Transcript CVPR2003TBSM - Department of Statistics

Computer Vision and Pattern Recognition 2003
Medical Image Analysis
Tensor-based Surface Modeling and Analysis
Moo K.
1Department
123
Chung ,
Keith J.
4
Worsley ,
Steve
4
Robbins ,
Alan C.
4
Evans
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
5. Random fields theory
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.
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
2. Magnetic Resonance Images
where
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.
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.
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.
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.
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 p=p0, the LaplaceBeltrami 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:
References
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.
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. 18:198–213, 2003.
Kollakian, K., Performance analysis of automatic techniques for tissue classification in magnetic
resonance images of the human brain. Master’s thesis, Concordia Univ., Canada. 1996.
MacDonald, J.D. et al., Automated 3D Extraction of Inner and Outer Surfaces of Cerebral Cortex from
MRI, NeuroImage. 12:340-356, 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. 4:58-73, 1996.