An Area-Preserving

1
An Area-Preserving Cortical Flattening
Method Yuefeng Lu1, Moo K. Chung1,2,3, , Steve
Robbins4, Alan C. Evans41Department of
Statistics, 2W.M. Keck Brain Laboratory 3Departmen
t of Biostatistics and Medical Informatics
University of Wisconsin Madison,Wisconsin,USA 4M
ontreal Neurological Institute Montreal, Canada
Motivation Brain surface flattening is becoming
increasingly important in visualization and
analysis in brain imaging. For example, the flat
representation of the cortical surfaces allows
researchers to easily visualize the functional
activations within the highly convoluted brain
surfaces. We usually assume that the cerebral
cortex has the topology of a 2-dimensional
convoluted sheet that does not have any holes or
handles (Angenent et al., 1999a Davatzikos et
al., 1996). When we map the curved cortical
surfaces onto the plan, it is most desirable to
preserve the topological characteristics of the
surface. However only for special surfaces, there
exists a homothetic map that preserves both
relative distances and angles. In most cases, we
can only achieve a conformal mapping that
preserves the area of any region but it is not
possible to preserve area and angle at the same
time (Angenent et al., 1999b).age 16.1.
Area-preserving cortical flattening onto a sphere
showing automatic segmentation of gyri (white)
and sulci (red).
A typical highly convoluted brain surface. It is
hard to visualize what is happening in hidden
regions.
Tensor-Based Morphometry via Flat Mapping An
additional advantage of the flattening algorithm
is that it automatically gives a surface
parameterization, which can be used in
tensor-based morphometry on the cortex (Chung et
al., 2002). The inverse mapping
of the cortical flattening map is a
natural parameterization. Let be
the partial derivative vectors. Then the
Riemannian metric tensors of the surface
is given by
It enables us
to measure lengths, angles and areas on the
cortical surface. For example, a sulcus can be
represented as parametric curve
in the flattened surface. Then the
length of the sulcus is given by
Area-Preserving Flattening Algorithm We have
developed an optimal area-preserving flattening
algorithm. The flattening is optimal in the
least-squares sense that it tries to minimize the
overall fit of the sum of errors of the surface
areas between the original cortex and a flat map.
Our area-preserving approach works on the
triangulated meshes of any surface that is
topologically equivalent to a sphere. We start
with any planar map of the cortex that preserves
the connectivity of the original triangular mesh.
Let us denote to be n
triangles in the cortex . The
flattening map will deform the convoluted
cortex to a plane or a sphere.
Flattening onto the sphere is simpler because
there is no need to punch a hole or cut the
cortex as in the case of flattening onto the
plane. Let
be the deformed triangles and be
the area of . Then we want to flatten the
cortex in such a way that for some constant
. There are n equations but approximately 3/2n
deformation parameters to estimate so there will
be infinite number of area-preserving flat maps.
Among all solutions, we find one solution that
minimize the sum of squared errors Gradient
descent method is used to find the minimum. In
each iteration step, the position of every
vertices are slightly moved to locally minimize L
until it is not possible to minimize any
further.
An example of tensor computation reprinted from
Chung et al. (2002). The picture displays the
amount of bending of the surface based on the
thin-plate spline energy. The bending energy was
used in localizing the region of maximum
curvature change in a group of developing
children in Chung et al. (2002) .
References 1 Angenent, S., Haker, S.,
Tannenbaum, A., Kikinis, R., On the
Laplace-Beltrami Operator and Brain Surface
Flattening, IEEE Transactions on Medical Imaging,
18707-711, 1999. 2 Angenent, S., Haker, S.,
Tannenbaum, A., Kikinis, R, On Area Preserving
Mapping of Minimal Distortion, System Theory
Modeling, Analysis and Control, edited by
Djaferis, T., Schick, I., Kluwer, Holland,
1999. 3 Chung, M.K., Worsley, K.J., Paus, T.,
Robbins, S., Taylor, J., Giedd, J.N., Rapoport,
J.L., Evans, A.C., Tensor-Based Morphometry, TR
1049, Department of Statistics, University of
Wisconsin-Madison, http//www.stat.wisc.edu/mchun
g/papers/surface_tech.pdf. 4 Davatzikos, C.,
Bryan, N., Using a Deformable Surface Model to
Obtain a Shape Representation of the Cortex, IEEE
Transactions on Medical Imaging, 15785-795,
1996. 5 MacDonald, J.D., Kabani, N., Avis, D.,
Evans, A.C., Automated 3D Extraction of Inner and
Outer Surfaces of Cerebral Cortex from MRI,
NeuroImage 12340-356, 2000. 6 Schwartz, E.L.,
Shaw, A., Wolfson, E., A numerical Solution to
the Generalized Mapmakers Problem Flattening
Nonconvex Polyhedral Surfaces, IEEE Transactions
on Pattern Analysis and Machine Intelligence,
111005-1008, 1989.
Area-preserving cortical flattening onto a plane
showing segmented gyri (yellow) and sulci (blue).
The segmentation is based on surface-based
diffusion filtering of mean curvatures. For
surface-based diffusion filtering, see Chung et
al. (2002).