The goal of this step is to make retrieval invariant to 3D transformations (including scaling, translation, rotation, and reflection) and also to generate multiple views of 3D models in the gallery (and possibly the probe2) sets.
The positions computed in the MDS need to be anchored by a frame of reference as translation, rotation, and reflection in the 2-D plane do not affect them.
Deformation can include the operations of translation, rotation and reflection; bending and stretching the branches; and crossing or knotting the branches.
For this, grid symmetries due to translation, rotation and reflection of the embeddings were considered as well as vertex subsets which inner permutations didn't change the value of the objective function.
These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value.
This is the same as saying that (G, q) can be obtained from (G, p) by an isometry of ℝ2, i.e., a combination of translations, rotations and reflection.
The Procrustes approach matches one configuration to another and produces a measure of match, while seeking the isotropic dilation and the rigid translation, reflection and rotation needed to best match one configuration to another (Cox and Cox, 1994).
Various different textures will be gained to tessellate figures by mathematical transformations including translation, rotation, reflection and glide reflection on a surface.
Procedures were designed to yield features invariant to translation, scale, reflection, and rotation in the xy-plane.
We found in practice that this clearly outperforms the usual PCA alignment (see Figure 10 and Table 2) and makes the retrieval process invariant to several transformations including rotation, reflection, translation, and scaling.
