ﬁxed triangulation K of S

2

, typ-

ically chosen to have fairly uniform triangles. Then,

each φ

i

can be viewed as a mapping K → M

i

that re-

spects the mesh structures. The idea is to consider

reparametrizations of one the shapes to best match the

geometry of the other using the alignment of the out-

ward unit normal ﬁelds as criterion. We ﬁrst discuss

reparametrizations.

Given a parametric shape φ: S

2

→ M and a ro-

tation matrix R ∈ SO(3), we consider reparametriza-

tions of M of the form

p 7→ φ(R

T

p), (3)

Figure 4: Transferring texture from a sphere to a shape with a minimal-distortion parametrization.

which simply rotates the sphere S

2

before mapping it

to M via φ. We denote this new parametrization of

M by φ

R

. Note that the singular values of J

−1

φ

(x) and

J

φ

−1

R

(x) are the same, for any x ∈ M. Thus, if φ is

a minimal distortion parametrization, as measured by

(2), then φ

R

is also minimal.

Given a rotation matrix R and a pair of

parametrizations φ

1

and φ

2

R

, for each p ∈ S

2

, we

think of φ

1

(p) ∈M

1

and φ

2

R

(p) ∈M

2

as corresponding

points and wish to measure the average discrepancy

of the normal ﬁelds N

1

(p) and N

2

(p;R), for p ∈ S

2

.

Normal ﬁelds are insensitive to translations and scale,

but they do change under rotations and reﬂections of

a surface. Thus, before comparing the normal ﬁelds

of φ

1

and φ

2

R

, we ﬁnd the orthogonal matrix U ∈ O(3)

that minimizes

S

2

kN

1

(p) −U (N

2

(p;R)) k

2

dp. (4)

In practice, the most interesting triangulations K of S

2

are those that are fairly dense and uniform. For these,

the minimizer of (4) can be estimated as

U

R

= argmin

U∈O(3)

∑

v∈K

kN

1

(v) −U (N

2

(v;R)) k

2

, (5)

where the sum is taken over the vertices v of K. The

solution to (5) can be computed in closed form, as

follows. Order the vertices of K as v

1

,. . . ,v

ℓ

and form

the 3×ℓ matrices P and Q

R

, whose jth columns are

the normal vectors N

1

(v

j

) and N

2

(v

j

;R), respectively.

Let the singular value decomposition of PQ

T

R

be given

by

PQ

T

R

= UΣW

T

, (6)

with V,W ∈ O(3) and Σ diagonal with nonnegative

eigenvalues. Then,

U

R

= UW

T

. (7)

The reparametrization of φ

2

that best aligns the

parametric shapes φ

1

,φ

2

: K → S

2

is determined by

the rotation matrix

ˆ

R = argmin

R∈SO(3)

H(R) , (8)

where

H(R) =

S

2

kN

1

(p) −U

R

(N

2

(p;R)) k

2

dp. (9)

4 ALIGNMENT ALGORITHM

The alignment of M

1

and M

2

is implemented in two

stages: ﬁrst, we carry out a coarse estimation of

ˆ

R by

sampling the functional H over a large ﬁnite set of ro-

tations. Subsequently, a local reﬁnement is performed

using a gradient search. An alternative stochastic gra-

dient procedure will be discussed elsewhere.

4.1 Coarse Estimation

Let η

1

,. . . ,η

s

be a fairly uniformly distributed and

dense collection of unit vectors to be thought of as

determining axis of rotations of R

3

. An example of

such a set is the collection of vertices of the triangula-

tion K of S

2

used in the previous section. Divide the

interval [0,2π] uniformly into r parts to obtain angles

0 = θ

0

< θ

1

< . .. < θ

r

= 2π. We sample H over the

set R(η

i

,θ

j

), 1 6 i 6 s, 1 6 j 6 r, of rotations about

the axis η

i

by the angle θ

j

. As in (5), for a rotation R,

the calculation of the functional H uses the following

discretization over the vertices of the mesh K:

H(R) =

∑

v∈K

kN

1

(v) −U

R

(N

2

(v;R)) k

2

. (10)

The top results are recorded and a ﬁner search is car-

ried out around each of these candidates.

4.2 Local Reﬁnement

The space SO(3) of 3 ×3 rotation matrices R is a

3-dimensional manifold (as a matter of fact, a Lie

group). To perform the aforementioned gradient

search, we ﬁrst describe the tangent space to SO(3)

at R. Let

E

1

=

0 1 0

−1 0 0

0 0 0

, E

2

=

0 0 0

0 0 1

0 −1 0

,

E

3

=

0 0 1

0 0 0

−1 0 0

.

(11)

For any R ∈ SO(3), the matrices

E

R

i

=

1

√

2

RE

i

, (12)

1 6 i 6 3, form an orthonormal basis of the tangent

space to SO(3) at R. A geodesic deformation of R

along the direction E

R

i

is given by

R

i

(t) = Re

tE

i

/

√

2

. (13)

We estimate the partial derivatives of H in the direc-

tions E

R

i

, as follows:

∂

i

H(R) ≈

H(R

i

(ε)) −H(R)

ε

, (14)

with ε > 0 small. The gradient of H at R can be cal-

culated as

∇H(R) = RA

R

, (15)

where A

R

=

∑

3

i=1

∂

i

H(R)

√

2

E

i

. The (geodesic) update

rule for the gradient search is

R

j+1

= R

j

e

−δA

R

j

, (16)

with δ > 0 small.

5 SHAPE OF SURFACES

We employ the parametrization and alignment tech-

niques presented in Sections 2 and 3 to model

and quantify shape similarity and divergence within

a given collection of spherical meshes M

1

,. . . ,M

k

.

First, construct minimal distortion parametrizations

φ

i

: S

2

→ M

i

, 1 6 i 6 k. Choose a spherical trian-

gulation K of S

2

, which will be used to align and

discretize the parametrizations. Remesh all surfaces

with respect to K using the parametrizations φ

i

– we

abuse terminology and still refer to the new meshes as

M

i

. Each φ

i

can now be viewed as a mesh-preserving

mapping K → M

i

. To simplify the discussion, we

align each φ

j

, 2 6 j 6 k, with respect to φ

1

. This type

of alignment will be biased toward the ﬁrst shape, so

reﬁnements of the alignment will be discussed below.

An advantage of remeshing all surfaces over K is that

the alignment of the surfaces induce a natural corre-

spondence between the vertices of any pair M

i

and

M

j

, i 6= j. We exploit this fact to study the shapes of

the surfaces.

We order the vertices of K arbitrarily as v

1

,. . . ,v

n

and discretize each M

i

using the ordered point cloud

φ

i

(v

1

),. . .,φ

i

(v

n

). Since correspondences have been

established between the point clouds associated to any

pair of surfaces, we can resort to Kendall’s shape the-

ory to analyze the shapes of the surfaces (Kendall,

1984), (Kendall et al., 1999). We brieﬂy review the

model, which uses a Procrustean alignment of shapes

and a geodesic metric to quantify shape divergence.

5.1 Kendall’s Model

For a parametric shape φ: K → M, we write the co-

ordinate vectors of the vertices φ(v

1

),. . .,φ(v

n

) as the

columns of a 3×n matrix P. Ordered conﬁgurations

of vertices that differ by rigid motions or scale are to

be viewed as having the same shape. To eliminate

translational effects from the representation, the or-

dered point cloud is translated to have its centroid at

the origin. This amounts to subtracting the mean

µ

φ

=

φ(v

1

) + . . . + φ(v

n

)

n

∈ R

3

(17)

from each column of the matrix P. To ﬁx the scale,

a centered matrix P is normalized to have Frobenius

norm 1. Centered matrices of unit norm are referred

to as pre-shapes. Heretofore, we assume that all ma-

trices have been normalized to represent pre-shapes.

Unlike translation and scale, there is no standard

normalization that satisfactorily accounts for rotations

and reﬂections. This is due to the facts that rota-

tional alignment for shape comparison depends on

the shapes that are being compared in a more essen-

tial manner and the relevance of chirality tends to be

contextual. Given pre-shapes P,Q, the best alignment

in Kendall’s model is given by the orthogonal matrix

ˆ

U ∈ O(3) characterized by

ˆ

U = argmin

U∈O(3)

kP−UQk

2

. (18)

ˆ

U is the orthogonal transformation that places Q clos-

est to P as measured by the Frobenius norm. Using

a singular value decomposition, write PQ

T

= V

1

ΣV

T

2

,

with V

1

,V

2

∈ O(3) and Σ diagonal with nonnegative

eigenvalues. Then,

ˆ

U = V

1

V

T

2

. The shape distance is

the geodesic distance between the pre-shapes P and

ˆ

Q =

ˆ

UQ, which can be expressed as

d(P,Q) = arccos(trΣ). (19)

Moreover, if P 6=

ˆ

Q, the geodesic deformation be-

tween P and

ˆ

Q is given by

Λ(t) = cos(ωt)P+ sin(ωt)

ˆ

Q−(trΣ) P

k

ˆ

Q−(trΣ) Pk

, (20)

where ω = arccos(trΣ) and 0 ≤ t ≤ 1.

5.2 Algorithm

The following algorithmic steps compute the Kendall

geodesic between the shapes represented by the pre-

shapes P and Q:

1. Find a singular value decomposition PQ

T

=

V

1

ΣV

T

2

.

2. Set

ˆ

Q = V

1

V

T

2

Q.

3. If P =

ˆ

Q, the geodesic is represented by the con-

stant path Λ(t) = P. Else, it is given by (20).

Figure 5: A shape geodesic with a compatible texture map applied to the deformation.

5.3 Examples of Geodesics

An example of a geodesic interpolation between a

horse and a cow is shown in Figure 5. To calculate

the geodesic, we ﬁrst remesh the shapes to be interpo-

lated over a common spherical triangulation K of S

2

and align them using the algorithm of Section 4. Sub-

sequently, the point clouds given by the vertices of the

meshes are interpolated using Kendall’s model. The

point-cloud interpolation between the (ordered) ver-

tex sets is displayed in Figures 6 (a)–(e). At any stage

of the interpolation, the vertices of K are in correspon-

dence with the point cloud. Thus, the mesh structure

on K can used at all stages of the geodesic path to

obtain meshes as indicated in the last row of Figure

6. As exempliﬁed in Figure 5, these aligned spherical

parametrizations allows us to apply texture maps to

the entire path in a compatible manner.

6 FR

´

ECHET MEAN SHAPES

To illustrate the usefulness of the algorithm for calcu-

lating shape geodesics in the study of shape statistics,

we indicate how one can deﬁne the notion of mean

shape using a natural extension of Fr

´

echet means in

Kendall’s theory (Kendall et al., 1999). If s

1

,. . . ,s

r

are spherical shapes, a mean shape will be deﬁned as

follows. First, parameterize all shapes over a com-

mon spherical mesh K and align them all with respect

to, say, the ﬁrst one. This ﬁxes a parametrization for

each of the shapes. Let P

1

,. . . ,P

r

be the pre-shapes

associated with the vertices of the meshes induced by

these parametrizations over K. Then, a mean of the

family is a shape represented by a pre-shape P that

minimizes the scatter function

V(P) =

1

2

r

∑

i=1

d

2

(P,P

i

). (21)

Techniques for calculating the mean in Kendall’s the-

ory are discussed, e.g., in (Karcher, 1977). Figure 9

shows the mean of 4 horses. As well documented in

the literature, in other contexts, once the sample mean

of a collection has been computed, one can estimate

probability models for the family using the exponen-

tial map at the mean and tangent-space statistics (Dry-

den and Mardia, 1998).

7 SUMMARY AND CONCLUSION

We developed techniques to produce parametric rep-

resentations φ: S

2

→ M of spherical surfaces that re-

ﬁne the methods of (Praun and Hoppe, 2003). The

parametrizations are constructed so as to minimize the

overall distortion in geometry. A method for align-

ing parametric surfaces to best match their unit nor-

mal ﬁelds was introduced. The procedure is fully

automated and can be used to quantify shape simi-

larity and divergence, and to model shape variations

observed within a class of objects. In conjunction

with Kendall’s shape theory, the alignment technique

yields a metric for shape comparison and geodesic

shape morphing technique. Applications to the con-

struction of compatible texture maps for a family of

surfaces and the calculation of mean shapes were dis-

cussed.

Figure 6: A geodesic interpolation between ordered point clouds. The ﬁrst and last frames display the shapes to be interpolated

and the intermediate frames show 3 stages of the deformation.

Figure 7: Close-up view of the evolution of the mesh structure associated with the point clouds in Figure 6.

Alignment of shapes is based on reparametriza-

tions of a shape by rotations of the sphere S

2

. This

is natural in the proposed setting since the action of

rotations preserve the minimal-distortion property of

a parametrization. In future work, the alignment tech-

nique will be reﬁned to an elastic alignment to bet-

ter match the geometric features of the surfaces. The

Kendall model of shapes was applied to a dense point

cloud representing a surface. While this is already

very useful and attractive from a computational stand-

point due to the simplicity of the model, the shape

metric fails to incorporate important higher order sur-

face geometry. The shape metric and interpolation

technique will also be reﬁned to a model that can be

computed efﬁciently and which takes higher order ge-

ometry into account.

ACKNOWLEDGMENTS

This work was supported in part by NSF grants CCF-

0514743 and IIS-0307998, and ARO grant W911NF-

04-01-0268.

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Figure 8: An example of a shape geodesic morphing a horse into another horse.

Figure 9: Four horses and their Fr

´

echet mean shape, which is shown on the bottom-right panel.