A Novel Distribution for Representation of 6D Pose Uncertainty

The 6D Pose estimation is a crux in many applications, such as visual perception, autonomous navigation, and spacecraft motion. For robotic grasping, the cluttered and self-occlusion scenarios bring new challenges to the this field. Currently, society uses CNNs to solve this problem. The CNN models will suffer high uncertainty caused by the environmental factors and the object itself. These models usually maintain a Gaussian distribution, which is not suitable for the underlying manifold structure of the pose. Many works decouple rotation from the translation and quantify rotational uncertainty. Only a few works pay attention to the uncertainty of the 6D pose. This work proposes a distribution that can capture the uncertainty of the 6D pose parameterized by the dual quaternions, meanwhile, the proposed distribution takes the periodic nature of the underlying structure into account. The presented results include the normalization constant computation and parameter estimation techniques of the distribution. This work shows the benefits of the proposed distribution, which provides a more realistic explanation for the uncertainty in the 6D pose and eliminates the drawback inherited from the planar rigid motion.

On the Connections between Thermodynamics and General Relativity

<p>In this thesis, the connections between thermodynamics and general relativity are explored. We introduce some of the history of the interaction between these two theories and take some time to individually study important concepts of both of them. Then, we move on to explore the concept of gravitationally induced temperature gradients in equilibrium states, first introduced by Richard Tolman. We explore these Tolman-like temperature gradients, understanding their physical origin and whether they can be generated by other forces or not. We then generalize this concept for fluids following generic four-velocities, which are not necessarily generated by Killing vectors, in general stationary space-times. Some examples are given. Driven by the interest of understanding and possibly extending the concept of equilibrium for fluids following trajectories which are not generated by Killing vectors, we dedicate ourselves to a more fundamental question: can we still define thermal equilibrium for non-Killing flows? To answer this question we review two of the main theories of relativistic non-perfect fluids: Classical Irreversible Thermodynamics and Extended Irreversible Thermodynamics. We also take a tour through the interesting concept of Born-rigid motion, showing some explicit examples of non-Killing rigid flows for Bianchi Type I space-times. These results are important since they show that the Herglotz–Noether theorem cannot be extended for general curved space-times. We then connect the Born-rigid concept with the results obtained by the relativistic fluid’s equilibrium conditions and show that the exact thermodynamic equilibrium can only be achieved along a Killing flow. We do, however, introduce some interesting possibilities which are allowed for non-Killing flows. We then launch into black hole thermodynamics, specifically studying the trans-Planckian problem for Hawking radiation. We construct a kinematical model consisting of matching two Vaidya spacetimes along a thin shell and show that, as long as the Hawking radiation is emitted only a few Planck lengths (in proper distance) away from the horizon, the trans-Plackian problem can be avoided. We conclude with a brief discussion about what was presented and what can be done in the future.</p>

On the Connections between Thermodynamics and General Relativity

<p>In this thesis, the connections between thermodynamics and general relativity are explored. We introduce some of the history of the interaction between these two theories and take some time to individually study important concepts of both of them. Then, we move on to explore the concept of gravitationally induced temperature gradients in equilibrium states, first introduced by Richard Tolman. We explore these Tolman-like temperature gradients, understanding their physical origin and whether they can be generated by other forces or not. We then generalize this concept for fluids following generic four-velocities, which are not necessarily generated by Killing vectors, in general stationary space-times. Some examples are given. Driven by the interest of understanding and possibly extending the concept of equilibrium for fluids following trajectories which are not generated by Killing vectors, we dedicate ourselves to a more fundamental question: can we still define thermal equilibrium for non-Killing flows? To answer this question we review two of the main theories of relativistic non-perfect fluids: Classical Irreversible Thermodynamics and Extended Irreversible Thermodynamics. We also take a tour through the interesting concept of Born-rigid motion, showing some explicit examples of non-Killing rigid flows for Bianchi Type I space-times. These results are important since they show that the Herglotz–Noether theorem cannot be extended for general curved space-times. We then connect the Born-rigid concept with the results obtained by the relativistic fluid’s equilibrium conditions and show that the exact thermodynamic equilibrium can only be achieved along a Killing flow. We do, however, introduce some interesting possibilities which are allowed for non-Killing flows. We then launch into black hole thermodynamics, specifically studying the trans-Planckian problem for Hawking radiation. We construct a kinematical model consisting of matching two Vaidya spacetimes along a thin shell and show that, as long as the Hawking radiation is emitted only a few Planck lengths (in proper distance) away from the horizon, the trans-Plackian problem can be avoided. We conclude with a brief discussion about what was presented and what can be done in the future.</p>

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

The fundamental model of any solid crystalline material (crystal) at the atomic scale is a periodic point set. The strongest natural equivalence of crystals is rigid motion or isometry that preserves all inter-atomic distances. Past comparisons of periodic structures often used manual thresholds, symmetry groups and reduced cells, which are discontinuous under perturbations or thermal vibrations of atoms. This work defines the infinite sequence of continuous isometry invariants (Average Minimum Distances) to progressively capture distances between neighbors. The asymptotic behaviour of the new invariants is theoretically proved in all dimensions for a wide class of sets including non-periodic. The proposed near linear time algorithm identified all different crystals in the world's largest Cambridge Structural Database within a few hours on a modest desktop. The ultra fast speed and proved continuity provide rigorous foundations to continuously parameterise the space of all periodic crystals as a high-dimensional extension of Mendeleev's table of elements.

Some Issues on Crystal Plasticity Models Formulation: Motion Decomposition and Constitutive Law Variants

In this paper, kinematic relations and constitutive laws in crystal plasticity are analyzed in the context of geometric nonlinearity description and fulfillment of thermodynamic requirements in the case of elastic deformation. We consider the most popular relations: in finite form, written in terms of the unloaded configuration, and in rate form, written in terms of the current configuration. The presence of a corotational derivative in the relations formulated in terms of the current configuration testifies to the fact that the model is based on the decomposition of motion into the deformation motion and the rigid motion of a moving coordinate system, and precisely the stress rate with respect to this coordinate system is associated with the strain rate. We also examine the relations of the mesolevel model with an explicit separation of a moving coordinate system and the elastic distortion of crystallites relative to it in the deformation gradient. These relations are compared with the above formulations, which makes it possible to determine how close they are. The results of the performed analytical calculations show the equivalence or similarity (in the sense of the response determined under the same influences) of the formulation and are supported by the results of numerical calculation. It is shown that the formulation based on the decomposition of motion with an explicit separation of the moving coordinate system motion provides a theoretical framework for the transition to a similar formulation in rate form written in terms of the current configuration. The formulation of this kind is preferable for the numerical solution of boundary value problems (in a case when the current configuration and, consequently, contact boundaries, are not known a priori) used to model the technological treatment processes.

Improving the robustness of MOLLI T1 maps with a dedicated motion correction algorithm

Myocardial tissue T1 constitutes a reliable indicator of several heart diseases related to extracellular changes (e.g. edema, fibrosis) as well as fat, iron and amyloid content. Magnetic resonance (MR) T1-mapping is typically achieved by pixel-wise exponential fitting of a series of inversion or saturation recovery measurements. Good anatomical alignment between these measurements is essential for accurate T1 estimation. Motion correction is recommended to improve alignment. However, in the case of inversion recovery sequences, this correction is compromised by the intrinsic contrast variation between frames. A model-based, non-rigid motion correction method for MOLLI series was implemented and validated on a large database of cardiac clinical cases (n = 186). The method relies on a dedicated similarity metric that accounts for the intensity changes caused by T1 magnetization relaxation. The results were compared to uncorrected series and to the standard motion correction included in the scanner. To automate the quantitative analysis of results, a custom data alignment metric was defined. Qualitative evaluation was performed on a subset of cases to confirm the validity of the new metric. Motion correction caused noticeable (i.e. > 5%) performance degradation in 12% of cases with the standard method, compared to 0.3% with the new dedicated method. The average alignment quality was 85% ± 9% with the default correction and 90% ± 7% with the new method. The results of the qualitative evaluation were found to correlate with the quantitative metric. In conclusion, a dedicated motion correction method for T1 mapping MOLLI series has been evaluated on a large database of clinical cardiac MR cases, confirming its increased robustness with respect to the standard method implemented in the scanner.