active shape models
Active shape models (ASMs) are often limited by the
inability of relatively few eigenvectors to capture the full range
of biological shape variability. We present a method that overcomes
this limitation, by using hierarchical formulation of active shape
models. The hierarchical formulation achieves two objectives. First,
it is robust when only a limited number of training samples is available.
Second, by using local statistics as smoothness constraints, it
eliminates the need for adopting ad hoc physical models, such as
elasticity or other smoothness models, which do not necessarily
reflect true biological variability.
- Global/local spatial partition of the shape
We define a deformable contour as a collection of
segments, each comprising
points. Let ,
, be a vector
corresponding to the -th
segment, and formed by concatenating the coordinates of all points
in that segment. Let, also,
be the center of mass of the points on the -th
segment. In one of its simplest forms, a hierarchical ASM can be
represented by two levels. At the bottom level, the covariance matrix
of is used
as prior for the deformation of the -th
segment, thus imposing some sort of local smoothness constraint.
At the top level, statistics of the vector
are collected from the training set, and are used as prior for more
global shape properties.
Schemetic drawing for global/local spatial partition
of the shape.
- Hierarchical formulation using the wavelet tranaform
Instead of the heuristic partition of the spatial domain described
above, the wavelet transform provides an elegant way to perform
a scale-space decomposition. In this work, we use a logarithm tree
2-band wavelet packet to divide the space-frequency domain. For
a level wavelet packet,
we divide the frequency domain into
frequency bands. The following is a 3-level logarithm tree 2-band
A wavelet transform is applied to the parametric functions
representing a deformable contour. The resulting wavelet coefficients
are then grouped into B bands (in the case of above figure, B=8),
and the joint distribution of each band is estimated from the available
training samples, via its mean and covariance matrix. This effectively
transforms the covariance matrix of the full joint distribution
into a matrix that is close to, but not necessarily exactly, a block
diagonal matrix. The submatrix corresponding to Band-1 reflects
global shape characteristics, whereas the submatrix corresponding
to Band-8 reflects local shape characteristics at a particular segment.
Band-7 reflects local shape characteristics of a neighboring segment.
(Left) Midsagittal MR image. (Right) Automated segmentation
obtained using the standard Active Shape Model (ASM) trained on
99, and 5 samples (top and middle image, respectively), and the
automatical segmentation obtained using the hierarchical active
shape model with global/local spatial partition trained on 5 samples
(bottom). The ASM performs well when enough training samples are
available, relative to the variability of the structure, but starts
to fail when relatively few training samples are available. But
the hierarchical active shape model can achieve good results even
with as few as 5 training samples.
Top row: midsagittal images of the corpus callosum of
three subjects. Second to fifth rows: Segmentations obtained using
ASM trained on 5 samples, ASM trained on 99 samples, Method 1, and
Method 2 trained on 5 samples. It is clear that ASM needs to be
trained on a sufficiently large number of samples in order to capture
the finer details of individual shapes, which is not the case with
the hierarchical methods.
1. Christos Davatzikos, Xiaodong Tao
and Dinggang Shen, "Hierachical Active Shape Models: Using the Wavelet
Transform", IEEE Trans. Med.
Imag. , 22(3):414-423, March, 2003
2. Christos Davatzikos, Xiaodong Tao
and Dinggang Shen, "Applications of wavelets in morphometric analysis
of medical images", Proceedings of SPIE
Vol. #5207 Wavelets: Applications in Signal and Image Processing
X , San Diego, CA, Aug. 2003