Tensorbased Surface Modeling and Analysis

1
Computer Vision and Pattern Recognition 2003
Medical Image Analysis
Tensor-based Surface Modeling and Analysis Moo K.
Chung123, Keith J. Worsley4, Steve Robbins4, Alan
C. Evans4 1Department of Statistics,
2Department of Biostatistics and Medical
Informatics, 3W.M. Keck Laboratory for functional
Brain Imaging and Behavior University of
Wisconsin, Madison, USA 4Montreal Neurological
Institute, McGill University, Montreal, Canada
1. Motivation We present a unified tensor-based
surface morphometry in characterizing the gray
matter anatomy change in the brain development
longitudinally collected in the group of children
and adolescents. As the brain develops over time,
the cortical surface area, thickness, curvature
and total gray matter volume change. It is highly
likely that such age-related surface changes are
not uniform. By measuring how such surface
metrics change over time, the regions of the most
rapid structural changes can be localized.
5. Random fields theory Statistical analysis is
based on the random field theory (Worsley et al.,
1996). The Gaussianess of the surface metrics is
checked with the Lilliefors test. The isotropic
diffusion smoothing is found to increase both the
smoothness as well as the isotropicity of the
surface data. For a paired t-test for detecting
the surface metric difference, we used the
corrected P-value of t random field defined on
the manifold which is
approximately where is the 2-dimensional
EC-density given by and is the
total surface area of the template brain
estimated to be 275,800mm2. The validity of our
modeling and analysis was checked by generating
null data. The null data were created by
reversing time for the half of subjects chosen
randomly. In the null data, most t values were
well below the threshold indicating that our
image processing and statistical analysis do not
produce false positives.
2. Magnetic Resonance Images Two T1-weighted
magnetic resonance images (MRI) were acquired for
each of 28 normal subjects at different times on
the GE Sigma 1.5-T superconducting magnet system.
The first scan was obtained at the age 11.5 years
and the second scan was obtained at the age 16.1
years in average. MRI were spatially normalized
and tissue types were classified based a
supervised artificial neural network classifier
(Kollakian, 1996). Afterwards, a triangular mesh
for each cortical surface was generated by
deforming a mesh to fit the proper boundary in a
segmented volume using a deformable surface
algorithm (MacDonald et al., 2000). This
algorithm is further used in surface registration
and surface template construction (Chung et al.,
2003).
Top Thin-plate spline energy functional computed
on the inner cortical surface of a 14-year-old
subject. It measures the amount of folding of the
cortical surface. Bottom t-statistic map showing
statistically significant region of curvature
increase between ages 12 and 16. Most of the
curvature increase occurs on gyri while there is
no significant change of curvature on most of
sulci. Also there is no statistically significant
curvature decrease detected, indicating that the
complexity of the surface convolution increase.
6. Morphometric changes between ages 12-16
Gray matter volume total gray matter volume
shrinks. Local growth in the parts of temporal,
occipital, somatosensory, and motor regions.
Cortical Surface area total area shrinks. highly
localized area growth along the left inferior
frontal gyrus and shrinkage in the left superior
frontal sulcus. Cortical thickness no
statistically significant local cortical thinning
on the whole cortex. Predominant thickness
increase in the left superior frontal sulcus.
Cortical curvature no statistically significant
curvature decrease. Most curvature increase
occurs on gyri. No curvature change on most
sulci. Curvature increase in the superior frontal
and middle frontal gyri.
4. Surface data smoothing Beltrami flow To
increase the signal-to-noise ratio and to
generate smooth Gaussian random fields for
statistical analysis, surface-based data
smoothing is essential. Isotropic diffusion
smoothing or Beltrami-flow is developed for this
purpose. It is not the surface fairing of Taubin
(1995), where the surface geometry is smoothed.
We solve an isotropic heat equation on a manifold
with an initial condition. where the Laplacian
is the Laplace-Beltrami operator defined in terms
of the Riemannian metric tensor g We estimate
the Laplace-Beltrami operator on a triangulated
cortical surface directly via finite element
method (Chung, 2001). Let F(pi) be the signal on
the i-th node pi in the triangulation. If
p1,...,pm are m-neighboring nodes around pp0,
the Laplace-Beltrami operator at p is estimated
by with the weights
where  and are the two
angles opposite to the edge pi - p in triangles
and is the sum of the areas of m-incident
triangles at p. Then the diffusion equation is
solved via the finite difference scheme
Left The Gyri are extracted by thresholding the
thin-plate energy functional on the inner
surface. Middle Right Individual gyral
patterns mapped onto the template surface. The
gyri of a subject match the gyri of the template
surface illustrating a close homology between the
surface of the individual subject and the
template.
References Chung, M.K., Statistical Morphometry
in Neuroanatomy, PhD Thesis, McGill University,
Canada Chung, M.K. et al., Deformation-based
Surface Morphometry applied to Gray Matter
Deformation, NeuroImage. 18198213,
2003. Kollakian, K., Performance analysis of
automatic techniques for tissue classification in
magnetic resonance images of the human brain.
Masters thesis, Concordia Univ., Canada.
1996. MacDonald, J.D. et al., Automated 3D
Extraction of Inner and Outer Surfaces of
Cerebral Cortex from MRI, NeuroImage. 12340-356,
2000. Taubin, G., Curve and surface smoothing
without shrinkage. The Proceedings of the Fifth
International Conference on Computer Vision,
852-857, 1995. Worsley, K.J., et al., A unified
statistical approach for determining significant
signals in images of cerebral activation, Human
Brain Mapping. 458-73, 1996.
Left Individual cortical surfaces (blue
interface between the gray and white matter,
yellow outer cortical surface). Right The
surface template is constructed by averaging the
coordinates of homologous vertices.
3. Tensor geometry Based on the local quadratic
surface parameterization, Riemannian metric
tensors were computed and used to characterize
the cortical shape variations. Then based on the
metric tensors, the cortical thickness, local
surface area, local gray matter volume,
curvatures were computed.