Detection and analysis of statistical differences in anatomical shape
Introduction
Anatomical shape, and its variation, remains an important topic of medical research. Understanding morphological changes caused by a particular disorder can help to identify the time of onset of a disease, quantify its development and potentially lead to better treatment. Other examples of morphological studies include investigating anatomical changes due to aging by comparing different age groups, and studies of anatomical differences between genders. Originally, image-based statistical studies of morphology were based on simple measurements of size, area and volume. Shape-based analysis promises to provide much more detailed descriptions of the anatomical changes due to the biological process of interest. In this paper, we present a computational framework for performing statistical comparison of populations based on complex shape descriptors. The analysis considers the entire set of shape features simultaneously and yields an assessment of how much the shape of the organ differs between the two populations, as well as a detailed description of the identified differences.
Image-based shape analysis typically consists of three main steps. First, quantitative measures of shape are extracted from each input image and are combined into a feature vector that describes the input shape. The set of feature vectors is then used to construct either a generative model of shape variation within one population or a discriminative model of shape differences between two populations. This is followed by interpretation of the statistical model in terms of the original shape and image properties. Such interpretation is necessary for visualization and improved understanding of detected shape differences. In this section, we describe each of the three stages of the analysis, provide a review of related work and outline our approach.
Shape analysis starts with extraction of shape features from input images. A great number of shape descriptors have been proposed for use in medical image analysis. They can be classified into several broad families, such as landmarks (Bookstein, 1997, Cootes et al., 1992, Dryden and Mardia, 1998), dense surface meshes (Brechbühler et al., 1995, Kelemen et al., 1998, Shenton et al., 2002, Staib and Duncan, 1992, Székely et al., 1996), skeleton-based representations (Bookstein, 1979, Fritsch et al., 1994, Golland et al., 1999, Golland and Grimson, 2000, Pizer et al., 1996), deformation fields that define a warping of a standard template to a particular input shape (Christensen et al., 1993, Davatzikos et al., 1996, Martin et al., 1994, Machado and Gee, 1998) and distance transforms that embed the outline of the object in a higher dimensional distance function over the image (Golland et al., 2000, Leventon et al., 2000). The choice of shape representation depends crucially on the application. For statistical modeling, the two most important properties of a shape descriptor are its sensitivity to noise in the input images and the ease of registration of the input examples into a common coordinate frame.1 These determine the amount of noise in the training data, and therefore the quality of the resulting statistical model. In this work, we choose to use an existing approach based on distance transforms for feature extraction, mainly because of its simplicity and its smooth dependence on the noise in the object’s boundary and its pose. The focus of this paper is on the later steps of the analysis that produce an interpretation of the statistical model, and not on the shape representation per se. Section 6 offers a discussion on employing other shape descriptors in conjunction with the statistical analysis tools presented in this paper.
Once the features have been extracted from the images, they are used to construct a statistical model of differences between the two groups of feature vectors. One approach is to treat the features as independent variables and to perform a simple hypothesis test on each feature separately (Bookstein, 1997, Machado and Gee, 1998, Yushkevich et al., 2001). If the features are local (e.g., voxels or boundary segments), the interpretation of the resulting model becomes straightforward, as we can create a mask in the image domain indicating the features that were deemed significantly different in the two groups. Unfortunately, it can be difficult to assess the significance of the entire pattern from the individual statistical tests. Moreover, if the features are global, such as Fourier coefficients of the outline curve, the interpretation of the detected differences becomes difficult. Alternatively, we can estimate a statistical model based on the entire vectors and possible dependencies among the features. In the generative case, this is typically done by applying principal component analysis (PCA) to estimate the mean and the covariance structure of the training set (Cootes et al., 1992, Cootes et al., 1999, Kelemen et al., 1998). Earlier work on shape differences between populations employed PCA for dimensionality reduction, followed by training a simple (linear or quadratic) classifier in the reduced space (Csernansky et al., 1998, Martin et al., 1994). In this work, we use the support vector machines (SVMs) algorithm (Burges, 1998, Vapnik, 1995, Vapnik, 1998) to estimate the optimal classifier function directly in the original feature space while explicitly controlling its complexity. In addition to the theoretical reasons for its asymptotic optimality, Support Vector learning has been empirically demonstrated to be robust to overfitting and to generalize well even for small data sets. Furthermore, the algorithm provides a principled way to explore a hierarchy of increasingly complex classifier families, trading-off the training error and the complexity of the model.
To be useful in the clinical context, the resulting statistical model (eigenmodes or a classifier function) must be mapped back to the image domain, i.e., analyzed in terms of the input shape or image properties in order to generate a comprehensible description of the structure in the training data that was captured by the model. In the generative case, this is often done by sampling the implied Gaussian distribution with the mean and the covariance estimated from the data or by varying one principal component at a time. We previously used a similar approach for interpretation of a linear classifier by translating the original feature vector along the projection vector of the classifier function (Golland et al., 1999). More commonly, however, the resulting classifier is used only to establish statistical significance of morphological differences between the classes, and the generative models based on PCA are employed for visualization of the shape variation within each group (Csernansky et al., 1998, Gerig et al., 2001, Martin et al., 1994), and approach that does not provide a direct comparison of the populations based on the estimated discriminative model.
In this paper, we demonstrate how to obtain a description of shape differences captured by the classifier function that was constructed in the analysis step. To understand the differences implicitly represented by the classification function, we study the function’s sensitivity to changes in the input along different directions in the feature space. For every input example, we solve for the direction in the feature space that maximizes the change in the classifiers value while introducing as little irrelevant changes into the input vector as possible. We derive the sensitivity analysis for a large family of non-linear kernel-based classifiers. The results can be represented in the image domain as deformations of the original input shape, yielding both a quantitative description of the morphological differences between the classes and an intuitive visualization mechanism. Thus, in addition to the statistical descriptors, such as test error and confidence bounds, we also provide a mechanism for explicit visual interpretation of the detected shape variation between the two populations.
The remainder of this paper is organized as follows. In the next section, we explain our choice of the distance transforms for extracting shape features and introduce a local parameterization of the distance transform space which allows us to represent and visualize changes in the distance transform as deformations of the corresponding boundary surface. This is followed by a brief review of the Support Vector learning and a derivation of the discriminative direction as a description of differences between two classes captured by the classification function. We then combine shape description with statistical analysis and demonstrate the technique on a simple artificial example, as well as real medical studies. The paper concludes with a discussion of the lessons learned from the presented experiments and future research directions.
Section snippets
Shape representation
We chose to work with volumetric descriptors in order to avoid the implementation difficulties associated with establishing a common coordinate frame on the surface of such relatively smooth objects as subcortical structures. Our main interest lay in further developing the interpretation step of the analysis that provides a representation and visualization of the statistical model, leading to a choice of a very simple shape descriptor. And since the analysis assumes that the representation
Statistical analysis
Once feature vectors have been extracted from the input images, they are used to construct a classifier for distinguishing between the two example groups. We use the SVMs learning algorithm to estimate the optimal classifier function. The classifier function constructed during the training phase implicitly encodes the differences in data between the two classes and we are interested in understanding the nature of the these differences. If expressed in terms of the original images or shapes,
System overview: a simple example
Before presenting the experimental results for the real medical studies, we explain how the components of the analysis described in the previous sections are combined into a system. We will illustrate the steps of the algorithm on a simple simulated shape study that contains 30 volumetric images of ellipsoidal shapes of varying sizes. The width, height and thickness of the shapes were sampled uniformly out of a ±10 voxel range centered at 20, 30 and 40 voxels, respectively. We randomly divided
Experimental results
In this section, we consider studies of two different anatomical organs: the hippocampus–amygdala complex in schizophrenia and corpus callosum in the first episode affective disorder. The hippocampus study includes two separate studies, as we compare the right and the left hippocampus separately.
Discussion
In this section, we reflect on our experience with the technique, unexpected problems that arose in the experiments and the insights they provided into the nature of the statistical shape analysis.
Conclusions
The focus of this paper is the interpretation of the classifier function constructed to distinguish between two populations. We present a novel technique for classifier analysis in terms of the input features in the general context of the statistical learning theory and instantiate the technique for shape analysis by establishing a locally linear parameterization of the distance transform space using deformations of the corresponding boundary surface. Such parameterization yields a
Acknowledgements
Quadratic optimization was performed using PR_LOQO optimizer written by Alex Smola.
This research was supported in part by NSF IIS 9610249 grant. The authors would like to acknowledge Dr. Shenton’s Grants NIMH K02, MH 01110 and R01 MH 50747 Grants, Dr. Kikinis’s Grants NIH PO1 CA67165, R01RR11747, P41RR13218 and NSF ERC 9731748 Grants.
The authors thank the anonymous reviewers, whose careful proofreading and thoughtful comments and suggestions helped us to substantially improve the clarity of
References (35)
The line skeleton
CGIP: Computer Graphics and Image Processing
(1979)Landmark methods for forms without landmarks: morphometrics of group differences in outline shape
Medical Image Analysis
(1997)- et al.
Parameterization of closed surfaces for 3-D shape description
CVGIP: Image Understanding
(1995) - et al.
The multiscale medial axis and its applications in image registration
Patter Recognition Letters
(1994) - et al.
Skeletonization via distance maps and level sets
CVIU: Computer Vision and Image Understanding
(1995) - et al.
Amygdala–hippocampus shape differences in schizophrenia: the application of 3D shape models to volumetric MR data
Psychiatry Research Neuroimaging
(2002) - et al.
Segmentation of 2D and 3D objects from mri volume data using constrained elastic deformations of flexible fourier contour and surface models
Medical Image Analysis
(1996) A transformation for extracting new descriptors of shape
A tutorial on support vector machines for pattern recognition
Data Mining and Knowledge Discovery
(1998)- Christensen, G., Rabbitt, R.D., Miller, M.I., 1993. A deformable neuroanatomy textbook based on viscous fluid...
Training models of shape from sets of examples
A unified framework for atlas matching using active appearance models
Hippocampal morphometry in schizophrenia by high dimensional brain mapping
Proceedings of the National Academy of Science
A computerized method for morphological analysis of the corpus callosum
Journal of Computer Assisted Tomography
Statistical Shape Analysis
The Jacknife, The Bootstrap, and Other Resampling Plans
Shape differences in the corpus callosum in first psychotic episode schizophrenia and first psychotic episode affective disorder
American Journal of Psychiatry
Cited by (87)
Distance-based tests for planar shape
2021, Journal of Multivariate AnalysisStatistical Shape and Appearance Models for Bone Quality Assessment
2017, Statistical Shape and Deformation Analysis: Methods, Implementation and ApplicationsA Bayesian framework for joint morphometry of surface and curve meshes in multi-object complexes
2017, Medical Image AnalysisCitation Excerpt :There are several examples of mesh based morphometry in the literature. Most of them use a single-object approach since they select and analyse only one particular structure of an organ (Golland et al., 2005; Niethammer et al., 2007; Davies et al., 2010; Hufnagel et al., 2009; Kurtek et al., 2011; Savadjiev et al., 2012; Cury et al., 2015). This strategy limits the extent of the clinical conclusion to the chosen object, thus neglecting the information given by the surrounding structures.
Statistical shape analysis: From landmarks to diffeomorphisms
2016, Medical Image Analysis