Spectral Methods for Morphometry

Methods from shape analysis are used in morphometry, which is the quantitative analysis of macroscopic anatomical features. We assume that anatomy is flexible, and this brings us to the first problem of resolving how ``shape'' should be represented if it is allowed to bend. We are motivated to use representations of intrinsic geometry, which, for example, does not distinguish a flat sheet of paper from a rolled sheet. The spectral embedding (``heat kernel representation'') as a representation of intrinsic geometry has many desirable features for computational anatomy and other areas of shape and data analysis. Several breakthroughs are made toward understanding and applying this representation. A novel shape representation is also considered and used for classification of control vs. affected groups. One goal of morphometry is to make statistically objective comparisons. Hence, once a suitable representation of shape is chosen, the second problem is to compare shapes. Shape comparison may occur at many levels of scale. The simplest comparisons are made with global features: volume, length, etc. Finer comparisons may occur at regional levels. A finest level of comparison can be made after matching all homologous points, that is, after finding a one-one correspondence between points on shapes. A point correspondence is found by a registration algorithm. A method for unsupervised shape registration is presented and applied to localize differences between control and affected groups. We focus on the 3D case, where imaging has made anatomical surface data readily available, yet the analysis challenging. Structural MRI of living persons is currently used to study the macroscopic effects on anatomy by neurodegenerative disease (e.g. Alzheimer's). In the earliest stages of Alzheimer's disease (AD), certain brain structures have been observed to have reduced volume, in autopsy and in vivo, including the hippocampus, putamen, and thalamus. Our methods will be applied to these surfaces.