Hierarchical active shape models

1. Overview

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.

2. Method

  • Global/local spatial partition of the shape (Method 1)

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 (Method 2)

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 wavelet packet.

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.

3. Results

(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.

References

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