Limitations of the principal-axes theory

The classical principal-axes transformation (PAT) has been used in numerous publications for three-dimensional (3-D) reconstructions by sequential alignment of histological sections. However, the PAT can determine at most 1/2n(n + 1) parameters (scaling-rotation) in n dimensions. Distortions (shearing) of histological sections can be described by an affine transformation with n2 parameters. An analytical model is devised for calculating rotational and scaling errors which can be determined by relating the transformation parameters of the PAT to the exact solution of a singular value decomposition (SVD) of the perturbation matrix. The results show that form and deformation of the form are intertwined and that these results can be transferred to real data. The model is important for assessing the quality that can be expected with the PAT for 3-D reconstructions if no multimodality reference is available (rigid transformation) and reveals the misalignment in terms of rotational and scaling errors resulting from the PAT as derived in a previously published paper.