2013 Fourth International Conference on Emerging Security Technologies

Tissue Surface Model Mapping onto Arbitrary Target
Surface Based on Self-organizing Deformable Model
Shoko Miyauchi∗ , Ken’ichi Morooka∗ , Yasushi Miyagi† , Takaichi Fukuda‡ , Tokuo Tsuji∗ and Ryo Kurazume∗∗ Graduate School of Information Science and Electrical Engineering, Kyushu University

Motooka 744, Nishi-ku, Fukuoka 819-0395 Japan
Email: miyauchi@irvs.ait.kyushu-u.ac.jp, morooka@ait.kyushu-u.ac.jp† Department of Neurosurgery, Kaizuka Hospital, Fukuoka 812-0053 Japan‡ Graduate School of Medical Sciences, Kumamoto University, Kumamoto 860-8556 Japanis realized by selecting suitable target surface according to
the tissue shape. However, there are few mapping methods
satisfying the three requirements.

Abstract—This paper proposes a new method for mappinga tissue surface model onto an arbitrary target surface while
preserving the geometrical features of the tissue surface. In our
method, firstly, the tissue model is roughly deformed by using
Self-organizing Deformable Model. Since the deformed model
may contain folded patches, the folded patches are removed.
Moreover, by Free-Form Deformation (FFD), and the area- andangle-preserving mapping, the model is mapped onto the target
surface while preserving geometrical properties of the originalmodel. From several experimental results, we can conclude thatthe proposed method can map tissue models onto arbitrary target
surface without foldovers.

I.

In this paper, we propose a new method for mapping atissue surface model onto an arbitrary target surface while
preserving the geometrical features of the original tissue
model. The proposed method uses Self-organizing Deformable
Model (SDM) [3], [4], which is a deformable model guided
by competitive learning and an energy minimization approach.
The SDM allows to move some vertices included in the SDM
toward specific points on the target surface and to choose
arbitrary shapes for both the SDM and target surface.

I NTRODUCTION

On the contrary, when the SDM is applied to the brain
surface model, multiple patches in the model may be projected
onto the limited area of the target surface. When the model
contain such overlap, called foldover, the mapping between
the model and the target surface is not bijective. To avoid thefoldover, the original SDM algorithm [3] introduces certain
techniques, however, does not necessarily guarantee foldoverfree mapping. Matsui et al. [4] incorporated Least-squares
Meshes (LSM) into the SDM algorithm. In the LSM method,a matrix is employed to represent the point correspondence
between the model and the target surface. The model isdeformed by the inverse matrix. Since the matrix consistsof all vertices in the model, the computation of the inverse
matrix increases exponentially when the model has a large
number of vertices. Moreover, to perform LSM, a user needs
to specify point correspondences between the model and thetarget surface. In case of the brain surface with complex shape,
many point correspondences are needed to avoid the foldovers.

Recent medical imaging devices can provide high resolution medical images. Tissue models generated by the images
are used in support systems for diagnosis and treatment [1].
One technique using the models is to find the relationshipand similarity between the models of a target tissue. Eachtissue has fairly consistent shape while the shape varies acrossindividuals. By analyzing many shape patterns of the tissue, astatistical shape model (SSM) is build to identify the considerable natural variability of the tissue. Because of the shapeprior information which SSM provides, recent works show that
SSM-based techniques have obtained considerable success inthe tissue detection from medical images [2].
The fundamental process for building SSM is to establish
the correspondence between the models. Generally, triangularmesh models of the tissue have different number of vertices
and different topology. The correspondence problem becomescomplex in the case of the tissue with complex shape suchas the human brain. One solution for this problem is to map
the models onto a target surface with simple shape including
a plane and a sphere. Since the target surface is described bya simple parametric function, such mapping-based approachallows to easily determine the correspondences on the target
surface through the parameters in the function.

To solve the problems, in our method, the tissue modelis roughly deformed by using the original SDM deformation,and folded patches are removed from the deformed model.
Moreover, by Free-Form Deformation (FFD), and the areaand angle-preserving mapping, the model is mapped onto thetarget surface while preserving geometrical properties of theoriginal model.

To achieve this, the mapping method needs to guarantee
one-to-one correspondence between the model and the target
surface while preserving the original geometric properties
as far as possible. To compare the models effectively, it isdesirable to control the mapping so that anatomical features
involved in the model are constrained to lie at specific locations on the target surface. Moreover, if the mapping deals
with arbitrary shape, the effective mapping of tissue models

978-0-7695-5077-0/13 $26.00 © 2013 IEEE
DOI 10.1109/EST.2013.34

II.

T ISSUE MODEL MAPPING BASED ON SDM

This section explains the detail of the proposed mapping
method. Here, we describe several definitions used in our
method. Given a vertex vi , the 1-ring region Wi of vi is defined(i)(i)
by a set of the patches wk (k = 1, 2, · · · , Np ) which have
79

(i)

the vertex vi . Np denotes the number of patches containedin the 1-ring region.

Lattice point|| ∗ − ∗ ||lattice space

A. Self-organizing Deformable Model
The SDM is a deformable mesh model represented bytriangular patches. A target surface is represented by a set ofpoints on the surface, called control points. Arbitrary shapes
can be chosen for both the SDM and target surface, and thesize of a target surface is sufficient to cover the initial SDM,
provided that the SDM and the target surface have the same
topological type. Feature vertices are the vertices which correspond to specific locations on the target surface, and the control
points close to the locations are the corresponding points of thefeature vertices. In the SDM algorithm, a given mesh model
is deformed to fit to the target surface by competitive learningand minimizing an energy function. See the ref [3] in detailof the SDM algorithm.

1
deformed brain surface

1

Fig. 1. Movement of feature vertices using FFD.Red and blue points are thefeature vertex and its corresponding point.

The use of FFD enables the local surface deformation without foldovers. Owing to this reason, the feature vertices are
efficiently moved to their corresponding points by applying
FFD with the local area where the feature vertices and theirneighbor vertices exist.

B. Foldover removal

To apply FFD, for each feature vertex v ∗ , 3D control latticearound v ∗ and its corresponding point p∗ is generated asshown in Fig. 1. The local FFD coordinate system is definedby three vectors S, T , U which are orthogonal each other. The
axis S is defined by S = l(p∗ − v ∗ ) (l : an odd number). The
axis T is the vector selected from the vectors orthogonal to
S. The axis U is obtained by the cross product of S and T .
The magnitude of T and U is set to (l − 1)(p∗ − v ∗ ).

When a brain surface model is mapped onto a given target
surface by using the original SDM algorithm, the deformed
brain model may contain a foldover. The foldover is removed
based on the method proposed by Athanasiadis et al. [5]. Since
the method in [5] dealt with only a spherical surface as a target
surface. We extend the method in [5] to apply to target surface
with arbitrary shape.
The foldover occurs when a vertex is not included inits 1-ring region. Considering this, the foldover is removedby moving the vertices to within their 1-ring region. This
movement is made by moving each vertex vi forward thelocation computed by∑(i)( k A(i)wk C k )vi = ϕ
(1)
∑ (i)
k Awk

To facilitate manipulation of the FFD coordinate system,
the axes S, T, U are uniformly divided into a grid by the
distance between v ∗ and p∗ . The control lattice comprises
l grid points along the S axis, and l − 1 grid points along
the axes T and U (Fig. 1). The global coordinate K(a, b, c)(0 ≤ a ≤ l, 0 ≤ b, c ≤ l − 1) of the (a, b, c)-th grid point isgiven byabcK(a, b, c) = X0 + S +
T+
U
(3)
l
l−1
l−1
where X0 is the origin of the FFD coordinate system, andits location is defined satisfying v ∗ = K(h, h, h) and p∗ =K(h + 1, h, h) (h = l−12 ) as follows:(1)
1
1
X0 = v ∗ − h S +
T+
U .
(4)
l
l−1
l−1
The local coordinate (s, t, u) of an arbitrary point v̂ in the gridis expressed by(i)(i)
where Awk and Ck is, respectively, the area and centroid of a(i)patch wk included in the 1-ring region of the vertex vi . The

function ϕ is used to map the moved vertex onto the target
surface. In our method, the vertex v is moved to the control
point p closest to the vertex, and ϕ is formulated by
ϕ(v) =

∥p∥
v
∥v∥

(2)

where ∥y∥ is the norm of a vector y.

v̂ = X0 + sS + tT + uU .

In the process of removing the foldover, all vertices are
moved by Eq.(1) and (2). The removal process is repeateduntil all vertices are not moved.

(5)

Using the grid points, v ∗ is moved to coincide with its
corresponding point p∗ . Since v ∗ = K(h, h, h) and p∗ =K(h+1, h, h), the movement is made by the FFD deformation
using only K(h, h, h):

C. Movement of feature vertices with Free-Form Deformation
By using the original SDM algorithm, feature vertices are
mapped to the locations of their corresponding points on thetarget surface. However, in the foldover removal described inSec. II-B, all the vertices are moved to achieve foldover-freedeformation of the tissue model. After the removal process, thefeature vertices may be far from their corresponding points.
To correct the positions of the feature vertices, Free-Form
Deformation(FFD) [6] is used which deforms a mesh modelby deforming the space in which the model is embedded.

K(h, h, h) ← K(h, h, h) +

1
S;
lB

B = Ihl (sv∗ ) × Ihl−1 (tv∗ ) × Ihl−1 (uv∗ );( )mm
In (λ) =(1 − λ)m−n λn
n

(6)
(7)
(8)

where (sv∗ , tv∗ , uv∗ ) is the location of v ∗ in the FFD coordinate system. The movement using Eq.(6) causes the local
80deformation of the model included in the lattice concurrently.
The positions of the vertices in the lattice are updated by
∑
∑
∑
v̂ =
Ial (s)Ibl−1 (t)Icl−1 (u)K(a, b, c).
(9)
a

c

b

temporal lobe

The algorithm for moving the feature vertices is as follows:
[F1]

[F2]
[F3]

Fig. 2.

For each feature vertices,a) Generate the FFD coordinate system and thegrid points.
b) Move all vertices within the grid by Eqs. (6)and (9).
Map the vertices on the target surface by Eq. (2).
If all feature vertices is satisfied with ∥p∗ − v ∗ ∥ <τ (τ：threshold), the process is terminated. Otherwise
return Step.F1.

central sulcus

A simple brain surface:(left: front, middle: top, right: left side)dwhere θw
is one angle of the patch w.

The distortion metric E is formulated as a linear combination of Earea and Eangle :∑E(σ) = µEarea + (1 − µ)Eangle ≈e(i, σ);(16)ie(i, σ) = µψearea (i, σ) + (1 − µ)eangle (i, σ)III.

G EOMETRICAL FEATURE PRESERVING MAPPING

where ψ is a scaling factor to adjust the ranges of the two
metrics. When a weight coefficient µ (0 ≤ µ ≤ 1) increases,
the property of the distortion metric E change from angle toarea-preservation. Therefore, the area- and/or angle-preserving
mapping is determined by minimizing the distortion metric in
Eq. (16).

After the movement of feature vertices process, the tissue
model is deformed to preserve the original geometric properties as far as possible. This deformation is based on an areaand angle-preserving mapping. The definition of each mappingis described by the example of mapping one triangular meshmodel DW onto a given parameter domain Ω∗ . Here, themapped model on the domain Ω∗ is denoted as D∗ .

Applying a greedy algorithm with Eq.(17), the minimization is the optimization problem of positioning the vertices in
the model by moving them repeatedly. Practically, for eachvertex vi , the suitable next position of vi is selected from its(k)(i)
candidates vi (k = 1, 2, · · · , Np ) obtained by∗

An area-preserving mapping is the mapping σ : D → D
if the area of each patch Aw in DW is the same as that of thecorresponding patch σ(Aw ) in D∗ . When the total area of
DW is the same as that of D∗ , the area-preserving mappingis determined by minimizing the distortion metric of the area
[7]:∑Ẽarea (σ) =(σ(Aw ) − Aw )2 .
(10)
W

(k)

vi

The algorithm for area- and angle-preserving mapping is
as follows:
[A1]

i

(i)

wk ∈Wi[A2]σ(Aw(i) )k(i)wk ∈Wiσ(Aw(i) )
k

[A3]

Aw(i)

−∑

k

(i)

wk ∈WiAw(i). (13)

[A4]

k

[A5]

An angle-preserving mapping is the mapping σ : DW →D∗ if the angle of each pair of intersecting arcs in DW isthe same as that of the corresponding arcs in D∗ [7]. In
our method, the angle-preserving mapping is determined byminimizing the distortion metric of the angle:∑
Eangle (σ) =eangle (i, σ);
(14)
3
∑ ∑
(i)
wk ∈Wi d=0

d
d
|σ(θw
(i) ) − θ (i) |.
w
kChoose randomly the vertex from all the vertices
except the feature vertices.
Compute the position candidates of the chosen vertex
by using Eq. (18).
Move the vertex to the candidate with minimum error
in Eq.(17).
If all vertices don’t moved, go to Step. A1. Otherwise,
go to Step. A5.
Map the vertices on the target surface by Eq. (2).
IV.

E XPERIMENTS

To verify the applicability of our proposed method, we
made the experiments using 6 brain surface models and atarget surface shown in Fig. 2. The target surface, called asimple brain, has the simplified shape of a human brain. In theexperiment, 11 feature vertices are selected from longitudinal
fissure of cerebrum, lateral sulcus and central sulcus. The
parameters τ in Step.F3 and α in Eq. (18) is set to τ = 0.01
and α = 0.1, respectively.

ieangle (i, σ) =(18)

movement of vi is limited within its 1-ring region. Among thecandidates, vi is moved to the candidate with minimum error
in Eq.(17).

However, in our method, the surface area of the model DW is
not always equal to that of the mapped model D∗ . Instead ofEq. (10), we define a objective function by∑
σ(Aw )
Aw
Earea (σ) =
−
(11)
W
Σ
σ(A
)
Σ
w
w∈D
w∈D W Aww∈DW∑≈
earea (i, σ);
(12)

∑

(i)

= vi + α(uk − vi )(i)where α (0 ≤ α < 1) is a coefficient, and uk is the vertex
connected to vi by one edge. To prevent the foldover, thew∈D Wearea (i, σ) =
∑

(17)

(15)

k

81

(a)

(b)

(c)

(d)

(e)

Fig. 3. (a) brain surface model and (b-e) the resulting model by the proposed method: (b) the movement of feature vertices; (c) the area- and (d) angle-preserving
mapping; (e) the area- and angle-preserving mapping. Red and blue points are the feature vertices and their corresponding points. Green points denote the feature
vertices completely coincided with their corresponding points.

Area errorAw
1 σ(Aw )+− 2;
σ(Aw ) S Aw(b) angle error

Fig. 4. Error distributions in areas and angles by the proposed mapping
method. black: without geometrical process, red: µ = 0, blue: µ = 0.5,
green: µ = 1.

foldovers. Moreover, the model is deformed to preserve area
and angle before and after mapping. From the experimental
results, our method can map a brain model on a target surface
while both controlling feature vertices position and keeping
geometrical features without foldovers.

Σw∈W σ(Aw ).
Σw∈W Aw(19)
The error function ϵangle for angle-preservation is calculatedby the sum of the absolute difference between the angles ofthe patches in the mapped model and the original. When thevalues of the error functions are close to zero, the mappings
preserves their corresponding geometric properties. Fig. 4(a)and (b) show the distributions of ϵarea and ϵangle , respectively.
In these figures, three colors correspond to the setting ofµ: 0.0(red), 0.5(blue), and 1,0(green). Compared with the
distribution (black line in Fig. 4) using the model obtainedby FFD deformation Fig. 3(b), our mapping method (red, blueand green lines) reduces the errors on areas and angles. In
the case of µ = 1 (green line), our mapping is the areapreserving mapping, and the distribution of ϵarea has thehighest peak at zero. The distribution of ϵangle becomes narrow
with decreasing the value of µ. For µ = 0 (red line), ourmapping is the angle-preserving mapping. In this case, thedistribution of ϵangle is steep, and has the single peak at zero.
From these results, our method can achieve the preservationof the geometrical features.
V.

Angle error(a) area error

To evaluate our area- and/or angle-preserving mapping,
we studied the error distrubtions in areas and angles by thethree types of the maapings. The error function ϵarea for areapreservation is formulated by
ϵarea = S

DensityDensity

The mapping results of the brain surface model are shownin Fig. 3. Fig. 3(b) show the resulting model by performing
the original SDM algorithm and the processes of removing
the foldover (Sec. II-B) and moving the feature vertices by
FFD (Sec. II-C). All the mapped brain models are completely
fitted to the target surface without foldovers. while locating
the feature vertices at their target position correctly. Our areaand/or angle-preserving mapping is applied to the model inFig. 3(b). Fig.3(c) and Fig. 3(d) show the resulting models by
the mapping which preserve area and angle alone, respectively.
Fig. 3(e) shows the resulting model obtained by the distortion
metric (Eq.16) with the parameter µ = 0.5. In this case, themapping leads to the preservation of areas and angles.

S=

ACKNOWLEDGMENT
This work was supported by JSPS KAKENHI Grant
Number 23500244, 24390345, 24103708 and the Ministry of
Health, Labour and Welfare(201024171A).
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C ONCLUSION

In this paper, we proposed the method of mapping of thebrain surface model with complex shape onto an arbitrarytarget surface. The model is mapped roughly on the target
surface by SDM method. After removing the foldover, the local
surfaces including the feature vertices is deformed by FFD tomove the feature vertices to their corresponding points without82