A Minimum Description Length Approach to Statistical Shape Modeling

Introduction

This paper presents an algorithm for generating statistical shape models by addressing it as a correspondence problem of finding the parameterization of each shape in the training set, instead of manually annotating a set of “landmark” points for each image in the training set. The authors demonstrate the robustness of the algorithm by applying it to a variety of training image sets - infarcts, kidneys, knee cartilages, hand outlines, hip prostheses, and left ventricles. The proposed minimum description length model leads to good compactness, specificity and generalizes well, outperforming the contemporary gold standard - manual landmarking. Moreover, the authors also show that this model can be extended to work with 3-D images.

Method

Given a training set of $n_s$ shapes, a set of $n_p$-dimensional shape vectors ${x_i}$ is sampled, which is to be modeled using a multivariate Gaussian model. Changing the coordinate system to align with the principal axes of the dataset, we have $$ X_i = \bar{x} + \sum_{m=1}^{n_s-1} \boldsymbol{p}^m b_i^m $$ where $\boldsymbol{p}^m$ represents the $n_s-1$ eigenvectors that span the subspace containing the training set. Ordering the eigenvectors by their non-decreasing eigenvalues, we construct the orthonormal set

$$ \boldsymbol{\tilde{p}}^m = \frac{\boldsymbol{p}^m}{|\boldsymbol{p}^m|}, \ \text{for $m = 1$ to $n_s-1$} $$

The corresponding new set of coordinate axes are defined as

$$ y^m \equiv x . \boldsymbol{\tilde{p}}^m $$

Now for each direction $\boldsymbol{\tilde{p}}^m$, let the set of values transmitted by $Y^m \equiv {y^m_i : i = 1,…n_s}$. Since the coordinate axes are aligned with the principal axes of the data, each direction is modeled as a 1-Dimensional Gaussian. Also, let $R$ represent the strict upper-bound of the range of the data and $\Delta$ represent the estimated value for the quantization parameter.

Replacing the original data values $Y^m$ with the respective quantized values, we get $\hat{Y}^m$. Now, $\hat{\mu}^m$ amd $\sigma_m^2$ represent the mean and the variance of the quantized data along each direction. The lower bound of the modeled variance is represented by $\sigma_{min}$ and is given by $\sigma_{min} \equiv 2\Delta$. The description length $D_m$ can now be represented as

$$ D_m = \begin{cases} ln\big(\frac{R}{\Delta}\big) + D^{(1)} (\hat{Y}^m, R, \Delta), & \text{if}\ \sigma^m \geq \sigma_{min} \\ ln\big(\frac{R}{\Delta}\big) + D^{(2)} (\hat{Y}^m, R, \Delta), & \text{if}\ \sigma^m \le \sigma_{min}, \hat{Y}^m \ge \Delta \\ ln\big(\frac{R}{\Delta}\big), & \text{otherwise}
\end{cases}. $$

Set $n_g$ and $n_{min}$ to be the number of directions for which the first and the second of the above criteria hold valid. Then the total description length for the training set can be written as

$$ D = (n_s-1) ln\big(\frac{R}{\Delta}\big) + \sum_{p=1}^{n_g} D^{(1)} (\hat{Y}^p, R, \Delta) + \sum_{q=n_g+1}^{n_g+n_{min}} D^{(2)} (\hat{Y}^q, R, \Delta) $$

The first term in the objective function above represents the cost of transmitting the mean shape. However, since the values of $R$ and $\Delta$ remain constant for a given training set, it can be dropped.

$D^{(1)}$ and $D^{(2)}$ can be represented as functions of only $R$, $\Delta$, and $n_s$ and the objective function is well defined.

For selecting correspondence points, a parameterization function $\phi_i$ explicitly defining how the points are sampled on each shape is defined for each shape. Then the training shapes are represented parametrically as curves in two dimensions. It is important that in order to constrain point ordering, the parameterization function $\phi_i(t)$ must be a monotonically increasing function of $t$ for each shape $i$. Therefore, each $\phi_i(t)$ must be an one-one, onto, and invertible.