Atlas-building from population data is widely used in medical imaging. In this work, we focus on the statistical characterization of the population data, once spatial alignment has been achieved.
We introduce and propose the use of the weighted functional boxplot. This allows the generalization of concepts such as the median, percentiles, or outliers to spaces where the data objects "Box plot definition statistics of sexual immorality" functions, shapes, or images, and allows spatio-temporal atlas-building based on kernel regression. In our experiments, we demonstrate the utility of the approach to construct statistical atlases for pediatric upper airways and corpora callosa revealing their growth patterns.
We also define a score system based on the pediatric airway atlas to quantitatively measure the severity of subglottic stenosis SGS in the airway. This scoring allows the classification of pre- and post-surgery SGS subjects and radiographically normal controls.
Experimental results show the utility of atlas information to assess the effect of airway surgery in children. Atlas-building from population data has become an important task in medical imaging to provide templates for data analysis.
Numerous methods for atlas-building exist, ranging from methods designed for cross-sectional, longitudinal, and random design data. These approaches typically estimate a representative data object Wang and Marron, e. A limitation of all these methods is that they result in a single summary representer and discard much of the population for subsequent analysis.
For instance, a single point is used to summarize the entire population on the manifold, when one summarizes it using an atlas or a median.
For regression, a single curve summarizes the population without carrying forward any information from the local distribution of data around the curve. These are restrictive representations that limit the capability of the model to present confidence bounds, quantile measurements or to identify outliers. In the literature, the limitation of the single summary representers has also been acknowledged.
For instance, Aljabar et al. In another study, Gerber et al. Another strategy to retain population variation information is to represent additional aspects of the full data distribution, such as percentiles, the minimum and maximum, variance, confidence regions and outliers as captured by a boxplot for scalar-valued data.
The functional boxplot Sun and Genton, is an effective tool to represent such statistics for functions.
The main goal in this paper is to generalize the notion of functional boxplots to summarize variabilities within population of entities such "Box plot definition statistics of sexual immorality" shapes and images see Fig. This can provide a simple and generic method to augment atlases with additional population information while avoiding restrictive point-wise analyses of data-objects.
Note that we focus in this paper on augmenting atlases with statistical information, and assume a given spatial alignment of data objects. The method can also be extended to build order statistics from low-dimensional manifold embeddings where point-wise analysis becomes meaningful as each point then represents a full data object. Illustration of boxplots for points, functions, shapes and images. Median middle black lineconfidence region magenta and the maximum non-outlying envelope two outward blue lines.
The gray dashed lines are the outliers. Another goal of this work is to generalize the notion of confidence bounds to the estimates of regression using functional boxplots. As subject data typically has associated individual Box plot definition statistics of sexual immorality e. For example, given a subject at a particular age we want to compute subject age-specific confidence regions to assess similarity with respect to the full data population. The method described in this paper is an extension of the preliminary ideas we presented in a recent conference paper Hong et al.
This paper offers more details of our proposed method, additional experiments for quantitative assessment, and more validation on synthetic and real data.
This section introduces a weighted variant of the functional boxplot and extends it for use with kernel regression of functional data. We first cover the preliminaries on kernel regression and later present the concept of weighted band-depth essential to defining the weighted functional boxplot. The proposed method is applicable to the analysis of function, shape, and image populations to create non-parametric regression models with associated subject characteristics.
As an example, we consider subject age and demonstrate the effectiveness of weighted functional boxplots and kernel smoothing Wand and Jones, to build a spatio-temporal atlas. Given spatially aligned data objects we want to capture population changes, e. Spatial alignment refers to a pre-processing step that transforms all data objects to common coordinates for further analysis.
The type of alignment depends on the objectives of a particular study. For instance, this alignment may be a rigid transformations when the statistical analysis needs to be performed modulo translations and rotations only. Kernel-based methods exhibit a bias near the boundary of the data. This is usually attributed to the asymmetric averaging of limited information at the boundaries.
Many solutions have been proposed to address this issue Schuster, ; Jones, ; Marron and Ruppert, If the target age for the atlas is located within the interior part of the observed population, no boundary effects exist. However, for studies involving models for growth, aging or memory decline, we often build atlases for very young or very old subjects.
This usually requires averaging kernel weights with respect to an age near the Box plot definition statistics of sexual immorality.