Computing minimal deformations: application to construction of statistical shape models

Nonlinear registration is mostly performed after initialization by a global, linear transformation (in this work, we focus on similarity transformations), computed by a linear registration method. For the further processing of the results, it is mostly assumed that this preregistration step completely removes the respective linear transformation. However, we show that in deformable settings, this is not the case. As a consequence, a significant linear component is still existent in the deformation computed by the nonlinear registration algorithm. For construction of statistical shape models (SSM) from deformations, this is an unwanted property: SSMs should not contain similarity transformations, since these do not capture information about shape. We propose a method which performs an a posteriori extraction of a similarity transformation from a given nonlinear deformation field, and we use the processed fields as input for SSM construction. For computation of minimal displacements, a closed-form solution minimizing the squared Euclidean norm of the displacement field subject to similarity parameters is used. Experiments on real inter-subject data and on a synthetic example show that the theoretically justified removal of the similarity component by the proposed method has a large influence on the shape model and significantly improves the results.