Changizi Neda, Hamarneh Ghassan
Medical Image Analysis Lab, Simon Fraser University, Canada.
Med Image Comput Comput Assist Interv. 2010;13(Pt 3):563-70. doi: 10.1007/978-3-642-15711-0_70.
Several sources of uncertainties in shape boundaries in medical images have motivated the use of probabilistic labeling approaches. Although it is well-known that the sample space for the probabilistic representation of a pixel is the unit simplex, standard techniques of statistical shape analysis (e.g., principal component analysis) have been applied to probabilistic data as if they lie in the unconstrained real Euclidean space. Since these techniques are not constrained to the geometry of the simplex, the statistically feasible data produced end up representing invalid (out of the simplex) shapes. By making use of methods for dealing with what is known as compositional or closed data, we propose a new framework intrinsic to the unit simplex for statistical analysis of probabilistic multi-shape anatomy. In this framework, the isometric log-ratio (ILR) transformation is used to isometrically and bijectively map the simplex to the Euclidean real space, where data are analyzed in the same way as unconstrained data and then back-transformed to the simplex. We demonstrate favorable properties of ILR over existing mappings (e.g., LogOdds). Our results on synthetic and brain data exhibit a more accurate statistical analysis of probabilistic shapes.
医学图像中形状边界存在多种不确定性来源，这促使人们使用概率标记方法。尽管众所周知，像素概率表示的样本空间是单位单纯形，但统计形状分析的标准技术（例如主成分分析）已应用于概率数据，就好像它们位于无约束的实欧几里得空间中一样。由于这些技术不受单纯形几何形状的约束，最终产生的统计上可行的数据代表的是无效（超出单纯形）形状。通过利用处理所谓的成分数据或封闭数据的方法，我们提出了一种新的框架，该框架是单位单纯形固有的，用于概率多形状解剖结构的统计分析。在这个框架中，等距对数比（ILR）变换用于将单纯形等距且双射地映射到欧几里得实空间，在该空间中，数据以与无约束数据相同的方式进行分析，然后再逆变换回单纯形。我们证明了ILR相对于现有映射（例如对数几率）具有良好的特性。我们在合成数据和脑数据上的结果展示了对概率形状更准确的统计分析。
