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2.2 Circle Packing
In (Hurdal et. al., 1999) Hurdal a method of build-
ing such a conformal map using circle packing is sug-
gested. This relies on the ability to approximate con-
formal structure on surfaces by circle packings. The
authors use this method for MRI brain images and
conformally map them to the three possible models
of geometry in dimension 2 (i.e. the 2-sphere, the
Euclidian plane and the hyperbolic plane). Yet, the
method is applicable for a surface which are topolog-
ically equivalent to a disk whereas the brain cortex
surface is not. This means that there is a point of the
brain (actually a neighborhood of a point), which will
not map conformally to the plane, and in this neigh-
borhood the dilatation will be infinitely large. More-
over, it must be assumed that the surface triangulation
is homogeneous in the sense that all triangles are equi-
lateral. Such triangulations are seldom attainable.
2.3 Holomorphic 1-forms
Gu et al. ( (Gu, X. and Yau, S. T., 2002) and a series of
consequent papers), are using holomorphic 1-forms
in order to compute global conformal structure of a
smooth surface of arbitrary genus and arbitrary num-
ber of boundary components. Holomorphic 1-forms
are differential forms that depict conformal structure.
As such, this method can be applied to colon unfold-
ing. Yet, implementation of this method is extremely
time consuming.
2.4 Angle Methods
Sheffer et al. (Sheffer, A. de Stuler, E. ) parameterize
surfaces via an angle-based method that minimizes,
in a way, angle distortion while flattening. However,
the surfaces are assumed to be approximated by cone
surfaces i.e. one considers surfaces that are composed
of cone-like neighborhoods.
2.5 The Proposed Method
We propose yet another solution to the problem of sur-
face unfolding, with a special focus on its applications
in medical imaging. The method proposed herein is
based on theoretical results obtained by Gehring and
V
¨
aisal
¨
a in the 1960‘s and refereed to in (Gehring-
V
¨
aisala, 1965). Gehring and V
¨
aisal
¨
a investigated the
existence of quasi-conformal maps between Riemann
manifolds. The basic advantages of the proposed
method is its simplicity. It is straightforward to im-
plement this method and it is computationally effi-
cient. An additional advantage is that it is possible
to guarantee that the distortion will not exceed a pre-
defined bound, which can be as small as desired with
respect to the extent of localization one is ready to ac-
cept. Moreover, since, in contrast to other methods,
the algorithm makes no use to derivatives, it is highly
suitable for analysis of noisy data and for the study
of surfaces with folds and ridges, such as the colon
wall and the cortex. The proposed algorithm is most
able in cases where the surface is complex (high and
non-constant curvature) such as colon wrapping.
The paper is organized as follows: In the next sec-
tion we provide a review of related works. In Sec-
tion 3 some theoretical background to the fundamen-
tal work of Gehring and V
¨
aisal
¨
a. In the following sec-
tion we describe our algorithm for surface flattening,
based on their ideas. In Section 5 we present some
experimental results of this scheme and in Section 6
we summarize the paper and discuss further improve-
ments.
3 THEORETICAL BACKGROUND
Definition 1 Let D ⊂ R
3
be a domain. A homeo-
morphism f : D → R
3
is called a quasi-isometry (or
a bi-Lipschitz mapping), iff
1
C
|p
1
− p
2
|≤|f(p
1
) −
f(p
2
)| <C|p
1
−p
2
| , for all p
1
,p
2
∈ D; 1 ≤ C<∞.
C(f ) = min{C | f is a quasi − isometry} is called
the minimal distortion of f (in D). (The distances
considered are the induced intrinsic distances on the
surfaces.)
If f is a quasi-isometry then K
I
(f) ≤ C(f )
2
and
K
O
(f) ≤ C(f)
2
, where K
I
(f) and K
O
(f) repre-
sent the inner and outer dilatation of f , respectively
(see (Caraman, P., 1974) and references therein). It
follows that any quasi-isometry is a quasi-conformal
mapping (while, evidently, not every quasi-conformal
mapping is a quasi-isometry). Quasi-conformal is the
same as quasi-isometry where distances are replaced
by inner products between tangent vectors.
Definition 2 Let S ⊂ R
3
be a connected surface. S
is called admissible iff for any p ∈ S, there exists a
quasi-isometry i
p
such that for any ε>0 there exists
a neighbourhood U
p
⊂ R
3
of p, such that i
p
: U
p
→
R
3
and i
p
(S∩U
p
)=D
p
⊂ R
2
, where D
p
is a domain
and such that C(i
p
) satisfies: (i)sup
p∈S
C(i
p
) < ∞
and (ii)sup
p∈S
C(i
p
) < 1+ε .
Let S be a surface, n be a fixed unitary vector, and
p ∈ S, such that there exists a neighbourhood V ⊂ S,
such that V D
2
, where D
2
= {x ∈ R
2
||x|| ≤ 1}.
Moreover, suppose that for any q
1
,q
2
∈ S, the acute
angle (q
1
q
2
,n) ≥ α. We refer to the last condition
as the Geometric Condition or Gehring Condition.
Then for any x ∈ V there is a unique representation
of the form: x = q
x
+ un, where q
x
lies on the plane
through p which is orthogonal to n and u ∈ R.We
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