Rigid transformation

In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a proper rigid transformation, or rototranslation. Any proper rigid transformation can be decomposed into a rotation followed by a translation, while any improper rigid transformation can be decomposed into an improper rotation followed by a translation, or into a sequence of reflections. Any object will keep

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On the benefit of concurrent adjustment of active and passive optical sensors with GNSS & raw inertial data

Davide Antonio Cucci, Kyriaki Mouzakidou, Jan Skaloud

In airborne laser scanning a high-frequency trajectory solution is typically determined from inertial sensors and employed to directly geo-reference the acquired laser points. When low-cost MEMS inertial sensors are used, such as in lightweight unmanned aerial vehicles, non-negligible errors in the estimated trajectory project to the final point-cloud, resulting in unsatisfactory accuracy on the ground. There are different multi-sensor fusion approaches to correct the point-cloud errors caused by an imperfect trajectory determination. Mismatches between different optical observations and/or in the overlapping regions of the point-cloud can allow the correction of the final point-cloud, either directly, by means of rigid transformations, or indirectly, via improving the scanner trajectory estimation. In this work we propose to fuse lidar and cameras in a single adjustment based on dynamic networks, considering 2D tie-points from the imagery and 3D tie-points from overlapping point-cloud sections. On a challenging corridor mapping scenario, we show that considering either 2D or 3D tie-points, along with inertial and GNSS observations, results in a remarkably accurate point-cloud, even when low-cost inertial sensors are employed and in presence of challenging surface textures, such as high vegetation. Furthermore, since the distribution of the 2D and 3D tie-points is complementary, considering them together further increases the robustness of the adjustment due to higher redundancy. By employing the proposed approach within this controlled example, we were able to improve the final point-cloud accuracy by more than three times with respect to conventional geo-referencing methodology and to reduce the magnitude of the errors.

2022

Normalization of Transcranial Magnetic Stimulation Points by Means of Atlas Registration

Mathieu De Craene, Benoît Macq, Jean-Philippe Thiran, Dominique Zosso

Transcranial magnetic stimulation (TMS) is a well-known technique to study brain function. Location of TMS points can be visualized on the subject’s Magnetic Resonance Image (MRI). However inter-subject comparison is possible only after a normalization i.e. a transformation of the stimulation points to a reference atlas image. Here, we propose a generic and automatic image processing pipe-line for normalizing a collection of subjects’ MRI and TMS points. The normalization uses a loop of rigid-transform followed by Basis-Splines registration. The used reference atlas is the common Montreal National Institute (MNI) brain atlas. We show preliminary results from 10 subjects. Those normalized points were compared to the SPM normalization and validated by TMS experts.

2006

On Matrix Rigidity and the Complexity of Linear Forms

Mahdi Cheraghchi Bashi Astaneh

The rigidity function of a matrix is defined as the minimum number of its entries that need to be changed in order to reduce the rank of the matrix to below a given parameter. Proving a strong enough lower bound on the rigidity of a matrix implies a nontrivial lower bound on the complexity of any linear circuit computing the set of linear forms associated with it. However, although it is shown that most matrices are rigid enough, no explicit construction of a rigid family of matrices is known. In this survey report we review the concept of rigidity and some of its interesting variations as well as several notable results related to that. We also show the existence of highly rigid matrices constructed by evaluation of bivariate polynomials over finite fields.

2005

Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body

Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but i

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PHYS-426: Quantum physics IV

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