Pre-processing involved temporal alignment of volume slices (to correct for asynchronous slice acquisition), normalization within each scanning run to a fixed image intensity value, and then correction for motion using a rigid-body rotation and translation algorithm [19], [20].
Registrations were performed using Elastix software [ 18] using a rigid transformation model (translations and rotations).
The set of localizations collected from the Alexa 647 channel was transformed, using a polynomial warp transform to account for differences in magnification, rotation, shear, etc., and then the images were aligned using a rigid translation, based on the bead positions in the image.
However due to rigid body displacements, an optimal superposition may not be obtained using a rigid body superposition method.
The PCA is a rigid rotation of the Procrustes shape coordinates that maximizes the variation among individuals using a spectral decomposition of a covariance matrix [ 39, 40].
Therefore, (F_{f}) is a rigid rotation on ((0,vartheta_{max})).
The rigid bodies, named 1 and 3 in Fig. 4, are characterized by a rigid translation along the horizontal and vertical direction, respectively, while, the blocks named 2 move with a rigid translation u and a rigid rotation φ.
One considers a perturbation of a rigid rotation of a circle, or a system of coupled oscillators.
If (n=0), that (F_{f}) is a rigid rotation on ((a_{1},vartheta_{ast})) immediately follows from the fact that there is no rapid phase at all.
The result of Ortega shows that it is always possible to choose some such that the Poincaré matrix (of the period ) of (4.43) is a rigid rotation (4.44).
