Section 2.3.2 introduced segmentation by posterior optimization of probabilistic deformable models. The segmentation framework of m-reps described in [Pizer et al., 2003] also follows the posterior optimization approach, i.e. deforming a template model by minimizing or maximizing an objective function that consists of a log prior term (a geometric typicality term) and a log likelihood term (a geometry-to-image match term). The geometric typicality measures how probable the shape of the deforming template model is in the population of the object, and the geometry-to-image match measures how likely the target image values relative to the deforming template model are in the population of the image.
The geometric typicality is a Mahalanobis distance (2.14) in a tangent space to the sym- metric space M = Gd as shown in section 2.4.3.1 when PGA is used to estimate the shape prior. The Mahalanobis distance (2.14) is simplified to the sum of coefficients along all princi- pal geodesics. During optimization the geometric template model (usually mean)is deformed by varying these coefficients.
The geometry-to-image match comes from the regional intensity summary based match method developed by Broadhurst et al. [2006]. The image regions are determined by object- relative coordinates that training m-reps produce since object-relative coordinates of m-reps provide correspondence to intensities of training gray-scale images. The method represents each regional intensity collection by the curve of intensity values versus quantile called an RIQF (regional intensity quantile function) [Pizer et al., 2005a, Broadhurst et al., 2005]. An RIQF is the inverse of the cumulative distribution function of image intensity in the region. Broadhurst et al. [2006] showed that 1) QFs do not suffer from quantization effects like most intensity histogram-based methods, 2) the analysis of RIQFs can be captured by linear statistics on DRIQFs (Discrete sampling of RIQFs), 3) Because the Earth-mover’s distance on histograms (probability distributions) is equivalent to a Euclidean distance on quantile functions (QFs) [Levina, 2001], QFs form a linear space that PCA can be applied to, and 4) DRIQFs of a continuous distribution function that is parameterized by locations and scales form a 2-d linear space. In practice, DRIQFs also suffer from the HDLSS situation, so PCA is used to estimate the probability distribution of DRIQFs. Then the geometry-to-image match is a Mahalanobis distance of a DRIQF obtained from a target image region, which is the sum of coefficients along all principal directions in the region.
M-rep based segmentation is inherently multi-scale starting from the scale of multi-object complexes or a single figure down to the scale of medial atoms. As mentioned in section 2.2.3.1, the relevant shape properties at each scale and the relation to the neighboring geometric entities (figures or section of medial atoms) need to be taken into account in the geometric typicality term. One main topic of this dissertation is to estimate the probability shape distribution of multiple objects, which is related to the segmentation in the scale of multi-object complexes.
Segmentation of single figures is implemented in multi-scale. Initially a template model is placed into the target image by matching a few either manually or automatically chosen
Figure 2.4: Segmented kidneys, aligned via landmark, figure stage, and atom stage result (This figure is from the medial book. [Siddiqi and Pizer, 2008])
landmarks. The segmentation proceeds through a number of stages at successively smaller levels of scale. Fig. 2.4 shows the segmentation results of a kidney at each stage: model alignment by landmarks, the figure stage, and the medial atom stage.
Chapter 3
Evaluation of Statistical Shape Models
1Principal component analysis (PCA) has become a very popular method used to analyze shape variability. Cootes et al. [1995], Bookstein [1999] were early users of PCA for shape analysis. The usefulness of PCA is two-fold: 1) decomposition of population variables into an efficient reparametrization of the variability observed on the training data and 2) dimension reduction of population variables that allows a focus on the subspace of the original space of population variables. Especially, #2 is a major advantage of PCA in shape analysis because most shape representations presented in the literature have very high dimensional feature spaces due to the complexity of object shape. On the other hand, available training samples are limited due to the cost and time involved in the manual segmentation of images. This kind of data is called high dimension, low sample size (HDLSS) in statistics. The measure we propose in this chapter applies to statistical shape models that use PCA as their method to describe shape variability.
Given a set of training samples from a population, PCA allows us to extract important directions (features) from these training samples and to use these features to describe new members of the population. The predictability of statistical shape models refers to this power of statistical shape models to predict a new member in the population.
There exist in statistics several criteria to judge the appropriateness of any dimension reduction technique. However, we will mainly concentrate on the criteria of predictability in view of the many practical applications of statistical shape models as explained in section 1.1.2.
1
In addition, we will touch on the questions of the interpretability and the stability of the extracted directions that are equally important as the predictability: Does the direction have a meaningful interpretation or are they mere mathematical objects?; How do these directions differ from sample to sample, and how many training samples do we need to get a stable estimate of the important directions?