Computing minimal deformations: application to construction of statistical shape models

Abstract

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.

Approximated Curvature Penalty in Non-rigid Registration Using Pairwise MRFs

“…Additionally, a block matching strategy which is mainly based on a translational search most likely fails to recover large rotations or scaling. However, in practice it is likely that a certain amount of linear transformation is still present when starting the non-rigid alignment [17]. Again, we generate two target images from a source image, both with a 25 • rotation (cp.…”

Semantically steered visual analysis of highly detailed morphometric shape spaces

“…For landmark data it is often sufficient to examine the principal axes of the PCA to find correlations between shape and investigated trait. However, for higher dimensional data a PCA is often ambigious and the correlation under investigation is not captured by a single axis but spread over several axes (see Sec.2.3.1 in [30] for an illustrative example). This renders examination of warps along the principal axes alone inadequate.…”

Extracting a Purely Non-rigid Deformation Field of a Single Structure

“…With initial rigid registration and final non-rigid-registration, we have obtained the linear and non-linear part of the overall transformation of the abdominal aorta. However, according to Zikic et al [6], deformation fields that are computed by a local nonlinear registration algorithm, still contain a linear transformation T local = T linear • T nonlinear . In order to obtain a pure non-linear deformation field, the authors suggest to minimize the norm of the displacement field V of the nonlinear component T nonlinear with respect to the linear transformation T linear .…”