# Deformation-Based and Tensor-Based Morphometry

DBM [13, 17] is one of the methods to detect and analyze brain shape differences among the brains of different populations. The deformation vector in each voxel is computed and a deformation vector matrix or field throughout the brain is generated by nonlinear spatial normalization, in which a brain image is transformed to a template image, and is used to describe global or local differences in brain shape.

A deformation field can be considered as a continuous 3D vector field denoted by three elements * (x, y, z)* at each voxel. When the same template (reference) image is used for the transformation of a series of image datasets, we obtain a series of deformation fields—one for each image—by which we can compare the differences between the images. For example, let us assume that there are two groups of subjects, A and B, with different characteristics (i.e., male and female, young/old, healthy/ill). Using the same template (reference) image, we can compute a series of deformation vectors for the two groups. To test the significance level in different deformations, the p value for the significance test is calculated using multivariate analysis of covariance (ManCova) and canonical variate analysis (CVA). DBM is fully automated (operator independent) and reproducible.

To localize structure differences between subject groups, tensor-based morphometry (TBM) is used to produce a statistical parametric map of regional shape differences. A deformation field that maps one image to another can be considered as a discrete vector field. The Jacobian matrix of the deformation field is generated, in which each element is a tensor describing the relative positions of the neighboring elements. Morphometric measurements derived from this tensor field can be used to localize regions with different brain structure. This is most often used to measure regional volume increase/reduction [83, 103]. Statistical parametric maps of these determinant fields can be used to compare the anatomy of subjects’ groups. Other measurements are derived from the tensor fields.